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Topics on critical point theory Fang, Guangcai 1993

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Topics on critical point theorybyGuangcai FangB.Sc., Anhui Normal University, 1982M.Sc., Tianjin University, 1986A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Mathematics)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly of 1993© Guangcai Fang, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of M atiA-e.01441'The University of British ColumbiaVancouver, CanadaDate  :nay 17 073DE-6 (2/88)ABSTRACTMany questions in mathematics and physics can be reduced to the problem of findingand classifying the critical points of a suitable functional on an appropriate manifold. Inthis thesis, we will be concerned with the problems of existence, location and structure ofcritical points by building upon the well known min-max methods that are presently usedin non-linear differential equations. The thesis consists of two parts:In the first part, we exploit the new powerful mountain pass principle of Ghoussoub andPreiss and its higher dimensional extensions by Ghoussoub to classify the critical pointsgenerated by Min-Max methods. The functionals under study are only assumed to be C1and therefore the classical Morse theory is not available. In order to do this, we isolatevarious topological indices that can be associated with certain critical sets and points. Ifthe functionals are C2 and the critical points are non-degenerated, these indices can thenbe used to recover the standard results on Morse indices. The study in this direction wasfirst initiated in the case of the mountain pass theorem by Hofer and was expanded laterby Pucci-Serrin and Ghoussoub-Preiss. We shall extend and simplify all the previouslyobtained structural results in this setting, but more importantly, we consider the case of thesaddle point theorem and various other higher dimensional settings.In the last part, we construct an almost critical sequence (xn). by min- max proceduresfor a C2-functional co on a Hilbert space with some Morse type information. Actually weobtain some analytical (second order) properties concerning the Hessian d2co(x,i) which canbe viewed as the asymptotic version of the information on the Morse index of the limit of(x„)„ whenever such a limit exists. As noted by P.L. Lions in his studies of the Hartree-Fock equations for Coulomb systems, this type of additional information about an almostcritical sequence can sometimes be crucial in the proof of its convergence and in solvingthe corresponding variational problems. Examples are given as applications of the generaltheory developed in the thesis.TABLE OF CONTENTSAbstractTable of ContentsAcknowledgementINTRODUCTION^ 1Chapter 1 Preliminaries 111.1 A strong form of the min-max principle^ 111.2 Lipschitz and smooth perturbed minimizations^ 151.3 Combinatorial, topological and approximation lemmas^19Chapter 2 Structure of the critical set in the mountain pass theorem 242.1 Introduction^ 242.2 Two topological lemmas^ 262.3 Main theorems^ 34Chapter 3 Structure of the critical points obtained viathe higher dimensional min-max principle (n > 2)^ 433.1 Preliminaries^ 433.2 The homotopic case 453.3 The cohomotopic case^ 533.4 The homological case 573.5 Application to standard variational settings^ 61Chapter 4 Morse indices of min-max critical points^ 634.1 Morse indices of min-max critical points—the nondegenerate case 634.1.1 The Morse lemma^ 634.1.2 The main results 674.1.3 Morse indices in the saddle point theorem^ 704.2 Morse indices of min-max critical points—the degenerate case^714.2.1 Statement of the theorems^ 724.2.2 Restrictions of a homotopy-stable class to a neighborhood^734.2.3 The Marino–Prodi perturbation method^ 754.2.4 The reduction to the non-degenerate case 804.2.5 Application to standard variational settings^85Chapter 5 Morse-type information on Palais-smale sequences 865.1 Introduction and statements of the main results^ 875.2 A proof of the upper estimate in the homotopic case 935.3 A proof of the lower estimate in the cohomotopic case^1035.4 A proof of the two-sided estimates^ 120Bibliography^ 129ivAcknowledgementThe author would like to take this opportunity to thank his Ph.D. thesis supervisor,Professor Nassif Ghoussoub. Without his professional guidance, encouragement and advice,the thesis would not have the present form. He owes much to Jun Zhu for his help inthe algebraic topology. Many thanks are also due to his many fellow students, the facultyand staff of the Department of Mathematics and especially the supervisor's family for thepleasant scientific and living environment.Special thanks go to his thesis committee members and examiners, Dr. U. Haussmann,Dr. D. Rolfsen, L. Rosen, specifically Dr. L. Nirenberg and Dr. J. Petkau for their correc-tions, critical remarks and comments.The author is indebted to the University of British Columbia and NSERC for theirimportant financial support.His family support, especially his wife's every effort (not to mention the professionalinvolvement in the thesis) makes all possible. He dedicates the thesis to his wife GuolingWang and their lovely daughter Celestia Fang.vIntroductionCritical point theory has a long history and is still undergoing many changes and de-velopments. The formulation of laws of nature in terms of minimum principles is the veryearly form of the theory known today. Minimum principles were used, refined and gener-alized in early days by many illustrious mathematicians like Fermat, Maupertuis, Euler,Lagrange, Legendre, Gauss, etc. In the first half of this century, new theories and methodswere developed for dealing with the existence of unstable or saddle-type critical points. Thewell-known ones among those are Morse theory and the min-max methods (or the calculusof variations in the large) introduced by G. Birkhoff and later developed by Ljusternik andSchnirelmann. Currently, they are being actively refined and extended in order to overcomethe limitations to their applicability in the theory of partial differential equations that arecaused by the infinite dimensional nature of the problems and by the prohibitive regularityand non-degeneracy conditions that are not satisfied by present-day variational problems.In this thesis, we will be studying the existence, location and structure of critical pointsby building upon the well known min-max methods that are presently used in non-lineardifferential equations. Our main contributions fall in two categories:A. The structure of variationally generated critical setsWe study the structure of the critical set generated by various homotopic, cohomotopicand homological min-max theorems Here is the novelty in our results.1) We first isolate various topological indices that can be associated to certain criticalsets and points. These properties of critical points can be identified outside the restrictivesetting of Morse theory, since they make sense for functionals that are only Cl. Actually, thesame holds for functionals that are only locally Lipschitz but we shall not expand here onthis aspect. If the functionals are 0 and non-degenerate, these indices can then be used torecover the standard results on Morse indices. This direction of study was first initiated inthe case of the mountain pass theorem by Hofer [22] and was expanded later by Pucci-Serrin[32], [33], [34] and Ghoussoub-Preiss [21]. We shall extend and simplify all the previously12^ INTRODUCTIONobtained structural results in this setting, but more importantly, we consider the case of thesaddle point theorem and various other higher dimensional settings.2) Secondly, we manage to find such critical points arbitrarily close to any prescribedsequence of min-maxing sets and any prescribed sequence of max-mining dual sets. As shownin the monograph of Ghoussoub [20], such information on the location of critical points canbe extremely useful both in the theoretic aspects as well as in the applications to partialdifferential equations.3) On the other hand, all these results are proved under a compactness condition onthe functional that was systematically used by Ghoussoub and which is much weaker thanthe standard Palais-Smale condition. Again, this weakening of the Palais-Smale conditionis crucial in various problems, especially the ones involving the critical Sobolev exponent.(See Tarantello [41], [42] and Ghoussoub [20]).B. Second order information on almost critical sequencesAssume now that 0 is a C2-functional that is bounded below on a Hilbert space H.One can then construct an almost critical sequence (xk)k for 0 (i.e. d0(xk) -4 0) that isminimizing (i.e. 0(xk) -4 c :. infH 0) and which also satisfies the following second ordercondition:(d20(xk)w, w) .> — -k-1 1142 for any w E H.As noted by P.L. Lions [27] in his studies of the Hartree-Fock equations, the additionalinformation about almost critical sequences can sometimes be crucial in the proof of theirconvergence and therefore in solving the corresponding variational problem.If now c is an ii-max level of 0 over a homotopic family .F of dimension n as in themountain pass theorem, it is then well known that if 0 verifies a Palais-Smale type conditionand if d20 is Fredholm, then one can find critical points at the level c whose Morse index isat most n. In chapter 5, we study what happens when the compactness and non-degeneracyconditions on 0 are not satisfied. More specifically, we want to construct an almost criticalsequence (xk)k that carries the topological information given by T, in addition to the prop-erties concerning the closeness of (xk)k to any prescribed sequence of min-maxing sets in .FINTRODUCTION^ 3and any prescribed sequence of max-mining sets that are dual to the class Y. Actually, weare looking for an analytical (second order) property concerning the Hessian d20(xk), whichcan be viewed as the asymptotic version of the information on the Morse index of the limitof (xk)k whenever such a limit exists. For example, in the context of the mountain passtheorem, we obtain an almost critical sequence (xk)k and a sequence of subspaces (Ek)k ofcodimension one such that the sequence (xk)k satisfies, in addition to the normal propertiesmentioned above, the following condition:(d20(xk)v, v) _?_ —1711142 for any v E Ek-In other words, for each k E N, d20(xk) has at most one eigenvalue below —1/1c, whichclearly implies that any potential cluster point for (xk)k will be a critical point of Morseindex at most one.We shall now present, for the sake of non-specialists, two simple examples that showcritical point theory at work in solving partial differential equations.Consider the following Dirichlet problem.1 Au = u3 + f (x)^in It(1)t u = 0^on anwhere S2 C R3 is a bounded domain and f is in the dual H-1 of the Sobolev space H :=Hj(f2). Refer to [1] for details about Sobolev space Hj (12). The weak solutions of (1) arethe elements u in H such thatifi (vu,vv) dx + I u3v dx + I f (x)v dx = 0n^tfor all v E H. In other words, they are the critical points of the functional co defined on Hbyco(u) = -21 LI v ur dx + IL u4 dx + h f (x)u(x) dx.^(2)It is easy to see that cp is C1 and bounded below on H. Actually, for any u E H, we haveso(u) ? -21 Al I v ur dx — ill f (x)u(x)Idx ? -21 iluli2H — lifilH-111u11H.^(3)4^ INTRODUCTIONSince any global minimum of co is a critical point, therefore to find a critical point it is enoughto show that any minimizing sequence (u,i)fl is convergent in H. To do that, we note thatby (3), the sequence (u„,)„ is bounded in H, hence it is weakly precompact and we maytherefore assume that /in --+ it weakly for some it in H. The latter is necessarily a minimumsince V is weakly lower semi-continuous on H. By a slightly more involved argument, onecan actually show that the sequence actually converges in the norm topology.The above process of finding solutions by minimization is also known as the direct methodin the calculus of variations. Obviously, it cannot be used in the case when the functional isindefinite i.e. when co is neither bounded from below nor from above. Sometimes, a min-maxprocedure can be used to handle such functionals. Here is the most intuitive example:Consider a function co E Cl(R2,11t) and view co(x,y) as the altitude of the point(x, y, co(x, y)) in V. Suppose that there exist points ul, u2 E II/2 and a bounded open neigh-borhood ft of u1 such that u2 E R.2\1-2 and v(u) > max{co(ui), co(u2)} whenever u E aft (thisis the case for example if u1 is a strict minimum and u2 is a point with co(u2) < co(u1)). Ifwe look at the graph of cc in a topographical way, we then see that (ui, co(ui)) is in a valleysurrounded by a ring of mountains pictured by the set {(n, cc()); u E as2} and (u2, co(u2))is located outside of the mountain ring. To go from (u1, co(ui)) to (u2, v(u2)) in a way whichminimizes the highest altitude of the path, we must cross the mountain range through thelowest mountain pass. The projection on III2 of the top of the mountain pass will providea critical point of cc and the height of the top is the critical value. For example this is thecase for 0 = x2 + y2 — x3 — y3. Clearly (0, 0,0) is in a valley in the following graph of 0 .INTRODUCTION^ 5More precisely, let (p E Cl (X, R) where X is a Banach space and for two points ul, u2 EX, we setrtel,; = {f ([0, 1]);! E C([0,1], X) and f(0) = ui, f (1) = U2}.If there is a closed set F C X with ul, u2 0 F and maxlco(ui), w(u2)} < inf co(F) andF separates 74, u2 i.e. A n F 0 0 for any A E 1'uu'2, then c = infAEr:1 maxx€A So(x) is apotential critical value and the set K, = Ix E X; ç(x) = c,co'(x) = 0} is not empty moduloa compactness condition. This is known as the mountain pass theorem of Ambrosetti andRabinowitz [3]. Many generalizations and applications of this theorem have been givensince it first appeared. The saddle point theorem of Rabinowitz [35] is one of its higherdimensional generalizations.Here is a simple example where the mountain pass theorem is applicable.Consider the Dirichlet problem.1 - A u = luIP-2u + f in S2(4)t u = 0^on af22N^where S-/ E RN(N > 3) is a bounded domain and 2 < p <^= 2*. Let H = Hd(12)- N-2and assume that f E H-1 as usual. When p = 2*, the critical exponent in the Sobolevembedding, the above equation has an important origin. It originates from the study of socalled Yamabe problem in differential geometry [44]. It is also interesting to note that (4)does not necessarily have nontrivial solutions if f = 0 in this case (see [40]).As before, the weak solutions for this problem are the critical points of the functional1^1co(u) = — I I v u12 — — I kir — f fu2 n p SI^11defined on H. It is easy to see that co is not bounded below or above. To use the mountainpass theorem and to make things easy, let us assume that 2 < p < -2NN-2 and f = 0. First wenote that for any u E Hco(u ) = lliull - -1ilur >- 1114 - CliullPH2^p P 26^ INTRODUCTIONwhere C is a constant. Since p> 2, there is a r > 0 and a > 0 such that co(u) > a for allU E H with 11u11H = r. On the other hand fix a u 0 and let v = Ru we have thatco(Ru) = —21 R211u1131 1113Since p > 2, we see that there exists u2 E H with 11u211 > r such that co(u2) < 0 < a.Clearly the remaining thing to do is to check the (PS) compactness condition i.e. anysequence (x„)„ E H verifying limn,,, cp(xn) = c for some c E R and limn„ yd(xn) = 0 hasa convergent subsequence. This can be done in the following general fashion. Let (un)n bea (PS) sequence i.e. verifying limn,„, co(xn) = c for some c E R and limn, cd(xn) = 0.Then we may assume that 1v(un)1 < M and ii(P'(un)li < 1 for some M and all n > 1. Hencewe have1M^Co(n)^(un),u)1 1= —2Ilun112H — 111unlIC, — —111un1131 + -lungP^P= (-1 — —1nunlI2H-2 pHence (1 — 1)iiu 112H —111unI1H^M. Since 12p np^—< 2 > p, we see that un is bounded in H. Thenwe proceed as before to see that co verifies (PS) condition. Now by mountain-pass theoremwe see that (4) has a nontrivial weak solution.The above simple example has already demonstrated the beauty and the power of themountain pass theorem. One thing should be noted though. The strict inequality inf yo(F) >max{co(ui), yo(u2)} is always assumed. What can one say when it is an equality? In 1989,Ghoussoub and Preiss [21] generalized the mountain pass theorem to include the "limitingcase" where the above mentioned equality occurs. In the mountain pass theorem setting,their mountain pass principle says that when inf cp(F) = c, then there exists a sequence(xn)n in X verifying(i) limn, cp(xn) = c;(ii) lim,0 cd(xn) = 0;(iii) limn„ dist (xn , F) = 0.INTRODUCTION^ 7Precisely, a (PS) sequence with extra information is found. If co verifies (P S) Fc condition,i.e. every sequence verifying (i) (ii) (iii) above has a convergent subsequence, then IC, nF 0 0. As it is pointed out in [21], one can think the general mountain pass principle asthe counterpart of Ekeland's variational principle in the 1-dimensional min-max setting.This principle, besides giving a positive answer to the "limiting case" of mountain passtheorem and generating a (PS) sequence with extra topological information, also gives moreinformation about the location of critical points captured. This makes the study of thestructure of critical set generated by general mountain pass principle possible.Let us go back to the example of finding solutions of (4). When p = 2* and f 0 0, ithas been shown recently by Tarantello [41] [42] that it has at least two different nontrivialsolutions under some mild extra assumptions. The first solution is found by minimizing caover manifold M = Ix E H;(cp'(x),x) =01. It is important to note that the (PS) sequenceobtained in this way has the property of being close to the set M. As it turns out co verifies(PS)m,,. By the mountain pass theorem, she was able to find another critical point u2 for co.By the known structure of critical set generated by the mountain pass theorem, she actuallyfound a saddle point (of mountain pass type). Hence two different solutions are found for (4).For more details, see [41] or [20]. As in the mountain pass theorem case, the general mountainpass principle can be generalized to higher dimensions. This was done by Ghoussoub in [19].In doing so, a (PS) sequence with further topological properties is obtained. In the abovegeneral mountain pass principle setting, for any sequence (Fn)„ of closed sets separatingu1, and u2, and sequence An E I':21, if limn,„„ max (A) = c . limn_+. inf co(Fn) andJim dist(Fn, Iui , u21) >0, then there exists a sequence (xn)n in X verifying besides (i) andn—>oo(ii) above the following:(iii)' dist(xn, Fn) = 0;(iv) dist(xn, An) = 0.This result carries the idea of finding a (PS) sequence with additional information furtherand has many applications [20]. It also makes possible the study of the structure of thecritical set generated.8^ INTRODUCTIONWe now look at another interesting example. Consider the following boundary valueproblem.1 — A u — Au = g(u)t u = 0on 1/on ao(5)where 12 C RN is a smooth bounded domain and g(0) = 0. When g(u) = lu IP-2u withp = A, then the problem originates from the Yamabe problem as mentioned above. Theproblem is interesting as it is found by Brezis and Nirenberg [8] that the (PS) condition mayhold up to only a certain level c. Their work actually rekindles the interests in nonlinearelliptic equations that correspond to the critical Sobolev exponent. That case was leftaside mainly because of a dramatic counterexample due to Pohozaev (see [40]). Howeverby choosing suitable relevant set F, it is shown that co verifies (PS)F,„ with ci. > c. Thisis done by Cerami-Solimini-Struwe [10]; they actually found a sign changing solution asopposed to the first positive solution. Another interesting thing to note is that Tarantello[42] presented another method of finding the second sign changing solution. In her methodthe second order information satisfied by the obtained (PS) sequence is exploited.For a general g, however, one has to use the general min-max procedure and the knownproperties of the critical points generated by the min-max procedure to show the existenceof non-trivial solutions of (5). For more details, see [20] or [2].In chapter 1 we present a strong form of the min-max principle of Ghoussoub [19]. Someconcepts and notations are also included. Here one can find the Ekeland principle andthe Borwein-Preiss smooth variational principle which are mentioned above in their generalforms without proof. Some of the corollaries which are needed later are also stated andproved. In the last section of this chapter, we shall recall some topological results and provesome combinatorial and approximation lemmas. Those lemmas are needed in Chapter 5and they are of independent interest as well.In chapter 2, we give a detailed description of the critical set generated by the generalmountain pass principle of Ghoussoub-Preiss [21]. In the process, we extend and simplifythe structural results of Hofer [22] and Pucci-Serrin [32], [33], [34] in the context of theINTRODUCTION^ 9classical mountain pass theorem of Ambrosetti-Rabinowitz [3]. We will not be using Morsetheory and the functionals are only supposed to be Cl. The main result states that ifthe two "base points" are not critical, then one of the following three assertions about thecorresponding critical set holds true:(a) The set of proper local maxima contains a closed subset that separates the two basepoints;(b) There is a saddle point of mountain-pass type;(c) There is at least one pair of nonempty closed disjoint sets of saddle points that are alsolimiting points of local minima. These sets are connected through the set of local minimabut not through the set of saddle points.In Chapter 3, we continue to study the structure of critical set generated by the min-maxprinciple under the condition that go is CI on a Banach space and n > 2. In order to doso, we first introduce concepts of (weak) saddle type and co-saddle type points of order kin analogy to Hofer's critical point of mountain pass type. We then show that this type oforder of critical points generated by the min-max principle are related to the dimensions ofhomotopic or cohomotopic dimensions of the stable families used.In Chapter 4, we try to further study the topological properties of critical points gener-ated by the general min-max principle under the assumption that the functionals are C2 ona Hilbert space where the regular Morse theory is applicable. In the nondegenerate case, wefirst find the relations between the topological indices of critical points introduced in Chap-ter 3 and the standard Morse indices associated to such points. Hence we find the relationsbetween the Morse indices of the critical points generated and the homotopic (cohomotopic)dimension of the homotopy stable families used. For the degenerate case, we shall refine theperturbation methods introduced by Marino-Prodi [28], Solimini [38] and Ghoussoub [19]to get the appropriate estimates under suitable Fredholm type of conditions.In Chapter 5, we shall mainly concentrate on constructing (PS) sequences with Morsetype of information in both the homotopic case and the cohomotopic case. As mentioned10^ INTRODUCTIONabove, this is done in the minimization. This further information for (PS) sequence some-times is crucial to ensuring its precompactness as noted by P.L.Lions in his study of Hartree-Fock equations for coulomb systems. When the necessary (PS) condition is available, theresult implies the result obtained in Chapter 4 without the Fredholm type of conditions orthe non-degeneracy needed there.Chapter 1PreliminariesIn this preliminary chapter, we lay down the necessary background for the rest of thisdissertation. We first state a strong version of the min-max principle due to Ghoussoub,including some notations and definitions. We then state without proof Ekeland's variationalprinciple and Borwein-Preiss' smooth variational principle. After that we recall some neededtopological results and prove various combinatorial and approximation lemmas.1.1 A strong form of the min-max principleDEFINITION 1.1 Let B be a closed subset of a complete metric space (X, d). We shallsay that a class Y of compact subsets of X is a homotopy-stable family with boundary Bprovided:(a) every set in .7" contains B;(b) for any set A in .7- and any n E C([0, 1] X X; X) satisfying ?A x) = x for all (t, x) in({0} X X) U ([0,11 X B) we have that cil x A) E T.The above definition as well as all the statements proved below are still valid if theboundary B is empty, provided we follow the usual convention of defining sup(0) = —oo. Inthis case, we will just say that .7- is a homotopy-stable family.DEFINITION 1.2 Say that a closed set F is dual to Y if F verifies the following:(F1)^F n B = 0 and F n A 0 0 for all A in .F.1112^ 1. PRELIMINARIESDenote by .7.* a family of closed sets that are dual to .F and we say that .T* is a dual familyto T. Note that for such a dual family, we readily have thatc* := sup inf co(x) < inf max (x) .: C.FEY* wEF^AE.F xEAThe following theorem covers many of the standard results like the mountain pass andthe saddle point theorems. For its comprehensive applications, further generalizations andits variants, we refer to [19] and [20] for details.THEOREM 1.3 Let co be a C1-functional on a Banach space X. Consider a homotopy-stablefamily T of compact subsets of X with a closed boundary B and a dual family .7-* of T.Assume thatsup inf co(x) = inf max (x) = CFEY" xEF^AEF xEAand is finite. Then for any sequence of sets (An)n in .7" and a sequence (Fn)n in ,F* such thatlimn supxEA. cp(x) = c = limn infzEF„ Co(x) and Jim dist(Fn,B) > 0, there exists a sequencefl 00(xn)n in X such that(i) limw(xn) = c;(ii) limildcp(xn)11 = 0;(iii) lip dist(x„, Fn) = 0;(iv) limdist(xn, An) = 0.DEFINITION 1.4 A sequence (Fn)n in T* is said to be a suitable max-mining sequence inT* if limn inf yo(F) = c* and Jim dist(F„, B) > 0. A sequence (An)n in T is said to ben—>oomin-maxing in .7" if limn supxEA„ so(x) = c = c(V, -7.) •DEFINITION 1.5 Say that cp verifies (PS), (resp. (PS)F,,) (resp. (PS)F,, along a min-maxing sequence An E .7") (resp. (PS), along a min-maxing sequence An E .7" and asuitable max-mining sequence Fn E .F*) if every sequence (xn)n that verifies (i) and (ii)(resp. (i), (ii) and (iii) with Fn = F E .7.* ) (resp. (0, (ii), (iii) with Fn = F E T* and (iv))(resp. (i), (ii), (iii) and (iv)) above has a convergent subsequence.1.1. A strong form of mm-max principle^ 13Throughout this thesis, we shall denote by itl„„ the setA°. = Ix E X; limdist(x, An) = 0}and by Foc, the setI'm = Ix E X; limdist(x, F) = 0}.We shall denote by Kc the set of critical points at level c, i.e.,.K, = Ix E X; so(x) = c, clyo(x) = 0}.COROLLARY 1.6 Let X, co and .F be as in Theorem 1.3 and consider a family of sets .7'that is dual to T. Assume thatsup inf so(x) = inf max (p(x) = CFEY.* xEF^AEY xEAand is finite. If c9 verifies (PS), along a min-maxing sequence (A„)„ in .7- and a suitablemax-mining sequence (F)„ in .7', then there exists a sequence (x„)„ with x„ E X thatconverges to a point in A, n Foo n K, 0 0.Roughly speaking, the above corollary implies that, whereas the min-max procedure on.7. determines the critical level of so, the max-min procedure on .F* locates the critical pointson that level.COROLLARY 1.7 Let X, yo and Y be as in Theorem 1.3 and consider a family of sets ..T*that is dual to .T. Assume that F E .F* verifying:(F2)^inf cp(F) ?_ c = c(so, .7") = AIS InxEalc 99(4If co verifies (PS), along a min-maxing sequence (A,,)„ in .T, then there exists a sequence(x„),,, with x„ E X that converges to a point in Ac,„ n F n K, 0 0.The above Theorem 1.3 is actually a corollary of the following quantitative version ofTheorem 1.3.14^ 1. PRELIMINARIESTHEOREM 1.8 Let X, co, B, c and .7. be as in Theorem 1.3. Let F be a closed set satisfying(F1) and(F2)8^ inf cp(F) > c — (5.Suppose 0 < 8 < A dist2(B, F), then for any A in .7- satisfying max co(A) < c ± S, thereexists x8 E X such that(i) c — 8 _< yo(x8) < c + 98;(ii) lith,o(x8)II _< 18-13;(iii) dist(xs, F) < 5-V-S;(iv) dist(xo, A) < 3V75.DEFINITION 1.9 Suppose a = Ua -Fa where each Ya is homotopy stable with boundary Ba.We then say that a is generalized homotopy stable with boundary Q1 = UaBa. We say aclosed set F is generalized dual to a if F verifies the following:F n Bo =0 for all a and F 11 A $0 for all A E a.Denote by a. a family of closed sets that are generalized dual to a.Now we have the following generalized min-max principle which follows directly fromTheorem 1.8.THEOREM 1.10 Let yo be a C'-functional on a Banach space X. Consider a generalizedhomotopy-stable family a = lia.ra with boundary Q3 = UaBa and a generalized dual familya. of t. Assume thatsup inf cp(x) = inf max 92(x) = cFEa. xEF^AE xEAand is finite. Then for any sequence of sets (An)„ in a and a sequence (Fn),, in such thatlimn sup v(An) = c = limn inf v(Fn) and inf„ lim dist(Fn, Ba) > 0, there exists a sequencen-ioo(xa)a in X such that(i) lim cp(xn) = c;1.2. Lipschitz and smooth perturbed minimizations^ 15(ii) urnIldv(x„)11 = 0;(iii) dist(x„, F) = 0;(iv) lim dist(x„, An) = 0.DEFINITION 1.11 A sequence (Fn)n in a. is said to be a generalized suitable max-miningsequence in a. if limn,0 inf v(Fn) = c* and infa Jim dist(Fn, Ba) > 0.n—oo1.2 Lipschitz and smooth perturbed minimizationsThe following theorem is the well-known Ekeland's variational principle [13].THEOREM 1.1.2 Let (X, d) be a complete metric space and consider a function go : X(—oo,±oo] that is lower semi-continuous, bounded from below and not identical to +co.Let e > 0 and A > 0 be given and let u E X be such that co(u) < infx^c. Then thereexists v, E X such thatgo(v)^W(n);(ii) d(u,v,) <11A;(iii) For each w ve in X, co(w) > co(vc) — cAd(vE,w).The proof of this theorem can be found in many papers and books [13] or [20]. We shallnot give it here. We shall rather mention a few corollaries that will be useful to us later on.COROLLARY 1.13 Let X be a Banach space and let co : X -- • t be a function boundedfrom below and differentiable on X. Then, for each minimizing sequence (uk)k for Co, thereexists a minimizing sequence (vk)k for co such that co(vk) < co(uk), limk link — vkll = 0 andlimb = 0.Proof: If (uk)k is a minimizing sequence for go, takeV(uk) — infx^if co(ub) — infx > 0=1/k^if co(nk) — infx go= 0.16^ 1. PRELIMINARIESClearly for each k, we have that ek > 0 and (p(uk) < infx co + Ek• By Theorem 1.12 withViT,A = 1/Nf^ geT, we can find vk E X such that co(vk) < uk), liuk — vkll <^andv(w) > so(vk ) — N/le II vk — wit^ (* )for all w 0 vk in X. By applying (*) to w = vk + th with t > 0, h E X, VIII = 1, weget co(vk + th) — Co(vk) > —1/FT,t. Dividing both sides by t and letting t^0, we obtain—Are; 5_ ((p'(vk), h) for all h E X with 011 = 1. This proves the corollary. •The following corollary show that the (PS) condition is quite restrictive. In particular,it forces the function to be coercive„ i.e., 1iminf11u11,,3 w(u) = oo.COROLLARY 1.14 Let w be C1-functional on a Banach space X.(i) If co is bounded below and verifies (PS), with c = infx w, then every minimizingsequence for w is relatively compact. In particular, w achieves its minimum at a point inK.(ii) If d = liminfiltill_,. co(u) is finite, then w does not verify (PS)d•Proof: (i) follows from Corollary 1.13. For (ii), we shall show the existence of a sequence(un)„ in X such that Ijunil --4 cc, ç(n) —* d and 11(p'(un)11 -- 0.For that, define for r > 0 the functionm(r) = inf co(u).11uPrClearly m(r) is nondecreasing and limr,,, m(r) = d. For e < 1/2, find ro > 1/c such thatd — e2 < m(r) for r > 7.0, then choose uo with Iluoll > 27-0 such thatCo(u0) < m(2r0) + e2.Apply now Ekeland's theorem in the region D = {Hull > rol, to find vo with Poll > ro suchthatw(vo) 5_ w(u) — Ellu — volt for all u E D.1.2. Lipschitz and smooth perturbed minimizations^ 17It follows thatd — €2 < m(ro) < co(vo)^co(uo) — Elluo yolkHence Iluo — voll < 2E and Poll > ro. Since vo belongs to the interior of the region D,the argument in Corollary 1.13 gives that 11cd(v0)11 < E. This clearly proves (ii) and thecorollary. •Next we state a so-called smooth perturbed minimization principle which is due toBorwein-Preiss [7]. First, for (X, d) a complete metric space, consider the class 0 of allreal-valued functions on X of the form008(x) = 1/2^pnd(x, Vn)2n=1where pn > 0,^= 1 and where (vn)n is some convergent sequence in X.THEOREM 1.15 Let (X, d) be a complete metric space and let co : X -4 (— oo , +oo] be alower semi-continuous function that is bounded from below and not identical to +co. Fix> 0, A> 0 and assume that u E X satisfies co(u) < infx go + 1. Then there exist 8 E 0and II, such that(i) co(v,) < infx + c;(ii) d(u, vf) < A;(iii) For all x E X,co(x) + 2E/ A20(x) go(v) + 26/ A20(ve).Moreover, if X is a Banach space with a smooth norm (away from the origin) and if cp isdifferentiable, then licel(v,)11 < 2E/A.REMARK 1.16 In general for a reflexive Banach space, the perturbation 8 can be taken tobe of the form 0(x) = 1/211x — w112 for some w usually not equal to v€.The following corollary then is immediate.COROLLARY 1.17 Let go be a C2-functional that is bounded below on a Hilbert space H.Then, for each minimizing sequence (uk)k of v, there exists a minimizing sequence (vk)k ofgo such that18^ 1. PRELIMINARIES(i ) limk i I uk — vk II = 0;(ii) limk kg (vk)11 = 0;(iii) liminfk(e(vk)w,w) > 0 for any w E H.For the proof of Theorem 1.15, we refer to [19]. Next we recall the well-known deforma-tion lemma.LEMMA 1.18 (Deformation Lemma) Let X be a real Banach space and let co E Cl (X, R)and satisfy (PS), where c E R. If T > 0, and 0 is any neighborhood of IC", (0 may be emptyif K, = 0), then there exists an e E (0,0 and 77 E C([0,1] X X, X) such that1° 77(0,u) = u for all E X;2° ?At, U) = u for all t E [0,1] if co(u) c$ [c — "e, c + E];30 g(t,u) is a homeomorphism of X onto X for each t E [0,1];4° II71(t, u) —u 5_ 1 for all t E [0,1] and u E X;5° co(ij(t,u)) _< cp(u) for all t E [0,1] and u E X;6° 77(1,Gc+E\O) C Gc--e;70 If IC, = 0, n(i,Gc+) c Ge_c•As a corollary to the above deformation lemma, we have the following.COROLLARY 1.19 Let D be a subset of a Banach space X and co E Cl (X, IR). Let 6 < 1/5be a positive real number and let -6, c E IR with è + 63 < c. Assume that for any y E Xwith dist(y,D) < 6, either Iv(y) — cl < 6 or Ildco(Y)ii > 6. Then for any D1 C X with(D1)28C D, there is a homeomorphism n : X —> X such that(1) co(i(x)) 5. c,o(x) and dist(x, ri(x)) < 6 for all x E X;(2) 77(x) = x for all x E Ix; co(x) 5_ 61 U (X\D);(3) co(77(x)) 5_ max{co(x) — 0, c — 63} for all x E D1 such that v(x) < c + 6.1.3. Combinatorial, topological and approximation lemmas^ 191.3 Combinatorial, topological and approximation lemmasIn this section, we isolate some of the topological tools that are needed later on. First, werecall the following results from dimension theory which can be found in the book of Nagata[30].DEFINITION 1.20 The topological dimension (or covering dimension) of a metric space D(in short, topdim D) is the least integer m such that the following property holds: for anyfinite open covering 0 of D, there is an open covering 01 refining 0 such that any p E Dbelongs to at most m -I- 1 elements of 01.The following theorem summarizes the properties of topological dimension that will beneeded in the sequel.THEOREM 1.21 Let X be a metric space. Then the following holds:i) topdim Xi < topdim X for any subspace X1 of X;ii) If X has a finite covering consisting of closed sets {Xi; i E NI with topdim Xi < m,then topdim X < m;iii) topdim Rm = m.The following basic theorem is well known. It relates the topological dimension of aspace to certain extension properties for non-linear mappings into euclidian spheres.THEOREM 1.22 A metric space X has a topological dimension at most m if and only if forevery dosed subset Xi C X and every continuous mapping f of X1 into Sm (the standardm—sphere in Rm ) there is a continuous extension f off to all of X.The following corollary will be frequently used in the sequel.COROLLARY 1.23 Let V be a Banach space, K a closed subset of Rn and let 9 be a contin-uous mapping from K into the unit sphere Sv of V. If n < dim V, then 9 can be extendedto a continuous mapping from RI' into Sy.20^ I. PRELIMINARIESThe following combinatorial result will be crucial for the proofs in Chapter 5.LEMMA 1.24 For each n E N, there is an integer N(n) < (203---1- 1 + 2)" such that forany compact subset D C IV and any > 0, there exists a finite number of distinct points{x1;1 < i < k} with the following properties:(i) D ç u1B(x1;e/4) c Neo(D);00 The intersection of any distinct N(n) elements of the cover ( .6.- (Xi, 12))!F_i is empty;iii) For any compact set D1 c IV containing D, there are points tyi; 1 < i < kil suchthatD1 C U1B(x1;E/4) U UiB(Yi; ER) g. Ne/2(D1).Moreover, the intersection of any distinct N(n) elements of the cover (B(xj,6/2))i and(R(yi,e /2))i is empty.Proof: Let Q = =^E Zn C^Clearly, the family of balls of the form{B(2^ l, i); / E Q} will cover R". By the compactness of D, there are xi, xk E R" suchthat D C 4_1.13(xi;e/4) C NE/2(D). Note now that for each / E Q, B^i) intersectsat most (2 711---1- 1 + 2)" distinct balls of that type. Assertion (iii) follows from the sameconstruction.^ •The following notions turned out to be relevant for the proof in the cohomotopic case inChapter 5.DEFINITION 1.25 Let (X, 6) be a metric space.i) Say that a finite sequence of points (ui)1 in X is (C, r)-metrically ordered for someconstants C > 0 and r > 0 if for any 1 < ii <i2 < <z <M, such that 6(uim,ui„,+,) <rfor all 1 < m < k — 1, we have that (5(ui„ui,) < Cr.ii) Say that (X, 6) has C -orderly pavings for some C > 0, if for any compact subset D C Xand any r1 > r2 > 0, there is a (C,ri) -metrically ordered finite sequence ul, ..., um E D suchthat D C eiB(ui , r2).1.3. Combinatorial, topological and approximation lemmas^ 21Note that any ultrametric space (X, (5) (i.e., 5(x, z) < max{(5(x, y), 5(y, z)}) clearly has 1-orderly pavings. However, we are interested in the sequel in Hilbert spaces and the followinglemma turns out to be crucial for our proof of the lower estimate in the cohomotopic case.We would like to thank Guoling Wang for some insightful ideas regarding its proof.LEMMA 1.26 For each n E N, there is a constant C(n) < 2"+' ( \Ft + 2) such that IR" hasC(n)-orderly pavings.Proof: For r1 > r2 > 0, take 0 < a < and consider the lattice L containing 0 and suchthat a is the distance between two neighboring points. Clearly ir c u,E,B(t, r2/2) since(2# < r2/2. If D is a compact subset of IR", there exist then M distinct points (ua)„Ei inL such thatD C U„,E1B(u, r2/2) and D n r2/2) 0 0 for every a E I.We will show that (u„ )au can be ordered to become a (2n+1(Vii,+1), 2r1)-metrically orderedsequence ?LI, .., um. Once this is done, we can just pick any u E B(Iti, r2/2)nD (1 < < m.),to get that D c r2) while u'm is now (2n+2(V12+ 2), ri)- metrically ordered.To proceed with the ordering, we first associate to each t E L, its coordinates (iia, ..., Ea a)where fi E N. For simplicity, we shall write (els, ..., t) for the components of the element u.Start by letting u1 be the point such that er < tr` for all a E I and 1 < i < n. Letnow r be the smallest integer such that r > 211 and consider for any n- tuple of integers— a—, in) E N", the sets^E L;^+ jnir < in, <^+ (jr„ + 1)r for 1 < m < n}.Clearly ua E^joErsin^for all a E I and^fl^= 0 unless ji = j: for all1 < i < n.We now order =^ in the following way:For el = (4' ,^ten' ) and e2 = (42, ...,^in the same Lii,...,in , we shall say that el^e2if and only if tei </2 where J is the first integer j (1 < j < N) such that P.,' 0 £2.22^ 1. PRELIMINARIESNext if el E L31^ and e2 E^then say that el^e2 if and only the followingholds at the level p defined as the first integer i (1 < i < n) where ji X:(i) Either 4 is odd and jp is even,(ii) or both have the same parity and 4 > jp.This is clearly a total order on r, and therefore we can label (ua)„Ei in such a way thatit is increasing for that order, so that we obtain a sequence ul, ..., um such that ui uiwhenever 1 < i < j < M.We now show that this sequence is 2n+1(Aiii+1),2r1)-metrically ordered. Note first thatfor any el, e2 E^we have that lel —e21 < (r — 1)a Agi < 2riVT-t, while if el Ee2 E^and if — > 2 for some 1 < i < n, we then have lei — e2I > (r + 1)a > 2r1.Suppose now luim — uim+, I <2r1 for all 1 < m < k — 1. Since we also have ui,uik, then either ui„^ui, belongs to the same L31 ^in this case lui, —I <—^< 2r1 /Ti, or we may assume that for some q (1 < q < k),ui„^E^and nig+, EIt is clear that lui —N1_11 < 2r1fri+ 2r1. We now start again with the sequence {uit; q +1 <t < k} until we get uik E We need to estimate how many steps are needed and forthat, we start by noting that ji, j„ cannot all be odd, unless it is the last block, i.e., theone containing wik• Indeed, otherwise j , ...,A in the next one will all be odd, which meansthat IA — j4 > 2 for some 1 < i < n, so that luil — n1 > 2ri which is a contradiction.This observation, combined with the (lexicographical) nature of the order, yield that thereare at most 2n — 1 sets of the form between and Ljc,...,g• It follows thatlui, — uik I < (2n — 1) (2ri^+ 2r1) = 2+1( /+ 1)r1which proves the lemma.The last lemma of this section is concerned with the rate of approximation of Lipschitzfunctions by C3-functions in a finite dimensional space.1.3. Combinatorial, topological and approximation lemmas^ 23LEMMA 1.27 For each integer n, there is a constant N := (n) such that for any subsetF C1Rn and p> 0, there exists a C3-function SF :^ k that satisfies:{1 x E F(i) SF(x) =0 x E R" \Art,(F)(ii) 114(x)I1^Ili, I 14,(x)11^-11+72-,1157(x)11^13.Proof: Let a(x) = dist(x , F). Set v = p/3. The function0(x) =^v^1}}a(x) (1)equals 0 if (x) < v; 0(x) = 1 if a(x) > 2v, and 0 < p < 1. Let j(x) be a non-negative C°°function with support in the unit ball and such thatj(x)dx = 1.^ (2)For some constant C, depending only on nf^f vi, f liml^ (3)The functiong(x) =^j(x^ )13(y)dy = Jj(z)/3(x — vz)dz^(4)is in C°° and it satisfies:0 < g < 1, andJ g(x) = 0 if x E Fg(x) = 1 if dist(x, F) > 3v = p.Furthermore, by the first expression in (4),Ig' l^Ig' l^Ig' lNow take SF =1 - g. This clearly proves the lemma.^ •Chapter 2Structure of the critical set in the mountain pass theoremIn this chapter we study the structure of the critical set generated by the general mountainpass principle of Ghoussoub-Preiss. In the process, we extend and simplify the structuralresults of Pucci-Serrin in the context of the classical mountain pass theorem of Ambrosetti-Rabinowitz.2.1 IntroductionWe start by recalling the statement of the mountain pass theorem of Ambrosetti-Rabinowitz[3].THEOREM 2.1 (Mountain Pass Theorem) Let co be a real-valued C'-functional on a Banachspace X that satisfies the Pal ais-Smale condition. Suppose that there exist two real numbersb and R> 0, such that cc(x) > b whenever IA = R while co(0) <b and v(e) < b for somee E X with Hell > R. Then, with F denoting the class of continuous paths joining 0 and e,the numberc = inf max yo(g(t))gEr tEN,11is a critical value of yo, i.e. the set K, = Ix E X; w(x) = c, co'(x) = 0} is nonempty.Note that the above hypothesis implies that c > b> a = max(co(0), v(e)). To cover thecase when c = b = a, Ghoussoub and Preiss established in [21] a general mountain passprinciple which extends the above theorem. Before stating it, we shall recall the followingdefinition.242.1. Introduction^ 25DEFINITION 2.2 A closed subset H of a Banach space X is said to separate two points uand v in X if u and v belong to disjoint connected components of X\H .We also denote by r the set of all continuous paths joining u and v, that is:rvi = fg E C([0, 11; X); g(0) = u and g(1) = v}where C([0, 1]; X) is the space of all X-valued continuous functions on [0, 1].THEOREM 2.3 (Genera) Mountain pass principle) Let yo : X —> R be a continuous, Gateaux-differentiable function on a Banach space X such that cp' : X -4 X* is continuous from thenorm topology of X to the weale-topology of X*. Take two points u and v in X and considerthe numberc = inf max cp(g(t)).gErL. o<t<1Suppose F is a closed subset of X such that F n Ix E X; (p(x) > c} separates u and v. Thenthere exists a sequence (x„)„ in X verifying the following:(i) limn dist(xn, F) = 0;(ii) limn cp(x,i) = c;(iii) limn Ilcd(x.) II = 0-Moreover, if co verifies (PS)F,,, then F n Ke 0 0.Besides giving a positive answer to the "limiting case", the above result gives someinformation about the location of the critical points. By making appropriate choices forthe separating set F, Ghoussoub and Preiss [21] used their existence theorem to extendand simplify some results of Hofer [22] and Pucci-Serrin [32], [33] about the structure of thecritical set. Observe also that both Theorem 2.1 and Theorem 2.3 are corollaries of Theorem1.3 in Chapter 1. In this chapter, we shall simplify and extend the other finer results ofPucci-Serrin [34] by using the methodology of Ghoussoub-Preiss. Throughout this chapter,we denote by B(x, e) the open ball in a metric space (X, d) which is centered at x and withradius E > 0.26^2. STRUCTURE OF CRITICAL SET IN MOUNTAIN PASS THEOREM2.2 Two topological lemmasThis section is solely devoted to the proof of two topological lemmas which are needed inthe proof of the main result of this chapter. We start with the following lemma. We give aproof for completeness since we can not find a reference.LEMMA 2.4 Let M be a subset of a metric space (X, d). Suppose M = M1 U M2 andn M2 = 0. If M1 is both open and closed relative to the subspace M, then there existopen sets DI, D2 Of X such thatM1 C D1^M2 C D2,^n D2 = O.Proof: Since M1 is both relatively open and closed, so is M2. Hence there exist open setsE2 such thatPutThenM1 C H1,^M2 C H2,^n = O.Next, since E1 is open, for each x E aE2 n E,3 Ex > 0 such that the ball B(x, cx) centeredat x with radius Ex is contained in El. LetThen ex > cz > 0. PutSimilarly, we have= dist(x, °El n E2).=^U B(x,ex/4)).xEaE211E1D2 = H2 U^U B(y,414)).yEaEinE2M1 C El^M2 C E2^(El n E2) n m = O.Hi = E1\ (E1 n E2^H2 E2 \(E1 n E2).2.2. Two topological lemmas^ 27Clearly,c^M2 C D2and Di D2 are open. We now claim that Di n D2 = 0. Indeed, if not, say z E D1 fl D2, thenthere exist B(x, ex /4), B(y, ey / 4) such thatz E B(x, cis/4) n B(y, ey/4).Sod(x, y) < d(x, z) d(z, y) ex/4 + 4/4 < max(ex, ey).On the other hand,ex < d(x, y),^ey d(x, y)which imply d(x , y) > max(,, ey). A contradiction which completes the proof of thelemma.^ •For the sequel, we shall need the following concept:DEFINITION 2.5 For A, B two disjoint subsets of X and any nonempty subset C of X, wesay that A, B are connected through C if there is no FCCUAUB relatively both closedand open such that A C F and F n B = (11.When A and B are connected through C, we also say C connects A and B or spaceCUAUB is said to be connected between A and B. We refer to [25], p.142-148 for details.LEMMA 2.6 Let Si (i = 1,2,- • • , n) be n mutually disjoint compact subsets of a metricspace (X, d) and let M be any nonempty subset of X. If for all i, j (i j i j = 1, 2, • • , n),the sets Si n M and Si n m are not connected through M, then there are n mutually disjointopen sets Ni (i = 1, 2, • • , n) such thatM u(u1Si)cu1Ni and Si C Ni for all i = 1,2, • • • , n.^(L1)ThenandOinPi=0,^M8uMco1uPi.Put for each i (i = 1, 2, • • • , n)oi = oin(nj=1(L3)(L4)(L5)(L6)28^ 2. STRUCTURE OF CRITICAL SET IN MOUNTAIN PASS THEOREMProof: For each i (i = 1, 2, • • • , n), we denote by Mi the compact set Si n M. Since by^assumption none of the pairs Mt, Mi (i j^j = 1,2, • • • , n) are connected through M,there exist by Lemma 2.4 open sets Oi; and Pii(Oii = P1 , i^j i , j = 1, 2, • • • , n) suchthatmi c oia,^mi c Pii,^Oq n Pij =^(i5Ej i,j =1,2,• • • , n)andMi U M U^C Oij U Pii^(i j i,j = 1,2, • • • , n).For each i (i = 1, 2, • • • , n), let^Oi = n Oii,^=^Pi;^ (L2)andThen by (L2)—(L5), we havemi c oi,^0in03=0^(i^= 1,2, • • • ,n).^(L7)It is not generally true that M, U M CU1Oi. In order to prove the lemma, we letM' = (M, u M)\(u10t),^m" = (M, u M) nThenmsum=itrum",^m'nm"=0.^(L8)2.2. Two topological lemmas^ 29By (L5) and (L6), we see that M" is both open and closed relative to Ms U M. Again byLemma 2.4, there exist two open sets D' and D" such thatC D',^M" C D",^D' n D" = 0.^(L9)Now for each i (i = 1, 2, • • • ,n) put OiD = Oi n D". By (L7) and (L9), thenD' n (u,o)L) = 0,^mi c ob,^ob n^=^( i^= 1,2,• • • ,n). (L10)By the compactness of Si and Mi, we may introduceai = dist(Mi, X\OiD) > 0,1(51 = —2 minIdist(Si,Si); i j i,j=1,2,•••,n} >O.(L11)Let 52 = min{ai,Si; i = 1,2,• • • , n} andQi = fx E X; dist(x, Mi) < S21,^S = Si\Qi.^(L12)ThenQçO ,^sign1 7 = O.By (L11), we see thatdist(Sqi,^> dist(4^— 6.2 > 36.2•By the compactness of Si l we may also introduceqPut1bi = — dist(Si' M) > 0,4^qS3 = min{b1,(52; i = 1,2,• • • ,n} >0.andThenP = {x e x; dist(x, M) <83}= Qi U (Ob n P),^R' = D' n P.M' C R',^M" C30^ 2. STRUCTURE OF CRITICAL SET IN MOUNTAIN PASS THEOREMBy (L10), we have that R' n (Li1Ni) = 0 and Ni n N = (i^j i,j = 1, 2, • • • , n).Furthermoredist(Sqi, P) > dist(Sqi,^— (53 > 36.^ (L16)Hencedist(Sqi, R') > dist(Sqi , P) > dist(S , M) — 53 > 353.^(L17)By (L14) and (L16), we also have thatdist(Sqi, NJ) > minfdist(Sig, Qi), dist (Sqi , P)} > min(362, 353) > 353.^(L18)Now letN1 =N1 U E X; dist(x,Sql) < 631 U R',Ni =^U^E X; dist(x, Sig) < (531 (i^1 i = 1,2, • • • ,n).By (L8), (L12) and (115) it follows thatScN,^M C^(L19)By (L10), (L17) and (L18), we see thatNi n Ni =^(i j i,j =^,n).^(L20)So (L19) and (L20) imply that Ni satisfy (L1) and this completes the proof of the lemma. •LEMMA 2.7 Let (X, d) be a locally connected metric space and let F0 be a closed subset ofX that separates two distinct points u and v. Let Zi (i = 1, 2,•• • , n) be n mutually disjointopen subsets of X such that u,v Let G be an open subset of X\F0 and denoteby Yi = ZIG. Then the following is true:(1) The setF1 = [Fo\(vizi)] U (UTI-10Yi)^(LL1)separates u and v.2.2. Two topological lemmas^ 31(2) If Ai (i = 1,2, ••• ,n) are n nonempty connected components of G and Ti (i^1, 2, • • • , n) are relatively open subsets of Zi n OA, such that Ti n^0 for any connectedcomponent L of G with L Ai, thenF2 = [F0VU7-1Z)] U U7-1(.911\71)^ (LL2)also separates u and v.Proof: (1) Since G C X\Fo, we have[F0vu07i)] u^ (LL3)Suppose F1 does not separates u and v. Then there exists a component of X\Fi, say W,which contains both u and v. Clearly either W C Yi for some i (1 < i < n) or W CXVU07). Since u, v U137i, so W C XVU:Li-Z). By (LL3), we see that W C X\F0.Clearly W is a connected subset of a component of X\F0. This contradicts the initialassumption that F0 separates u and v. (1) is proved.(2) Denote by Uig (a E Ei i = 1, 2, • • • ,n) all the components of XV1 such thatuicnTio^(LL4)and Uy(v E r) all the components of X\FI such that-07,117i=0^for all i = 1, 2, ••• , n.^(LL5)Clearly Ti c aYi, F1 = F2 U (UiTi) and F2 n (U:LiTi) = 0. So we have that= (141_1 U U1j U U^(LL6)EE;^vErandX\F2 =^(Ti U U Uig) U U Uv.^ (LL7)aEE;^vErSince X is locally connected and X \FI is open, so Uv, Uig (a E Ei, v E r) are all open. Nextfor each i (i = 1,2, • • , n), let Vi be the union of all the components of X \FI whose closure32^ 2. STRUCTURE OF CRITICAL SET IN MOUNTAIN PASS THEOREMintersect 7, that is Vi = UcrEE,Uiff . We also let^(1 < i1 < n 1 = 1,2, • • • , Li) be all those7i's(1 < j < n) which intersect 1-7. and letvLi =^u (uf'21 1 T11).We shall first prove the following claims:(a) For each Uiff(o E Ei^= 1, 2, • • • , n), either Uiff C Yi\aYi or Ai C Ui„ and Uiff^=0. Hence we may denote by U1ff,1 the component Uiff for which Ai C Uic and by Ui,,2 thecomponent Uig, for which Ui„ C YAOYi.(b) For each v(v E r ) , U1, is a component of X\F2.(c) For each i (i = 1, 2, • • • , n), VLi is also a component of X\F2. Moreover, U1,,17L1 (v EI' i = 1, 2, • • • ,n) are all the components of X\F2.Proof: (a) By (LL4), we know that U T 0. Let xi E E7 ^T, CZ1 n aAi. Since Ti isopen relative to Zi n 0,41, there exists a ball B(xi, co) (€o > 0) such thatB(xi, €0) C Zi,^B(x1 , co) n OA, cWrite Tzi = B(xi, co) n aAi. We then have thatB(x1,c0)\Txi = [B(xt , co) n (z1\-47)] u [B(x1 , co) n Ai].Hence Uiff must intersect either Zi\T or A. If Ai nui, 0, then Ai C Uiff . Since Ai nYi = 0we see that vi, n = 0. If Ai n ui, = 0, then Ui„ n B(xi, co) n (Z1\71.;) 0 0. Put U1 =n B(x1 , co) n If U1 n X\G 0, then Ui n (ZAG) 0 0. Hence U1 flY2 0.So U1 c YAOYi. We are done. If Ui C G, then Ui C G\Ai. So there is component Lof G with L Ai such that L n U1 0 0. Hence either U1 C L or Ui n az, 0. SinceU2nT2=ri:nT1o0 and OL nTi = 0 by assumption, we see ulna 00. HenceUi n X\G 0 i.e. Uia c (a) is proved.(b) From (LL4), (LL5) and (LL7), we see that for each v(v E r)u, n (x\F2)^n [u1(Ti u ayEi Ulu) U Uv] = Uv.vEr2.2. Two topological lemmas^ 33Since U„ is open, so U„ is both open and closed relative to X\F2. Thus U,, is a componentof X\F2.(c) We prove (c) in two steps.Step 1: VL, is connected for each i (i = 1, 2, • • • ,n).By (LL4) and the definition of VL,i, we know that= v u (uLiTi,) (U ui0) u (ULITit)*aEEiBy (a), we can write VL,, as follows:denoteVLi = U 147,1)^U Uia,2) U^= Vi,1 Vi,2 (UL1Ti1).^(LL8)aEE;^crEEiAgain from (a), we know that Ai C U, ,1 for all a E Ei. So Via_ is a component of X\Fi.Since n Ti, 0 for all 1 = 1, 2, • • , L2 by definition of V , we see that C Via. From(LL8), we see that VL, is connected. Also by (a) we have that Yi,2 CStep 2: VL, is a component of X\F2 for all i = 1, 2, • • • , n.In this step, we first show the following:For all j i i,j = 1, 2, • • • , n, either VL„fl Vi,, = 0 or VL, = V.Li •^(LL9)To see this, as in (LL8), we can write VL; as follows:LiVLj= U UIf VL., n VL, 0, then we have the following possibilities:(i) n V^0, or V , i n V ,20 0, or^fln^0 0 for some 1 < 11 < L.(ii) Vj,2 n V ,1 0, or Vj,2 n Vi,2 0 0) or Vj,2 n^O 0 for some 1 < 12 < L.(iii) There are l' ,1" 1 < < Lj 1 <1" < Li such that 71, n Ti,„ 0 0.Case (i): If Via fl Via^0, then Via = Via since Via and Via are both open. HenceVL = VL; by the connectness of VL., and VL,. If Via fl Ti,i^0 for some 1 < 11 < Li, thenC Vi,,, by (a). So^= Via.  Hence VL., = VL,. Observe Via and Vi,2 are also open.Since Via n^= 0 and Vi,2 c (Yi\81'),37, n^= 0 by (a), we see that Via fl V,2^0 isimpossible.34^ 2. STRUCTURE OF CRITICAL SET IN MOUNTAIN PASS THEOREMCase (ii): As in Case (i), 172 n Vo 0 0 is impossible. Since 1'i,2 C (Y.A0Yi) andV1,2 C (Y1\ O)) by (a), so Vi,2fl Vi,2 0 is also impossible. If Vi,2 n Ti 0 0, then we havej = i,2 . Hence Vi,2 = Vii2 ,1 • So we have VL„ = VLi•Case (iii): Since the n^= 0, this is impossible except when^= Ti,„. But thenAjo = Ai,„ C^n V1, by (a). So Vi,i = V1,1. Hence VL, = VL,. In all the cases, wehave VL„ = VL; if t' ^0 0. So (LL9) is true. Next we let V be the union of all thecomponents of X\F2 given in (b), i.e. V = U,,ErUy. By (LL7) we have thatX\F2 = V U U1(Vi U Ti) = V U UlVLI.^(LL10)By (LL9), we havevLi n (x\F2) = (vL, n v) u [n (u.7=1v0] =So VL, is closed relative to X\F2. Note that by (LL6), we have V n (x\Fi) = V. Hence by(LL5) and (LL7) we have thatn (x\F2) = V.So V is closed relative to X\F2. Since (X\F2)\VL1 is a union of V and a finite number ofVLi's, we see that (X\F2)\VL, is also closed relative to X\F2. So VL, is also open relative toX\F2. Hence VL, is a component of X\F2. It is clear from (LL10) that U, VL , (v E i =1, 2, • • • , n) are all the components of X\F2. (c) is thus proved.Suppose F2 does not separates u, v. Then by (b) and (c) or (LL10), we see that theremust be a VL,, a component of X\F2, which contains both u and v. As in the proof of(c), we get from (LL8) and (a) that u, v E Vo. Because V1,1 is a component of XVI, thiscontradicts the fact that F1 separates u, v. Lemma 2.7 is proved. •2.3 Main theoremsIn this section, we are going to present the main result of this chapter. Let us first introducesome notations. Throughout this section, X will be a fixed real Banach space and co :2.3. Main theorems^ 35X -4 R a Gateaux-differentiable function on X. As before we denote by Nf(F) = Ix EX; dist(x, F) <c}. We also let^G, =^E X; go(x) < c},L, = fx E X; C0(x)^=^E Kc; x is a local minimum of yob= fx E Kc; x is a proper local maximum of co,that is x is a local maximum of cp and x Epcm^{x E^x is a local maximum of 4^Sc =^E Kc; x is a saddle point of co, that is in each neighborhoodof x there exist two points y and z such that co(y) < cp(x) < co(z)}Following Hofer [22], we say that a point x in IC, is of mountain-pass type if for anyneighborhood N of x, the set {x E N; co(x) < c} is nonempty and not path connected. Weshall need the following theorem established in [21].THEOREM 2.8 Let X, go, {u, v},c and F be as in Theorem 2.3. Assume co verifies (PS)F,c•If F n P,,,„ contains no compact set that separates u and v, then:(a) Either F n M, 0, or(b) F n ice contains a saddle point.We shall also need the following corollary.COROLLARY 2.9 Let X,v,{u,v},c and F be as in Theorem 2.3. Assume go verifies(PS)N,(FuKc),, and that u,v M. If P, does not contain a compact subset that separatesu and v, then S, 0.Proof: First observe that IC, is the disjoint union of Sc, Mc and P. By the (PS)Ne(Fulco,ccondition, we know that K, is compact. Suppose 5', = 0. For each x E Mc, there exists a36^ 2. STRUCTURE OF CRITICAL SET IN MOUNTAIN PASS THEOREMB(x, ex) such that B(x, ex) C L. LetN= U B(x,c2).xEMcThen McCNCL,. Since u, v M, and M, is compact, we may assume that u, v N.Now putF0 = (F\N) U ON.It is clear that infrEF0 co(x) > c and that F0 separates u, v. Moreover, F0 fl (M, U Sc) = 0.By Theorem 2.8, P, n F0 and P, must contain a compact subset that separates u, v. This isa contradiction. •We are now ready to state and prove the structure theorem about the critical set of themountain pass principle.THEOREM 2.10 In Theorem 2.3, we further assume u,v K, and that cc verifies (PS),.Then one of the following three assertions concerning the set K, must be true:(1) P, contains a compact subset that separates u and v;(2) IC, contains a saddle point of mountain-pass type;(3) There are finitely many components of Gc, say Ci (i = 1,2, • • • , n) such thats,= _ 1u7s!, Sr-S = O^(i^<where SI = ScrICi. Moreover there are at least two of them SI1 , S"(ii i2 1 <i1, i2 < n)such that the sets A/1,n s!i , mc n S!2 are nonempty and connected through Mc(see Definition2.5 in Section 2.2).All the known results about the structure of K, up to date can be deduced from Theorem2.10. In particular, we have the following corollary which was established in [34].COROLLARY 2.11 In Theorem 2.1, assume that K, does not separate 0, e. Then at leastone of the following two cases occurs:(a) K, contains a saddle point of mountain-pass type;2.3. Main theorems^ 37(#) Mc intersects at least two components of Sc.Moreover, if M, has only a finite number of components, then either(a') K, contains a saddle point of mountain-pass type, or(a') at least one component of M, intersects two or more components of Sc.PROOF OF THEOREM 2.10: Suppose assertions (2) and (3) are not true. In order to provethe theorem, we need to show that assertion (1) holds true.As in the proof of Corollary 2.9, we know that IC, is the disjoint union of Se, Mc and P.Also by the (PS), condition, IC, is compact. It is also clear that Se is closed and compact.We will assume that Se 0 0 since otherwise we conclude by Corollary 2.9. We start withthe following:Claim 1: There exist finitely many components of Gc, say Ci (i = 1, 2, • • • , n) and ni > osuch thatGe n fx; dist(x, Sc) <7]i} C u1C2.^ (Ti)Indeed, if not, we could find a sequence xi in S, and a sequence (C1)1 of different com-ponents of G, such that dist(xi, Ci) —> 0. But then any limit point of the sequence xiwould be a saddle point for co of mountain-pass type, thus contradicting our assumptionthat assertion (2) is false. Claim 1 is hence proved. We clearly may assume that Ci 0 forall i = 1, 2, • • , n.Next for each i = 1, 2, . • • , n, let S = Sc^Clearly they all are compact and mutuallydisjoint. Also we have thatSc = U1Sic.^ (T2)Claim 2: There are n mutually disjoint open sets Ni (i = 1, 2, . • , n) such that u, vtfiNi andU MCU1N2 and ,.5" C Ni for all i = 1, 2, • • • , rt.^(T3)Indeed, we have two cases to consider.38^ 2. STRUCTURE OF CRITICAL SET IN MOUNTAIN PASS THEOREMCase 1: Mc = O.This is a trivial case. By the initial assumption that u, v Ke, for each i (i = 1, 2, • • • , n)there exists an open neighborhood Ni of S! such that u, v Ni. Since the S's are mutuallydisjoint compact sets, we may take the Ni's in such a way that they are also mutuallydisjoint. This proves Claim 2 in Case 1.Case 2: Mc 0 O.In this case we are in a situation where we have n mutually disjoint compact sets 51 (i =1, 2, • • • , n) and a nonempty set M. Moreover all the pairs S! n Mc, sl n Mc (i j i,j =1, 2, • • • , n) are not connected through Mc since assertion (3) is assumed false. ApplyingLemma 2.7, we can then find n mutually disjoint open sets Ni such that (T3) is verified.Since u, v I-Cc, we may clearly assume that u, v U:LiNi. Claim 2 is proved in both cases.In order to finish the proof of Theorem 2.10, we still need the followingClaim 3: There exists a closed set P such that P separates u, v whileinf co(x) > c and P n^u me) = 0.^(T4)xEFTo prove claim 3, we first let for each i (i = 1, 2, • • • , n)Yic = Ni\G,.^ (T5)Then for each i (1 < i < n), for any x E SI, there must by B(x, cx)(cx > 0) such that forany connected component U of G, with Ci U, B(x, cx) n U = 0. Otherwise x is a saddlepoint of mountain-pass type and this contradicts that assertion (2) is assumed false. PutTic = u„si3(x, ex/2) n aci n Ni n {x E X; dist(x, S!) 70.^(T6)ClearlyS!C Tic,^Tic C Ni n aCi^ (T7)and Tic is open relative to Ni n aci. Also Tic n au = 0 for any component U of G, withU Ci. Now letP = [(F n Le) \ (Li1Ni)] u (u7._1aYie\7'f).2.3. Main theorems^ 39Then clearly, infxci, co(x) > c. Since F n L, separates u, v and in view of Claim 1, Claim 2,(T5) and (T7), we see that we can apply Lemma 2.7 with Ai = Ci, G = Gc, = Ni,Yi =Yic, = Tic for all i = 1,2, • • • , n to conclude that P separates u, v. On the other hand,since M fl (G,\Gc) = 0, we have by (T3) and (T5), that OYie n Al, = 0. Therefore by (T2)and (T6), we have U1 (O\7)\Tic) n u itie) = 0. Hence fr n u sc) = 0 and Claim 3 isthus proved.Finally by Theorem 2.8, we see that P n /:), and hence Pc must contain a compactsubset that separates u, v which implies assertion (1). This clearly finishes the proof of thetheorem. •REMARK 2.12 Theorem 2.10 is clearly still true if yo only verifies (PS)N,(Fuicc),, instead of(PS),.REMARK 2.13 The proof of Theorem 2.10 shows that its statement is still true if we replacethe condition u, v (I IC, by u,v S UAs a corollary, we have the following which first appeared in [21].THEOREM 2.14 In Theorem 2.3, we assume that cp verifies (PS)N,(F),, for some c > 0.Then either F fl M, 0 0 or F fl K, contains a critical point of mountain-pass type.Proof: We shall prove it by contradiction. Suppose F fl IC, contains no critical point ofmountain-pass type and F fl M = 0. Let P F fl L. Then we claim that:There exist finitely many components of G,, say C1, • • • , Cn and ft1 > 0 such thatG, n ix; dist(x,P Ice) < pil CC1 U C2 U • • • U^ (R1)Indeed, if not, we could find a sequence xi in Sc and a sequence (C1)1 of different componentsof G, such that dist(xi, Ci) -4 0. But then any limit point of the sequence xi would be asaddle point for yo of mountain-pass type, thus contradicting our assumption. The claim ishence proved. We clearly may assume that Ci 0 for all i = 1,2, • • • , n.40^ 2. STRUCTURE OF CRITICAL SET IN MOUNTAIN PASS THEOREMClearly for all i,j (i,j = 1, 2, ••• ,n i j), we have(fr n K, n)n (P n ice n -0;) = (fr n Ke) n (-07 n)= 0.^(R2)Indeed, otherwise P and hence F will contain a critical point of mountain-pass type. Put= n Kc n = FnLcmiccnUi.By the compactness of ,t and (R2), we may find for each i (i = 1, 2, • • • , n) an open set Nisuch thatNi,^Ni n^= 0 for all i,j = 1, 2, • • • , n i5 j.^(R3)Since F n M = 0 and u, v F, we may assumeM n (u1N1) = 0^u,v^ (R4)Next for each i (1 < i < n), for any x E it there must be B(x, ex) such that B(x, Ex) n = 0for any component U of Gc with U C. Puttic =^ex/2) n aci n Ni.Then letYic = IVi\Gc and P = [P\(t.iNi)] u [utit - (aYie 77)].^(R5)Clearly, infxEp w(x) > c. Since Tic is open relative to Ni n aci and SC Tic by (R4), wesee that we can apply Lemma 2.7 to conclude that P separates u, v. By (R3) and (R5), wemay assume that u137 c N(F). Hence by Theorem 2.3, we have P n IC, 0 0. On theother hand by (R1), (R4) and the assumption that F n mc = 0, we have P n K = 0. Thisis a contradiction. •It is important to know the number of critical points. Rather surprisingly, we have thefollowing corollary concerning the cardinality of the critical set Kc generated by Theorem2.3.2.3. Main theorems^ 41COROLLARY 2.15 Suppose dim X > 2. Then under the hypothesis of Theorem 2.10, oneof the following three assertions must be true:(1) .1<, has a saddle point of mountain-pass type;(2) The cardinality of Pc is at least the same as the continuum;(3) The cardinality of Mc is at least the same as the continuum.Proof: If If, does not contain a saddle point of mountain-pass type, then either assertion(1) or assertion (3) in Theorem 2.10 is true. Let us first assume that assertion (3) is true.Then there exist two disjoint nonempty closed subsets of Ke, say, Mc' and V which areconnected through M. Clearly dist(Mel, V) = d> 0. For any 0 < a- < d, letM, = Ix E X; dist(x, X) <a}.Then Ma n V=0,Vc Ma. We claim that OM, n .M, 0. Otherwise, there will be twodisjoint open sets M, and X\M, such thatXcma, mc,nm!=0, Me U g- Li M C M, U (X\M,).This contradicts that V, V are connected through M. Now let ma E ax n M. Thenwe have a map f from (0,d) to Mc defined as:f : a E (0,d) —4 ma E Mc.Clearly f is injective. Hence assertion (3) in Corollary 2.15 is true. If instead, assertion (1)in Theorem 2.10 is true, then we can argue similarly that assertion (2) in Corollary 2.15 istrue. Let d = dist(u, v). Then for all (7(0 < a- < d), we have 49.B (u, a) n Pc 0 0. Otherwise,as above, we will see that Pc will not separate u, v. Define a map f from (0, d) to Pc asfollows:f : a E (0,d) —4 ina E Pc n (9 B (u, a-).Clearly f is injective. Hence assertion (2) is true and this completes the proof the corollary..As an interesting application of the above corollary, we have the following.42^ 2. STRUCTURE OF CRITICAL SET IN MOUNTAIN PASS THEOREMCOROLLARY 2.16 Suppose cp has a local maximum and a local minimum on a Banach spaceX. If cp satisfies (PS) and if dim(X) > 2, then necessarily yo has a third critical point.Proof: Suppose u1 is a local maximum and u2 is a local minimum. If yo is not boundedbelow, then we have a mountain pass situation with u2 as an initial point and Corollary2.15 applies to give either an infinite number of critical points or a saddle point of mountainpass type which is necessarily distinct from u1 and u2.If, on the other hand, cp is bounded below then, since it satisfies (PS),, Corollary 1.14yields that cp cannot be bounded above. Hence we have a mountain pass situation for ..—cpwith n1 as an initial point. Again Corollary 2.15 applies to yield our claim. •Chapter 3Structure of the critical points obtained viathe higher dimensional min-max principle (n > 2)As in the last chapter, we shall study the topological properties of the critical points gen-erated by min-max procedures where the function under study is only supposed to be Cland therefore without the use of classical Morse theory. But here we shall deal with thecase of "n-dimensional" homotopy-stable families when n > 2. In order to do this, we firstintroduce the concepts of (weak) saddle - type point and co-saddle point of order k which canbe seen as the higher dimensional analogue of Hofer's points of mountain pass type. Weshall see in the next chapter that these notions are closely related to the classical Morseindices whenever these indices can be defined; that is when co is a C2-functional and whenthe critical points are non-degenerate. For more detail, refer to [15].3.1 PreliminaryWe shall always assume in this chapter that X is a Banach space and let B be a fixed closedsubset of X and Sk be a standard k-sphere in Rk+1. We shall adopt the following definitionsfrom [20].DEFINITION 3.1 A family F of subsets of X is said to be hornotopic of dimension n withboundary B if there exists a compact subset D of Rn containing a closed subset Do and acontinuous function u from Do onto B such that= {A C X; A= f (D) for some f E C(D; X) with f = a- on Dol.4344^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESDually, we can introduce the cohomotopic classes. For that, fix a continuous map a*B -4 Sk and for any closed subset A of X containing B, set-y (A; B , a*) = inf{n; 3f E C (A; Sn) with f = a* on B}.DEFINITION 3.2 A family .7 of subsets of X is said to be cohomotopic of dimension n withboundary B if there exists a continuous a* : I3 .4 Sn such that.7- = {A; A compact subset of X, A DB and -y(A; B, a*) > n}.DEFINITION 3.3 A family .F of subsets of X is said to be a homological family of dimensionn with boundary B if for some non-trivial class a in the n-dimensional relative homologygroup Hn(X, B) we have that= {A; A compact subset of X,A DB and a E 110:1, )}where i'," is the homomorphism ii,l, : Hn(A, B) .4 Hn(X, B) induced by the immersioni : A 4 X.Suppose now that F is a closed subset of X that is disjoint from B. It is readily seenthat F is dual to .F(a) if and only if a st /m(i) where i. : Hn(X \ F, B) -4 fin(X , B). Weshall only use singular homology with rational or real coefficients.DEFINITION 3.4 A family a . u„,.7a of sets of X is said to be generalized homotopic (resp.cohomotopic) (resp. homological) of dimension n with boundary 93 = UaBa if for each a,..Ta is homotopic (resp. cohomotopic) (resp. homological) of dimension n with boundaryBa.For convenience, we also introduce the following notation.DEFINITION 3.5 A compact subset L of .K, is said to be an isolated critical set for go in Kif it has a neighborhood in which cp has no critical points at the level c other than the onesthat are already in L.3.2. The homotopic case^ 45Next we recall some facts from general topology. We refer to [31] and [24] for moredetails.DEFINITION 3.6 Let U = {U-117 e r} be a collection of closed subsets of a Hausdorfftopological space T. If the union of any subcollection of U is a closed subset of T, then wesay that U is a discrete closed collection in T.DEFINITION 3.7 A Hausdorff topological space T is called collectionwise normal if for everydiscrete closed collection {U7 1-y E r} in T, there is a disjoint open collection {071-y E 11such that U7 C 07 for all 'y E T.THEOREM 3.8 A metric space is collectionwise normal. A Hausdorff topological space T iscollection wise normal if and only if for every continuous map of every closed set U of T intoa Banach space can be continuously extended over T.We shall show in the next few sections that certain topological properties of a criti-cal point or critical set generated by a min-max procedure are related to the topologicaldimensions (defined above) of homotopy-stable families (homotopic, cohomotopic and ho-mological) under consideration.3.2 The homotopic caseRecall thatlf, = Ix E X; cp(x) = c, '(x) = of L, = {x E X; co(x) c} G, = X\Lcand that sup w(0) = —oo by convention.DEFINITION 3.9 Let co be a C1 functional on a Banach space X and let K be a subset ofK. We say that K is a weak saddle-type set of order k if k is the least integer such thatthere is a neighborhood N of K verifying that for any open sub-neighborhood M C N of46^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESK and any co > 0, M n G,_, is not (k — 1)-connected for some 0 < < co. We shall thenwrite w-sad(K) = k.If the above holds for co = 0, we then say that K is a saddle-type set of order k and wewrite sad(K) = k.If K is a singleton {x} we shall then say that x is a weak saddle-type (resp a saddle-type)point of order k.Clearly from the definition, sad(K) > w -sad(K).REMARK 3.10 By convention we say that a set is —1-connected if it is nonempty. Hence acritical point x of mountain-pass type is a critical point with sad(x) = 1. x is a minimumif and only if x has sad(x) = 0 which holds if and only if w-sad(x) = 0.In the case where regular Morse indices are defined, we shall see in Chapter 4 a criticalpoint x has Morse index k if and only if sad(x) = w-sad(x) = k.We shall prove the following result which roughly speaking, implies that a homotopicfamily ,F of dimension n will necessarily generate a weak saddle-type critical point of orderat most n.THEOREM 3.11 Let cc be a C1-functional on a Banach space X and consider a homotopicfamily .7" of dimension n with dosed boundary B. Let .F* be a family dual to „T such thatc := sup inf (p(x) = inf max co(x)FEY* xEF^AE.F xEAand is finite. Assume that cc verifies (PS), along a min-m axing sequence (Ak)k in ,F and asuitable max-mining sequence (Fk)k in .7"*. Suppose k, := K n Fcc, n A„„ is isolated in K.Then, for any neighborhood N of kc, there is a connected component M of N such thatM n ke 0 and w-sad(M n k,) < n .Moreover, if we assume that ke consists of isolated critical points, then there is x E k,with w-sad(x) < n.3.2. The homotopic case^ 47If we assume that k, consists of isolated critical points and Fk = F for all k > 1, thenwe have the following corollary.COROLLARY 3.12 Let co be a C'-functional on a Ban ach space X and consider a hornotopicfamily Y of dimension n with closed boundary B. Suppose that c := c(co,Y) is finite andthat F is dual to F with inf cp(F) > c. If co verifies (PS)F,, along a min-maxing sequence(Ak)k and if the set IC, (1 A„,„, n F consists of isolated critical points, then there exists x inKcnFnA,,,, with sad(x) n.If we suppose that sup AB) < c, then the above applies to the dual set F = {co >and we get the followingCOROLLARY 3.13 Let cc be a 0-functional on a Banach space X and consider a hornotopicfamily .1 of dimension n with closed boundary B. Suppose that c := c((p, .7) is finite andthat sup AB) < c. If co verifies (PS), along a min-maxing sequence (Ak)k and if the setK,n Acc, consists of isolated critical points, then there exists x in K, n A„,„ with sad(x) < n.The following corollary of Theorem 1.3 will be crucial in the proof of the main resultsof this chapter. For any set V, we shall denote by N5(V) = 1u E X; dist(u , V) < 81 itsö-neighborhood.COROLLARY 3.14 Under the hypothesis of Theorem 1.3, assume cc verifies (PS), along amm-maxing sequence (A.)„ in .F and a suitable max-mining sequence (Fn)n in F*. Supposeke := K n Foo fl A„, is isolated in K, and let f > 0 such that N€(K) n K = ke. PutFL = Fk U (Lc, n NE(k)) where ck = min cp(Fk). Then for any S > 0 and any k0> 0, thereexist A E F and a F1, with k > ko such thatA C (X\FI:) U Ar8(F,, n ,400 nx,).Proof: If not, then for some 6 > 0 there is an increasing sequence ni such that the set= ,\1\78(Foo n A n Kc) are dual to F for all i. Since 1im1^inf cp(Fr,",) = c, we haveby Theorem 1.3, that F:0 n Aao nifc00 which is absurd. •48^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESREMARK 3.15 (a) If we apply the above corollary to F„ = L, for all n, we get the existenceof an A E such thatA C G, U Na(Ke)•(b) Under the classical condition: sup co(B) < c, we obtain the well known result aboutthe existence of A E .7- with A C G, U N8 (Kr).The proof of Theorem 3.11 needs some algebraic topological tools. We shall recall andprove first some results in algebraic topology. As in general, for a simplicial complex K,We denote by KI its underlying topological space and for simplexes s and t we writet < s (t < s) if t is a ( proper) face of s. For a simplex s, we denote s° to be the opensimplex of s. Here is a lemma from [23] (p108-125).LEMMA 3.16 Let D C IV be a compact subset. Then for any 6 > 0, there is a finitesimplicial complex K of Rn such thatD C tK C N(D).We shall also need the following lemma. Since we can not find a reference for it, we givea proof for completeness.LEMMA 3.17 Let K be a finite simplicial complex of Rn. Then there is a simplicial sub-complex L of K such that ILI 014Proof: We assert that for any a E K with 1a01 n aim^0 then lal C OIKI. Note firstthat in = dim a < n — 1 if a° n 011C1 0 0. We prove the assertion by induction on mdownward. It is clear that afC 01K1 if lain allfi 0 0 and in = n — 1. Suppose that itis true for all m with k < m < n — 1, we need to show that it is true for m = k — 1. Foreach x E a0 n awl, since K is a finite simplicial complex there is an n-dimensional ballB(x, Ez) with B(x, €) n C la°1 such that for any b E K if B(x, ex) n Ibl $ 0 then eitherb = a or a < b.3.2. The homotopic case^ 49If there is abEK with a < b and lb°1 n aim 0, then by the induction assumption,lb1 g aim, hence lal g ibi c aim. If not, we will have 1a01 C 01K1 i.e. 1a1 C aim aswell. To see this, we note that B(x, Ex)\1a01 is connected since dim a < n — 2 and thatfor any path joining y, z with y EIntIK1 and z E Illn \WI, then the path must intersect014 So B(x, ex)nInt1K1 = 0 i.e. B(x, ez) n la°1 C B(x, Ex) n 1K1 C 014 This shows thatlain 01K1 is open in lel, also closed since 01K1 is closed. But 1a01 is connected, therefore1a01 n awl = 1a01 i.e. 1a01 =14Finally we putL = {a; a E K, lal c aixi}.Clearly L is a simplicial subcomplex of K, by the assertion established above, we have thatILI = 014 This proves the lemma. •Next we recall an elementary lemma from obstruction theory in algebraic topology. LetK be a CW complex and L be a CW subcomplex of K. Let Km be m-dimensional skeletonof K and km = L U Km. Here is the lemma [23] (pp 174-179).LEMMA 3.18 Let K, L, Km , km be as above and let g : L -4 Y be continuous. If Y isan m-connected topological space for some m > 0, then g has a continuous extension overkm-Fi .It is well known that there is a natural way to identify any simplicial complex as a CWcomplex.COROLLARY 3.19 Let K C Rn be a finite simplicial subcomplex and f : aim -4 Y becontinuous. If Y is path connected for n = 1 and each path connected component is (n —1)-connected for n> 1, then f has a continuous extension over IKI.Proof: For n = 1, the corollary follows directly from Lemma 3.18. For n > 1, we observethat 1K1 has only finite path connected components and f maps each path connected com-ponent into a path connected component of Y. Then applying Lemma 3.18 on each pathconnected component of 1KI, we see that the corollary is proved. •50^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESFor any x E X, E > 0, we let B(x, = E X; Ilx — yll <LEMMA 3.20 Let G, B, M be subsets of X with B compact and G open. Let D0, D becompact subsets ofRn with Do C D. Assume mns = 0 and choose 0 < v < 1/2 dist(M, B).Let f : GUBUM be continuous such that 1(Do) = B and suppose there is a subsetG' of G with G' n N(M) = G n N(M) such that each of its path connected component is(n — *connected, then there is g: X such thatg(D) GUB and g(x) = f (x) for all x E Do.Proof: Let U = f (D) n AVM). If U is empty, then the lemma is true. Otherwise letV = f (U). By Theorem 3.8, Rn is collectionwise normal. We have then an extension f:Rn -4 X of f. Clearly there is an open neighborhood DI of D such that (D1) C N( f (D)).Since f(Do) = f (D0) = B, V is compact and B n Arv(m) = 0, there is > 0 such thatNo(v) ç Di\Do, i(N5(v)) c N(M).By Lemma 3.16 and Lemma 3.17, there is a finite simplicial complex K of Rn and a simplicialsubcomplex L of K such thatILI = alKI, V c N12(V) c jK c N(V).ClearlyAILD c (N(M)) n (G u B) c (G' n (N(M)) C G'.By Corollary 3.19, we have : IKI^G'. Now defineg(x) = f_.(x) if x E D\IK1f(x) if x E IKI.Then g(x) :^G U B and g(x) is continuous with g(x) = f(x) on x E Do.^•PROOF OF THEOREM 3.11: Since ke is compact and N is a neighborhood of kc, we have afinite number of connected component (Mi):11 of N such that k, c U7l1M1 and kcnmi 03.2. The harnotopic case^ 51for all 1 < i < m. Let g kc 11 M. Clearly M is a neighborhood of the compact set gfor 1 < i < m. Hence there is r> 0 such thatN4,(k,) n B = 0 and N41-(g) c mi for all 1 < i < m.^(3.2.1)Since we suppose that k, is isolated in Kc, we may assume thatN47-(k,) n 'Cc = kc.^ (3.2.2)Let Sk c — inf co(Fk) and FL = Fk U Ulli(Lc-Sk fl N4,-(g)). Clearly FL is dual to 1" and610 -+ 0 as k^oo.Suppose the theorem is not true. Then for each g, there exist ci > 0 and an open sub-neighborhood ki c N4(M) of 11//. such that each path connected component of Mi n Gc-€is (n — 1)-connected for all 0 < c < ci. Take c = mini<i<m ci and 0 < a < T small suchthat N4(M) C ki for all 1 < i < m. Let ko > 0 such that Sk < E for all k > ko. Nowwe may assume that g* is given explicitly as in Definition 3.1 with D, D0 and a. Note thatB C X\Ft for all k > 1. Then by (3.2.1), (3.2.2) and Corollary 3.14, there exist f : D -4 Xcontinuous with f(x) = cr(x) on Do and a FL with k > ko such that(D) ç (X\FL) U Ar(k).^ (3.2.3)Note that(x\FD n N4,-(m) =Gck n N47(M)•Now we shall prove that there is g : D X with g(x) = u(x) on Do such thatg(D) ç (x\FD^ (3.2.4)which is clearly a contradiction since FL is dual to F. By induction and starting with g° = f,we shall construct (gi)1 : D .4 X continuous with g1 (x) = a(x) on Do such thatgi (D) C Gi^ (3.2.5)where the sets (Gi)ill are defined as:Gm = X\FL, Gi = (X\FL) U Urin_i+A(Mg) for all 1 < i < m — 1.^(3.2.6)52^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESFor i = 1, by (3.2.3) we have thatg° (D) C G' U Na(X).^ (3.2.7)Put G' = (XVI) n /41 C GI. Note that G' =^n Gc_sk since 141 C N4T(g). Note alsothat dist(Na(M),B) > 3a. Hence we have thatn N2a(Mel) = n N2.(X)•On the other hand, each path connected component of G' is (n — 1)-connected by as-sumption and (Sk < f. Hence we can apply Lemma 3.20 with this G' and G = G1 to havegl : D -4 X continuous with g1 (x) = o(x) on Do such thatg1 (D) C G1which is asserting (3.2.5) for i = 1. Next, suppose we have constructed (gi)f=i for 1 < i < /(1 < / < m) so that (3.2.5) is verified. Note that(D) c c G/1-1 u N(M1)and dist(Nce(MI+1),B) > 3. Put G" = (X\F) n /1-P+1 C Gm. Then we haveG" n N2a(M,I+1) = n N2a(McI+1).Again by Lemma 3.20 with G' = G" here and G = G1+1 we have gm such thatg1+1(D) c G1+1which verifies (3.2.5) for i = /^1. This finishes the inductive construction of (gi)111.Finally, g = gm gives the required map and Theorem 3.11 is proved.^•REMARK 3.21 The above proof actually shows that for n > 2 there exist an M such thatfor any co > 0 and any open sub-neighborhood M C M of Mn kc, one of the path connectedcomponents of .11.1 n Gc_, is not k — 1-connected for some 2 < k < n and 0 < e < co.3.3. The cohomotopic case^ 533.3 The cohomotopic caseIn this section we study the topological properties of the critical points generated by themin-max procedure in the cohomotopic case. For convenience, we introduce the followingnotation. For any subset D of X and a functional cp. on X, we letL9,(D) = {f E C (X , X); of cp, (D) gr) and f (x) = x on X\D}.We shall drop the subscript cp when no confusion arises in the sequel.DEFINITION 3.22 Let cp be a CI-functional on a Banach space and let K be a subset ofthe critical set of co at level c. We say that K is a co-saddle type set of order k if k isthe least integer such that for any neighborhood N of K, there exist a sub-neighborhoodM C N of K and f in L(N) such that topdim f (M) < k. We then write sad*(K) = k.If K is a singleton {x} we shall then say that x is a co-saddle type point of order k.Here is the theorem which basically says that a cohomotopic family F of dimension nwill necessarily generate a co-saddle type critical point of order at least n.THEOREM 3.23 Let cp be a C'-functional on a Ban ach space X and consider a cohomotopicfamily ,7" of dimension n with closed boundary B. Let .7** be a family dual to ,F such thatc := sup inf cp(x) = inf max '(x)AEY xEAFEY* rEFand is finite. Assume that cp verifies (PS), along a min-m axing sequence (Ak)k in F, and asuitable max-mining sequence (Fk)k in .7. Suppose that k := K n Foo n Acc is isolatedin K. Then, for any neighborhood N of kc, there is a connected component M of N suchthat M n ke is not empty and sad*(K, nm)>n.Moreover if kc consists of isolated critical points, then there exists x E k, with sad* (x) >n.If we suppose that sup co(B) < c, then the above applies to the dual set F = {cp > c}and we get the following:54^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESCOROLLARY 3.24 Let co be a C'-functional on a Banach space X and consider a coho-motopic family .7. of dimension n with closed boundary B. Suppose that c := c(cp, .7-) isfinite and that sup co(B) < c . If co verifies (PS), along a min-maxing sequence (Ak)k andif the set IC, n Acc, consists of isolated critical points, then there exists x E K n Acc withsad*(x) n.The proof of Theorem 3.23 needs the following easy lemma which singles out an importantstability property enjoyed by cohomotopic families.LEMMA 3.25 Let ,F be a cohomotopic family of dimension n with boundary B in a metricspace X. Then, for any A E F, any continuous function f : A X with (x) = x on B andany open set U such that U n B = 0 and topdim f (U) < n — 1, we have that f (A\U) EProof: Suppose that f (A\U) does not belong to Y. Then there exists a continuous maph : f (A\U) --+ S"-1 such that h = a (the boundary data) on B. Let h' be the restrictionof such a map to f (A n au). Since topdim f (U) < n — 1, Theorem 1.22 applies to yield anextension h" of h' from f (A fl U) into Sn-1. It is now clear that the maph(x) if x E f (A\U)ii(x) =h"(x) if x E f (A n U)is a continuous map from f(A) into Sn-1 that is equal to a on B. In other words,-y(f (A); B, a) < n — 1, which is a contradiction since (A) E F.^ •PROOF OF THEOREM 3.23: Since k, is compact and N is a neighborhood of kc, we have afinite number of connected component (M1)111 of N such that ke c U1M1 and kcnmifor all 1 < i < m. Let 11/I! = k n Mi. Clearly Mi is a neighborhood of the compact set gfor 1 < i < m. Hence there is r> 0 such thatN4,(k) fl B = 0 and N4T(g) C Mt for all 1 < i < m.Since we assume that k, is isolated in Ife, we may assume that(k) (I K = kc.(3.3.1)(3.3.2)3.3. The cohomotopic case^ 55Let Sk = c - inf cp(Fk) and FL = Fk U Ui(Lc_ok 11 N4T(g)). Clearly FL is dual to .F and6k -.> 0 as k -4 oo.Suppose the theorem is not true. Then for each g, neighborhood N4,(Mci), there exista sub-neighborhood fii c N4(M) of g and fi E L(N4T(g)) such that topdim fi(Mi) <n - 1. By taking sub-neighborhood of g inside of ./Cii if necessary, we may assume that /C/iis closed. Note that B C X\FL for all k > 1. By (3.3.1), (3.3.2) and Corollary 3.14, thereis A E .7. and FL such thatA C (X\FL) U U711/1-4.i.Note (X\11) fl N4(M) = Gc_8, n N4r (Ai) . Let f = fff, o fm_i o • • • o fi and A = f (A\ UT-1X(i). Clearly A\ Ur_i Mi C X\FL. Since co o f < co, f (x) = x on X\ U711 N4(M) and(X\FD fl N4r(g) = Gc-8k fl N4,-(g) we have that A c x\F,',. On the other hand, wehave that A E .7. by Lemma 3.25. But this is a contradiction since FL is dual to T. •Now we can combine the previous results to get some two-sided information about thecritical points generated by min-max principles.THEOREM 3.26 Let (09 be a CI-functional on a Banach space X and consider a homotopicfamily .7" (resp. a cohomotopic family .P) of dimension n with closed boundary B. Let ,T*(resp. .-F*) be a family dual to .7. (resp. ,--7.) such thatc := sup inf co(x) = inf max co(x)FEy• xEF^AET TEA(resp.E:= sup inf co(x) = inf max cp(x))ACT xEAPE.p. xEFand is finite. Assume that co verifies (PS), along a min-maxing sequence (Ak)k in .7" and asuitable max-mining sequence (Fk)k in 1-*. Suppose that k, := 'cc n Foo n A„.„ is isolated inK. If .7" C .P, c =E and Fk is dual to .P for all k > 1, then for any neighborhood N of kc,there exists a connected component M of N such that w-sad(M n k) < n < sad* (M n kc) •Moreover if we assume that ke consists of isolated critical points, then there existsx E k, such that w-sad(x) < n < sad*(x).56^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESIf we assume that Fk = F for all k > 1, we then have the followingCOROLLARY 3.27 Let co be a C1-functional on a Ban ach space X and consider a homotopicfamily .7- (resp. a cohomotopic family .f"") of dimension n with closed boundary B. Assumethat c := c(cp, (resp. E := c(co, Y)) is finite and that F is dual to F with inf c,o(F) >c. Suppose that co verifies (PS)F,, along a min-maxing sequence (Ak)k and that the setn Ao„ n F consists of isolated critical points. 1fF C F, c = E and F is dual to P, thenthere exists x E K n F n A„,, such that sad(x) < n < sad*(x).If we suppose that sup co(B) < c, then again the above applies to the dual set F = {c9 >c} and we get the followingCOROLLARY 3.28 Let co be a C1-functional on a Banach space X and consider a hornotopicfamily .7- (resp. a cohomotopic family :) of dimension n with closed boundary B. Supposethat c := c(co, (resp. E := c(cp,Y)) is finite and that sup (B) < c. Assume that wverifies (PS), along a min-maxing sequence (Ak)k and that the set K, n A00 consists ofisolated critical points. If F c and c = E, then there exists x in .1C, n A0„ such thatsad(x) < n sad*(x).PROOF OF THEOREM 3.26: Since k is compact and N is a neighborhood of ke, we have afinite number of connected component (0)111 of N such that k c LiMi and kcilMifor all 1 < i < m. Let /1// = K n M. Clearly M is a neighborhood of the compact set Mcifor 1 < i< m. Hence there is r > 0 such thatN4,-(k) n B = 0 and N47(Mci) C Mi for all 1 < i < m.^(3.3.3)Since we assume that ke is isolated in Kc, we may assume thatN4, (k,) n IC, = kc.^ (3.3.4)Let bk = c — inf y,(Fk) and PI Fk U Ur_i(Lc-a, fl N4T(g)). Clearly II is dual to F and—> 0 as k oo.3.4. The homological case^ 57Suppose the theorem is not true. Then without loss of generality, we may assume thatfor 1 < i < I <m and each g, there exist ei > 0 and a neighborhood MC N4(M) ofM such that ki n Gc_e is (n — 1)-connected for all 1 < < ci. Also for all I +1 < i < m,each g and neighborhood N4T(g), there exist sub-neighborhood .A./Y1 c N4(M) of 114.and fi E L(N4,(Mci)) such that topdim f (k) < n — 1. Take E mini<i<i ei and < a < Tsmall such that N4(M) Ci2t1 for all 1 < i < m. Next we may assume that .F is givenexplicitly as in Definition 3.1 with D, Do and cr. Note that B C X\Fk for all k > 1. Letko > 0 such that 5k < E for all k > ko. Then by (3.3.3), (3.3.4) and Corollary 3.14, thereexist f : D X continuous with f(x) o(x) on Do and a FL with k > ko such thatf (D) c (x\FL) u Na(kc).Note that(x\FL) n N4(M) Gc_h n N4,-(g).Now just as in the proof of Theorem 3.11, we will have a continuous map g : D -+ X withg(x) = c(x) on Do such that thatg(D) ç (X\FL) U U71/+iNa(g).Put A = g(D) and note that g(D) E f" since P C P. Let f =fm o • o fr+i andA = f(A\141/44Na(g)). Since by assumption topdim < n-1 for all I+1 < i < m,we have also that topdim f(U71/4.1./Cli) < n-1. So topdimi(U71mNa(M)) < n-1. Thenas in the proof of Theorem 3.23, we have that A E by Lemma 3.25. Next we haveA C X\F/: since that w(f(x)) < (p(x) and f(x) = x on X\ Urili+i N4,(Mci). This is acontradiction since by assumption that FL is dual to P. •3.4 The homological caseLike the homotopic and cohomotopic cases, a homological family .7" of dimension n will alsonecessarily generate a critical point with some topological properties. To describe theseproperties, we introduce the following concept.58^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESDEFINITION 3.29 Let w ba a C'-functional on a Ban ach space X and K be a subset of Kc,the critical set at level c. We say that K has the weak bi-saddle property of order k (k > 1)if there a neighborhood N of K such that for any co > 0 and any open sub-neighborhoodM C N of K with Hk(M) = 0, we have that Hk-i(G,--, n m) # 0 for some 0 < c < fo•If the above holds for € = 0, we say that K has bi-saddle property of order k.Here is the main result of the section.THEOREM 3.30 Let w be a C1-functional on a Banach space X. Consider a homologicalfamily .7- of dimension n with boundary B. Let .F* be a family dual to .7- such thatc := sup inf w(x) = inf max v(x)FE.T* xEF^AET xEAand is finite. Assume that w verifies (PS), along a min-maxing sequence (Ak)k in .F, anda suitable max-mining sequence (Fk)k in F*. Suppose k-, :. .K, n Foo n A,,,, is isolated inK. Then for any neighborhood N of ic-c, there exists a connected component M of N withM n k , 0 0 such that M n -K, has the weak bi-saddle property of order n.Moreover if k, consists of isolated critical points, then there is an x E ke having theweak bi-saddle property of order n.If we assume that k consists of isolated critical points and Fk = F for all k > 1, thenwe have the following corollary.COROLLARY 3.31 Let w be a C1-functional on a Banach space X and consider a homologi-cal family „F of dimension n with closed boundary B. Assume that c := c(cp, .F) is finite andthat F is dual to .7. with inf (p(F) > c. If w verifies (PS)F,, along a min-maxing sequence(Ak)k and if the set IC, n F n Aoo consists of isolated critical points, then there exists x inIC, n F n A,, having the bi-saddle property of order n.If we suppose that sup cp(B) < c, then the above applies to the dual set F = {w > c}and we get the following corollary.3.4. The homological case^ 59COROLLARY 3.32 Let cp be a C1-functional on a Banach space X and consider a homo-logical family .T of dimension n with closed boundary B. Set c = c(cp,.F) and assume thatsup(B) < c. If co verifies (PS), along a min-maxing sequence (Ak)k and if the set K, n A„.„consists of isolated critical points, then there exists x in K, n A°. having the bi-saddleproperty of order n.PROOF OF THEOREM 3.30: Since k, is compact and N is a neighborhood of ke, we have afinite number of connected component (M1)1 of N such that k, c U:1110 and kcnmi 0for all 1 < i < m. Let g = kc n Mi. Clearly Mi is a neighborhood of the compact set gfor 1 < i < m. Hence there is T > 0 such thatN4r(k,) n B = 0 and N4,(Mci) C Mi for all 1 < i < m.^(3.4.1)Since we assume that kc is isolated in Kc, we may assume thatN4T(ke) n Kc kc.^ (3.4.2)Let Sk c— inf (p(Fk) and F, = Fk U Ur_i(Lc-sk n N(M)). Clearly FL is dual to ,F and8k^0 as k^oo.Suppose now that the theorem is not true. Then for the neighborhood Mi of g andthe sub-neighborhood N4(M) of At there exist if) > 0 and an open sub-neighborhoodC N4(M) of g such that H(Mt) =^n^0 for all 0 < E < Co. Sinceki is open, we have the following Mayer-Vietoris exact sequenceH(X\FL, B) e H„(14')^H„((X\11) U^, B) --+ 11„_1((X\FL) n it"4-‘2).Since (5k -4 0 as k^oo, we have that there is ko > 1 such that 0 < Sk < for all k > ko.Since W N4T(g), we have that (X\Fk) fl X//‘ = Ge_sk n /ai. By assumption we have forall k > 4, that H1((X\FI) n f/i) = 0. So for k > ko, we have that: H„(X\F, B)^11,2((X\F;,) U60^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESis onto where j,, is induced by the inclusion j : (X\FL, B)^((X\F4) U JtP, B). Hence wehave that the set (FAU:liffi) is dual to J for all k > ko. Since limk-+,0 inf cp (F AUT._^) =c, we have by Theorem 1.3 that (F\ U711 Mi) n floc, n If, 0 0. This is a contradiction. •We shall end this section by stating the following theorem which is the corollary ofTheorems 3.11, 3.23, 3.26 and 3.30.THEOREM 3.33 Let be a C'-functional on a Banach space X. Consider a generalizedhomotopic family a u,„.ra (resp. cohomotopic family (resp. homological family -t) ofdimension n with boundary 93 = UctBa. Let a. (resp. -**) (resp. *) be a generalized dualfamily such thatc := sup inf co(x) = inf max (x)FEa` zEF^AEa xEA(resp.sup inf co(x) = inf max co(x))FEr xEF^AE xEA(resp.sup inf co(x) = inf max (x))frEt. xEF^AEa xEAand is finite. Assume that yo verifies (PS),, (resp. (PS)) (resp. (PS)) along a mM-maxing sequence (Ak)k in a (resp. (11k)k in^(resp. (Ak)k in -t..) and a generalized suitablemax-mining sequence (Fk)k in a. (resp. (Pk)k in^(resp. (Pk)k in 't*) and assume thatk, :=K n Fa, n A„„ (resp. k := KEnPoo n A0,0) (resp. ke.:=KinPoonAo,o) is isolatedin K., (resp.^(resp. K). Then the following holds.(i) For any neighborhood N of k ,, there exists a connected component M of N withM n k, 0 such that w-sad(M nk) < n. Moreover if k, consists of isolated criticalpoints, then there is an x E k, with w-sad(x) < n;(ii) For any neighborhood N of ke, there exists a connected component A-4- of N withfankeo 0 such that sad* (M n kE) > n. Moreover if k e consists of isolated critical points,then there is an Eke with sad*() > n;(iii) For any neighborhood N of kz, there exists a connected component M of N withM n ke- 0 0 such that M n ke has the weak bi-saddle property of order n. Moreover if3.5. Application to standard variational settings^ 61ke consists of isolated critical points, then there is an EK having the weak bi-saddleproperty of order n;(iv) If a^a, c^-e and Fk is generalized dual to for all k > 1, then for anyneighborhood IT of kc, there exists a connected component M* of AT such that w-sad(f4 nke) < n < sad* (la n kc). Moreover if k, consists of isolated critical points, then there isan t E ke with w-sad() < n < sad*().3.5 Application to standard variational settingsLet E=YeZ with dim(Y) = n and consider the following class= {A; 3h : By E continuous, h(x) = x on Sy and A = h(By)}.It is clear that is a homotopic class of dimension n with boundary Sy. Let now= {A; A compact, A D Sy and 0 E 1(A) whenever f E C (A; Y)and (x) = x on Sy 1.is clearly a cohomotopic class of dimension n and with boundary Sy. Note also that.T c P.Regard now a = [Sy] as the generator of the homology Hn_i(Sy, 0) and let^EHn (E, Sy) such that 0,,i3 = a where a. is the map in the exact sequence-4 Hn(SY ) Hn(E) -4 Hn(E , Sy) xn_1(sY) -+.Consider = ....t(13) to be the corresponding homological family. Since a 0 in Hn_i(E\Z),it follows that Z is dual to the class F.COROLLARY 3.34 Let cp be a C2-functional on the Hilbert space E such thata := inf co(Z) > 0 sup (Sy).Let c = c(co , .7"),^c(, P) and = c(cp,^Assume that cp verifies (PS) and that d2cp isnon-degenerate on the critical set. The following holds:62^3. STRUCTURE OF CRITICAL POINTS IN GENERAL MIN-MAX PRINCIPLESIf 0 <E, then1) There exists x1 in If, with sad(xi) < n;2) There exists x2 in Ke with sad*(x2) > n;3) There exists x3 in Ke having bi-saddle property of order n;4) If c = e, there exists x4 in ICc with sad(x4) < n < sad*(x4)•Chapter 4Morse indices of min-max critical pointsIn this chapter, we shall assume that c,o is C2-functional on a Hilbert space E. We shall usethe last chapter's results to relate the topological properties of the homotopy-stable class .Fto the Morse indices of those critical points obtained by min-maxing over .7- and which arelocated on an –a priori– given dual set. We shall be able to find one-sided relations betweenthe Morse index and the homotopic (resp. cohomotopic) dimension of the class, while forhomological families, two-sided estimates are available. We do that in the non-degeneratecase by simply finding relations between the topological indices of critical points introducedin Chapter 3 (saddle-type point, etc) and the standard Morse indices associated with suchpoints. For the degenerate case, we shall refine the perturbation methods introduced byMarino-Prodi, Solimini [38] and Ghoussoub [19] to get the appropriate estimates undersuitable Fredholm type conditions of degeneracy. We refer to [11] for more about infinitedimensional Morse theory and its applications.4.1 Morse indices of min -max critical points—the nondegenerate caseIn this section, we will always assume E, a Hilbert space with inner product (, ) and normJill, Co E C2(E,R). For any u E E, we let D2co(u) denote the unique bounded self-adjointlinear operator T : E -4 E such that e(u)(v)(w) = (Tw, v) for all u, v, w E E. We shallwrite m(v) for the Morse index of the nondegenerate critical point v.4.1.1 The Morse LemmaWe shall first recall some basic concepts of Morse theory. The following lemma is standard.6364^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTSLEMMA 4.1 Assume cp is a C2-functional on a Hilbert space E. If vo is a non-degeneratecritical point for cp (i.e. if d2cp(vo) is invertible), then there exists a Lipschitz homeomorphismH from a neighborhood W of 0 in E onto a neighborhood M of vo with H(0) = vo in sucha way thatw(H(z)) = W(vo) + liz+112 - iiz-112where z -> (z_, z+) corresponds to the decomposition of E into the positive and negativespaces E+ and E_ associated to the operator d2v(vo). The Morse index of vo will be thedimension of E_.Proof: Assume without loss of generality that vo = 0. Write co" for d2cp. We need toprove the existence of a Lipschitz homeomorphism H from a neighborhood W of 0 into aneighborhood M of 0 satisfying H(0) = 0 and for every z E Ecp (H (z)) = cp (0) + -21 (co"(0)z, ). (4.1.1)For that define, near [0,1] x {0}, the functionF(t, z) = (1 - 00,0(0) + -21(p"(0)z, z)) + t(z)and the vector fieldf (t, z) ={-Ft(t, z)I1Fz(t, z)II-2 Fz(t, z)^if z 0 00^ if z = 0.10(z) = v(z) - co (0) - (co"(0)z, z)and note that 0(0) = 0, ii/(0) = 0 and 71/0) = 0. Consequently,1^ 17,b(z) = jo (1 - s)(0"(sz)z, z) ds and 7i(z) = Jo 011(sz)z ds.Thus, for each E > 0, there exists O(e) > 0 such that for lizil < 5(e),10(z)i^Ellz112 and 11Y(z)11^6'114We need to show that f (t, .) is Lipschitz on a suitable neighborhood of O. For that, definethe function(4.1.2)4.1. Morse indices of min-max critical points—the nondegenerate case^ 65Since 0 is non-degenerate, there exists K> 0 such that for all z E E,K -1114 - IV (0)4 5_ KPH-It follows that for z 0 0, we havef (t, z) = —b(z)licon (0)z + t'cb' (z)II-2 (co" (0)z + to' (z)).Let e = el?, and use (4.1.2) and (4.1.3) to obtain for lizil 8(4I fit , z)I 2K (K + *PH.Since f(t,0) = 0, f is continuous. Let p E (0,5(E)) be such that for all lizil < p,Ike(z)II .. 1.(4.1.3)(4.1.4)(4.1.5)Using (4.1.2), (4.1.3) and (4.1.5) it is easy to find Ki. > 0 such that Ith(t, z)11 < K1 forPH < p and z 0 0. This coupled with the mean value theorem yields some constant C > 0such thatIlf (t, zi) — f (t, z2)II^ClIzi - z211for all z1, z2 in B.It follows that the Cauchy problemVno) = f (t, 77)= zhas a continuous solution n for z in some open neighborhood W of 0. Note thatd—dt F (t ' 77(t)) = Ft(t , ?Kt)) + (F z(t , 71(0), 7)(t)) = 0and in particularcp(0) + (cp" (0)z , z) = F(0, z) = F (1, n(i, z)) . v(77(1, z)which means that the homeomorphism H(z) = n(1, z) verifies the claim of the Lemma •66^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTSCOROLLARY 4.2 Let co be a C2-functional on a Hilbert space E and vo be a non-degeneratecritical point for yo with m(vo) = k. Then for any r > 0, there exist a neighborhood N ofvo with N C B(vo,r) and eo > 0 such that for all 0 < c < coN n Gv,(,,o)_, L-11 11+ X Sk-1 X (0, 1).^ (4.1.6)where B+ = lu+;u+ E E+ and Ilu+11 < 11.Proof: Let E,E+,E_,H,M and W be given as in the above lemma associated to vo. Letr1 be small such that B(0, ri) C W and put i,b(z) = yo(H(z)) - co(vo) = liz+112 - Ilz-II2. Weclaim that for any r2 and el with 0 < ci <r2 <r1 we have for all 0 < c < ci thatB(0, r2) fl 10(z); IP(z) < -cl'-"--='- B+ x Sk-1 x (0,1).Indeed, for any r2 > r3 > c > 0 and z_ E E_ with PA = r3 we have thatfz+;11z+11< r3 - c} c B(0, r3) n {z; 0(z) < -E}.Let t be small enough such that H(B(0, t)) C B(vo, r) fl B(vo, ri). Then N = H (B(0 , t))together with co = t/2 will verify (4.1.6) and the corollary is proved.^ •We also need the following lemma which is basically due to Lazer-Solimini [26].LEMMA 4.3 Let cp be a C2-functional on a Hilbert space E and v be a non-degeneratecritical point with Morse index n. Then for any r > 0, there are 0 < r', r" < r and acontinuous map f on E such that the following holds:(i) t f (x) -I- x(1 - t) = x on E\B(v,r') for all 0 < t < 1;(ii) co(t f (x) + x(1 - t)) < w(x) on E for all 0 < t < 1;(iii) f (B(v,r")) is homeomorphic to a subset of IV.Proof: Since v is a non-degenerate critical point for V) on E, let H be the change of variablesmap associated to vo by the Lemma 4.1 and write E = E_ e E. Choose r1 > 0 and r2 > 0small enough so that if B_ (resp. B+) denotes the closed ball in E_ (resp. E+) of radius4.1. Morse indices of min-max critical points—the nondegenerate case^ 67r1 (resp. r2) centered at 0, then 2B_ + 2B+ is contained in the domain of H. We may alsoassume 4r? + 4r < r2. Let a be a Lipschitz function from R to [0,1] so that a = 0 on(-00,0] and a = 1 on [1, +oo). Let n : E E be defined by^P^ 1)] +a (iiz+ii^1)]ri(z_ + z+) = z_ + z+ [a(i 1) [1 — a (i-z±1ri r2^r2and consider the following transformation f : E Eon E \ (2B_ + 2B+)f (x) IxH o no II-1(x) on 2B_ + 2B+Clearly f is a continuous map on E. Now take r' = 2Vr? + r3 and r" small such thatB(v, r") C B_ + B. Then (i) is obvious. For (ii), we first note that it is true when t 0.Then we need to note that for any z = z_ +z+ we have that t f (z) + (1— t)z = z_ + z+g(z,t)whereg(z,t) = t [ (a 112-=—^) [1— ari II 1^(Li—r211 — 1)] + a (iizr+211 1)1 + (1and that g(z,t) < 1 for all z and 0 < t < 1. (iii) follows obviously from the definition off^ •We shall need the following basic result from algebraic topology.LEMMA 4.4 For all n > 1, we have that r(S) = 0 if 0 < r < n and 11,(S) = 0 if r <nand r 0. Hence Sn is (n — 1)-connected. Moreover we have that H,.(S") = 0 if r > n.4.1.2 The Main ResultsFirst, we establish the following theorem.THEOREM 4.5 Let yo be a C2-functional on a Hilbert space E and let vo be a non-degeneratecritical point of co with m(vo) = k (k > 0). Then the following holds:(1) w-sad(vo) = sad(vo) = m(vo) and sad*(vo) = m(vo).(2) vo does not have the weak bi-saddle property of order n if n k for all n > 1.68^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTSProof: (1) If k = 0 this is trivial. For k > 1, we first proof that sad(vo) > w-sad(vo) > ni(vo)and sad*(vo) < m(vo). By Corollary 4.2, we see that for any neighborhood N of vo, thereexist an open sub-neighborhood M C N of vo and an co > 0 such that for all 0 < € < €o wehave thatm n G9,(,,0)_, e.-. /if x sk-4 x (3,1).By Lemma 4.4, we see that M n Gs00,0_, is k — 2-connected. Hence we have that sad(vo) >w-sad(vo) > k. As for sad*(v0), we first note that by Lemma 4.3, for any neighborhood Nof vo, there exist r1, r2 > 0 and f verifying the conclusion of that lemma. Clearly from (iii)of that lemma, we have that sad*(vo) < k. Next we show that m(vo) > sad(vo) > w-sad(vo)and sad*(vo) > m(v0). To do this we consider cp(z) = 114112 — ljz_112 on Hilbert spaceE = E+ 0 E_ where E_ :=2- Rie and z --+ (z_, z+) corresponds to the decomposition ofE into the positive and negative spaces E+ and E_. It is easy to set up a trivial min-max process for both homotopic and cohomotopic cases. Clearly co verifies (PS) conditionand 0 is the only critical point. Now by Corollary 3.13 and Corollary 3.24, we see thatm(vo) > sad(vo) > w-sad(vo) and sad*(vo) > m(v0).(2) If k = 0, since H(0) = 0 for all n E Z we see that vo does not have the weakbi-saddle property of order n for all n > 1. For k > 1, as above by Corollary 4.8, we seethat for any neighborhood N of vo, there exist a sub-neighborhood M C N of vo and anco > 0 such that for all 0 < e < co we have thatm n G)_, F--.a ifi+ x sk--1 X (0, 1).Now by Lemma 4.4 we see that vo does not have the weak bi-saddle property of order n ifn k for all n > 1. •THEOREM 4.6 Let yo be a C2-functional on a Hilbert space E and consider a homotopicfamily .7- (resp. a cohomotopic family Y) (resp. a homological family J.) of dimension nwith closed boundary B. Let .7.* (resp. .7-*) (resp. .-F*) be a family dual to .7. (resp. .F)(resp. J.) such thatc := sup inf co(x) = inf max (x)FEY** xEF^AET xEA4.1. Morse indices of min-max critical points—the nondegenerate case^ 69(resp.(resp.:= sup inf ca(x) = inf max co(x))FEfr. xEF^AEY xEAand is finite. Assume that cp verifies (PS),, (resp. (PS)e) (resp. (PS)) along a min-maxing sequence (Ak)k in T (resp. (Ak)k in ---1) (resP. (Ak)k in .'F') and a suitable max-mining sequence (Fk)k in .F* (resp. (Pk)k in .F*) (resp. (Pk)k in and assume thatkc^n Foo n A,, (resp.^n l n A00) (resp.^n Poo n A) consists ofnon-degenerate critical points. Then the following holds:(1) There exists an x1 E kc such that m(xi) < n;(1)' If B 0, then there exists a xl E kc such that min{2, n} < m(4) < n;(2) There exists an x2 E ke such that m(x2) > n;(3) If c^c = e and Fk is dual to .F for all k > 1, then there exists an x3 E k, suchthat m(x3) = n.(4) There exists an x4 E ke such that m(x4) n.REMARK 4.7 Note that the results in Theorems 4.6 remains valid if the min-max is takenover a union of homotopic (resp. cohomotopic) (resp. homological) families of the samedimension but possibly with different boundaries, provided these boundaries are all disjointfrom the set F. This follows from the proof below and Theorem 3.33.If we suppose that sup co (B) < c, then again the above applies to the dual set F =^>cl and we get the followingCOROLLARY 4.8 Let co be a C2-functional on a Hilbert space E and consider a homotopicfamily .F (resp. a cohomotopic family .-7;) (resp. a homological family -.F) of dimension nwith closed boundary B. Suppose that c := c(co,,F) (resp. e c(v, .F)) (resp."e" := c(co, J"))and that sup (B) <c (resp. sup (B) <) (resp. sup co(B) < 0. If co verifies (PS), (resp.(PS),-) (resp. (PS),-) along a min-maxing sequence (Ak)k and if the set K, n A°. (resp.:= sup inf co(x) = inf max yo(x))Fet. xEF^AE.T rEA70^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTSKe fl tiec) (resp. ICE n Ao„) consists of non-degenerate critical points, then the followingholds:1) There exists x E IC, n A°. with m(x) < n (resp. x E Ke n Ao0 with m(x) > n) (resp.x E Ke- n Aco with m(x) = n);ii) If.TcP and c = E, then there exists 53 in K, n Acx, with m(x3) = n.Similar results to the above corollary in the classical saddle point theorem setting wereobtained by Lazer A. C. and Solimini S. [26].PROOF OF THEOREM 4.6: (1) Let x be the point given by Theorem 3.11. Then by Theorem4.5 we have that m(x) < n.(1)' We may assume n> 1. Otherwise (1)' is a corollary of Theorem 2.10. When n > 2,we note, as in the proof of Theorem 4.6, that for any neighborhood N of x, there exist asub-neighborhood M C N of x and an co > 0 such that for all 0 < e < co we have thatM n G,, :4' B+ x Sm(x)-1 x (0, 1).Now note that each path connected component of S° is contractible. So we must have thatm(x) > 2 if n > 2 by Remark 3.21.(2) It is the direct corollary of Theorem 3.23 and Theorem 4.6.(3) It is the direct corollary of Theorem 3.26 and Theorem 4.6.(4) It is the direct corollary of Theorem 3.30 and Theorem 4.6.^ •4.1.3 Morse indices in the saddle point theoremLet E = Y ED Z with dim(Y) = n and consider the homotopic (resp. cohomotopic) (resp.homological) class .7. (resp. ,P) (resp. J") with boundary Sy described in Corollary 3.34.COROLLARY 4.9 Let cp be a C2-functional on the Hilbert space E such thata := inf co(Z) > 0 > sup co(Sy).4.2. Morse indices of min-max critical points—the degenerate case^ 71Let c = c(cp,Y), E = c(cp,Y) and 5 = c(cp, f'). Assume that cp verifies (PS) and that d2cp isnon-degenerate on the critical set. The following holds:If 0 < E, then1) There exists x1 in 1‘, with m(xi) < n;2) There exists x2 in KE with m(x2) > n;3) There exists x3 in KE with m(x3) = n;4) If c = E, there exists x4 in Ke with m(x4) = n.Moreover5) If E = a > 0, there exists x5 in Kb- 11 Z with m(x5) = n;6) If S = a > 0, there exists x6 in KE n Z with m(x6) = n.Proof: Clearly c > E > a > 0 and if c> 0, then 1), 2), 3), 4) and 6) follow immediatelyfrom Theorem 4.6. Suppose now E = a, then by Theorem 4.6, there exists x5 in ICE n Zwith n < m(x5). It remains to notice that since cp > a = E = cp(x5) on Z, we have thatm(x5) < codim(Z) = n. •4.2 Morse indices of min-max critical points—the degenerate caseIn this section, we shall refine yet another perturbation technique developed by Marino-Prodi, in order to reduce, in some instances, the general problem to the non-degeneratesituation where the results of the previous chapter can be applied.The proof consists of perturbing the function co in a neighborhood N of the critical setto obtain a non-degenerate function 0 on which the results of the previous section mayapply. Since we need 0 to have a critical point of the right index in that neighborhood, themin-max procedure needed to be taken on what we call the homotopic (resp. cohomotopic,resp. homological) restriction of the original family to the neighborhood N.Two extra complications are to be overcome during the application of the standardMarino-Prodi method. First, we need to preserve the location of the critical point on thelimiting set F. Secondly, we are only assuming the Palais-Smale condition to hold along a72^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTSfixed min-maxing sequence (A.),, and a suitable max-mining sequence (Fn). In total, twoperturbations of the function coupled with an appropriate change in the homotopy-stableclass will be necessary.4.2.1 Statement of the theoremsIn the sequel K(cc) will denote the set of critical points of the function co at the level c.When no confusion is possible, we will drop yo from the above notation. We shall deal withthe following kind of degenerate critical points. Assume vo E K, is such that d2co(vo) is aFredholm operator on the associated Hilbert space E, then E = E0 e El where E0 and E1denote the kernel and range of d2co(vo) respectively. Moreover, dim(E0) <+00 and Ap(vo)is invertible on El. From basic spectral theory, it follows that E1= E+ e E_ where E+and E_ are the positive and negative spaces associated to d2co(vo) restricted to El. Theaugmented Morse index of vo will bem*(vo) = dim(E0) + dim(E_).We start by stating the main results of this section.THEOREM 4.10 Let cp be a C2-functional on Hilbert space E and consider a homotopicfamily (resp. cohomotopic family .P) (resp. homological family P) of dimension n withclosed boundary B. Let 2** (resp. ) (resp. ) be a family dual to .F (resp. .F*) (resp.:7"*) such that^:= sup inf^(p(x)^inf max co(x)FE.r. rEF AEY xEA(resp.:= sup inf w(x) = inf max CO(X))^pEf. xEF^AEY xEA(respsup ma cp(x) = inf, mac (x))^E-Ej. xEP^AEiand is finite. Assume that cp verifies (PS), (resp. (PS)) (resp. (PS)) along a mm-maxingsequence (Ak) in .7. (resp. (Ak)k in .P) (resp. (ilk)k in .-t), and a suitable max-mining4.2. Morse indices of min-max critical points—the degenerate case^ 73sequence (Fk)k in .7-* (resp. (Pk)k in ..P* ) (resp. (Pk)k in .-t*). Suppose that d2v is Fredholmon either K, n Foo which is isolated in K ( resp. Ke n Fe° which is isolated in K) (resp.Ke- n Pc.,, which is isolated in Ke) or IC, n Ac„,, which is isolated in IC, (resp. Ken 4,, whichis isolated in K) (resp. Ke n Acc, which is isolated in Ke). Then the following holds:1) There exists x1 in K fl Foo n A„,, with m(xi) < n;2) There exists x2 in K,n P„,,nAoc, with m* (x2) > n;3) There exists x3 in Ke n Pe„ n Acc, with m(x) < n < m* (x3);4) If .7. C.P,c=E and Fk is dual to f' for all k > 1, then there exists x4 in Kcn Fi„,, n A0,3with m(x4) < n < m*(x4).If we suppose that sup co(B) < c, then the above applies to the dual set F = {co > c}since IC, fl F = K is obviously isolated in itself and we get the followingCOROLLARY 4.11 Let co be a C2-functional on a Hilbert space E and consider a homotopicfamily .F (resp. a cohomotopic family .f.) (resp. a homological family ..F') of dimension nwith closed boundary B. Assume that c := c(yo,,T) (resp. E := c(c p, -.7")) (resp.^6 := c(co,i))is finite and that sup yo(B) <c (resp. sup co(B) < (resp. sup co(B) < -6). If co verifies (PS)c(resp. (PS)) (resp. (PS)) along a min-maxing sequence (A„)„ and if dc' is Fredholm on(resp.^(resp. Ke). Then the following hold:1) There exists x E K fl Ao. with 712(x) < n (resp. x E Ke n Aec with m' (x) > n) (resp.x E KflA, with m(x) n m*(x));2) If C .F and c = E, then there exists x3 in IC, n A„,, such that m(x3) < n < m* (x3).4.2.2 Restrictions of a homotopy-stable class to a neighborhoodLet .7- (resp.^(resp. 5-) be a homotopic (resp. cohomotopic) (resp. homological) familyof dimension n and with boundary B. Let N be an open set such that Nn B = 0. Let C bea subset of^(resp. .F)(resp. .-F) consisting of sets A such that A \AT .F (resp. „P) (resp.We define the homotopic (resp. cohomotopic) (resp. homological) restriction of G toN in the following fashion:74^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTS1) Since A E .T is of the form f(D) with D being a subset in Rn and since A n N 0 0,let DA = f- ' (A n N), D61 = f-1(A n am, 0-A= fiD64 and consider the classLA = {C; 3 g : DA -+ E continuous withg = crA on Do4 and C = g(DA)}.Let L(N) = UAE.c LA. It is clearly a generalized homotopic family of dimension n and withboundary equal to Wee A n ON.2) Let now AELC P. Since A\N 0 ..T there is f : A\N —> Sn' continuous such thatf = a on B. Let aA = fonaN and consider the classZit = IC C E; C compact containing A n ONand -yG(C, A n aN,o-A) > n} .ELLet L(N) = UA LA. It is clearly a generalized cohomotopic family of dimension n andwith boundary UAEL A n ON.3) Let now AELC P. Since AVV 0 ..t, we get from the exact sequenceHn(A\ N, B) —> Hn(A,B) -4 Hn(A, A\ N)that H„(A, A\N) 0 {0} and since by excision, this class is isomorphic to 11(A n51, AnaN),we shall denote by (IA the non-trivial induced class in the latter. Again, we let Z(N) =UAEL-r(aA) where F(QA) is the n-dimensional homological family with boundary A n ONassociated to the class aA•The following lemma summarizes the above remarks and compares the homotopic, co-homotopic and homological restrictions to the set-theoretic restriction of L to N. That isLN = {A n N; A E L}. The proof is straightforward and is left to the reader.LEMMA 4.12 Let .7. (resp. .f.) (resp. ,-F") be a homotopic (resp. cohomotopic) (resp. ahomological) family of dimension n with boundary B. Let N be an open set such thatN n B = 0. Let G be a class of subsets of .7- (resp. f)(resp. fr) consisting of sets A suchthat A\N 0 .7- (resp. Y) (resp. ,-t). Then,4.2. Morse indices of min-max critical points—the degenerate case^ 75i) L(N) (resp. Z(N)( resp. L" (N)) is a generalized homotopic ( resp. cohomotopic)(resp. homological) family of dimension n and with boundary B' = UAEL A n ON;ii) LN C Z(N) (resp. G(N))( resp. Z(N));iii) For each C E L(N) (resp. .-C(N)) ( resp. Z(N)), there is A E L such that (A\MUC E.7- (resp. .f.) ( resp. J.).4.2.3 The Marino — Prodi perturbation methodWe summarize the properties of this perturbation in the followingPROPOSITION 4.13 Let cp be a functional in C2(E, IR) and let K be a compact subset ofIf, on which cl2co is a Fredholm operator. For b> 0 and E > 0 small enough, there existsa function 0 in C2(E, IR) that can be taken to be comparable to co and which verifies thefollowing:(i) 11C0 — 011c2 .< E and 0 can be chosen such that either 11) < cp or cp < 11);(ii) w(x) = V.,(x) if x V N28(K);(iii) All the critical points of IP in No(K) (if any) are non-degenerate and finite in number;(iv)do is a proper map on N46(K).Moreover, if K is an isolated critical set in IC then 5 and c can be chosen in such a waythat(v) 0 has no critical points in N38(K)W8(K) at any level in [c— E, C-I- EL More precisely,for 5> 0 small enough, there exists p = p(8) > 0 such that for E small enough, 0 can bechosen in such a way that for every x E Emax{114(x)11,10(x) —cl,dist(x,N38(K)\N8(K));x E El ?:_ p.For the proof we shall need the following two lemmas.LEMMA 4.14 If K is a compact subset of E, then for every it > 0 there exists a C'function t : E —> [0, 1] with all its derivatives bounded such thate(x) = 1 for all x E N(K)^ (1)76^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTSt(x) = 0 for all x E E\N2m(K).^ (2)Proof: Fix a finite number of points xl, X2, ...x,„ E K such thatK CWe get C'-functions on E with bounded derivatives such thatti(x) = 1 if x E B(xi —3 It)2ti(s) = 0 if x B(xj,2,u)by simply taking a real function a : R^[0, 1] such thata(s) = 1 if s<a(s) = 0 if s > 4and lettingei(x) = a (Ilx xill2it2^) •Then we take /3 : R^[0, 1] such that 13(s) = 0 if s < 0 and /3(s) = 1 if s > 1. The function: E IR defined byt(x) = 13(^ei(x))is the required function.^ •LEMMA 4.15 Assume that for every x in the compact set K, d2co(x) is a Fredholm operator.Then there exist a neighbourhood N of K and E > 0, such that every functional 7,b E C2,with II (P Olic2 < e has the gradient cl71) that is a proper map on N.Proof: It is clear that we can restrict ourselves to the case where K is a singleton {1}. Fora given e, take a closed bounded neighborhood N of .t so that dv— d2c,oW is a Lipschitz mapon N, with a Lipschitz constant less or equal than E. If E C2 is such that liv — 0116,2 < e,we consider the map r(x) = d0(x) — d2co(-X")(x) which is also a Lipschitz map on N with4.2. Morse indices of min-max critical points—the degenerate case^ 77Lipschitz constant less or equal to 2e. Since d2co() is a Fredholm map, we can thereforesplit the space E into the sum of two closed subspaces V and W such that V is finitedimensional and such that for some positive constant aIld2VM(w)II allwIl for all w E W.^(3)Suppose now (x,i)„ is a sequence in N such that d(x) is convergent. For every n, writexn = vn ± wn with vn E V and wn E W. Since Vn is finite dimensional we can assume thatthe sequence (vn)n is convergent. For integers n and m, we haveII wn — wm11^a-111(d2C0()) (wn — Wm) (I< ce-111r(xn) — r(xm)II + a-111(d2C0(-))(vn — vm)II+a-1 11 dtP(xn) — thP(xm ) II< 2ea-l(lvn — vmll + Ilwn — wm11)+a-1 II d2SoW II II vn — vrn II + a-ill d'IP(xn ) — d0(xm)II.So, if we take e < ia, we have from (3) that for a suitable constant c,liwn — wmil < calvn — vm II ± IldiP(xn) — d0(xm)11).^(4)Hence, since (vn)n and (dO(xn))n are two Cauchy sequences, we see from (4) that (xn)n isa Cauchy sequence. Therefore the map 7// is proper on N.^ •PROOF OF PROPOSITION 4.13: Assume that d2v(x) is a Fredholm map for every x E Kwhere K is a given compact subset of K. Let N = NI(K) (p > 0) be the neighborhoodof K given by Lemma 4.15, and let i be the function associated to co, K and p by Lemma4.14. Let M = sup{lixil; x E N2p(K)}. Fix E > 0. Since cico is Fredholm, we may use theSard-Smale theorem [36] to find y E E*, with HMI < eig such that —y is a regular value forthp.For any xo E N21(K), the function defined by0(x) = cp(x) + t(x )(Y,x — xo)78^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTSclearly verifies i) and ii) of Proposition 4.13. Moreover, since on No(K) we have thatdti) = thp -I- y, it follows that all critical points of 0 in N, (K) are non-degenerate.To obtain w < 0 or I,b < co, it is enough to choose the xo that minimizes or maximizesthe function x .4 (y,x) on N2i,(K).Suppose now that K is isolated in K. There is a p > 0 such that cp has no criticalpoints at level c in N3(K)\K. By Lemma 4.15, we can assume d(p is a proper mapping onN4 (K). It follows that the functionx -4 1 14 (41 + I co(x) — clis bounded below by a positive constant b on the set N34(K)\ATI,(K). It is now enough toapply the first part of the Proposition with e < .. to get assertion (v). •If now (A„)„ is a min-maxing sequence such that IC, n Ao„ is isolated in IC, or (11)„ isa max-mining sequence such that IC, n F00 is isolated in K,, we shall need to isolate the setF,„3nK,n A°. in K. The next proposition shows that we can do these provided we perturbthe function appropriately.Suppose that cl2cp is Fredholm on 11,0 n 1‹, or 1<", n A. Then we note that there is p > 0such that for functional ti) close enough to co in the C2 norm, d20, clO are Fredholm andproper respectively in N3p(Kc n Foo) or N3p(24„, n Kr).PROPOSITION 4.16 Let co be a C2-functional on a Hilbert space E and consider a homotopicfamily .7" of dimension n with closed boundary B. Let .7.* be a family dual to .7" such thatc := sup inf co(x) = inf max co(x)FE.r.xEF AE.F xEAand is finite. Assume that (p verifies (PS), along a min-maxing sequence (Ak) in .7., and asuitable max-mining sequence (Fk)k in F*. Suppose that d2(p is Fredholm on L := K, n Foo(or L := K, n A00) which is isolated in IC Let Li = A„Q n Fc,„ n K. Then for (5 > 0and E> 0 small enough, there exist (pi (or (p2 E C2) and an isolated compact subset Ki ofK,((p1) (or an isolated compact subset K2 Of Kc(CO2)) such that4.2. Morse indices of min-max critical points—the degenerate case^ 790 Li c Icc(p1)nFoonAc,3 c Ki c N28(L1) (or Li c Kc(c02)nFo.nA,„ c K2 C N28(14));ii)I1V1 — VII 5- E 1 Ca^C°1 (or 11W2 — Wii .< E, (dr) < (p2) on E and (p = (Pi (or (p = y(72) in aneighborhood of Li;iii) c(1,.F) = c(v,..7") (or c(v2, .71 = c(cp, .7)) and (An)„ is still a min-maxing sequencefor (pi (or (022) and (F)„ is still a suitable max-mining sequence for (pi (or (P2);iv) (pi (or co2) verifies (PS), along the min-maxing sequence (An)„ and the suitablemax-mining sequence (F,)ri). Moreover, (pi (or co2) is Fredholm on Ki (or K2).All the above statements are true respectively in cohomotopic and homological cases.Proof: We shall consider the case L = Foonicc only. The other case can be proved similarly.As above, we assume 61 > 0, € > 0 such that for any functional 0 with 110 — coll < c, then0 is Fredholm on N38, (Li). Take- > 6> 0 small and let M = Ix E L; 6 < dist(x , L1) <261. It is compact and far from Li, hence for 62 small enough we have that for large k,N282(M) n Ak = 0. We can also choose 62 in such a way that N282(M) C Ns, (Li). ApplyProposition 4.13 to M, 6, and 82 to get a perturbation (pi > (p and satisfying i), ii) and iii) ofthat proposition. Since ep' is proper on N38(L1), we can also assume that the critical pointsof (pi at level c in N28(L1) \N8(L1) are in finite number and contained in N82 (M). We cannow easily find an isolated subset K1 of Kc ((pi) satisfying i). ii) follows from Proposition4.13 while we get iii) from the fact that v = col on Fk for k large enough and hencec(tpi, .7) < liminf„ sup coi (An)= liminf„ sup (p(An)= c((p, .7") < c((pi,Y).Clearly (A„)„ is still a min-maxing sequence for (pi whilec > lim sup inf (pi(F„) > lim sup inf (p(F„) = C.n-100^ fl-4000iv) follows from the fact that ("1 is a proper map on the set where (pi is different fromV-^ •80^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTS4.2.4 The reduction to the non-degenerate casePROOFS OF THEOREM 4.10: Let An be a min-maxing sequence in Y and let L denotes theset Kc n Fno n A. Let s inf{ m(x); x E L} , t sup{m*(x); x E L} + 1 and let Al(x) <A2(x) < • • • < Ak(x) < • • • be the eigenvalues of d2co(x). Note that As(x) < 0 < At(x) forevery x E L. Since the latter is compact and the Ak's are continuous, we can find So > 0 sothat As(x) < 0 < At(x) for every x in a So-neighborhood of L. Choose E0 small enough sothat for cp, with II (P e (P11C2 < e < 607 the eigenvalues (At (x))k of d2yoe(x) will satisfyA(x) < 0 < A(x) for every x in N80(L).In other words, for every critical point x of co, in N8(L) we have^s < m(x) < t — 1.^ (1)The remainder of the proof will consist of finding such a cp, that will have a critical pointof the right index in N60 (L)First we consider the following situation.Case (1): L = K n Foo n Ano c K c N40(L) where K is a closed isolated subset of K, andco is Fredholm on K.We can then apply Proposition 4.13 to obtain a function with b < co and satisfying(i)—(v) of that proposition. We can also assume that (5 is such that N46(K) C Nso(L).Unfortunately, we cannot apply Theorem 4.6 to V) and the homotopic family Y(N28(K))because we cannot insure that the function at the boundary is strictly below the min-maxvalue. To remedy that, we let N = N28(K), R = N38(K)\N6(K) and we consider the classC {A E .T; lim sup //)(A n Fn) < c and lim sup //,((A \ Fn) n R)) < cl.n-ioo^ n-+ooWe distinguish two cases:^If C is empty then, since ti) < co we have that c(0,^c, the sequence (An)n is min-maxing and the sets F'n = F,, U R e .7" is a suitable max-min sequence for b and the classF. MoreoverKc(0) n^fl Ax, C [N U (K,((p) n Foo n A00)] u (i-c,(0) n^c N4.2. Morse indices of min-max critical points—the degenerate case^ 81and hence all the critical points of ti) on it are non-degenerate. Since /// is proper in N45(K),ii) verifies necessarily condition (PS)F,,c along (An)n. Theorems 4.6 then apply to 0, F' andthe same class .7" to yield the required result.Suppose now C is not empty and letc1 = inf sup 0(A n N) . c(p, cAT)•AECWe have thatC — 6 < C1 < C.Indeed, the second inequality is obvious while the first follows fromc < lim sup cp(A n Fn) < max{ lim sup (A n F„,\ N), sup cp(A n Fn fl)}n—roo^ n—roo(2)< max{ lim sup 0(A n Fn), sup (p(A n T)}.-400so that if A E C, then lim sup 0(AnF) < c and hence c < sup (p(AnN)). Since UV-0i Eln—roo(2) is proved.Let now S = N58/2(K) \ N38/2(K) in such a way thatON c S c Ar814(s) c R.We shall need the following deformation lemma. Note that in view (v) of Proposition 4.13,we can assume S, E and p chosen in such a way that 0 < p < 6/4, E < p3 and for everyx E Emax{W(x)11,10(x) — cl,dist(x, Nu (K)\N8(K)); x E El > p.LEMMA 4.17 There exists a homeomorphism n : E —> E satisfyingi) 11)(71(x)) < 0(x) for all x E E;ii) P(x, n(x))^P;iii) n(x) = x whenever x E E\(RUIr E X; 0(x) < el) where e E R such that e < c— p3;iv) n maps Ix E X; 0(x) 5_ cl n S into Ix E X; 0(x) < c — p3}.82^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTSPROOF OF LEMMA 4.17: Apply Corollary 1.13 to the function 0 and the disjoint setsD = R, D1 = S to find an appropriate deformation q.Back to the proof of the theorem. Consider the familyG = {n(A); A E C}and the associated homotopic classes that we shall all denote by L(.N'). We shall prove thatci = c(0, .C(N)) is a critical level for 0. First note that since £ N C L(N), we have^CI C2 = C(0) LN)'^ (3)We now show thatG C C and therefore c1 < c2.^ (4)Indeed, we have for any set A,fim sup t71)(7)(A) n Fn) < lim sup //)(A n 71-1 (Fn)n-oo^ n-+oo< max{ lia sup CA n Fn), sup 0((A\Fn) n Trl(Fn))}.If now A E C and since (rri(Fn)\Fn)AR = 0 for all n, we get that the above is less thanmax{ lim sup CA n Fe), lim sup tp((A\Fn) n R), lim sup 7,b((A\Fn) n 77-1(Fe) \ R)} < c.n-c3o^n->co^ n-400On the other hand, since n-1(R) c R, we have thatfirn sup 0((n(A)\Fn) n R) < sup 0(A n n-i(R)) < sup 0(A n R)n-oo< max{ lim sup(O(A n Fn ft R), iii sup 0((A\Fn) n R)} < c.n->oo^ n-+oc)This proves assertion (4). We now prove thatC2 = Ci •^( 5 )Since N C 11(N U S) we have thatsup 71)(77(A) n N) < sup1P[n(A n (N u S))]4.2. Morse indices of min-max critical points—the degenerate case^ 83< max{sup l,b(A n N), lim sup 0(7/(A n Fn n S), Jim sup 0(77((A \ Fn) n R n S))}.^n—>oo^ n—/ooIf A E C, we get in view of Lemma 4.17 thatsup ?/'(n(A) n^5_ max{sup 7/)(A n N), c — p3, c — p3} .On the other hand, (2) gives that sup 0(A n^> C1 > C—E> C— p3, it follows thatsup 0(77(A) n^.< sup (A n 7C7).This clearly proves (5). We now prove that(6)First note that by (3) and (5) we have that c' < c2 = c1. For the reverse inequality, letC E L(N) and assume without loss of generality that sup tP(C) < ci. By (iii) of Lemma4.12, there exists AELCC such that the set A' = (A \ N) U C e Y. We claim that A' E C.Indeed,Jim sup 1'(A' n Fn) < max{ Jim sup (A n Fn), sup 0(0}n—>oo^ n-400< max{ Jim sup0(A n Fn), Cl} < C.n—*cx)SimilarlyJim sup7P((A' \ Fn) n^.< max{ Jim sup(C(A \ Fn) n R), sup 11)(C)}n—>oo^ n-400< max{ Jim sup(0((A \ Fn) n R),ci} < c.n-4o3It follows that C fl N = A' n N E CN, which clearly implies that c' > c1 and hence we haveequality.Finally, we claim that we have the boundary conditionsup sup 11)(77(A) n aN) <Cf.AEC (7)84^ 4. MORSE INDICES OF MIN-MAX CRITICAL POINTSIndeed, since ON C n(S), we have in view of (iv) of Lemma 4.17sup 0(n(A) n ON) < supl,b(ri(A n S))< max{ lim sup 0(77(A n F„ n S), lim sup (A \ Fiz) n S)}fl-400^ fl- 00< c- p3 < C — e < ci = c I.But (7) means that the set {ti) > c'} verifies conditions (F1) and (F2) with respect to theclass L(N) and the function 0. Moreover, since c2 = c' and LN C .C(N), we can find a min-maxing sequence (C„)„ consisting of subsets of 7. Since chp is proper in N, tl) necessarilyverifies (PS),, along the sequence (C„)„. It follows that Coo c N and K(0)n Coo C N8(K)on which the critical points are non-degenerate. Theorems 4.6 and Remark 4.7 now applyand we get a critical point x for ti) in N80(10 with m(x) < n. This coupled with (1) abovegive the claims in Theorems 4.10.To finish the proof of these Theorems, it remains to deal with the case when we onlyassume thatCase (2): L = Keil Fc.0 (1 Aoo is not contained in a set which is isolated in K.Since co is Fredholm on Kc 11 Foo (or Kc n A00) which are isolated in Kc, we can useProposition 4.16 to isolate it in the critical set of an appropriate perturbation of the func-tional.Let so and E0 be as in the beginning of the proof and let Li = Foo n K. Then weapply Proposition 4.16 with 0 < Si < (50/4 and 0 < Ei < E0/4 to get a perturbationCol that verifies assertions i)—iv) of that proposition. Since foi is Fredholm on Ki andK,(01) n Foo n ,40„ C Ki C N281(Ko(c2' 1) n Foo n Ao0) with that Ki is isolated in K,(01),and since ciii verifies the same properties as co with respect to the class ,T, we can apply theconclusion of the first case to 01 to get x E K(ç1) n Foo n Aoo = L with m(x) < n. SinceKoi — yoli < E0, the claim follows from assertion (1). The proofs of the cohomotopic andhomological cases are identical. •4.2. Morse indices of min-max critical points—the degenerate case^ 854.2.5 Application to standard variational settingsWe let again E = Y ED Z with dim(Y) = n and consider the homotopic (resp. cohomotopic,resp. homological) class .7- (resp. .f", resp. -P) with boundary Sy described in Corollary3.34.COROLLARY 4.18 Let co be a C2 -functional on the Hilbert space E such thata := inf so(Z) 0 sup co(Sy).Let c =^ë = c(co,.P) and e = c(cp, -i""). Assume that co verifies (PS) and that d2cc, isFredholm on the critical set. The following hold:If 0 < e, then1) There exists x1 in Ke with m(xi) < n;2) There exists x2 in Ke with m*(x2) > n;3) There exists x3 in Ka with m(x3) <n < m*(x3);4) If c = E, there exists x4 in K, with m(x4) <m < m*(x4)•On the other hand5) If = a > 0 and if Ka n Z is isolated in Ka, there exists x5 in Ka n Z with m(x5) <n < m*(x5);6) If = a > 0 and if Ka n z is isolated in Ka, there exists x6 in Ka n Z with m(x6) <n < m*(x6).Proof: 1), 2), 3) and 4) follow immediately from Corollary 4.11, while 6) is a consequenceof Theorem 4.10. Suppose now E = a, then by Theorem 4.10, there exists x5 in Ke n Zwith n < m*(x5). It remains to notice that since cp > a = = 'p(x5) on Z, we have thatm(x5) < codim(Z) = n. •Chapter 5Morse-type information on Palais-Smale sequencesLet cp be a C2-functional on a Hilbert space H, and suppose c is an inf-max level of coover a homotopic family .7. of dimension n. We have seen in the last chapter that, if cpverifies a (PS)-type condition and if co" is Fredholm, then one can find critical points atthe level c whose Morse index is at most n. In this chapter, we study what happens whenthe compactness and non-degeneracy conditions on co are not satisfied. More specifically,we want to construct a Palais-Smale sequence (x k)k that carries the topological informationgiven by .F, in addition to the properties of being close to a min-max sequence (An)n anda max-mining sequence (F„)„ concerning the location of (xk)k obtained in Theorem 1.3. Inother words, we are now looking for a more analytical (second order) property concerningthe Hessian co"(xk), which can be viewed as the asymptotic version of the information onthe Morse index of the limit of (xk)k whenever such a limit exists. As shown in Section 2of Chapter 1, this can be done in minimization problems: one then gets an almost criticalsequence (xk)k that is minimizing and which satisfies(V"(xk)w, w) —ii 1142 for any w E H.This is done by establishing a smooth perturbed minimization principle which shows that anappropriate quadratic perturbation of (do actually attains its minimum. Analogously, one maytry to establish perturbed variational principles for problems not involving minimization.The idea is to construct C2-small perturbations so that the new functional has a true criticalpoint with the Morse index that is expected for the original one. We can then transfer thisinformation to the Hessian of the original function co at that point, even though it is onlypseudo-critical for that functional. Unfortunately, and unlike the minimization case, theproblems are quite involved and we run into serious difficulties while trying to execute this865.1 Introduction and statements of the main results^ 87program. The main difficulty lies in the improvement of the perturbations from being inthe space C" to C2 which is relevant for applying the results of Chapter 4.However, we shall now present a method for constructing directly without establishingthe perturbation result an almost critical sequence with the appropriate second order infor-mation, provided we have an additional, but acceptable, assumption of Holder continuityon the first and second derivatives. In return, and in the case where the (PS) conditionholds, we obtain all the results of Chapter 4 without the Fredholm-type assumption on yo".For more details about this chapter, we refer to [16] and [17].5.1 Introduction and statements of the main resultsIt is by now well known that if cp is a C2-functional on a Hilbert space H that satisfies acompactness condition a la Palais-Sm.ale while d2co satisfies a non degeneracy condition ala Fredholm, and if c is an inf-max level over a homotopic family (resp. a cohomotopicfamily P) of dimension n, then one can find at the level c, critical points whose Morse index(resp. augmented Morse index) is at most n (resp. at least n). This is done for examplein the last chapter. See also [26],[6], [20]. If now the compactness and non-degeneracyconditions are not satisfied, we shall try to construct a Palais-Smale sequence (xk)k as inTheorem 1.3, which also satisfies the asymptotic version of the information on the Morseindex of the limit of (xk)k whenever such a limit exists. For that, we need the concept ofapproximate Morse indices:For x E H and p> 0, we definern-f; (x) = supfdim(E); E subspace of H, (d2cp(x)w,w) < —P11w112 vw E Elandm(x) = inf{codim(E); E subspace of H, (d2 cp(x)w , w) > Pliw112 Vw E El.Note that if p = 0, then nt-,; (x) (resp. mp+(x)) coincide with the Morse index m(x) (resp.,the augmented Morse index m+(x)) of x.88^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESHere are the main results of this chapter. The following theorem will be proved in section5.2.THEOREM 5.1 (The homotopic case) Let cc be a C2-functional on a Hilbert space H suchthat dcp and d2cp are Holder continuous. Let ..T be a homotopic family of dimension n withboundary B and let .7F" be a family dual to .F such thatc := inf max cp(x) = sup inf cp(x)AEY xEA^FEYF" xEFand is finite. Then, for every min-maxing sequence (Ak)k in ..7- and every suitable max-mining sequence (Fk)k in F", there exist sequences (xk)k in H and (pk)k in IR+ with limk Pk -=0 such thati) limk cp(xk) = c;ii) limk Ildso(xk)il = 0;iii) xk E Ak for each k;iv) limk dist(xk , Fk) = 0;v) mp-k (xk) < n for each k.We note that the last assertion implies that for each k, d2co(xk) has at most n eigenvaluesbelow —pk (See Lemma 5.12. Moreover, any potential cluster point of the sequence (xk)kwill have a Morse index at most equal to n.The above theorem also yields, among other things, the following result concerning theexistence of some good paths. First, setGd = Ix E H; co(x) <d}, Ld = Ix E H; co(x) _?_ d}.IC (n, e) = Ix; c — e cc(x) 5. c + E 1 114(X)11 < E , n1; (X) 5_ n}K; (n) = {x; co(x) = c, dcp(x) = 0, m(x) < n}.andCOROLLARY 5.2 Let co be a C2-functional on a Hilbert space H such that dcp and d2y, areHOlder continuous and let .7" be a homotopic family of dimension n with boundary B such5.1 Introduction and statements of the main results^ 89that sup cp(B) < c(p,..F) =: c. Then, for every c > 0, there exist (5 (0 < (5< E) and A E .Fsuch thatsup co(A) < c + and A C Gc_j U^(n,c).Proof: If not, then there exists e > 0 such that for every (5> 0, the setF6 = Lc_s n (Ix; Ildcp(x)II^U {x; m; (x) > n})is dual to _F. Since lim inf (p(F8) = c, Theorem 5.1 yields a sequence (x8)8 and a positive5-Alfunction p((5) with lim p((5) = 0 such that lim ç(x) = c, lim d(x6) = 0 and m(5) (x5) < n8.-H)^ 8->0^8->0while at the same time approaching the set{x; Ildc,o(x)II^el U {x; m;"" (x) > n}.This contradicts the uniform continuity of dy and d2cp.^ •In view of Theorem 5.1, the following definition is in order.DEFINITION 5.3 A 0-function on a Hilbert space H is said to have the Palais-Smalecondition at level c, around the set F and of order less than n (in short (PS)F,c,n-), if asequence (xk)k in H is relatively compact whenever it satisfies the following conditions:limk cp(x k) = c, lirnk ilthgx k)II = 0, limb dist(xk, F) = 0 and there exists a sequence ofpositive reals (pk)k with limk Pk = 0 such that for every k, d2(p(xk) has at most n eigenvaluesbelow —Pk.For examples of functionals satisfying (PS),- but not the classical Palais-Smale con-dition, we refer to [16] and [17]. Now we can state the followingCOROLLARY 5.4 Under the hypothesis of Theorem 5.1, assume that .F* consists of a singleset F and that cp satisfies condition (PS) F,c,n- then IC- (n) n F is non-empty.Moreover, for any E > 0 and any neighborhood U of IC- (n) n F, there exists A E .F suchthat sup co(A) < c + e and A C (H\F) U U.In particular, if sup co(B) < c, then for any E > 0 and any neighborhood U of k(n),there exists A E .F such that sup co(A) < c + E and A C G, U U.90^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESProof: The first part follows from Theorem 5.1 and the fact that co satisfies (PS)F,c,n- •Let now U be any open neighborhood of I<"(n) n F and assume the second assertion nottrue for some e > 0. This means that the set F \U is dual to .7" and since inf co(F) > c, itfollows from the first part that ./C(n) n (F \ U) is non-empty, which is a contradiction. •In the setting of the classical saddle point theorem, Theorem 5.1 yields the followingCOROLLARY 5.5 Let cp be a real-valued C2-functional on a Hilbert space H such that dcpand d2cp are Holder continuous. Assume that H = H1 e H2 where dim H1 = n and thatmax(p(SH,) < inf cp(H2). Then, for some c > inf w(H2), there exist sequences (xk)k in Hand (pk)k in R+ with limk Pk = 0 such that0 limk (p(xk) = c;ii) limk Ild(10(xk)11 = 0;iii) for each k, d2(p(xk) has at most n eigenvalues below —pk.Moreover, if c = inf cp(H2), then (xk)k can be chosen so that it also satisfiesiv) limk dist(xk, H2) = 0.Proof: The subspace 112 is dual to the classF= {A; 3h E C(Biii; H), h(x) = x on SH, and A= h(BH,)}and the latter is a homotopic class of dimension n with boundary SHi. Therefore,max co(SH,) < inf cp(H2) < c(co, .7-) := c and Theorem 5.1 applies to yield our claim.^isWe now consider the cohomotopic case. Unfortunately, and unlike the homotopic setting,our proof requires the space to be finite dimensional. It can be found in section 5.3.THEOREM 5.6 (The cohomotopic case) Let cp be a C2-functional on a finite dimensionalspace H such that dcp and d2ce are Wilder continuous. Let .7" be a cohomotopic family ofdimension n with boundary B and let ..7-* be a family dual to 1. such thatc := inf max cp(x) = sup inf CO(X)AET xEA^FE.F* xEF5.1 Introduction and statements of the main results^ 91and is finite. Then, for every min-maxing sequence (Ak)k in .F and every suitable max-mining sequence (Fk)k in .T*, there exist sequences (xk)k in H, (Pk)k in IR+ with lirnk Pk = 0such that:0 limk co(xk) = c;ii) limk 114(xk)II = 0;iii) xk E Ak for each k;iv) limk dist(xk, Fk) = 0;v) wit (xk) > n for each k.As above, we setK(n,E) = Ix; c — e^CO (X)^C -I- 6 , ildcp(x)l{ < E. , 171,-,F (X)^n}and.1C-cf(n) = Ix; (p(x) = c, dcp(x) = 0, m+(x) ?_ n}.A cohomotopic analogue of Corollaries 5.2, 5.4 and 5.5 can then easily be established in thefinite dimensional case. In infinite dimensional situations, we have to settle for results ofthe following form. Here is the case of the saddle point theorem.COROLLARY 5.7 Under the hypothesis of Corollary 5.5, there exists > inf (p(H2) suchthat for every increasing sequence (Ek)k of finite dimensional subspaces containing H1 andspanning H, there exist sequences (xk)k in H and (pk)k in IR+ with limk Pk = 0 such thati) limk co(xk) = e;ii) xk E Ek for each k;iii) limk(VIEk)' (xk) = 0;iv) m-14 (xk) > n for each k.Proof: Consider the class.7- = {A; A compact, A D SHI and 0 E f(A) whenever f E C(A; Hi)and 1(x) = x on SHIL92^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESAlso, for each subspace Ek containing H1, we considerFk = {A C Ek; A compact, A D SIli and 0 E f (A) whenever f E C(A; H1)and 1(x) = x on SH, 1.It is clear that .-f- and each Pk are cohomotopic classes of dimension n with boundarySm. Moreover if e (resp. Ek) is the min-max value on .f* (resp. Pk) then 4 4, E. Theorem5.6 then applies to the restriction of co to each Ek to yield our claim. •This justifies the following notion.DEFINITION 5.8 A C2-function on a Hilbert space H is said to have the Palais-Smalecondition at level c, around the set F and of order greater than n (in short (PS)F,c,n+), if asequence (xk)k in H is relatively compact whenever it satisfies the following conditions withrespect to an increasing sequence (Hk)k of finite dimensional subspaces spanning H:limk (p(x k) . c, limk(Coixk)' (xk) = 0, limb dist(xk, F) = 0 and there exists a sequence ofpositive reals (pk)k with limk Pk = 0 such that mp+,(xk) > n for each k .Now we can state the followingCOROLLARY 5.9 Under the hypothesis of Corollary 5.7, assume that cp satisfies condition(PS),,n+ , then cp has at level c, a critical point of augmented Morse index greater or equalto n.In the case where a potential critical level is induced by a homotopic family and simul-taneously by a cohomotopic family, we then obtain two-sided second order estimates as thefollowing result asserts. The proof can be found in section 5.4.THEOREM 5.10 Let cp be a C2-functional on a finite dimensional space H such that dcp andethp are Holder continuous. Let .7 (resp. .P) be a homotopic (resp. cohomotopic) family ofdimension n with boundary B such that ,F c P and let .F* be a family dual to P. Assumethatc := inf max co(x) = inf max cp(x) = sup inf co(x)AEP xEA^AE.F rEA^FE.F* xEF5.2. A proof of the upper estimate in the homotopic case^ 93and is finite. Then for every min-maxing sequence (Ak)k in .T and every suitable max-miningsequence (Fk)k in 1', there exist sequences (xk)k in H, (pk)k in R+ such that:i) limk cp(xk) = c;ii) limk Ilthp(xk)II = 0;iii) xk E Ak for each k;iv) limk dist(xk , Fk) = 0;v) mp-k(xk) < n < mp+,(xk) for each k.5.2 A proof of the upper estimate in the homotopic caseIn this section, we establish Theorem 5.1. Actually, we shall prove the following morequantitative version. We use the following notation:For any subset A C H, (5 > 0, we write N(A) = fu E H; dist (u, A) < (5} for its(5-neighborhood.THEOREM 5.11 Let co be a 0-functional on a Hilbert space H. Suppose that for some0 < a < 1 and M > 1, we have for all xi, x2 E H,^licid (xi) — Cd(x2)11 5_ Milxi — x211° and 11(P"(xi) —  "(x2)11^Mllxi — x211°.Let .7" be a homotopic family of dimension n with boundary B such that c :=inf max (x) is finite, and let F be a dual set such that inf v(x) > c — E for some eAEY xEA^ xEFverifying:0 <€ < min {4- (4"+7)M - (n + 1) -2n, Mc* (il dist(B; F)).t 1where 0< al < 2(:+2) < 1. Then for any A e .T with max (x) < c+ E., there exist x, E HxEAsuch thati) c — 6 < co(x,) < c + E;ii) II(Pf (X e)II^3Eal ;iii) x, E A;iv) dist(xe, F) < ea' ;v) If (co"(xe)w,w) < —2saillw112 for all w in a subspace E of H, then dim E < n.94^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESAssertion v) of Theorem 5.11 implies in particular that ce(x,) has at most n eigenvaluesbelow -26-i. This is a consequence of the following standard lemma.LEMMA 5.12 Let A be a self-adjoint operator on a Hilbert Space H and let H1, 112 betwo subspaces such that H = H1 H2 dim H1 = k < oo and P2AP2 > 0, where P1, P2denote the orthogonal projections onto H1, 112 respectively. Then A has at most k negativeeigenvalues.Proof: For simplicity, assume that A is bounded. Multiplying, if necessary, A by a positiveconstant, we may assume that PiAPi > —Pi. Set A = -P1 + P2API + P1AP2 and notethat it is also self-adjoint, bounded and A < A. It is therefore enough to show that A hasat most k negative eigenvalues. Now, if A is an eigenvalue of A different from 0 and if x isa corresponding eigenvector, one hasP2x = 1—P2APix, and P1AP2AP1(Pix) A(A + 1)Pix.Observe that PIAP2AP1 is a nonnegative, self-adjoint operator on I/1. Moreover, to eachof its eigenvalues corresponds only one negative eigenvalue of A. This clearly proves theLemma. •We shall split the proof of Theorem 5.11 into several lemmas. For u E H and E > 0, weshall write as B(U,E) for the open ball centered at u, with radius E.LEMMA 5.13 Let co be a C2-functional on H, u0 a vector in H and fl> 0. Suppose thereis a non-zero subspace E of H and 6> 0 such that(v"(u)w,w) <-1311w112 for all u E B(uo, (5) and w E E.^(1)Let P be the projection from H onto E and set w1 (n) = ^ if Pp' (u)^0 andwi (u) = 0 otherwise. Then for any u E B(uo, g), the following holds(1) If wi(u) 0 0, then for all 0 < t < g we have049(u twi (u)) <^2cp(u) —^.5.2. A proof of the upper estimate in the homotopic case^ 95(2) If wi(u) = 0, then for all 0 < t < 4, we have13COOL tW) < CO(U) — -2-t2 for all w E E with 114 = 1.Moreover, for any 0 < to < J/2, there is Su with 0 < (Su < 4 - to such that for everyv E 1-3(u, Su) and any to < t < 4 we have(p(v tw) < co(v) — 14t2 for all w E E with iiwii = 1.Proof: (1) If w1(u) 0 0 then there is T (0 < T < 1) such that for all 0 < t < 4 we have+ twi (u)) = (u) t(C°1 (U) W1 (1))+ —t2(e(u + trtvi (Owl (u), (u))2P' (u)^t2< (u) + t(Pcp'(u),^)^IIP(Pqn) II^2t2< (p(u) —(2) If w1(u) = 0, then for 0 < t < 4 and w E E with iiwil = 1, there is T (0 < T < 1) suchthatt2 ",^„ t2 „W(lL tW) = V(u) t (VI (U)^± -2- (CO kIL trW)W^< p (it) -Moreover, there is & (0 < 5„ <4 — to) such that for all v E ri(u, JO we have thatII(p'(v) — (p'(u)II < It(i. Now again, there is T (0 < T < 1) such that for all v E (u,&),to < t < 4 and all w E E with JJwII = 1, we haveCp (v + tw) = v(v) + t(' (v), w) + t2 (cp'' (v + b-w)w, w)< (v) +^(v) — (u)Il —t2co(v) — fl •LEMMA 5.14 In lemma 5.13, we further assume that dim E > n+1. Let f be a continuousmap from a closed subset D of lit" into H and let K be a compact subset of D such96^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESthat 1(K) C B(u°,.(;), then for any v > 0 and 0 < to < 11, there is a continuous mapcr:RxD —> H such thati) o-(t,x) = f(x) if (t, x) E ({0} x D) U (R x D\N,(K));ii) cp(a(t,x)) < (p(f(x)) if (t,x) E [0,8/2) x D;iii) co(cr(t, x)) < W(i(x)) — It2 if (t, x) E [to, 6/2) x K;iv) licr(t,x) — f (x)II < t for all (t, x) ER x D.Proof: We can clearly assume that f (N,(K) n D) C B(u°, 6/2). LetT = Ix E Nu(K) n D; wi( f (x)) = 0}.According to Lemma 5.13, we can find for each y E T, an open ball B(y, vY) in IR" such thatfor all w E E with 114 = 1 and z E B(y, ,in) n D we have that13^8(f (z) + tW) < C5(f(Z)) — it2 to 5_ t < .Put 0 = UyETB(y, vY/2) and let g : R" —* [0,1] be a continuous function such that(1 x E Kg(x) .0 x E IV \Nu(K)Next consider the continuous map fi : D\O -4 E defined by fi(x) = wi(f(x)). ClearlyIlh(x)11 = 1 for all x E (N„(K) n D)\0. If the latter set is empty, we just let fi -a e forsome fixed e E E, Hell = 1. Since (N,(K) n D)\0 C IR" and dim E > n + 1, there exists byCorollary 1.23, a continuous map 12 : IR" -4 H such that 1112(x)11 = 1 for all x E IR" andf2(x) = f1(x) on (N„(K) n DAC0. Now let0-(t, x) = f(x) + th(x)g(x)on IR x D. Clearly cr(t,x) verifies the claims of the lemma.^ •LEMMA 5.15 Let co be a real-valued C2-functional on H and let f be a continuous mapfrom a closed subset D of IR" into H. Suppose that K is a compact subset of D with thefollowing property:5.2. A proof of the upper estimate in the homotopic case^ 97There exist two constants "6 > 0, # > 0 such that for all y E K there is a subspace Eyof H with dim Ey > n 1 so that for all x E B(f(y),(C), we have(V" (x)w, w) < gliwil2 for all w E E.^ (2)Then, for any 6 (0 < 6 < S) and v > 0 there is a continuous map 1: D H such that, ifN := N(n) is the number given by Lemma 1.24, we havei) 1(x) = (x) for x E D\N(K);ii) yo(f(x))^yo(f(x)) for all x E D;iii) If x E K, then cp(f(x)) < co(f (x)) — 166A2,2 ,iv) 11/(x) — f(x)II^6/2 for all x E D.Proof: Let 0 < 6 < 15 be fixed. For any y E K we can choose a ball By in 11171 suchthat ;f3y C Nv (K) and f(x) E B(f(y), ev) for all x E Tiy n D. Since K is compact, thereis a finite subcovering By1, • • • , By. of K. Choose 0 < T < v small enough such thatC Nv(K) and for 1 < i < m,f (x) E B(f (y,) —6 ) if x E C(Byi) n D.4NBy using Lemma 1.24, we may assume that any N possible distinct NT(By.)'s have an emptyintersection.We shall now define by induction, continuous functions fo, • • • , fm : D^H such thatfor all 1 < i <m we have that362(g.fi(x)) < (g.fi-i(x)) =16N2 if X E T3y. n D ( 3 )co(fi(x)) < cp(fi_1(x)) if x E D^ (4)0 x E D\NT(Bys)IlL(x) — f_1(x)l1 <(5)-21-v x E Arr(Ryi) n DLet fo = f and suppose that fo,^• • • , fk are well defined and satisfy (3), (4) and (5) fork < m. Clearlyi6f(x)I 2N if x E n1Nr(T3y1) n D98^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESSince any intersection of N distinct sets NT(By;) is empty, we have that— f(x)II^(5(N —1) . x E D.2NSince f : yk+1 nD^B(f (A+1), AT), we see that fk maps Byk+, n D into B(f (yk+i), go(1 —+^C B(f (yk+i), (5/2). By assumption (2), there is some subspace Elm+, of H withdim Eyk+, > n+1 such that for any u E B(f (yk+i), (5) and w E Hyk+, withlitvli = 1, we havethat (e(u), w, w) < —0. Hence, we may apply Lemma 5.14 with fk and any 0 < to <to obtain a continuous deformation cr(t, x) satisfying the conclusion of that lemma. Definenow fk+i(x) =^, x) to get a continuous function fk+i : D^1/- satisfying052Aik+i(x)) < Co(ik(x)) 16N2 for x E T3 yk+, n D(P(fk-}-1(x))^V(i(x)) for x E D0— f c (X)II^{ 82Nx E DVV,Ggyk+i)x E^n DBy induction we see that fo, • • • , fn„ are well defined. Clearly = fn., verifies the claims ofthe lemma.^ •LEMMA 5.16 Let ço be a real valued C1-functional on H and let f be a continuous mapfrom a complete metric space D into H. Suppose K is a compact subset of D such that fortwo constants .(5' > 0 and > 0, the following holds:1149' (11)11 > for all u E Ns(f (K)).^ (6)Then, for any (5 (0 <(5< (-5) and any v > 0, there is a continuous map : D H such thati) f(x) = f (x) for x E D \ N„(K);ii) co(j(x))^co(f(x)) for all x E D;iii)w(f(x)) cp(f (x)) — 135 for x E K;iv) Ilf(x) — f(x)ii^(5 for x E D.5.2. A proof of the upper estimate in the homotopic case^ 99Proof: Step 1: We first claim that there exists a with 0 < a < 5 such that i), ii), iii) andiv) hold with a replacing J.Indeed, For any y E K, let By be an open ball in D containing y such that for all x E By,f (x) E B(f (y), 1). Hypothesis (6) coupled with the continuity of cp' yields uy E H withiinyll = 1 and a ay > 0 such that for all v E H with 114 < cry, we have^(co' (f (x) + v), uy) <-3 for all x E B.^ (7)Since K is compact, there is a finite subcovering By1, , By, of K. We may assumeU1By1 C N(K) and let us define 1,Gi : D^[0,1]^= 1, 2, ... , k) bydist(x; D By.7)^(8)Oi(x) = dist(x; D By.).Clearly Oi is continuous on U1B andEll=i Oi(x) = 1 for x in UltLiByt. Next let 11, : D[0, 1] be continuous such that{1 x E K0(x) = 0 x E D 4_113y, •Take a = min{a-yl, • .., yk 45} and define(X) = 1(x) + a-0(x) E (X)Uy3j=1Clearly 11/(x) — f (x)II < a. On the other hand, by the mean value theorem, for any x E D,there is 7- (0 < < 1) such thatv(i(x)) - w(i(x)) = ((pi(i(x) + 70-0(x) E b(x)u,), a0(x) E oi(x)uyi)j=1^ i=1= 011(x) Elpi(x)((p(f (x) + TOV(X) E tp;(x)no, noi=1^J=1In view of (9), we see that verifies (i)-(iv) of the lemma with a instead of 5.Step 2: We shall now iterate the above process and construct by transfinite induction atransfinite family of continuous functions (fa)„ from D into H, and an increasing family ofreals (5a)a with 0 < S, < '5, such that for every a,(9)100^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESa) fa(x) = f (x) for x E D \ N(K) and co( fa(x)) < go( f (x)) for all x E D;b) cp(fc,(x)) < go(f(x)) — ,(38,„ for x E K;c) lifai-i(x) — fa(x)II 5_ 8.4.1 — 6,, for x E D;d) lkoi(u)11 > 0 for all u E Ns_8.(ia(K))•To do that, we start with Jo = 1.If a = -y + 1 and if (57 <6, apply step (1) to fy and 6—(57 to find cr7 with 0 < o-7 < i5— 67and the function 4, satisfying the above properties i), ii), iii) and iv). Set now 6c, = (5.7 ± cr.yand fc, = j..y.If now a is a limit ordinal: i.e a = limn ryn with -yn, < a, then iii) implies that (ly.)„ isa Cauchy sequence and we let fa be its limit. Also set 6„ to be the limit of the increasingand bounded sequence (51,.. This complete the construction.Now note that the increasing family of reals (6c,),„ is bounded above by S, hence there is7o before the first uncountable ordinal 12 such that (5a . 8 ^all a > yo. It follows that6.y. = -8 and therefore j = fy. verifies the claim of Lemma 5.16. •LEMMA 5.17 Let go be a real-valued C2-functional on H and let f be a continuous mapfrom a closed subset D of Rn into H. Let ,131 > 0, /32 > 0 and 6> 0 be fixed constants andsuppose K is a compact subset of D with the following property:For all x E K, either 11Cd(u)11 > 01 for all u E B (f (x), (-5) or there is a subspace Esof H with dim .E > n + 1 such that for all u E B(f(x), S) and w E Ex, we have that(co"(u)w, w) < — 13211w112 .Then, for any 6 (0 < 8 < I) and v > 0, there is a continuous map j from D into H suchthat, if N := N(n) is the number given in Lemma 1.24, we havei) j(x) = f(x) for x E D\N,(K);ii) v(j(x)) 5_ co(f(x)) for x E D;HO (P(/ (x)) < V(.f(x)) — p6 for x E K where p = minf i/31 , 16N2};iv) II j(x) — f(x)II <6 for xE Dn Nv(K).Proof: Let T1 = Ix E K; I(' ()II > fli. for all u E IV (x), 6.)} and T2 = K\Ti. Clearly, forany x E Ti we have that 1140'(u)11 > 01 for all u E B (1 (x),1÷3). Also for any x E T2, there is5.2. A proof of the upper estimate in the homotopic case^ 101a subspace Ex of H with dim Ex > n +1 such that for all it E B(f (x), W) and w E Ex with= 1 we have that (con(u)w, w) < —02. Note that T1 , T2 are compact and K =T1 u72.Apply now Lemma 5.16 with -j-6-(5 and 5 > o to obtain a continuous map g : D H suchthatg(x) = f(s) for x E DV\if (TO and Co(g(x)) cP(.f(x)) for x E D^(10)V(.4(x) < V(f(x)) — 7-6-601 for X E T1^(11)Ilg(x) f(x)II LS for x e NW]) n D. (12)But (12) yields that for x E T2, B(g(x), c B(f(x), lis). Apply now Lemma 5.15 with6 < and 5 to obtain a continuous map 1: D H such that1(x) = g(x) for x E D\A r .2.1 (T2) and 40(f(x)) 5- (9(x)) for x E D^(13)Co(f(x)) < V(g(x) 1/362N522 for X E T2^ (14)g(X)1(^for x E Arf (T2) n K. (15)Clearly j verifies the claims of the lemma.PROOF OF THEOREM 5.11: Suppose maxuEA co(u) < c^where A is a set in .T. Thereexists then a continuous function f from D C R" into H, which is equal to a on Do andsuch that A = 1(D).Let (5 := 5(E) = 1E22-^ji; and consider the closed setK = E D; Co(i (x)) c — e and f(x) E N28(F) n^(16)Since e < M71- (ldiSt(F; n al , we have that 26 < ldist(F; B) and hence K is a compactsubset of D\Do.Suppose now that the conclusion of Theorem 5.11 does not hold, then for all x E K, wehave that either lici/(f(x))11 > 3M6 or there is a subspace Er of H with dim Ex > n 1such that for all w E Ex, we have that (,o"(f (x))w , w) < _2s1 11w 1 12.102^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESLet v = dist(Do, K) in such a way that N(K) C 118.n \Do. In view of the Holdercontinuity assumption, the hypothesis of Lemma 5.17 are satisfied with :3 = 25(6), vdist(Do, K), 01 = M6 and ,(32 = EQ1. Hence, we can find a continuous map f : D --+ Hsuch that1(x) = f (x) for x E D\N,(K) and yo (/ (x)) 5_ co (f (x) for x E D,^(17)cp(i(x)) < (p(f(x)) — p6 for x E K,^(18)}where p = min{ iM5a, oaNi , (N given in Lemma 1.24) and^111(x)^f(x)ii^for all xe D .^ (19)Note that 1(D) E^and 1(D) C N25(A). To get a contradiction, we shall estimateinf v(f(D) n F). For any x E D such that /(x) E F, we have thatf (x) E N28(F) n A and cio(f(x)) V(f(x)) ?_ c — E.Hence x E K and by (17) we have that co(f(x)) < co( f (x)) — p6.Note now thatEal-1-2"2-16^— 64andEal^ 62> ^ 4,r-1iv/ a E a = ^m s a 4. 2÷^16N2^64N2^64N2Since M > 1, we have that>  1 m-21aEa1l-2a1 /ct^64^64N2and the latter is larger than 26, since the hypothesis combined with Lemma 1.24 yields thatE < 4—(4n+7)(n 1)-2nmzil <It follows that co(f(x)) < Cd-E —2E = C—E which contradicts the assumption that inf o(F) >c — E. The proof of the theorem is complete.^ •5.3. A proof of the lower estimate in the cohomotopic case^ 103In the next section, we shall tackle the problem of finding lower estimates for the di-mension of the "almost negative" eigenspace by min-maxing over a cohomotopic class. Onthe other hand, such an estimate can also be obtained by opting for a sup-inf procedure.Indeed, straightforward adaptations of the arguments used for Theorem 5.11, will yield thefollowing result. All what is needed is to reverse the direction of the deformations inducedby the first and the second derivatives in order to "push up" the critical level. The detailswill be left for the interested reader.THEOREM 5.18 Let co be a C2-functional on a Hilbert space H such that co' and co" areHolder continuous. Let .7. be a homotopic family of dimension n with boundary B and let.7.* be a family dual to ,F such thatc := sup inf ep(x) = inf sup co(x)AE.7- sEA^FEY xEFand is finite. Then, for every max-mining sequence (Ak)k in .7. and every suitable min-maxing sequence (Fk)k in .T*, there exist sequences (xk)k in H and (pk)k in IR+ such that:i) limk co(x) = c;ii) limk co' (x k) = 0;iii) xk E Ale;iv) limk dist(xk, Fk) = 0;v) for each k, mp+k(xk) > dim(H) — n.5.3 A proof of the lower estimate in the cohomotopic caseWe now establish the following quantitative version of Theorem 5.6. Here is the main resultof this section.THEOREM 5.19 Let cp be a C2-functional on a finite dimensional space H such that co' andco" are Milder continuous on H with HOlder exponent 0 < a < 1. Then, there exist aconstant A(co, H) > 0 and a positive decreasing function C(e) with limo C(e) = 0 such thatthe following hold:104^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESIf .7' is a cohomotopic family of dimension n with boundary B so that the level c :=inf max (x) is finite and if a dual set F satisfies inf yo (x) > c — s for some E verifyingAEY rEA^ xEFdist(B; F)24/cv0< E < min{A(co,H),^232/a^}then, for any A E .7' verifying In^there exists x, E H such that:Ea.ic W(x) < c ± E l0 C — (E.) 5. CI* e) 5 C -I- (e);ii) ii(id (x E)ii 5- e (E);iii) x, E A;iv) dist(x,, F)v) I f (ce(x,)w,w) _> (e.)11wil2 for all w in a subspace E of H, then necessarily codimE >n.In order to prove the above theorem, we need (for technical reasons) to consider first thecase where the standard boundary condition (sup co(B) < c) is assumed and when no dualset is singled out. The general result will be derived from this special case by a suitableperturbation of the function co and an appropriate change of the boundary. However, tobe able to carry out this program, one needs to prove the special case with estimates thatexhibit the size of the "tolerated" perturbations as well as the differential between theperturbed critical level and the possible values on the new boundary.The following theorem is the main step for proving Theorem 5.19.THEOREM 5.20 Let co be a C2-functional on a Hilbert space H with C-orderly pavings forsome C> 0 and suppose that for some 0 < a < 1 and M > 1, we have for all xi, x2 E H,W(xi) — Ox2)11^Milxi — x2110 and IIV"(xi) —49"(x2)11 5_ Mlixi — x211a.Let .7' be a cohomotopic family of dimension n with boundary B such thatsup (B) <c := inf max (x).AEY rEA5.3. A proof of the lower estimate in the cohomotopic case^ 105Then, for any E. such that0 <E < min 1 ( ^1 — (1/2)a ^\64(2Mlia + C)M2/a) 12/a, (c — sup c,o(B)4/a /and for any A E F with mEa:4xy9(u) < c + e, there is xe E H such thati) c — e < (O(XE) < C + 6;ii) IIV(X E)II 5- 6a/3;iii) xe E A;iv) If (co'i(acE)w,w) > 66a2 /4(1+a) Ille II 2 for all w in a subspace E of H then codimE > n.We shall split the proof of Theorem 5.20 into several lemmas.LEMMA 5.21 Let cp be a real-valued C2 functional on a Hilbert space H and consideru E H, a subspace Eu of H, p > o and 8 > 0 such that:(Co"(v)w, ,w) > 131142 for any v E B(u,S) and for all w E E.^(1)Let Bu(ri) (resp. Bu(r2)) be the closed ball in Eu (resp. En') centered at 0 and with radiusr1 (resp. r2). Assume0 < r? + 44 < (52^ (2)and that11(p'(v)II 5_ e for all v E u + Bu(r2)•^ (3)Then, for any v ,- u + v1 + v2 with Th. E aBu(ri) and u2 E Bu(r2), the following holds:i) (P(u + vi + v2) > go(a + v2) + gr? — riE;ii) yo(u + v2)^co(u) — 611 1)211.Proof: There exist 0 < 7-1 <1,0 < 72 < 1 such thatCo(u + vi + v2) = Co(u + v2) + (co' (u + v2) , vi) + -((loll (u + v2 + Tivi)vi , v 1)#> (p(u + v2) — diva + -2-11v11120 2= CP(U + V2) — Eri + —2 ri106^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESOn the other hand, we haveca(v + v2) = v(u) + ((pi(v + T2v), v2) _?_ co(v) — E11v211,which proves the lemma.We shall use the following notation. With the assumptions of Lemma 5.21, we writeDu (ri , r2) = u + Bu (ri ) + Bu (r2) and Du' (ri , r2) = n + Bu (ri ) + 2Bu (r2) •We also let Pu, Qu be the orthogonal projection from H onto Eu, Eui respectively. For anysubset D C Eul and any continuous map g from D into Eu, we shall write gp = {(g(v), v); v ED} for the graph of g considered as a subset of H = Eu ED E. It is clear that gp is thenhomeomorphic to D.LEMMA 5.22 Under the hypothesis of Lemma 5.21, assume also that00 < E < -2 ri.(4)Then, for each v2 E 2Bu(r2), there is a unique 'Vi E IntBu(r0 such thatinf^co(v) = (p(u + f)i + v2).vEu+B. (*I )+v2Moreover, the map gu from 2.13u(r2) into Eu defined by gu (v2) = fli is continuous.Proof: Assumption (1) implies that v is strictly convex on the set u + Bu(ri) + v2, henceit is weakly continuous and therefore attains its minimum on the weakly compact set u +Bu(ri) + v2 at a point u + ih + v2 where f)i E Bu(ri). The strict convexity yields theuniqueness of iii, which in turn implies the continuity of gu. •Let now f : IR —> [0,1] be a continuous function satisfying f(x) = 0 if x < 0, f(x) = 1if x > 1 and such that f restricted to [0,1] is a homeomorphism. For each u E H verifyingthe assumptions of Lemma 5.22, we define a map Ku :H -4Has followsif v E H\D'u(ri,r2)if v E D'u(r 1 , r2) •Ku(v) = 1 : + v2 + vi f( :,11:p _ 1) + (i f(11;2211^1)) gu(v2)5.3. A proof of the lower estimate in the cohomotopic case^ 107Note that we have used here the canonical decompositionv = u + vi + v2 E U + Bu(ri) + 2Bu(r2) = Du' (ri , r2) .LEMMA 5.23 In addition to the assumptions of Lemma 5.22, assume that260 := —1 (—[3 ri — ris — 2r2e) > O.2 2 (5)Then, the map Ku satisfies the following properties:(0 Ku (Du (ri, r2)) = u ± g73.(r) ;(ii) Ku is the identity on H\IYu(ri, r2);(iii) co(Ku(v)) < co(v) for all v E H;(iv) Ku is continuous on HVu + °Hu(ri) + 2Bu(r2));(v) Ku is continuous on {v : co(v) < co(u) +Eol.Proof: (i) follows directly from Lemma 5.22, while (ii) follows from the definition of K.To check (iii), consider v = u + vi + v2 E D'u(r1,r2) and note that the convexity of co onu + Bu(ri) + v2 yields= (1+1)2 + MI( _Ilv2II^1) + (1— pliv211^1))gu(v2))Co(Ku(v)) V r2 r2^5_ (Au + vi + v2).f(-11v211 1) + (1 — f ( 11v2I1^1))C0(gu(v2) + v2 + u)r2^r25_ (P(u + vi + v2) = (v).Combining this with (ii), we get assertion (iii).Finally for (iv), let v = u + vi +v2 E D'u(r 1 , r 2) . If v E IntYu(ri,r2), then—Ku(v) = u+Qu(u— v)+Pu(u—v)f( 11Qu(v)11 1) + (1 f ( 11(2.(uOH 1))gu (Q ii(u — v))r2^ , 2is readily continuous. If now v = u + vi. + v2 E gz(ri,r2) with Ilvill < r1 and 11v211 = 2r2,then we only need to note that Ku(v) = v.Finally, Lemma 5.21 and assumption (5) yield that yo(v) > co(u) + 60 for all v E (u +aBu(ri) + 2Bu(r2)). Therefore (v) follows from (iv).^ s108^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESThe above lemma provides a way to collapse a neighborhood of H into a set that ishomeomorphic to a subset of the (lower dimensional) subspace E. In the sequel, weimplement a procedure for collapsing a union of such neighborhoods. Evidently, the difficultyarises from the overlapping pieces. First, we handle the case of two neighborhoods.LEMMA 5.24 Let cp be a C2-functional on a Hilbert space H and suppose that for some> 0 and 9 > 0, there is ui E H and subspaces Eui (i = 1,2) such that:(cp"(v)w,w) > (311w1I2 for any v E B(ui, (5) and for all w e En,. (6)Suppose r1,r2,s are scalars verifying (2), (4) and (5) and let Bui (ri) , Ku, , Du, (ri r2) (i =1, 2, j = 1,2) be the objects associated to u1, u2 by the preceding lemmas. Assume inaddition thatEu, C E2^ (7)cid (v) —< € for all v E (ui Bu, (r2)) U (u2 + Bu2(r2))•^(8)Then, the following hold:(i) Ku, (Du2(ri r2) n {v; (v) 5_ co(u2) + sol) g Du2(ri r2);(ii) Ku2 0 Kul is continuous on the set Iv; co(v) < min(co(ui),(P(u2)) +o};(iii) If uo E H, D is a closed subset of EiLL, and if g is a continuous map from D intoEu, such that Ku2 is continuous on uo gm then there exist a closed subset C of Euli anda continuous map h from C into Eu, such that Ku2(uo + 9n) = u2 + hc •Proof: To prove (i), it is enough to check the image of the overlap between the twoneighborhoods. For that, let us consider u2 +q+v3 E Du2(ri , r2) such that 112411 <r1,11411 < r2 and ca(2/2) +60 > co(u2 +24 + 4). Also assume that u2 +24. + ui + vf +24with 11411 <r1 and liva < 2r2. By the definition of Ku„ we have thatKu, (u +^= u gui ) 24 = u2^+ v4 gul (24 ) —Note that gu' (24) — v E Eu, C E„,, so that v? + gui(vD - v E E2.and that5.3. A proof of the lower estimate in the cohomotopic case^ 109Suppose now that Ilv? +gu' (4) — vfll > r , then there is 0 < to < 1 such that II + i3 —vi = r1 where fyl = gul (v)to + (1 — to)v1. Since co is strictly convex on u1 + 4 + Bui (ri),we haveco(ul + 4 +^< cp(ui + 4 + gui (4))to + w(u + 4 + vp(i — to) < co(ul + 4 +On the other hand, by Lemma 5.21, we have that(P(u2^+ ./31 -I- — v1) > (2(u2) + Eo > (p(u2 + +^= v(ui + +v2'),so thatyo(ui + v + v1) > co(u1 + i3 + 4) co(u2 + + i — n +^> co(ul + + 4)which is a contradiction. Hence (i) is proved.For (ii) it is enough to combine (iii) and (v) of Lemma 5.23.To establish (iii), write for any uo + g(u) + u E uo g D,uo + g(u) +u^+ g(u) + P„(u + uo u2) + QU2 (U + U0 u2).Put V = Q2 (u Up — u2), thenK„(uo + g(u) + = u2 + (g(u) + P„ (u + uo — u2)) f ((—II" — 1) + (1 — f ( 11'711 — 1)) g'2 (V) + .r2^r2Clearly for u2 + v1 + v2 E Du2(ri, r2), we have thatK 2 (u2 + v1 -I- v2) = U2 + v2 + g" (v2).Let now C be the image of D by the map G : u^Q „(Ku2(uo + g(u) + u) — u2). Let= {u E D; + g(u) + u E Du2(ri , r2)} and 6.1‘ = G(13).Since f is a homeomorphism on [0,1], we see that G is a homeomorphism from D \ 15onto its image C \ . Define now h on C \ e byh(v) = Ku2 (G-1 (v) g(G-1 (V)) + u0) V u2110^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESand on C byh(v) = gu2 (Q u2 (v)) + Q2(v) — v.Note that for u E D, we have that u2 h(v) v = Ku2(uo g(u) u) where v = G(u) =Qu1(Ku2(u0 + g(u) + u) — u2). This clearly proves (iii).^ •LEMMA 5.25 Suppose T is a compact subset of H with C-orderly pavings for some C > 0and let 6, ,r1,r2 be positive scalars such thatr1 > r2 and (C 1)ri < 6.^ (9)Assume that for every u E T, there exists a subspace Eu of H with codimEu < n — 1 suchthat (cp"(v)w,w) > 011w112 for all w E Et, and all v E B(u, 6).Then there is a sequence ul, ..., tiM E T and subspaces Eu,(1 < i < M) of H withcodimEu, < n — 1 such that:T c^ (10)(cp"(v)w, w) > flflwfl2 for all v E B(uhri) and WE Eu,and^C Euiwhenever i < j and B(uhri) n B(ui, ri) 0 0.^(12)Proof: The hypothesis on T yields a finite sequence u1 , ..., um E T that is (C, ri)-metricallyordered and such that T C UiB(ui, r2). In order to construct the appropriate subspacesEu, (1 < i < M) of H, we shall proceed by (backward) induction.First, we define for 1 < I < M, the class C1 consisting of all (M — I + 1)-tuples(Eu1,...,Eum) of closed linear subspaces of H, such that for all I < i < M, they verify:codim Eu, < n — 1.^ (13)^(e(v)w, w) > Oilwir for all v E B(ui, ri) and w E Eli;^(14)andEu, C Eu,whenever I < i < i < M and B(ui, ri)11 B(uj, ri) 0 0.^(15)5.3. A proof of the lower estimate in the cohomotopic case^ 111Each non-empty class CI is clearly a Zorn class for the partial order induced by inclusionand therefore every element in C/ is contained in a maximal element.We now start the induction by choosing EM to be the maximal subspace of H verifying(13) and (14). We then suppose that for some I, (1 <I < M), we have chosen subspaces{E„,, Eum} such that the (M — I ± 1)-tuple (Eul, Eum) is a maximal element in theclass CI. Let now E be a subspace of H satisfying codim < n — 1 and(co"(v)w, w) > /311w112 for all v E B(7.4_1, (5) and WE E.^(16)We shall show that {E, E1, Eum} is an element in the class C1_1. Indeed, consider theset={i; I — 1 < i M and / — 1 =^i2^ik i for some k > 1}where i j means that i < j and B(ui, ri) n B(ui, ri) 0 0.Define for i E {/,...,M} the spaces ti = Ei if i U1_1 and ti = sWi(Ei U t) ifi E^We claim that (E, E1, •••, kum } belongs toIndeed, note first that since (uk)k is (C, 7-0-metrically ordered, we have that ui EB(ui_i,Cri) for every i E U1_1. Since (C + 1)ri <^(16) implies then that{(E,^kum satisfies (13) and (14).To check (15), consider i,j such that / — 1 < i < j < M and B(ui, ri) n B(uj, 7-1) 0 0.If i E then necessarily j E so that ki c ki (or E c .k.; if i = I — 1) by theirdefinitions and the induction hypothesis. If now i U1_1 and j Ur_, then ti = E cEi C ki by the induction hypothesis. Suppose finally that i U/_1 but j E thenagain ki = E1 C E C kj. Hence (E, E1, •.., km) satisfies (15) and therefore belongs toBut the maximality of (En1, Eum) in CI that is assumed in the induction hypothesisyields that E. Ei for all I < i < M. Hence (E, E„,, Eum) E C1_1 and is thereforecontained in a maximal element of Cr_i which is necessarily of the form (G, Et,/ •••,Eum)•It is now enough to set Et,1_, := G to finish the inductive step. •112^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESLEMMA 5.26 Let F be a cohomotopic family of dimension n with boundary B and letT be a compact subset of H with C-orderly pavings for some C> 0. Let 6,13,E, r1, r2 bepositive constants such that:0 < 2r2and160 := —2(C1 < r < —2 dist(T,+ 1)\/-2-ri < 6,2(-2ri — rie — 2r26)B),> 0.(17)(18)(19)Assume further thatsup co(T) — inf co(T)^E / 2 ,and that for every u E T,(20)(Pi (V)^6 for all v E u^2B(r2) (21)and there is a subspace Eu of H with codimEu < n —1 such that(e(v)w,w) > #11w112^for all w E Et, and all v E B(u, 6). (22)Then, there is 6' > 0 and a map K:^such that the following hold:(i) K is continuous on {u; co(u) < inf co(T) + 64;(ii) co(K(x)) < cp(x) for every x E H;(iii) K(x) = x on H\NvIri(T);(iv) K(AVV8,(T)) E Y for any A E .7. satisfying max co(A) < inf (p(T) + Eo.Proof: Apply Lemma 5.25 to get a finite sequence ul, ..., M E T and subspaces Eui (1 <i < M) of H such that:T C Ur_113(u1,r2)^ (23)(49"(v)w, w) > /31142 for all v E B(ui, f2-r1) and w E Eu.^(24)andEui C Et,iwhenever i < j and B(ui, \hr1) n B(ui, -4-ri) 0 0.^(25)5.3. A proof of the lower estimate in the cohomotopic case^ 113Now for each ui and Eui we can associate .13(ri)(j = 1,2) and Ki as defined in Lemma5.23. Put Ki = Kum o o Ku, and K = Km = Kum o o Kt„ . In view of Lemma 5.23, itis clear that K verifies (i),(ii) and (iii).For the rest, choose (5' > 0 such that^No,(T) C 4.1_1B(u1, r2)^ (26)and^sup cp(N8,(T) _< sup (T) +^ (27)To prove (iv), first note that by (20) and (27), we have for 1 < j < M thatDu,(ri, r2) n Ny(T) g Du, (ri, r2) n Iv; (p(v)^inf (p(T) + 601.But Lemma 5.24 implies that for 1 < i < j < M,Kui(Du,(ri, r2) n {v; '(v) < inf cp(T) + Eon^Du, (ri, r2) n Iv; v(v)^inf co(T) +E0}.HenceK(Nsi(T)) c K (UiDui(ri,r2) n N 81 (T)) LJIKi(Dui(ri, r2))•We need to prove the followingClaim: For each 1 < i < M, the topological dimension of the set Ki (Du, (7-1, r2)) is at mostn — 1.To see this, let i = i1 < i2 < < ik < M be the uniquely determined sequence suchthat for any fixed 1 <m < k, we have B(uim, .\&i) n (U13111B(ui,, N&I)) 0 0 while for anyim-1 < < im, B(ut, -‘&1) n N/7-1) = 0. By Lemma 5.23 (ii), we see thatKi (Dui (ri , r2)) = Kik 0 .• • 0 Kii (Dui (ri r2))-By Lemma 5.24, there is a closed subset Di C Eut and a continuous map gi : Di^Euisuch that Ki(Du,(ri, r2)) = nik +giDi and the claim is therefore verified since the dimensionof EiL is at most n — 1..By Theorem 1.21, we see that the topological dimension of K(Ny(T)) is at most n — 1.If now A E P is such that max (A) < inf yo(T) + Eo, then K is continuous on A andK (A)\K (1V5, (T)) E .7. by Lemma 3.25. •114^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESLEMMA 5.27 Let .F be a cohomotopic family of dimension n with boundary B in a Hilbertspace H that has C-orderly pavings for some C > 0. Let 803,61,62, r1, r2 be positiveconstants such that0 < 2r2 <^ (28)(C + 1) V1.1 < 8, (29)eo := 1^ _( rn2 2 1 ri-2 — -• 2,2^ (30)ande1r2 < 60/2.Let D be a compact subset of H such that D n B = 0 anddist(B; Ix; licp' (x)ii^62, 40(x) ?. sup CO(D)^61r2}) > 2r1.Assume further that for every u E D, eitherIP' (OH > El for all v E B(u, r2)OrII Cd(V) II <f62 for all v E B(u, 3r2)and there is a subspace E. of H with codim Eu < n — 1 such that(W" (Ow , w) > filind12 for all v E B(u,28) and all w E E.Then, there is a map K: H H and an open set 0 with O n B = 0 such that thefollowing hold:(i)ic is continuous on fu;cp(u) < inf yo(D) + Sal;(ii) yo(k(x))) co(x) for every x E H;(iii) k(x) = x on B;(iv) sup co(k(D\O)) < sup (p(D) — Eir2;(v) k(A\0) E Y for any A E .7. satisfying max co(A) < inf yo(D) E-2a.(31)(32)(33)(34)(35)5.3. A proof of the lower estimate in the cohomotopic case^ 115Proof: Consider the setD1 = {u E D;i1401 (011^Ei for all v E B(ul r2)}and set D2 = DWI.Use Lemma 5.14, with 6 = r2 and v < dist(D,B), to find a continuous map G :1-1 -* Hsuch that the following holds:G(x) = x for x E H \ Nv(Di); (36)co(G(x)) < co(x) for all x E H; (37)W(G(x))^Co(x) - 60.2 for x E D1; (38)IIG(x) - xil < r2 for all x E H. (39)Let now T = fu E G(D2); CO(u) sup (D) - Eir2}. We claim that it verifies theassumptions of Lemma 5.26 (with c = e2). Indeed, (18) and (19) are clear, while (21) and(22) follow from (34) and (35) combined with (39). On the other hand (17) follows from(34) and (32). Finally (20) is a consequence of (37), (31) and the definition of T. Wecan therefore find 6' and K that verify (i)-(iv) of that lemma. Let now k -=-KoG and0 = G-1(Arp(T)). Using (31), it is easy to check that they verify the claims of the Lemma. •PROOF OF THEOREM 5.20: Let ,F be a cohomotopic family of dimension n with boundaryB and suppose that H has C-orderly pavings for some C > 0. Let c = inf max co(u) andAEF uEAassume that max (B) < c. Let 6(e) = M-11"e4(4) , where12/a( ^1 — (1/2)a E < 64(2Mil + C)M2la)(40)ande < (c - sup yo(B)4/Q.^ (41)Let A E ..T be such that max yo(u) < c + E, we claim that there exists tie E A such thatu€A(i) c - e _<(p(uE) < c+ E;(ii) 11('(20I1116^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCES(iii) if (cp" (ue)w,w) > 664(11-7) 11w112 for every w in a subspace E of H, then necessarilycodimE > n.Indeed, if the claim is not true, then for any point u in the setD := Itt E A; co(u) ?_ c —we either have 11(P'(ue)11 > ea/3 or that liccAue)11 < Ea/3 and there is a subspace Eu of H withcodim E < n — 1 such that (cp"(v)w,w) > 6E-71=H:01142 for all w E E.Let L := 2M11" + C and take1^ 1 1/37-1 = 4—/, =^a E40+0,, and r2 = -LE .4LThe Holder continuity of co' and ca" gives that D verifies assumptions (33), (34) and (35) ofLemma 5.27, with.2= 4E111-77TH El = (1 —^)6a/3 and 62 = 26a/3.We now show that D satisfies the rest of the assumptions of that lemma. First, (28) and(29) are clearly satisfied. We also have1 13 2= — (-7-1 —7-1E2 — 2r2E2)2 21 .2^62^6^41/3)=^2E7(7:Fa7 2 16L2 2L-2^„ -1N.1 or 9::+2c_t^NI a a A.  a^2 ail= _e 4(1+a ) —6 3 4(1+a) —^316L2^4LSince by (40), < HO 4+a , we have thatsm a,M a o_42.^M a yjar±60 > _640+a) ^ no+a)16L2 2L^1 ^)12/aBut again, e <^(8L^, hence we obtain that2Al— a c_2_0160 > ^_E40+a)32L2Eo5.3. A proof of the lower estimate in the cohomotopic case^ 117On the other hand, we have1— .g- c.„^2±^1111^,_ --t a2+2a ^61r2 = ^6 3 < —E. 3 <L^64L22/a )12(14-a)/(a2-1-2a+4)since, by (40) we have that < ( M6-4L3/(2-a)(LILL )^Furthermore, since e <^, we have< 60/2,^(42)1 — 1/2' .41 1 — A ail26<^6* 3 < ^E 3 = 61T2,which establishes (30) and (31).To verify the remaining hypothesis (32), Let— A=f a.±.1S := fu E H; (p(n) c 1L La^6 3 7 II Vi (U) II^26'43}and note that for any y E S and x E B(y,r), we haveIco (x ) — Co(Y)I^(26a/3 +^xlia)11Y xil < (2e"13 Mr`irand—w(x) ?_ yo(y) — (26a/s Mra)r > c 1 LLa^— (26a/3 + Mra)r,so that if c — sup yo(B) > 1-1,* e°41 (26a/3 Mr' )r, then necessarily dist(S, B) > r. Toprove (32), it is therefore enough to show that this is the case for r = 27-1. But note that1—^4±1 8.-sp^m-1/aEl la/24 m-1/aEct/4^E 3^4rie/3 + M(27-1)1+" L (2L)1+a1 ail^1^1^ ea/4Ze 3 + LMlia Ea/4 + 2LM1ia< Ect/4and the latter is below c — sup co(B) by assumption (41). Therefore, condition (32) issatisfied.We can now apply Lemma 5.27 to get a map k : H^1-/ and an open set 0 with0 fl B = 0 such that the following hold:(i) k is continuous on { u; yo(u) < inf yo(D)118^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCES(ii) v(k(x))) < co(x) for every x E H;(iii) k(x) = x on B;(iv) sup go(k(D\O)) < sup yo(D) —6.17'2;(v) k(A\0) E .F for any A E .7. satisfying max (A) < inf co(D) + 1.Note now that sup yo(A) < c + 6 < inf co(D)+ 2E < inf co(D) + so/2. Hence by (v), wehave that k(A\0) E .F and therefore sup cp(k(A\0)) > c. But if x E A\O is such thatco(k(x)) > c, it follows from (ii) that ga(x) > c and hence that x E D. But (iv) then impliesthat cp(k(x)) < c + E — 2E, which is clearly a contradiction. The proof of Theorem 5.20 iscomplete.PROOF OF THEOREM 5.19: Again let co be a C2-functional on H such that for some a (0 <a < 1) and M > 1 we have for all u1,u2 E HIl vi (ui) — V(u2)11 .. mllui — u211a11(10" (ui ) — v"(u2) 1 1^m ll u i — u2 11a .Let .F be a cohomotopic family of dimension n with boundary B. Set c = lig max co andlet F be dual to .7. with dist(B,F) > 0. Put M = M + N where N(n) is the constantassociated to H in Lemma 1.27 and let C be the constant associated to H by Lemma 1.26.Suppose inf co(F) > c — e where 6 is such that1 — (1/2)" ^) 12/a diSt(B, F)  140 < E < min / (64(2/alia + C)/-42/a^'^232•(43)We shall prove that for any A E .F with sup 'p(A) < c + e, there is u, E A such that(i) c — 2e < v(uE) < c + 6€ + 2€a/8;(ii) 11VOLE)11 5. ca" + (2E + Ea/8)2/35r;(iii) dist(uf, F) < (2€ + Ea/8)1/3;(iv) If (co"(u,)w,w) > (6€1-n2-77, + (2€ + fa/8)1/35r) 11w112 for every w in a subspace E ofH, then necessarily codimE > n.5.3. A proof of the lower estimate in the cohomotopic case^ 119To do that, let AO = (2€ + ea/8)1/3 and use Lemma 1.27 to construct a function SFH -* [0,1] such thatSF (x) =0 x E H\Nf(c)(F).Since € < 2-ia dist(F, .13)2cr% we have that 2f (€) < dist(F, B) and B n N2f(,)(F) = 0.Now put 0(x) = co(x) + f3 (E)SF (X), in such a way that for all u1, u2 E H,IW(ui) - o'(u2)II 5_ ftllui — u211'iie(111) — O"(u2)II^Milui — udia-We shall show that 0 and 6 verify the conditions of Theorem 5.20.Indeed, set d = inf max0(u). Since max yo(A) < c + c, we have thatNET uENd < max1P(A) < c + e + f3(f).^ (44)Since F is dual to .F and inf go(F) > c — e, we see thatc — e + f3(€) < inf 0(F) < d.^ (45)It follows thatmaxi,b(B) < max 0(A\Npo (F)) < max cp(A) 5_ c+ c < d + 2€— f3(c) = d — c"I8 (46)andd _< max 0(A) < max (p(A) ± f3(€) < c + 2c + Eam < d + c.^(47)Moreover, since E < (fa18)4/a < (d _ max ip(B))41« , we conclude that '0, A and E verify theconditions of Theorem 5.20. Hence, there exists u, E A such that:(i)' d — E 0(20 < d + c;OW II IP ' (1 4)11^fa 13 ;a3(HO' If (0"(u,))w, w) > 6€40+011w112 for every w in a subspace E of H, then necessarilycodimE > n.Since sup 0(A\Nf(c)(F)) < d — ca/8 < d — 2€, we have that u, E Nf(,)(F) n A. So, wefinally get{ 1 x E F120^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCES(i)" d — — f3 (E) V(LE) d + + f3 (E);(ii)" dist(uE,^< AO;(iii)" livi(u,)11^fa/3 ± IPE5IN = e/3 + N(2e -I- ea/8)2/3;(iv)" If (e(u()w, w) > (6c4(') N(2€ + eq8)1/3)1Iw112 for every w in a subspace E ofH, then necessarily codimE > n.The proof of Theorem 5.19 is complete.5.4 A proof of the two-sided estimatesIn this section, we establish two-sided estimates on the approximate Morse indices in thecase where a homotopic and a cohomotopic level coincide. The following is a quantitativeversion of Theorem 5.10.THEOREM 5.28 Let co be a C2-functional on a finite dimensional space H such that cp' andcp" are Holder continuous with Milder exponent 0 < a < 1. Then, for any integer n, thereexist a constant A(cp,H,n) > 0 and a positive decreasing function (E.) with lime,o (e.) = 0such that the following hold:If .7" (resp. .P) is a homotopic (resp. cohomotopic) family of dimension n with boundaryB such that F c f" and c := inf max (x) (resp. E := inf max (x)) is finite and if aAE.F xEA^ AE.T xEAdual set F to „f" satisfies inf co(x) > E — 6 for some E withxEF0 2(C — < < min{A(co, H, n), (^4dist(B; F)  ) 24 /a},then, for any A E .T verifying max co(x) < c E/2, there exist ue E H such that:x€Ai) C — (E) G V(LE) G + e(E);ii) 11(Pi(ue)11^(e);iii) ue E A;iv) dist(uE, F)^(s);v) If (cp"(ue)w,^<— e(E)liwir for all w in a subspace E of H, then necessarily dimE <n;5.4. A proof of the two-sided estimates^ 121vi) If (co"(ue)w, w) > (6)11w112 for all w in a subspace E of H, then necessarily codimE >n.In order to prove the above theorem, we need as in the proceeding section to consider firstthe case where the standard boundary condition (sup ca(B) < c) is assumed and when nodual set is singled out. The general result will again be derived from this special case by asuitable perturbation of the function co and an appropriate change of the boundary.The following theorem is the main step for proving Theorem 5.28.THEOREM 5.29 Let co be a C2-functional on Hilbert space H with C-orderly pavings forsome C> 0 and suppose that for some 0 < a < 1 and M > 1, we have for all xi, x2 E H,Ilcd(xi) — V(x2)II 5_ Milxi — x211a and 11(P"(xl) — Co"(x2)II^Milxi — x2iia.Let .7" be a cohomotopic family of dimension n with boundary B such that sup co(B) < :=inf max yo(x) and let a be a continuous map from a compact subset Do of Rn into H suchAEY xEAthat 1(Do) = B. If N := N(n) is the integer given by Lemma 1.26 and L = 6(M1/a + C),then for any E such thatn 1 (  1 — (1/6)a )12(11-ava0 <s < mm64L2N2M2/a^, — sup AB))4/1 ,for any compact set D with D0CDCR" and any continuous map f : D -- H with f = aon Do such that A = f (D) E ..T and mEaf cp ( u ) .e + e, there is ue E H such thati)E— E < CO(Ue) <E+ e;ii) 11(P'(tte)11iii) Ile E A;iv) If (co"(ue)w,w) < —2Ea/2(2+a)11w112 for all w in a subspace E of H then dim E < n;v) If (e(ue)w,w) > 66-2/4(1+-)11142 for all w in a subspace E of H then codimE > n;We shall need the following lemma:122^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESLEMMA 5.30 Let 1" be a cohomotopic family of dimension n with boundary B in a Hilbertspace H that has C-orderly pavings for some C > 0. Let 5,61,62, /31) 021 7'1, r2 be positiveconstants and let N := N (n) be the integer given by Lemma 1.26. Set p = min{ lie], igiv2r2}and assume that0 < 2r2 < ri, (C -1- 1)V2ri < (5,1 (02 2—ri — r1e2 — 2r2E2 > 0,2 2and,ur2 < 60/2.Let f be a continuous map from a closed subset D of Ifin into H such that f (D) E 1- and1(D0) = B for some closed subset Do of D. Suppose b is a compact subset of D withf (b) n B = 0, such that for all x E b, one of the following holds:a) IIV(v)11 > el for all v E B(f (x),2r2);b) There is a subspace Ex of H with dim Ex > n +1 such that for all v E B(f (x),2r2),we have (co"(v)w,w) < —01 II w112 for all w E Eu;c) IIV(v) II 62 for ally E B(f(x), 6r2) and there is a subspace Hx of H with codimHx <n — 1 such that for all v E B(f (x), 2(5), we have (cp"(v)w,w) > 13 2iiw IF for all w E H.Assume further thatdist(B; Iv; licd(v)11 5_ E2, yo(v) _>. sup co(b) — pr21) > 2r1.^(4)Then, there is a continuous map f: D -3 H, a map K: H —> H and an open set 0 with6 n B = 0 such that the following hold:(i) f(x) = f (x) for x E Do;(ii) (p(j. (x))) < cp(f(x)) for every x E D;(iii) K is continuous on 1u; co(u) < inf co(b) -1- 11;(iv) co(K(x)) < yo(x) for every x E H;(v) K(x) = x on B;(vi) sup co(K(i(b)\0)) < sup co(f(b)) — Pr2;(vii) K(A\0) E .7. for any A E .-7-- satisfying max co(A) < inf (10(f (b)) + 1-.5.4. A proof of the two-sided estimates^ 123Proof: Consider the setbi = Ix E b; x satisfies a) or b) } and set 1)2 = b \iii.Use Lemma 5.17, with -4 = 2r2 and v < dist(b, Do), to find a continuous map f : D —* Hsuch that the following hold:(i)' 1(x) = f(x) for x E D\Nv(b1);(ii)' (p(i(x)) _< cp(f(x)) for all x E D;(iii)' Co(f(x))^Cc)(f(x)) — pr2 for x E b1;(iv)' 111(x) — f(x)II 5_ r2 for all x E D.Let now T = fu E 1 (b2);(p(u) _?_ sup so(b) — pr2}. We claim that it verifies theassumptions of Lemma 5.26 (with e = E2 and 13 = 132). Indeed, (18) and (19) are clear,while (21) and (22) follow from (c) and (iv)'. On the other hand (17) follows from (4) and(c). Finally (20) is a consequence of (ii)', (3) and the definition of T. We can therefore find(5' and K that verify (i)-(iv) of that lemma. Let now 0 = N81(T) . It is easy to check thatf and K verify the claims of the lemma. .PROOF OF THEOREM 5.29: Let f" be a cohomotopic family of dimension n with boundaryB and suppose that H has C-orderly pavings for some C > 0. Let E = inf max (p(u) andAE. -7. uEAassume that max (p(B) < E. Suppose(  1 — (1/6)a )12(1+a)/a6 < 64L2N2M2/a^ (5)andE < (e — sup p(B))41"^ (6)where L = 6(Mila + C), and let f be a continuous map from a closed subset D of Rn intoH such that f(D0) = B for some closed subset Do of D. Suppose A = f(D) E . -f. andmax cp (u) < E + E, we claim that there exists ue E H such that:u€A(i) E — E < V(tte) < e + e;OD 1149/(uE)11(iii) u, E A;124^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCES(iv) If (w" (u Ow , w) < —26.12(2-1-.) I. wil2ii ii for all w in a subspace E of H then dim E < n;(v) If (yr," (uE)w , w) > 66a2/4(1+a)11W112 for all w in a subspace E of H then codim E > n.Indeed, if the claim is not true, we shall show that the setb := fx E D; '(f (x)) c -satisfies the hypothesis of Lemma 5.30 Indeed, leta^1^_1j(E) = m-1/-640+0, 7.1 =^= _m4L 4Land r2 = 161/3.The Holder continuity of cp' and yo" gives that b verifies assumptions (a), (b) and (c) ofLemma 5.30, witha2^ M2'^= 62^02 = 4E-4(1^61 =^— )6"13 and 62 = 26a13.Let us show that b satisfies the rest of the assumptions of that lemma. The inequalities in(1) are clear. We also have1 (02 2= — —r1 - riE2 - 2r2622 21 (^a2^(52^ 461/3)= —2 2e'71--F7.7 16L2 L Ea /3 a/3„. —2^„ — 1M a 0 2+20M a a a^2 cail640+a) —^E 3 •TT—TFa —^316L2^4L2211±21Since by (5), 6 < (-1r) 4+a , we have that8M aA. —2^„. —1M a 21.1±.^M a 70+402 60 > --6 4(1+a) —612(1+a)16L2^2L1But again, since 6 <^-IT\ 12/a/ , we obtain that, 2— a 02-1-20 60 > 32L2 4(1+0)We now need to show that6026 < pr2 < so/2.^ (7)5.4. A proof of the two-sided estimates^ 125First, we have7^ m-^,,,2+2c,E*1.1*2 =^16L^ 64L27(1. ML2:  sala < 1^< —E40+a) < so/2,2( At_2/a )12( 1 4- a)/(a2+2a+4)since, by (5) we have that E < 64L . Hence (2) is satisfied.Next we haveM-2/a a2+2a ^E3("+2) < EV 2 ,64L2, N2 \ 4/3(a+2)since by (5), we have E < ,ri7 )^. The second inequality in (7) is therefore verified.For the first one, note that^1 — 1/6ft stil^7(1 — ilf2A,c, ) .4- 1^72E <^E 3 < E 3 = i511'23L^16L(L_IL.61)3/(2-a)since by (5), we have E < k 6L^.,^\ (6a+12)/(4-a)^On the other hand, since E < ( 32N12L2/i , we also have226 < ^ 6(c. +2) =^r16N2L2^16N2 2which establishes (7) and in particular (3).To verify the remaining condition (4) in Lemma 5.30, Let^S := 1u E H;co(u) c — eL-E1^(u)II 2e/31and note that for any y E S and x E B(y, r), we have^icP(x) V(Y)i 5- (25a/3 +^xil _< (26'13 + Mra)rand^cp(x)> (,o(y) — (26.13 + Me)r c —^— (25a/3 + Mra)r,so that if c — sup (p(B) >^+ (25a/3 + Mra)r, then necessarily dist(S, B) > r. To prove(4), it is therefore enough to show that this is the case for r = 2r1. But note that if p =thenp^ 1^1 a^a^1 aL E3 + (26'13 + M(2rir)2ri < F-L E1+013 +^4(1+a) + —2LET <01^2 _r 1 5 <67(4+28)1.6N2 2 — 16N2L2126^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESOn the other hand, if p =then16/V 2 rz17...sa8_^1 2 +a^1 a^aLEI + (2ell" M(2r1)')2r1^16N2L2^E 6(ar-F2) ± _E. 3 4(14-a) + -SI < 4 .2LSince by assumption (6), we have that Ecf < — sup w(B), the verification of (4) is complete.We can now apply Lemma 5.30 to get a continuous map : D H, a map K : H Hand an open set 0 with O n B = 0 such that the following hold:i(x) = 1(x) for x e Do;(ii)' (J (x)))^cp( f (x)) for every x E D;(iii)' K is continuous on {It; cp(u) < inf co(b) +(iv)' (K (x)) < cp(x) for every x E H;(Ar)' K (x) = x on B;(vi)' sup co(K(i(b)\0)) < sup cp(f (D)) — [0'2;(vii)' K (A\0) E for any A E .7. satisfying max (A) < inf co(f (b))Note now that sup cp( f (D)) < c + e < inf ca(b) +2s < inf yo(b) + 60/2. Hence by (vii)',we have that K(j(D)\0) E and therefore sup (p(K (f (D)\0)) > c. But if x E ADA° issuch that co (K (x)) > c, it follows from (iv)' that v(x) > c and hence that x E D. But (vi)'implies then that co(K(x)) <c + E pr2 <c + E — 2E, which is clearly a contradiction. Theproof of Theorem 5.29 is complete. •PROOF OF THEOREM 5.28: Let yo be a C2-functional on H such that for some a (0 <a < 1)and M > 1 we have for all u1, u2 E H^(ui ) — co'(u2)11^Mk]. — 11211"^11c0"(ui ) — (p"(u2) 11^— u211.Let Y. (resp. Y) be a homotopic (resp. cohomotopic) family of dimension 71 with boundaryB such that ,F C Y. Set c = inf maw, E = inf maw and let F be dual to Y withAE.F A^AEF Adist(B, F) > 0. Put fa = M + N where ST is the constant associated to H in Lemma 1.27and let C be the constant associated to H by Lemma 1.26.5.4. A proof of the two-sided estimates^ 127Suppose inf cp(F) > c — c where E is such that1 — (1 /6)^12+aa ) a^i diSt(B, F) ‘ 242(c — e) < s < min / (64 . 36(11/1/a + C)2N2A-12/.)^'^4^)7 •^(8)We shall prove that for any A E .7' with sup cp (A) < c + c/2, there is u, E A such that(i) c — 5E/2 < cp(u,) < c + 6c + 2ca/8;(ii) 11Cd(uE)11^ea/3 + (2€ + €"/8)2/35r;(iii) disque, F) < (2€ + ca/8)1/3;(iv) If (cp" (ete)w,w) < — (26-a12(2+0) + (2€ + ca/8)1/3*-AIV) 1111/112 for all w in a subspace E ofH then dim E < n;.2(v) If (cp" (uf)w , w) > (6c4) (2€ + ea/8)h/3N) 11w112 for every w in a subspace E ofH, then necessarily codimE > n.To do that, we proceed as in the last section and consider the function f(i) = (2E+Ea/8)1/3and then use Lemma 1.27 to construct a function SF : H —> [0,1] such that1 x E FS F(x) = 10 x E H\N f(,)(F).Since c < 4-+, dist(F,B)21, we have that 2 f (€) < dist(F , B) and B n N2f(e)(F) = 0.Now put 7,b(x) = w(x) + f3(E)SF(x), in such a way that for all 2/1, u2 E H,110' (u1) — I"(u2) I I^Milui — u2llaii0"(ui) — O"(u2)II 5_ Milui — u2ila.We shall show that 0 and E. verify the conditions of Theorem 5.20.Indeed, set d = inf max0(u). Since AEYCY and max co(A) < c+ €12 <e-I-E, weA'E.7. uEA!have thatd < max 0(A) < e + € + f3(e).Since F is dual to .7" and inf yo(F) > E — E, we see thate — c + f3(€) < inf IP(F) < d.128^ 5. MORSE-TYPE INFORMATION ON PALAIS-SMALE SEQUENCESIt follows thatmax cp(B) < max 0(AVV/(,)(F)) :5_ max cp(A) < E ± c < d +2c — f3(c) = d — earnandd< max 0(A) 5_ max (p(A) + /3(0 < E ± 26 + ca/8 < d + €.Moreover, since 6 < (Ea/8)4/a < (d _ max o(B))4/a, we conclude that //), A and e verify theconditions of Theorem 5.29. Hence, there exists u, E A such that:Or d — c <0(14) < d + E (ii)' II Otte)11 -- Ec'l 3 ;OW If (49"(ue)w, w) < —20/2(2+6)1142 for all w in a subspace E of H then dim E < n;(iii), If (vir(now,w) > 66a2/4(i+c)liwil2ii ii for all w in a subspace E of H then codimE > n.Since sup tP(A\Nf(,)(F)) < d— ("18 < d— 2c, we have that u, E Nf(f)(F)n A. So, we finallyget(i)" d — E — f3 (E) cp(ue) <d + c + f3(€);(ii)" dist(u„F) < f(c);(iii)" ilV (24)11 5_ cal3 + fi;(:)) N = &13 + N (2f + eq8)213;(iv)" If (co"(ue)w, w) ^_ (26.«/2(2+a) ± (2€ + Ea/8)1/3N) 11w112 for all w in a subspace Eof H then dim E < n;(v)" If (co"(u,)w, w) > (6€41*--.42-4 + N(2€ + E0/8)1/3)11w112 for every w in a subspace E ofH, then necessarily codimE > n.The proof of Theorem 5.28 is complete.^ IIBibliography[1] Adams R. A., Sobolev space, Academic Press, New York-San Francisco- London(1975).[2] Amann H. and Zehnder E., Nontrivial solutions for a class of nonresonance problemsand applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa (4),7 (1980) pp. 539-603.[3] Ambrosetti A. and Rabinowitz P. H., Dual Variational methods in critical point theoryand applications, J. Funct. Anal., Vol.14 (1973) pp. 349-381.[4] Aubin J. 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