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On non-homogeneous quasi-linear PDEs involving the p-Laplacian and the critical sobolev exponent Yuan, Chaogui 1998

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O n Non-homogeneous Quasi-linear P D E s Involving the p-Laplacian and the Crit ical Sobolev Exponent B y • Chaogui Yuan M . Sc. (Mathematics) Sichuan University, Chengdu A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES Department of . MATHEMATICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA 1998 © Chaogui Yuan, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Mathematics The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1Z1 Abstract This thesis is devoted to the study of some quasi-linear PDEs involving the p-Laplacian. This type of problem represents a model case for the general quasi-linear elliptic equa-tions. These problems arise from the Euler-Lagrange equations associated to various geometric problems and from topics like Non-Newtonian Fluids, Air Dynamics, Non-linear Biological Population, Reaction-Diffusion Problems etc. The difficulties in- these problems come either from the lack of compactness of the approximate solutions or from the lack of symmetry in the corresponding energy functionals. The thesis has basically two parts. The first part is in Chapter 3, where we consider a problem related to Yamabe's prescribed curvature conjecture in Riemannian Geometry. This problem is critical in the sense that it involves the critical exponent in the Sobolev embedding. In particular, we show the existence of sign changing solutions by using the duality methods introduced by N. Ghoussoub. In the second part of the thesis, chapters 4 - 7, we consider the associated non-homogeneous problems which arise from either a linear second member or from non-homogeneous Dirichlet boundary conditions. Because of the lack of symmetry, the tradi-tional equivariant variational principles do not apply here. To overcome those difficulties, we extend and use Bolle's method as well as Ekeland-Ghoussoub's virtual critical point theory to a Banach space setting. ii Table of Contents Abstract ii Acknowledgment v Introduction vi 1 Sobolev Spaces and the p-Laplacian Preliminaries 1 1.1 Sobolev Space and Its Properties . . . 1 1.2 Properties of the p-Laplacian Operator 2 1.3 Some Regularity Results 4 1.4 Best Sobolev Constants ' 5 2 General Variational Principles and the Critical Groups 7 2.1 Strong Min-max Principles . 7 2.2 Genus and Some Associated families 8 2.3 Critical Groups ' . . 11 3 Sign Changing Solutions for PDEs Involving the p-Laplacian and the Critical Sobolev Exponent 17 3.1 Introduction ; 17 3.2 Some Elementary Properties . . : . . . 18 3.3 The First Solution in the Sub-critical Case 20 3.4 The First Solution of the Critical Problem 21 .3.5 The Second Solution in the Sub-criticalCase . 28 iii 3.6 The Second Solution for the Case of Critical Non-linearities . . . ; . . . . 31 3.7 The Case q = p 41 4 Mult iple Solutions for P D E s Involving the p-Laplacian and Generic Linear Non-homogeneities 44 4.1 Introduction 44 4.2 Properties of the Energy Functional 45 4.3 Proof of the Main Theorem : : . 53 5 P D E s with Non-homogeneous Boundary Value Conditions 58 5.1 Introduction 58 5.2 Preservation of a Min-max Critical Level Along a Path of Functionals . . 59 5.3 Multiplicity of Solutions for Non-homogeneous Boundary Value Problems 62 6 V i r t u a l Solutions for Non-homogeneous P D E s Involving the p-Laplacian 70 6.1 Introduction 70 6.2 Duality Maps and Uniformly Convex Banach Spaces 70 6.3 Virtual Critical Point Theory 72 6.4 Virtual Solutions for Non-homogeneous p-Laplacian Equations . 78 7 Min-max Principles for Convex Sets and Applications to p-Laplacian 83 7.1 Introduction . . ' 83 7.2 A Minimax Principle on Closed Convex Subsets of Banach Spaces . . . . 83 7.3 Applications of the Minimax Principle to p-Laplacian 86 Bibliography 91 iv Acknowledgment It is a pleasure to thank my supervisor, professor Nassif Ghoussoub, not only for sug gesting my dissertation topic, but also for providing invaluable advice, support and en couragement which led to the completion of this work.' • Introduction In this thesis, we will use variational methods to study non-linear elliptic equations of the following form: -Apu := -div(\Vu\p-2Vu) = F'(u) in ft v v ' (P) u = 0 on 9ft V where 1 < p < n and ft will always denote a bounded smooth domain in Rn, and (Fl) F' is increasing, F(0) = 0, F regular, (F2) \F'(u)\ < C ( l + \u\r), for some r > 0. By setting p* = the Sobolev embedding theorem yields that ^ ( f l ) c L p * . We say that problem (P) is sub-critical, critical or supercritical if r < p* —1, r = p* — 1 or r > p* — 1 respectively. If p > n, then problem (P) is always sub-critical. For p = 2 this problem has been studied extensively in many contexts, especially in Riemannian Geometry^ where the Yamabe problem corresponds to the critical case r — 2* — 1. In recent years there has been an increasing interest in looking at these problems in the more general framework of quasi-linear elliptic equations for which (P) is a model case. In general, such problems can be seen as the stationary counterpart of evolution equations with nonlinear diffusion, see J. I. Diaz [17]. The techniques used to deal with the above problem are mostly variational. Critical point theory has proved to be a very effective method in dealing with a wide range of non^ linear partial differential equations. Finding critical points by optimization is as ancient vi as the least action principle of Fermat and Maupertuis, and the calculus of variations has been an active field of mathematics for almost three centuries. However, in this thesis, we will be dealing with three types of complications: • We will be searching for sign-changing solutions which are usually unstable extrema. • We are concerned with the problem of multiplicity of solutions even when the symmetry is broken by non-homogeneous terms or boundary conditions. • We will be mostly dealing with equations involving the critical Sobolev exponent, hence in settings where the lack of compactness undermines the effectiveness of the usual existence theories. In general, there are two approaches for dealing with the existence of such critical points: Morse theory and the min-max methods (or the calculus of variations in the large) introduced by G. Birkhoff and later developed by Ljusternik andSchnirelmann in the first half of this century. Currently, both theories are being actively refined and extended in order to overcome the limitations to their applicability in present-day variational prob-lems. In this thesis, we shall follow the approach of Ghoussoub [28] to overcome some of' the difficulties mentioned above. This method will be described in Chapter 2 right after Chapter 1 where the necessary analytical background about Sobolev spaces and the p-Laplaciari is recalled. We shall state these variational principles in full generality and therefore some non-elementary topological concepts (basic Homology theory) are used. We do that in order to include in Chapter 2, a result that is independent of the rest of the thesis: That Ghoussoub's version of the Min-Max principle actually yields the Morse inequalities. Moreover, a C1-version of these inequalities hold as long as the Morse indices are replaced by the critical groups introduced by K. C. Chang. However, for the rest of vii the thesis, an elementary form of these principles will be used: a version that will use only basic topological results like Brouwer's and Borsuk-Ulam theorems. In Chapter 3, we investigate the following problem -Apu = X\u\q-2u + \u\p'-2u in Q ( P A ) „ u\dn = 0. For p = 2, it originates from the study of the so called Yamabe problem in differential geometry. In the case where p = q = 2 and A is below the first eigenvalue of the Laplacian, Brezis and Nirenberg [11] constructed a positive solution by proving that, even though the problem is critical, there is some compactness below a certain energy level of the corresponding functional Ex(u) = - I \Vu\pdx - - j \u\qdx - — f \ufdx. p Jn q Jn p* Jn In Chapter 3, we deal with more general values of p, q, A and we construct second solu-tions that are sign-changing and with other qualitative properties, by using Ghoussoub's duality approach that allows to push back the threshold of non-compactness to a higher energy level as long as the approximate solutions belong to an appropriately chosen dual set. This approach has been used by Cerami-Solimini-Struwe [14] and Tarantello [48] in the case where p = 2. We establish this result in two cases: • when p>2, m&x{p,p* — ^} < q < p* and A > 0. • when p > 2, q = p, p3 — p2 + p < n and 0 < A < A x where Ai is the first eigenvalue of the p-Laplacian. In chapter 4, we start dealing with non-homogeneous problems, the first one being: —Anu=\u\q~2u+f(x) in fi, (Pf) u — 0 on dtt vm For p = 2 and / = 0, it is well known and easy to see that this problem has infinitely many solutions. When / is not. zero, the situation is much more complicated and the problem of multiplicity is still wide open, even though some partial results have been obtained (see below). On the other hand, Bahri [3] had shown that the result is true if one is allowed to perturb the non-homogeneous term. More precisely, If 2 < q < 2*, then the set of f G H~l such that the above problem has an infinite ' number solutions is a dense residual in H"1, In chapter.4, we essentially adapt the non-trivial methods of Bahri to establish a similar result for the case of the p-Laplacian (p > 2). In chapter 5, we deal with the case when we also have non-homogeneous boundary values. The main result of this chapter is the following: Let u0 e C2(Cl, R) and assume 2 < p < q < "ffi*_^(< P*), then problem —Avu = \u\q~2u + f (x) in Q, u = uo on dQ, has infinitely many solutions. Note that if p = 2 and u0 = 0, Bahri and Lions [5] get the same conclusion under the better hypothesis q < |2* . But it is still an open problem whether this remains true for all q up to 2*. It is not even clear whether this problem has a single solution for any / i n i ? - 1 . For 2 < p < q and u0 = 0, Azorero and Peral [25] adapted the above method to get the result for < w ^l p ) — 1. In this chapter, we establish the above by adapting and using a new method introduced by Bolle in [9] and developed in [10]. We also note that when UQ = 0, this method will also yield the result of Azorero and Peral. However, the ix latter is still not satisfactory since it does not yield the Bahri-Lions result in the case where p = 2. In chapter 6, we continue our study of the problem in chapter 4. But this time, we first extend the theory of virtual critical points of Ekeland and Ghoussoub [23] from a Hilbertian setting to certain Banach spaces. We then show that the above problem \Pf) has infinitely many virtual solutions. In chapter 7, we present a new -but not satisfactory- approach to the Ekeland-Ghoussoub result. We establish the min-max principle of Ghoussoub for convex subsets of a Banach space and then we use it to obtain one virtual solution for problem (Pf). x Chapter 1 Sobolev Spaces and the p-Laplacian Preliminaries In this chapter, we introduce the necessary background about Sobolev spaces and the p-Laplacian that will be used throughout this thesis. 1.1 Sobolev Space and Its Properties We will work in the Sobolev space HQ'p(CI) equipped with the norm (1 < p < oo) The following theorem is of particular importance in the variational approach to solving differential equations as it gives us some control over the nonlinear terms. Theorem 1.1.1 (Sobolev-Rellich-Kondrachov Theorem) Let p* = be the critical ex-ponent and assume that 1 < p < n, then. . . , : (1) the inclusion Hl'p(£l) c—>• If* (Vi) is continuous, (2) for any q < p*, the inclusion Ho'p(fl) M - L9(fl) is compact. We say that a Banach space is uniformly convex, if for any e > 0, there exists 0 < S = 5(e) < 1 such that for any x, y e X with = 1, ||y|| = 1 and \\x — y\\ > e, we have | | ^ | | < 1 - 5. Let X be a Banach space and S be a closed subspace, we use S1- = {<p G X*; <f>(x) = 0, for x G S} to denote its annihilator. We need the following 1 Chapter 1. Sobolev Spaces and the p-Laplacian Preliminaries 2 Proposition 1.1.1 With the above notation (1) ([6] p. 194) 'X is uniformly convex, ihenX/S, the quotient space, is also uniformly convex. (2) ([21] p.110) The quotient space X*/S± is isometrically isomorphic to S*. Let X = H-^'itt) be the dual of HQ'p(Q). then X is a reflexive Banach space and X* = HQ'P(Q,) is a uniformly convex Banach space. Actually, if we let N = £o<|a|<i be the number of indices a satisfying 0 < \a\ < I. For 1 < p < oo, let LPN = YljLi the norm of u = (u\, • • •, uN) in LPN being given . by = ( Z ) l l u i l l p ) ? -Then LPN is also a Banach space that is separable, reflexive and uniformly convex. Let us suppose that the iV-indices a satisfying 0 < |a| < 1 are linearly ordered in some convenient fashion so that to each u G H Q , p ( Q . ) we may associate the well-defined vector Pu in LVN given by Pu - (Dau)0<|a|<i. • • • Since | | P « ; L ^ | | = ||it|| f fi,P, P is an isometric isomorphism of iJo'p(Q) onto a subspace W C LPN. Since Hl'p{Vt) is complete, W is a closed subspace of LPN. Thus W is sepa-rable, reflexive and uniformly convex [1]. The same conclusions must therefore hold for Hl'p{Sl) = P-\W). Since, the dual space of #0 1 , P(Q), H'1*'{Q) = W* = L^/W-1, and because LPN(Q,) is uniformly convex, is also reflexive and uniformly convex. 1.2 Properties of the p-Laplacian Operator The p-Laplacian operator is defined, for 1 < p < oo, by Apu = div(|Vu| p - 2 Vw) Chapter I. Sobolev Spaces and the p-Laplacian Preliminaries 3 The p-Laplacian is the paradigmatic example of a degenerate/singular quasi-linear elliptic operator. Notice that if p = 2, the p-Laplacian is the classical Laplace operator. In this section, we will introduce some well known properties which will be needed through out this thesis. The main reference for this section is [41]. Theorem 1.2.1 ([41]) For any f e there exists u <E H^.tfl) such that [ {(\Vu\p-2Vu, V</>) - f(j)}dx = 0, ' € HQ'p(Q). Jn i.e. — A p is invertible as a map from HQP(Q) toH~1,P'(Q). The main properties of — Ap and ( — A p ) _ 1 are summarized in the following theorem. Theorem 1.2.2 ([41]). The following conclusions are true. (1) Ap : HQ,P(Q,) -> i y _ 1 ' p ' ( f 2 ) is uniformly continuous on bounded sets. (2) ( - A p ) ' - 1 : H~1'P'(Q) ->• Ho'p(Q) is continuous. (3) The composition operator ( - A p ) " 1 : ff-1,p'(fti) -> HQ'P(Q,) <->• LQ(Q) is compact if 1 < q < (4) For p > 2, —Ap is strongly monotone in the sense that [ (\Vu\p-2Vu -\Vv\p~2Vv) - .(V« - Vv)dx > Cp\\u - v\\php. Jn Ho With this theorem, we have the following proposition. Proposition 1.2.1 ([25] [38]) Let {un}n G HQ'p(Q) be a bounded sequence, then there exists a subsequence, without loss of generality, still denoted by {un}n, such that (a) un u weakly in HQ'p(Q),-Chapter 1. Sobolev Spaces and the p-Laplacian Preliminaries 4 (b) un^tu in L r ( f i ) ifl<r<p*, (c) un —> u almost everywhere, (d) u.. u weakly in Lp* [Q), (e) un\u u\u\p* weakly inLp*-1(Q). Now we consider the following weighted eigenvalue problem (1 < p < oo): < — Apu — Xb(x)\u\p 2u, ; ue # 0 1 , p(fi), u^O. We will say that A e R is the eigenvalue and u € #o'p(fi), u ^ 0 is the corresponding eigenfunction of the above eigenvalue problem if holds for any ip e # o ' p ( f i ) . Theorem 1.2.3 ([41] [33] [19]) Assume that b(x) > 0 , b(x) G Lq(fl), and \{x e fi : b(x) > 0}| 7^  0 ; where q > 1 if p > n, q > 1 if p = n, q > ^ > 1 otherwise. Let X^mfUnW; fnb(x)\v\p = 1}, then (1) Ai > 0 is the first eigenvalue; (2) Ai is simple and there exists precisely one pair of normalized eigenfunctions cor-responding to Ai which do not change sign in fi. Here, v being normalized means that fnb{x)\v\p = 1. 1.3 Some Regularity Results Consider the problem (*) -Apu = - d i v ( | V u | p - 2 V u ) = f(x,u) in fi, k u\aa = 0, Chapter 1. Sobolev Spaces and the p-Laplacian Preliminaries 5 where 1. < p < n and / satisfies - : (HI) . | / ( . r ,n ) |<C( l + |ur) (H2) r + 1 <p*. Theorem 1.3.1 ([41] [49]) Let u e #o'P( f i) be a solution of (*). /// verifies (HI) and (H2), thenueCl'a(U), for somea>0. ' 1.4 Best Sobolev Constants If 1 < p < n, by the Sobolev embedding theorem, there exists a constant C such that C\\u\\pp. < \\\7u\\pp for all u e Hl0'p(Cl). We use S(£l) to denote the best Sobolev constant, i.e. the largest constant C satisfying the above inequality for all u e HQ'P(Q). S(Q.) is given by " S (n )=- inf ueH^p(Q),u^0 \\u\\p* We denote S(Rn) by S. L e m m a 1.4.1 Suppose, 2 < p < n, the following conclusion hold. (1) (Struwe[45[ P. 40) S(Q) is independent ofQ, and will henceforth be denoted by S. (2) (Struwe[45] P. 4%) The best constant is never achieved on a (smooth) domain dif-ferent fromHn. In particular the infimum is never achieved on a bounded domain. (3) (Talenti[47], Guedda and Veron[30]) S is attained when Vt = R n by the functions ya(x) = (a • n( -)p )~p^ (a + \x\p-1) p Chapter 1. Sobolev Spaces and the p-Laplacian Preliminaries 6 for some a > 0. Moreover the functions ya are the only positive radial solutions of -div( Vu|" V//.) =-a""-1 in R n . Hence, 5||f/all^=-||V.ya||? = | |ya| |?;p5?. The importance of the best Sobolev constant lies in the fact that it provides an energy level under which some compactness requirements for PDEs hold. Chapter 2 General Variational Principles and the Crit ical Groups 2.1 Strong Min-max Principles For Banach spaces X and Y, we use C(X, Y) to denote the space of all continuous maps from X to Y. Definition 2.1.1 Let X be a Banach space and B be a closed subset of X. We say that a class T of compact subsets of X is a homotopy-stable family with boundary B provided that ; (1) every set in T contains B. (2) for any set A in T and any r\ e C([0,1] x X; X) satisfying n(t, x) = x for all (t, x) in ({0} x X) U ([0,1] x B) we have that n{{\} x A) e T. ' We say that a closed set M is dual to the homotopy-stable family T if F verifies the following M n B = 0 and'M n A 0 for all A G T. Definition 2.1.2 Say that the functional f on Banach space X verifies the Palais-Smale condition at the level c (in short (PS)C), if for any sequence {xn}n satisfying lim„ f(xn) = c and l im n ||/'(rEn)|| = 0 has a convergent subsequence. ; In this thesis, we shall use the following weaker version,of the Palais-Smale condition. 7 Chapter 2. General Variational Principles and the Critical Groups 8 Definition 2.1.3 Say that the functional f on Banach space X verifies the Palais-Smale condition at the level c and around the set M (in short, (PS)M,c) if for any sequence {xn}n in X verifying limnf(xn) = c, l im n ||/'(xn)|| = 0 and limndist(xn, M) = 0 has a convergent subsequence.. We call point x0 G X a critical point of the functional / if f'(xo) = 0 i.e. for any y G X, (f'(x0),y) = 0. The importance of the critical point of a functional and the critical point theory lies in that the solutions of many PDEs are the critical points of the corresponding functionals. The following well known theorem is due to N. Ghoussoub [28]. , Theorem 2.1.1 Let f be a Cl-functional on X and consider a homotopy stable family T of compact subsets of X with a closed boundary B. Assume that max/(B) <c = c(f,T) = inf max/(a;) and let M be a dual set to T that satisfies inf / (M) > c. If f satisfies (PS)M,C, then M n Kc ^ 4>. Moreover, if there exists a compact set A 6 T such that max^e^ f(x) = c then A n M n i f c ^ | , where Kc is the set of all critical points of f at critical level c. Let XQ € X, be such that /(x 0 ) < 0, then it is clear that the family T = { 7 € C([0,1]; X); 7(0) = xQ, 7(1) + x0 and /( 7(1)) < 0} is a homotopy stable family with boundary B = {/ < 0}. This family will be used in Chapter 3. Sometimes, this family is also called a Mountain Pass class. 2.2 Genus and Some Associated families Let G be a group which acts on the Banach space X. A functional / on X is said to be invariant if for any x G X and g G G, we have f(gx) = f(x). A subset M of X Chapter 2. General Variational Principles and the Critical Groups 9 is G-invariant if for any x E X and g E G, we have that gx E M. A deformation 77 G C([0,1] x X;X) is said to be equivariant with respect to the action of G, if for any x G X, t G [0,1] and g E G, we have that r](t,gx) = gr](t,x). It is clear that i f G = Z 2 , then a G invariant functional is just an even functional. For even functionals, it is usually useful to work with Z2-homotopy stable families. These are those families consist of compact C-invariant subsets of X and are stable under equivariant deformations. We now describe a typical example of such family. For that, we first introduce the classical concept of genus due to Krasnoselskii, as a tool to measure in a convenient way the size of symmetric sets. Given- X a Banach space, we consider the class E = {A C X : A closed, A = -A}. Definition 2.2.1 The Krasnoselskii genus, jz2> ^s defined as 722 : E ->• N U {00} A -+ 1Z2{A) where • ' ' jz2(A) =• min{k E N : 3<p EC(A, R f c - {0}), <p(-x) = -<p(x)}. • If the minimum does not exists, then we define 7z2(A) = +00. Notice that if C C X is such that C n ( -C) = 0 and we define A by A = C U ( - C ) then A G E and moreover 7z2(^ ) = 1- ft is sufficient define <p(x) = 1 for a; G C and ¥>(a;) = - 1 for x G - C . In this way <p E C{A, R - {0}). This elementary remark shows how to use the genus. Chapter 2. General Variational Principles and the Critical Groups 10 In general, to calculate the exact genus of a set is difficult. Often it is enough to have some estimates which can be obtained by comparison with sets whose genus is known, for instance, with respect to spheres. In this way the following results are very useful. L e m m a 2.2.1 Let A, B G E . Then (1) If there exists f e C(A, B), odd, then JZ2(A) < 7 z 2 ( £ ) . (2) IfAcB,then1Z2(A)<jZ2{B). (3) If there exists an odd homeomorphism between A and B, then jz2(A) = lz2{B). (4) If Sn~l is the sphere in R n , then 7 Z 2 (S n ~ 1 ) = n. (5) 7 Z 2 ( A U 5 ) <iz2{A) + 1Z2{B). (6) If jZ2 (B) < +oo, then jz, (A\B) > 7 2 2 {A) - l z , (B). (7) If A is compact, then 7z2(^) < +'od, and there exists 5 > 0 such that ' Jy,AA)=lz.2(Ns(A)): where NS(A) = {x e X : d(x, A) < 5}. (8) If XQ is a subspace of X with codimension k, and jz2(A) > k> then A fl X0 ^ 0. Let Sp = {x G X, \\x\\ — p} be a sphere with radius p in X and consider the class H = {h : X —> X odd homeomorphism }. By the properties of the genus, the families Tn — {A : A compact symmetric with /yZ2(h{A) fl Sp) > n, for any h G H} for any n G N' are clearly Z2-homotopy stable families with no boundary. Chapter 2. General Variational Principles and the Critical Groups 11 Theorem 2.2.1 Let f be an even Cl-functional on X and consider a Z2-homotopy stable family T of compact subsets of X with a closed boundary B. Assume that max/(B) < c = c(f, T) = inf max f(x) At'T xCA and let M be a G-invariant closed subset of X which is dual to T and satisfy inf / ( M ) > c. // / satisfies (PS)M,c, then M D Kc ^ <f>. ' ' As we shall see in the sequel, the problems arise when we are dealing with Z2 homotopy stable family in a framework when the functional is not even. 2.3 Crit ical Groups Let X be a Hilbert space and let / G C1(X, R) be a functional on X. Suppose XQ is an isolated critical point and c = f(xo). Let U be any closed neighborhood of p that contains no critical point except x0, then the n-th critical group of / at XQ is defined to According to the excision property of singular homology theory, the critical groups are well defined; i.e. they don't depend on a special choice of the neighborhood U. For / G C2(X, R 1 ) , the Morse index of'/ at a non-degenerate critical point XQ (i.e. f"(xo) is invertible) is defined to be the number of the negative eigenvalues of f"(xo). If / G C2(X,R1), with Morse index j at a non-degenerate critical point XQ, then Morse lemma (Ghoussoub [28] p.. 169 or Chang [15]) yields that: Let F be any closed subset of X, suppose KcnF — {z\\ z2 • • • zm}, then by the excision property, we have, for e > 0 small enough be Cn(f,x0) = Hn(unfc,un (fc - {xo}))-Chapter 2. General Variational Principles and the Critical Groups 12 Hn(fc, fc\Kcn F) = Hn(fc n U™=1B(Zj, e), fc D \J%x(B{Zj,e) \ &})) = ®?=iCn(f, where B(x,e) is the ball centered at x with radius e. The following standard deformation lemma will be needed in the sequel. L e m m a 2.3.1 (Deformation lemma [15] p.21) Let f G C1(X,H) be a functional on X with continuous first derivative and satisfy (PS)C, Vc G [a, b]. If f has no critical value in (a,b], then fais a strong deformation retract of ft,, i.e. the inclusion i : fa fb is homotopic to the identity on relative to fa ( Dold [18] p. 39). A family T of subsets of X is said to be a homological family of dimension n with boundary B if for some non-trivial class a in the n-dimensional relative homology group Hn(X,B) we have that J7 =: J-(a) = {A; A compact subset of X, A D B and a G lm(if)} where if is the homomorphism if : Hn(A,B) —> Hn(X,B) induced by the inclusion i : A ^ X . It is clear that a homological family is homotopy-stable. Proposition 2.3.1 Suppose that A G F(a) and let (3 G Hn(A, B) be such that i*((3) = o>, where is the map from Hn(A,B) to Hn(X,B) induced by inclusion. Let \(3\'.be the support of f3, then U B G T(a). Proof: Let 7 G (3 be any representative , then 7 G Cn(A) with G Cn(B). Since the image of 7, im( 7 ) , is included in |/3|, 7 G Cn(\(3\ U B) and [7] G Hn(\/3\UB, B). Let j be the inclusion j : \(3\ U B A, then i o j is the inclusion \(3\ U B 4^- X. Since j o 7 = 7 is a representative of f3, we have (*° J)*[T] = ** 0J*([T]) = **(!>' °7]) = = a , Chapter 2. General Variational Principles and the Critical Groups 13 i.e. \P\UBe T{a). Theorem 2.3.1 Suppose X is a Hilbert space, a G Hn(X, B) is nontrivial and let T{a) be the homological family associated to a, Assume max / (B) < c = iniA&r(a) snPxeA f(x)> and let M be a dual set to T(a) such that inf f(M) > c. If f verifies (PS)C, then there exists X'Q G KCC\ M such that Cn(f,x0) ^ 0. • Proof: It suffices to prove that Hn(fc,fc \ (KC D M)) ^ 0. Let e be any positive real number, then there exists A G T{a) such that Ac fc+£. Let j3. G Hn(A, B) be such that i*(/3) = a, where i* is the map from Hn(A,B) to Hn(X,B) induced by inclusion. Let \P\ be the support of $, then U B G T(a). Let 7 = d/3 be the boundary of (3, then |7| G B C fC\Kcr\M, we shall prove that [7] G Hn^(fC\Kcn M) is nontrivial. Indeed, if [7] is trivial, then there exists a singular"n - chain r such that | r | C fc\KCD M and dr = 7. Note that |r| U B G JF(a). Let ,40 = \r\ U 5, then max/(A 0 ) =• c. Theorem 2.1.1 yields that M n ^ 4 0 n AT C -^ 0, which is a contradiction. Thus [7] G Hn(fc\KcnM) is nontrivial. Now consider the exact sequence: #n( / c+ £ , / c \ ^ n M ) F n _ ! ( / C \ i f c n M ) A i ? n - l ( / c + e ) Since ^([7]) = 6, [7] G keri* = Imd*, we have Hn{fc, fc\Kcr\ M) ~ Hn(fc+£, fC\KcC\ M) ^ 0. The theorem is thus proved. In the particular case, where max/(B) < c, we can consider M = (/ > c) as a dual set to the homological family and we obtain the following Corollary 2.3.1 Suppose a G Hn(X,B) is nontrivial, T{a) is the homological family associated to a, and max/(B) < c = inf 4 ^ Q) s u P z e A fix) • If f verifies (PS)C) then there exists XQ G Kc such that Cn(f,xo) 7^  0. Chapter 2. General Variational Principles and the Critical Groups 14 Another direct corollary deals with the Morse index of critical points associated to homological families (Ghoussoub [28]). Corollary 2.3.2 Suppose X is a Hilbert space, a G Hn(X,B) is nontrivial and,T(a) is the homological family associated to a.. Assume maxf(B) < c = miAe:F^supxeA f(x). Let M be a dual set to T(a) such that inf f(M) > c. If f G C2(X,R) verifies (PS)C and the critical points in KCDM are non-degenerate, then there exists x0 G KCC\M such that m(x0) = n, where m(xo) is the Morse index of f at x0. Proof: If for any x G Kc n M, m(x) ^ n, then for any x G Kc n M, Cn(f, x) = 0, this is a contradiction. Now we consider an application of the above result to a Morse type inequality. Theorem 2.3.2 Let X be a Hilbert space and suppose f G C1(X, R) verifies the Palais-Smale condition. Let B be a closed subset of X such that m&xf(B)< inf inf max f (x). Define Mn(X,B)= Y, dimCVU.xo), xo€K, f(xo)>maxf(B) where K is the set of all critical points of f. If the critical values of f are isolated then Mn(X,B)> dimHn(X,B). • Proof: Since the critical values of / are isolated, we may arrange the values in a sequence: • • • < c_2 < c_i < c0 < C i < c2 < • ••-Since / satisfies the Palais-Smale condition, each critical set KCi is finite. Without loss of generality, we may assume {CJ}^ 0 are all of those critical values greater than m a x / ( £ ? ) , i.e. max f(B) < c0 < cx < c2 < • • • Chapter 2. General Variational Principles and the Critical Groups 15 Consider the functional 9(x) = Q e x P ( i i r _ r ! i i ^ i f II x -xQ ||< 5 ||x-a;o|r-<52 0 otherwise then g G Cl(X, R) , with X -— X§ ctS the only critical point. Let x0 be a critical point of / , choose S small enough so that a ball centered at XQ with radius 8 will contains no other critical point of /• except x0. By a suitable choice of the constant a in the functional g and add g to / will change the value of / at x0 but cannot change the property of the critical points of / , i.e. will not change the critical points and will not add or reduce any other critical points. So by a suitable adjustment, we may assume that / has different critical value at different critical point, this means that the critical set at critical level c is a single point set. For any a G Hn(X,B), let ca = iniAejr^ m&xxeA/(#)• By Corollary 2.3.1, there exists x0 G KCa such that Cn(f, XQ) ^ 0. Since ca > maxf(B), ca = infzea niaxxe|Z| f(x). Let OJI, OJ2? • , cxm be m non-equivalent elements in Hn(X, B) such that c Q l = ca2 = • c Q m = c. Let e be small enough such that / has no critical values in (c, c + e]. Let Pi, P2, • • •, Pm be the representatives of those a^s respectively, such that maxXE\^A f(x) < c + e and \dPi\ C B. Due to the Deformation lemma (Lemma 2.3.1); there exists n G C(X,X) such that rj ~ id and• rj(fc+£) C / c . Let r{ = rj(pi), then Tj G Zn(fc,B) C -Zn(/c, fc\{p}), the relative cycle groups, where p G K C a . = Kc = {p}, for i = 1,2,. . . ,ra. If {TJ}-^! are equivalent, i.e. E-^ 1 n i r i = 0 for some integers ni, i = 1,2,... ,m. We may find a G Cn(fc \ {p}) and x G C n+i ( / c ) such that E ^ L ^ T i = a + 5 n + i a ; , where 9 n + i : C n + i ( X ) . —>• Cn(X) is the boundary operator and Cn(X) is the n-chain group on X. Apply operator dn to the above equality, we get: c J n E ^ n j T j = dna+dndn+ix — dna. Thus <9„a G C n_i ( 5 ) , so E ^ n i T j ~ 0 in Zn[X, B) and this further yields that Ti^L-^riiai = 0 in Hn(X,B), a contradiction. Thus we proved that d i m C „ ( / , p ) > ra. Chapter 2. General Variational Principles and the Critical Groups 16 So for any critical value, the corresponding critical group has dimension no less than the number of non-equivalent homologous classes which generate the same critical value. In total, Mn(X,B) > dim Hn(X,B). Chapter 3 Sign Changing Solutions for P D E s Involving the p-Laplacian and the Crit ical Sobolev Exponent 3.1 Introduction Let p* = be the critical Sobolev exponent and A 6 R, Consider the problem —A„u = X\u\q~2u + \u\p*~2u in 0 (Px) u\9n = 0. •' The weak solutions of this problem are. the critical points of the following functional Ex(u) •='•-/'\Vu\pdx - - I \u\qdx - — [ \ufdx. p Jn q Ja p* Jn We will use the dual methods developed by Ghoussoub [28] to study this problem. We first reformulate the first solution in a form that can be used in the study of the second solution. The main results of this chapter are the following: Theorem 3 . 1 . 1 Assume p > 2 and m&x{p,'p* — ~-^} < q < p*, A > 0. Then problem (P\) admits a non-trivial sign changing solution u = u(x) satisfying .[ (\u\p*-p + \\u\q-p)v(u)p-1u = 0, Jn where v(u) is the eigenfunction of the weighted eigenvalue problem . -Apv = fi(\uf-p+ \u\q-p)\v\p~2v onO, v = 0 on dQ. . The next result deals with the case q = p. 17 Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 18 Theorem 3.1.2 Assume p>2,q = p, p3 — p2 + p < n and 0 < A < A i ; where X1 = M{f \Vw\pdx; weH*'p{n), [ \w]"= 1} Jn Jn is the first eigenvalue of — A p ; corresponding to the following eigenvalue problem for the p-Laplacian: ' -div(\VU\P-2VU) = X\U\P-2U xen < " • u = 0 x 6 30,. . Then problem (P\) admits a non-trivial sign changing solution u = u(x) satisfying / (luf-f + X^iuY^u^O,' Jn where v(u) is the eigenfunction of the weighted eigenvalue problem -Apv = p{\uf~p + A)\v\p~2v on 0, v = 0 on dn. 3.2 Some Elementary Properties Let's first recall the result of Brezis-Lieb. Proposition 3.2.1 ([11]) Suppose fn —> f a.e. and \\fn\\p < C < oo for all n and for someO:<p<oo. Then i ^ { \ \ f n \ \ P p - \ \ f n - f \ \ P p } = \\f\\Pr Another result we shall need is the following Proposition 3.2.2 ([20] p.Ill or [39] p.290-293) Let {«„} be a bounded sequence in HQ'p(Q,) such that E\(un) —)• c and E'x(un) —> 0. Assume also that un —> u weakly in HQ,p(£1). Then there exists a subsequence, denoted again by {un}, such that Vun —> V n a.e. Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 19 and ,. \Vun\p-2Vun \Vu\p-2Vu weakly in [ L ^ ( Q ) ] n . With this proposition, we have Proposition 3.2.3 Assume {um}m is a sequence that satisfies the condition in Propo-sition 3.2.2, then we have the following (i) | | v « m | g : = , | | v ( u m - u ) j | f + - | | v « | | j + o(i), '.'(») I M l f = I K ~ U\\P; + \\u\\Pp: + 0(1), (3) fn(um\um\p*-2 - u\u\p*~2){um - u)dx = fn \um - u\p*dx + o(l), • (4) UVum\Vum\p-2 - Vu\Vu\p-2)(Vum -Vu)dx = / n \ V u m - Vu\pdx + o(l). Proof of Proposition: For (1), we may just apply the last two propositions. For (2), apply Proposition 3.2.1 and Proposition 1.2.1 For (3), apply Proposition 1.2.1 and (2), (um\um\p*~2 - u\u\p*~2)(um-u)dx = / (\um\p* - um\um\p*~2u)dx + o(l) Jn = f (\um\p* -\uf)dx + o(;l) Jii = / \um — u\p*dx+, o(l), Jn where o(l) —»• 0 as m —>• oo. For (4), we need Proposition 1.2.1 and (1),. / (Vum\Vum\p-2 - Wu\Vu\p-2)(Vum - Vu)dx Jn = f [\Vum\p + \Vu\p + (Apu)um + [Apum)u)dx = f (\Vum\p-\Wu\p)dx + o(l) Jn •• = I \Vum-Vu\p + o(l). Jn Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 20 This finishes the proof of this proposition. -3.3 The First Solution in the Sub-critical Case Let ps = p* — 5, where 0 < 6 < p* — q, and consider the functional E{(u) = - [ \Vu\p-- j \ u \ q - - [ \u\Ps. p Jn q Jn ps Jn L e m m a 3.3.1 For 2 < p < q and X > 0, Ex verifies (PS)C at any energy level c. Proof: Assume the sequence {um} satisfies Ex(um) —>• c for some constant c and that {E5x)'(um) ^ 0 in Ho^'iQ) strongly. Then {um} is bounded in #0 1 , p(fi). Without loss of generality, we may assume that um —> u weakly in HQ,P(Q) for some u. Now, since Ps < p*, um —> u strongly in LP8. Since ((Ex)'(u),v) = 0 for any v 6 HQ'P(Q), we have o(l) = (um-u,(E{)\um)) = (um-u,(E{y(Um)-(Esxy(u)) = (um,(E{)'(um)) + {u,(E5x)'(u)) + o(l) = / |V(u T O - u)\p - X j \um-u\q - f \um - u\ps + o(l). Jn Jn Jn By Proposition 3.1.1. So um -> u strongly in Hl,p{Sl). Consider now the following Mountain Pass class ?i'= (7 e C°([0,1]; Hl0'p(Q)) : 7(0) = 0; 7(1) ^ 0, and Esx(j(l)) < 0}. with B = {Ex < 0}, we have L e m m a 3.3.2 Let c\ = c(E{, 0F{) = inf A e ^ sup^g^ Ex(x), and set M[ = {ue Hi*(Sl); (u, (E{)'(u)) = 0}. Then Mf is dual to Tf and inf EX(M() = c{ in either one of the following cases: Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 21 (i) 2 < p = q and 0 < A < Ai, where X\ is the first eigenvalue of the p-Laplacian —Ap, (ii) 2 < p < q,< p* and A > 0. Proof: For the duality of Mf to Tf, we referthe reader to the next section as the proof is the same as that for the critical problem. Now we prove the minimizing equality. For any 0) 6 Mf, consider the straight line ^(t) = tu, we have . - . • - E{(tu) = - [ \Vu\p -—(• \u\q - — / \uf. p Jn q Jn ps Jn Since l im t _ > 0 0 -£'*(tu) = —oo, C\ < s u p 0 < t < o o E5x(tu). from ^ ^ = # - 1 ( I V I ' - A ^ - 1 f \u\i-t™-1 f \u\" at: Jn Jn Jn and — ^ - ^ = 0 we get / \Vu\p = tq-p • A / \u\q + tps~p [ \u\ps. Jn Jn Jn Thus, t = 1, and c\ = i n f u g M « E5x(u). Now, we can use Ghoussoub's min-max principle ( Theorem 2.1.1) to get a critical point u{ which is located in the dual set Mf, and it is the minimizer of the functional Ex on this set. . • 3.4 The First Solution of the Crit ical Problem In this section, we consider the critical problem (PA)- The case p = 2 was established by Brezis-Nirenberg [12]. The first solution can be derived by the constrained minimization principle. A version of this method was first used by Z. Nehari [35] in 1960 for ordinary differential equations. The equivalence of this principle and the Mountain-Pass Lemma was first noticed by W. Y. Ding and W.-M. Ni in 1986 and was reported explicitly in Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 22 [36] and [37]. We shall use Ghoussoub's duality method to study the first solution since the duality property of the this solution will be used in our study of the second solution. Define S to be the best Sobolev constant, i.e., S = in f { | |Vu | | £ : \\u\\p. = l,VueLp}. ' L e m m a 3.4.1 For 2 < p < q < p* and A >.0, the functional E\ satisfies, the Palais-Smale condition at any energy level c less than -SP . Proof: Let {um} in HQP be such that Ex{um) -+.$< -SP and'E'x(um) -4 0. n To show the boundedness of {um} in HQ,P, note first that P n o ( l ) ( l + | | w n » | | f f i . p ) + -S* > pEx{um) - (um,E'x(um)) ^ X(l-^)^\um\qdx + (l-^)fn\umfdx iip<q . (1 - fi)Jn\um\ptdx iip = q . > (1-^) f \um\p'dx>^-MlT-1([\um\qdx)i, p* Jn n Jn Since | | « m | | ^ i , P = pEx(um)-r & JQ \um\qdx + fn \ufdx, so {um} is boundedin Ho'p, and thus we may assume that um —>• u weakly in HQP and {un} satisfies the condition of Proposition 3.2.2 and Proposition 1.2.1. The Rellich-Kondrakov theorem then gives that um —¥ u strongly in Lr(£T) for all r < p*. In particular, for any v G 0^(0.) {v, E'x(um)) = f {\Vum\p~2Vum • Vv - X\um\q-2umv - um\umf-2v)dx which converges as m —> 0 to / (\Vu\p-2Vu • Vv - X\u\q-2uv - u\u\p'~2v)dx = (v,E'x{u)) Jn Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian Hence u £ HQ'P is a weak solution to (P\). Choosing v = u we have x 0 = (u,E'x(u)) = [(\Vu\p - X\u\Q - \uf)d. J Cl and Ex(u) = A( - - -) / \u\qdx + (--—) / \ufdx > 0. ' > q Jn' 1 > i>' Jn1 1 By Proposition 3.1.1, we have • Ex(uin) = L\{u) - EQ{um- u) -r 0(1). •'• . and o(l) = (u m - ii, £ A ( u m ) . = (wm - u, E'x{um) - E'x(u)) = / (\V(um-u)\p- \um-uf)dx + o(l). Jn So .E0(um-u) = (--\) [ \V(um-u)\pdx + o(l) = - f \V(um-u)\pdx + o(l) p p* Jn n Jn While for large m, we have that E0(um-u) = E\{um)- — E\{u) + o(l) < Ex{um) + o(l) <c<-S?. n Thus for such m um - u\\v i, p < nc < S't i f f 1 By Sobolev's inequality we finally get o ( l ) = / \V{um-u)\p- \um-ufdx Jn > f \v(Um-u)\p-s-^([ i vK- t i ) !" )^ " Jn Jn = \\um - u||^i,p(.l - S~^\\um - u\\pHZ£) Chapter.3. Sign Changing Solutions for PDEs Involving the p-Laplacian 24 So um ^ u in HQ'P strongly. To set up the min-max principle, we consider the following Mountain Pass family ^ = {7GC70([0,1]; Hp(Q))i 7(0) = 0;' 7 (1)Y 0, and £ A ( 7 ( 1 ) ) < 0}. whose boundary is B = {Ex < 0} and let ci = c(Ex,Fi) = inf supE x(x). ' Consider now the set Mx = {u e HQ'P(Q), U ^ 0, and (Ex(u),u) = 0}. We have the following L e m m a 3.4.2 Mi is a closed set dual to the family T\ and inf EX(M\) = c\, in either one of the following two cases. -(i) 2 < p = q and 0 < A < Ai, where Ai is the first eigenvalue for the p-Laplacian - A p , (ii) 2 '< p < q <p* and A > 0. Proof: By definition, (E'x(u), u) = Jq(\VU\P - X\u\q - \uf)dx. Note that B n M 1 = 0 . By equality (3.1) and (3.2) above, we have for every u E Mi •Ex(u) = X(- --) f \u\qdx + ( - - •—) f \ufdx >•(- - —) f \u\ptdx > 0. • p q in 1 > p*' V 1 ~ P p* Jn 1 Case (i): To prove theclosedness of M x , use Sobolev's embedding theorem to find a constant c such that / \uf <c(f |Vwj p )£ . Jn Jn From this we have (E'x(u), u) = A [ | V u | p - A / \u\p + (1 - A) / \Vu\p- [ \uf Ai Jn Jn Xi Jn Jn Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 25 > T~ f | V u | p - - ^ - / \Vu\p + ( l - ^ ) [ \Vu\p-c([ | V « | P ) Xi Jn Ai Jn Ai Jn Jn , p (1 A i )ll«llffi .p ( n) c l l u l l ^ ( n ) IP H X l f f n l i , W 1 = l |Mir„ i . B / ^ U - •-: c|l'u||P p 4'pWy Ai " " " " ^ " W ' Choose some (3 > 0 such that if < then 1 — j- — c\\u\\p*^p > 0. This means that 1 "o' we can find some constant (3 > 0 such that for any « G M b we have ||u|| > 0.' So M i is closed. To prove the intersection property, fix 7 E ^ joining 0 to v, where u / . O and E\{v) < 0. Note that since A < Ai, we have that (E'x(j(t)),7(*))•> 0 for.t close to 0 by the proof above about the closedness of M\. On the other hand, since n / 0, we have that (E'x(v), v) < pE\(v) < 0. It follows from the intermediate value theorem that there exists t0 such that 7(to) £ Mi. This proves the duality and consequently C i > mi{Ex(u) : u E Mi}. To prove the reverse inequality, set for each u E HQ'P(Q,), U / 0 Q / H V t i l l S - A l l u l l S S A ( M , S 2 ) = — •UIIP* and consider the straight path joining 0 to u. Since E\(tu) —» —0  as £ —>• 00, we have C i < sup E\ (tu) 0 < i « x > o S u p J ^ ( | | V u | | 5 - A N | J ) - ^ | | u | | j : ) -Sx(u,Q)p. . ' n On the other hand, for u E Mi, we have Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 26 Ex(u). In other words, we have proved that ci = i n f £ A ( M i ) where, SX(Q) = mi{Sx{u,Q), u # 0, u G Ho*(p)}. Case (ii): To prove the closedness of Mx, again use the Sobolev embedding, there exist constant c', c" such that Thus we have (E'x(u), u) > \\u\\p -c'\\\u\\q -c"\\u\\p* = \\u\\p{l - d\\\u\\q-p - c" | |« | | p *- p ) . Choose 7 such that for any < 7, 1 - CiA||w|| 9 _ p - c 2 | H | p * ~ p > 0. This means that for any u G M, \\u\\ > 7. So M is also closed. For the intersection property, consider any 7 G T\ joining 0 and v, since p < q < p*, again by the proof above of the closedness of M , we conclude that {Ex(j(t)), 7(t)) > 0 for t close to 0. On the other hand, since v ^ 0, we have that (E'x(v), v) < pEx(v) ^ 0. Then again, by the intermediate value theorem, we conclude that there exists to such that 7(^0) £ this proves the duality and the inequality Jn' ' ~ Vn / \u\q < c'{[ \Vu\p)i and / Jn u |p* < d •!([ | V u | p ) £ . ci > M{Ex{u), ue Mi}. Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian For the reverse inequality, for 0) G M i , consider the straight path j(t) Ex(tu) = - f \Vu\p-—f \u\q-—[ \uf. p Jn q Jn p* Jn Since limt^oo E\(tu) = — oo, we have that Ci < s u p 0 < t < o o E\(tu). From - 1 / \Vu\p - Xtq~l [ \u\q - tp*-x [ \uf Jn Jn Jn dEx(tu) ^fp_ dt and = 0, we get that / \Vu\p = tq-p-X f \u\q + tp*-p f I d * * : -Jn Jn Jn By A > 0 and p < q < p*, so for t > 1 / \Vu\p > X [•\u\q+ [ \uf Jn Jn Jn and for t < 1 / \Vu\p < X [ \u\q+ f \u\p\ Jn Jn Jn Since u G M i , we should have / \Vu\p = X [ \u\q+ [ \u\p*. Jn .. Jn • Jn Thus, t must be equal to 1, and we conclude that ci = in f„ G Mi E\(u). L e m m a 3.4.3 Assume A > 0, then Ci < ^S? in either one of the cases, (i) If p = q, 0 < A < -Ai and p2 < n, '. ' • (ii) If p < q < p*, and X larger than a certain A 0 > 0, (iii) If max{p,p* - ?/T} < q < p*. This lemma is proved in [25] and [26]. Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 28 Theorem 3.4.1 Under the condition of Lemma 3.4-3, problem (P\) has a least energy solutions u that is strictly positive everywhere onVt. The proof of this theorem follows immediately from Lemma 3.4.1, 3.4.2, 3.4.3 and Ghoussoub's min-max principle ( Theorem 2.1.1). By regularity result, we may assume the solutions belong to Cl'a(Q). 3.5 The Second Solution in the Sub-critical Case In this section, we seek another solution for the p-Laplacian in the sub-critical case. Let Sp = {u e Ho'p(Q); \\u\\ = p} and U = {h : #o'p(Q) -4 Hv'p(Q) odd homeomorphism}. Let 7z2 denote the Krasnoselskii genus, and consider the class F2 ~ {A; A compact symmetric with jZ2(h{A) (1 Sp). > 2, V/ i€ FL}. From chapter 2, this is a Z2-homotopy stable class. L e m m a 3.5.1 For 2<p<q<r<p*, A > 0 and u € Lr(Q), u / 0 ; there exists a unique v = v(u) E HQ'P(Q) such that (a) / n ( H r - p + A | « r p K = l , t ; > 0 ; (b) \\Vv\\p = m{{\\Vcu\\p : /„( |u | -" + A|u|«-')M' = 1}: Furthermore, the map Lr(tt) —> i fo ' p (f i ) , u —> v(u) is continuous. Remark: Clearly (px(u),v(u)) corresponds to the first eigenpair of the (weighted) eigen-value problem . -Apv = p(\u\r-p + X\u\Q-p)\v\p-2v in ft v = 0 on <9ft. Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 29 Proof: Let ip(u) — ||«||p = Jn |Vu\pdx, it is clear that xj) is weakly lower semi-continuous, coercive. Moreover, the constraint set C = {LO G H : f {\u\r~p + \\u\q-p)\u\pdx = 1} is weakly closed in HQ'P(Q) and is bounded below on C. Therefore, by the direct methods of calculus of variations (Struwe [45] p. 4), the infimum in (b) is achieved and this infimum is the first eigenvalue of (EG\) and thus is simple. Any function where such infimum is achieved is the eigenfunction corresponding to the first eigenvalue of (EG\). By Theorem 1.2.3, it cannot changes sign in O. This gives the uniqueness of v{u) and therefore its continuity for non-zero u. In order to prove the duality, we need the following elementary L e m m a 3.5.2 For s > p and x, y G R, we have -s(\x\s-\y\s)>^(\x\p-\y\p)\y\s-p: (3.3) Proof: Inequality (3.3) is equivalent to V - i ) > - ( * p - i ) ; v t>o , s p which is trivially satisfied. Let c52 = infAejr2 supA Esx and . Mi = M f n { « S HQ'P(Q,); [ (\u\Ps~p + Xlu^'^viuY^u = 0}. Ja. L e m m a 3.5.3 M | is dual to the class T2 and inf £?*(M*) = c2 Proof: For u ^ 0, let t's(u) > 0 be the unique value such that t§(u)u G Mf. Clearly, t5(u) = t5(\u\)= ts(-u) and E5x(ts(u)u) = maxE\(tu). Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 30 The uniqueness of tg(u) and its properties give that the map u —>• ts(u) is continuous for every u ^ 0 in Hl,p{Q). Consequently, the map u —>• U{u)u defines an odd homeomor-phism between Sp and M f which gives t h a t ' 7 £ 2 ( A n Mf) > 2, \/A £.T2, by the property of the genus. On the other hand, the map h : A n —>• R given by • h(u) = [ (\u\ps-p + Xlul^nviuY^udx Jn defines an odd and continuous map. So we get that 0 G h(A D Mf) which means that A n M | ^ 0 and M | is dual to In particular, cs2 > i n f u g M i E{(u). To prove the reverse inequality, take u G M | and let v(u) be such that •• / (\u\p'-p + \\u\q-p)v(u)p-1udx = 0. Let w(u) be a minimizer for the problem: p2 = inf{^(w); w G Hl'p, f {u\Ps-p + X\u\q-p)v(u)p-lco = 0, Jn f (\u\p*-p + X\u\q-p)\uj\p = 1}. Since u G Mf , we obtain M2 < - 1 / n ( | i*h + A|u|9) Define A = span{i;(w),u>(u)} G T2- Then clearly, l|Vw||5 • l > . M 2 > T 7 T - i " x • \i , , Vo;G A , o ; ^ 0 . For o>0 G A satisfying Esx(LU0) — sup^-E^ > c2, we have cu0 ^ 0 and w0 G Mf. From the above inequality, we derive . ]n(\u\ps-p + X\u\q-p)\<J0\p > ||Va;o||p. Which implies - / {\u\ps~p + X\u\q-p)(\co0\p - \u\p)> iLY^S _ .JIY^'. Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 31 Applying Lemma 3.5.2 with s =pg;q and t = p respectively, we conclude V w o l l S ' \\\7u\\p i [ for + ±[ -fo0r - - / w » - - / > . Ps Jn q Jn ps Jn q Jn V V that is, • • . E[(u) > E[{LO0) >c&2. This finishes the proof of the lemma. ' Applying Ghoussoub's min-max principle ( Theorem 2.2.1), we get the following Theorem 3.5.1 Assume 2 < p < q < ps and X > 0, then the sub-critical problem -Apu = \u\ps~2u + X\u\q~2u on Q u = 0 on dQ, admits a non-trivial solution u\ which is located in the set M2 and is therefore sign changing. " 3.6 T h e Second Solution for the Case of Crit ical Non-linearities In order to get the second solution, we will follow Tarantello's methods, which consist of finding a limit for the second solution us2 as 8 —> 0. The location of u2 on the dual sets M2 will be crucial for the compactness. L e m m a 3.6.1 c\ —> c± as 6 —> 0. Proof: It is not difficult to show that for 8 > 0 and.suitable constants C i > 0 and C 2 > 0, one has ' . • Cl<\\u{\\ps<C2 ' . ; ;;. . . Similar estimates also hold for | |Vuf| | p and Notice that for every a / 0, there exist unique tg(u) > 0 and t(u) > 0 such that /.(//)// € A/i and ts(u)u G .1/f. -Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 32 Furthermore, ts(u) ->• t(u) as 5 -> 0. Set s5 = t(u{), so ssu{ G M x . Since s^ -uf G M i , we have ci < J5A (S*«I ) = 1 / | V a , u ; r - - / \ssu{\«-\ [ \s5u{f p Jn q Jn p* Jn = ( - - - * ) [ | V S ^ r + A ( - i - h [\sSui\o. (3.4) p p* Jn p* q Jn Since u{.e Mf, we have A V 1 7 p Jn1 q Jn ps Jn = ( i - - ) / |Vu?|" + A ( i - - ) /" |U{|*. (3.5) Thus . C i < B A ( S 5 M ? ) . . = E . M ) + (i _!)«_,) / n| V u«r +(i-i) / n « Note that —> 1 as 5 —> 0, we have ci<4 + o{l). To obtain the reverse inequality, set ts = ts(ui) > 0. Thus, t^Mi G Mf, —>• 1 as <5 —> 0 and . . 4 < Ei(tsu1) = Ex(tsul) + -\\tsul\\pp--\\tsu1\\ll P p Ps = Ex(u1) + o(l) = cl + o{l). Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 33 This completes the proof of the lemma. Assume, without loss of generality, 0 6 0, and let Ue(x) = (e + \x\p^) V 1 Ue(x) is a function i n - i f ^ R " ) where the best constant in the Sobolev inequality is attained. It is well known that they are, modulo the translation and dilations, the unique positive ones, where the best constant is achieved. Let 0 < (f>(x) < 1 be a function in C£°(fl) defined as <j>{x) = 1 if \x\ < R .0 i f k l > 2 i 2 where B2R(0) C 0. We define ue(x) = (f>(x)Ue(x). For e —>• 0, the behavior of u£ has to be the same like Ue, and we can estimate the error we get when we take u£ instead of U£. L e m m a 3.6.2 By taking v£ = ^"jj t , then \\ve\\p* = 1 and in addition we have the following: (1) \\Wv£\\; = S + 0(e^), (2) fn\Vv£\<0(s^), (3) J n | V ^ | 2 < 0 ( e ^ ) , / o r p > 3 , (p-l)(n-p) U) / n l V ^ r ^ O ^ -(p-2)(n-p) . (5) fn\Vv£r2<0(e »2 ) , • . . . . (6) if q > p*(l - J ) , then d e ^ " - * ^ < \\ve\\* < c W ^ " " 1 ^ (V if q = p*{\-\),then C x e l o g e | < \\v£\\\ < C2e^~\loge|, (8) if q < p*{l - i ) , then C ^ ^ ~ < \\v£\\l <G2e~^~[, ' Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian (9) if p < q <p*, then \\v£\\Qq -> 0 (as e -» 0), where C\, Ci > 0 are constants. n — p Proof: Let k(e) = (e • n(^2-)p~l)~p* , then y£(x) = k(e)U£(x) is the extremal function the best Sobolev constant as pointed out in the first chapter, and furthermore, k(e)p\\Vu£(x)\\p = k(e)>'\\Ue(x)\$ = \\ye(x)\\pp: = S ? . " The gradient of ue(x) is given by p — n xcj>(x) Vue(x) = (e+lxlp-1) p V<j>(x) + P ~ l '(e+|a;.|^T) P \ x p-i if bl < R P 1 (e+\x\i£l)p\x\%=* 0 if Id > 2R. Thus we have / \Wu£\p = 0 ( i ) + / ( !L_E) Jn J\X\<R p - 1 (e 4- Ixl?- 1) 7 1^! p-1 •= 0 ( 1 ) + / ( 2 Z £ ) > fc i | x | < K p - 1 (e + |x|p-i) n = o ( D + / ( H ) P J7^ N p — 1 = 0(l) + | | W £ | | p \ x p-1 (e + |x|p-!) r From this we further get l i v e n s = v ue p \UR Q(i) + l lV^| |g w. e l l p * 0 ( l ) + g t A : ( £ ) - P llu ll p. II . e l i p * 0(1) +Spk(e)-p 0(l) + k(e)-PS& 0(k(e)p) + SP~™* = S + 0(e^). Chapter 3: Sign Changing Solutions for PDEs Involving the p-Laplacian 35 (1) is thus proved. For (2), (3), (4) and (5), let con denote the surface area of the n - 1 sphere Sn~l in R n , then / i v « e r = o(i) + f p—E)* Jn J\X\<R -n- 1 dx I I D * ry\ I ' . , , P . a n (p-2)g \x\<R.p-L {£+\X\1^)~P~\X\~P^~ J U v 1 [e + rp-1)pr p-1 The order of r of the integrand is an a(p — 2) (n — l)(p — .1 — a) a + n - 1 : — 2 - . - = - — - . p — \ p 1 p — 1 Since p > 2, for a = 1, p — 2, p — 1, the order is non-negative. For a = 2, since we assume p > 3 in this case, the order of r is still non-negative. So, by their respective assumptions, we have / \VuE\a < 0(1) Jn for a = 1, 2, p — 2, p — 1. From this, the conclusions of (2), (3),- (4) and (5) follows immediately. For (6), (7), (8) and (9), we refer to Azorero-Peral [25] and Peral [41]. In order to prove the next lemma, we need the following: Calculus Lemma For every 1 < q < 3, there exists a constant C ( depending on q ) such that for a, P &JZ we have \a + PI" - \a\* - \p\o - qaP(\ar2 + \Pr2) |< C\a\\P\o-1 if\a\>\P\, C\a\q-l\p\ if\a\ < \P\. For q > 3, there exists a constant C ( depending on q ) such that for a, P € 7£ we have | \a + P\i - \a\* -\P\* - q a P ^ ' 2 + |/3|9"2) |< C(\a\q-2p2 + a2\P\q-2). This lemma is proved in [13] (see Lemma 4). Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 36 L e m m a 3.6.3 Suppose max{p,p* — ^} < q < p* and X > 0, then there exists o > 0 and-5o > 0 such that 4 < 4 - ~S> - o for 0 < 5 < <j„. • (3.7) Proof: We prove this lemma by estimating sup^£ E{ where A£ = span{«i, v£} G T2 and Mi is the first solution of the critical problem. By the regularity result, we may assume that Ui, Vui G L°°. For e > 0 arid <5'> 0 small, applying the calculus lemma, without loss of generality, we assume p > 3, otherwise, we use the other inequality in the above calculus lemma. . E{{a ui + Bve ) = - f \V(aui + Pv£)\p - - f \am + (3v£\q - - f \am + (3v£\p* p Jn q Jn ps Jn • < - [ \V(aui)\p-- f \ a u i \ « - - [ louxl" p Jn q Jn ps Jn + ^ [ \V(0v£)\p--[ \/3v£\"--[ \0v£r-p Jn q Jn ps Jn + Ai[f [VauiWVpv^-1+ \Vaui\p-1\Vpv£\ . Jn + \Vaui\2\Vpv£\pr2+ \Vaui\p-2\VPv£\2} : + Bi[f l o j W i H ^ i ' - 1 + l a u i i ' - 1 ! ^ ! H- l a M x i 2 ! ^ ^ - 2 + lawxi ' - 2 !"^! 2 ] Jn + Ci[f lauiWPv^'1 + l a u i l ^ " 1 ! ^ ! + \aui\2\Pv£\Ps-2 + \aui\Ps-2\Pv£\2] Jn < '- [ \V(aui)\p-- f lauil"-- f \aui\ps p Jn q Jn Ps Jn , . + - / \V(Pv£)\p - - [ \Bve\< - - I \Bv£r p Jn q Jn Ps Jn + A2{\a\p + \f3\p)^ + B2(\a\q + \ p \ q ) e ^ { n - ( S ^ £ l ) + C2{\a\ps + \(3\Pi)e^n-{P*~T~P)\ ; Therefore, for e sufficiently small, lim E{(aui + Bve) = —oo. Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 37 So we may assume that a, (3 are in a bounded set Let us consider the functions tp r , „ - tpt xtq 9(t) = Ex{tv£) = - [\Vv£\p- — 7 \v£\q p Jn p* q Jn p Jn   and fP r +P* g(t) = L \Vve\p--. p Jn p* As usual, limt^.O0.g(t) = —oo, and sup t > 0 g(t) is attained for some t£ > 0, and Therefore So so 0 = g'(t£) = tp-\ f \X7v£\p - I?-* - Xtq~p [ \v£\q). Jn Jn [ \VvE\p = tpJ-p + xtQ-p [ \v£\q > €~p Jn Jn t£<([ | V i ; e | " ) ^ , Jn [ \Vv£\p <tpe'-p + X([ \Vve\p)^{j \v£\q). Jn Jn Jn Choose e small enough, by lemma 3.5.2 6 ~ 2 That is, we have a lower bound for t£, independent of e. Now,we estimate g(te). The function g(t) attains its maximum at t = (Ja \Vv£\p)p*-p and is increasing in the interval [0, (fn \Vv£\p)p*-p], by Lemma 3.5.2 we have A g(t£). = g{te)---n [ h\q q Jn < g((f \Vve\>)J*) - -tq£ f \v£\q Jn q Jn < -Sp+Ce p. ( - V - p / \v£\q. ~~ n qy2' Jn 1 1 Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 38 Now return to, the estimating of E{. Since q > p* — ~ > p* — 1, we have E5x(aul + (3v£) < E{(aul)+Ex(pv£) + ^ - - ^ [ \ve\" p* ps Jn , . ^ E=±(n_{l-1)("-P)) P-lfa ( p * - l ) ( n - p K , . + A 3 e ^ + B3e p { p > + C3e p 1 ' < c{ + - S p + C e ^ + / k ^ - - ^ ) ^ / k + A 3 e ^ + B3e v [ n p ' + C3e p ( n p ) c I n ra-p n - p p-1/ («-l)(»- ) < c? + - 5 ? + A 3 e ^ + B 3 e V ( " P n p* Ps Jn £ Choose s small enough such that , n^ £ 2=£ P-If (9-1)("-P)N ^ p-1/ (p*-l)(n-p)s p - l f a ff(n p) \ p + A 3 e ^ + 5 3 £ p ( n p ;J +• C3e p {n p > - CAe p ( n p ' < -2o for some constant o > 0. Now choose S0 > 0 small enough such that I P T - i /?r Jn \Ps <o. for 0 < 6 < <Jn. The proof of this lemma is now complete. Proof of Theorem 3.1.1: It is not difficult to see that c\ is bounded uniformly in S. Hence I I H P < K-i ^ G (0,5o), For a suitable constant K > 0. For x G ft, define (M^)"1"^) = max{u2(x),0} and (us2)~(x) = max{—u^),0}. Clearly and (u$)-(x) ^ 0 and both belong to #d'P(^)- I n addition, l l V ^ f H < K , V5e(0,5 0 ) . ' (3.8) Thus, we can find 5n —>• 0 as n —»• +ocyu +, w~ G #d'P s u c f l that ( u 2 n ) ± 7^  U ± weakly in H Q ' P a s n - ) +oo. Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 39 We claim that u+ ^ 0 and u ^ 0. To shorten notation, set = (w2n)±,c" = cf",pn = pSn,En = E{n and Tn =.Mf". Since un is the solution of the corresponding sub-critical problem, we have that ETn. In particular, EnUfJ > (3.9) From lemma 3.6.3, we also know that : En(u+) + Enfa) = En(un) = 4" <cl + - S * - a n for n large. Necessarily, En{u±) < -o (3.10) for n large. From (3.8) and the fact that € T n , we derive Ki< r< | ! ; ) . <K2 ' (3.11) with suitable positive constants Ki and K2. Arguing by contradiction, assume, for example, that u+ = 0. From (3.10) and the fact that G T n , we obtain l \ \ ^ < \ ^ - h < \ t < l ^ - a + ^l)' (3.12) P Pn ' t ' and , . l | V < | | p - | | < ! | P : = o(l). - '(3.13) Consequently, s\\<\\$. < \\vui\\i = \\uX:+o(i) < \ \ < \ \ z - p \ K \ \ w ^ + o(i) Since | |«^| | p * is bounded away from zero, we conclude \ K \ \ ; : - p > \ n \ P J ^ s + o(i). Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 40 That is, R U E > s ? + o ( i ) . But by (3.12) and (3.13), we have ±S? + o(l) < i | | u + | | £ = l\\Vu+\\l - 1||«+||E + 0(1) < ifif? - a + o(l). lo ft JJ HTl > Similarly, one shows that u~ ^ 0. Set u = u+ — u~. Clearly u changes sign in 0 and Un '•— usn —1 u weakly in HQP(Q). Now, we prove a subsequence of {un} converges to u strongly in HQ'P(Q) and conclude that u is a solution of the critical problem (P\). Since {E(un)} is. bounded and E'n(un) = 0, by the proof of Noussair-Suianson-Yang in [39] ( p.290-394 ), we may assume that the conclusions of Lemma 3.2.3 hold for the sequence {un}. • Note that u e Mi, hence E(u) > Ci. Set un = u + wn, with wn 0 weakly in HQ'P. We have cl + : - 5 t - o > En(u + wn) To = ijjii v < - hu%i - + V « , B i u - 1||«,B|IE+o(i) P Pn y P Fn > C l + ' I | | V u ; J 5 - l | | t i ; N | | E + o(l). -.-P Pn Since |c" — ci| = o(l), we derive - | | V t i ; B | | J - - | | u ; B | | E < . - 5 ' . - f f + o(I). : (314) p * Pn n Furthermore, 0 = (KC^),^). = <^ '(^ ),^ > + IIV I^I^  - | |^ | |^+o( l ) ; (3.15) Chapter 3. . Sign Changing Solutions for PDEs Involving the p-Laplacian 41 Now repeat the above argument, we can show from (3.14) and (3.15) that the sequence | |Vw n | | p cannot be bounded from zero. Hence.,{wn} must admit a subsequence which strongly converges to zero. 3.7 The Case q = p In this section, we consider the changing sign solution of the critical problem in the case of q = p. The work we still need to do is to check the duality with a suitable dual set and the energy level of the corresponding functional. For u G Hi,p(0), define again if(u) = ||Vw||^. we have L e m m a 3.7.1 For p < f < p*, 0 < A < Ai and u G Lr(Q), u ^ 0, there exists a unique v = v{u) e H^'(Q) such that (a) Jn(\u\r-p + X)vp = 1 and v > 0, (b) il>{u) = i r i f ^ H , w e H1^, fn(\u\r-p + A X = 1} = p(u). Furthermore j the map u —>• v(u) is continuous for u 7= 0. Remark: Clearly, (p(u),v(u)) corresponds to the first eigenpair of the weighted eigen-value problem ' . • -APv = fi(\u\r-p + X)\v\p'2v on Q, v = 0 on Oil. The proof of this lemma is the same as that of Lemma 3.4.1. L e m m a 3.7.2. Let M | = Mf n {u e-H^iQ); / n ( |w| w ~ p + A)u(u) p _ 1u = 0} and T2 he the homotopy stable class defined in Section 3.4, then M | is dual to T2 and mfE<X(M}) = 4= inf s u p ^ . Chapter 3. . Sign Changing Solutions for PDEs Involving the p-Laplacian 42 The proof of this lemma is the same as that of Lemma 3.4.3. L e m m a 3.7.3 In the case of q = p, assume ps — p2 + p < n, then there exists o > 0 and S0 > 0 such that 4<C{ + ^ S P - a for0<5 < 50. • Proof: We proceed in the same way as in the proof of Lemma 3.6.3. l / • • , „ , \ A r . ~ 1 E{ ( a U l +/3vE) = - [ | V ( a « ! + (3vE)\p - - f \aUl + [3vE\p - - [ \ a U l + Bv, p Jo. • p Jn p6 Jn < - f | V . ( a « i ) | p ~ - / | a « i | p - — / \aUl\Ps p Jn p Jn Ps.Jn - I \V(PvE)\p-- j \(3vE\p-- f \f3v, p Jn p Jn ps Jn Since and and Ps+ A2(\a\p + \P\p)e^ + B2(\a\p + | / ? | p ) e ^ < n - l E z i ^ > V— 1 / ("P* — 1) f Tl — p) \ + C2{\a\Ps + \ p i n e e ^ i n — r l ) . p - 1 ^ _ (P* - I ) ( " - P K _ (P P P PZ p - 1 _ (p-l)(n-p) = Pz±fa+ n - P ) p p p • n 9{P) < -S* + C e ^ - - ( f ) J = i j \vE\p, n p 2 Jn P where g((3) = E\((3vE) is defined in section 3.6. Thus E{{aux + 0Ve) < E{(aUl) + Ex(PvE) + - \ ^ L \vE\p* - l c _ i n-p\ ( P - 1 ) ( " - P ) + A 3 e ^ + J B 3 e E F i ( p + ^ ; + C3e~ p n p* ps Jn q 2 Jn Q el " - P P - 1 / i n-p\ ( P - 1 ) ( " ~ P ) + Aze^f + B3e * { p + - } + C 3 £ ? Chapter 3. Sign Changing Solutions for PDEs Involving the p-Laplacian 43 < c{ + - S P + C e V + A3e^ + B ^ ^ ^ n p* ps Jn Since p3 —p2+p < n and therefore It^E < p—1, we can choose e small enough such that n-p n-p p-1 /„ , n-y, (p-l)(n-p) C7e .^ + p 2 + £ 3 £ * ( p + » ' + C3e > . - W 1 < -2o for some constant o > 0. Now choose <50 > 0 small enough such that \/3\p* \/3\Ps f • ^ - - ^ - / b £ | P i < a for 0 < 5 < 80 p* Ps Jn The proof is thus complete. Remark: With this lemma, the proof of Theorem 3.1.2 follows in the same way as that of theorem 3.1.1. Substituting p = 2 into p3 — p2 +p < n, we get that n > 6. This means that the results here extend the case p = 2 on nodal solutions for the Laplacian only for n > 6. Because the uniform ellipticity of the p-Laplacian is lost at zeros of | V « | , the best regularity result we could have and we do have is that u 6 Cl'a(Cl) ( see Struwe [45] p. 5 or Tqlksdorf [49] p. 128 ). Since in the proof for p = 2 and n = 6, the boundedness of | A u | is needed, this can not be true in our proof, we don't recover the case when p = 2 and n — 6. Chapter 4 Multiple Solutions for PDEs Involving the p-Laplacian and Generic Linear Non-homogeneities 4.1 Introduction In this chapter, we are going to study the solutions for the sub-critical p-Laplacian: • f —Avu=\u\q~2u+f(x) in Q, < M (Pf) u = 0 on <9Q v where 2 < p < q < p*. The solutions of this problem will be the critical points of the functional 1(a) - . . ^ J^\Wu\pdx ~-qjn W\qdx - fudx for u € Hl'p{Q). For any v G #d'p(0) (I'(u),v)= [ \Vu\p-2Vu • Vv - [ \u\q-2uv - [ fv. Jn Jn Jn For any v, w £ H0l'p(Q), (I"(u)v, w) = (p-l) [ \Vu\p~2Vv -Vw-(q- 1) / \u\q-2vw. Jn Jn For p — 2, Bahri [3] proved that for any q up to 2* and / in a dense G$ subset of H~ l, problem (Pf) has infinitely many solutions. In this chapter, we consider the corresponding problem for the p-Laplacian. The main result is the following Theorem 4.1.1 For each n G N, there exists an open dense subset On of H~1,p'' (Q) such that for f G On, problem (Pf) has n solutions. In particular, for any f in the dense G$ set G = n „ e A r O n , (Pf) has infinitely many distinct solutions. 44 Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 45 We associate to the functional / defined on i?d'p(ft) a functional J defined on the unit sphere S of #01,p(ft): S = {UeH}*(n); ||u||^i,p ( n ) = 1} by the formula Set J(v) = sup /(Au);-. v e S.' • A£[0,oo) (4.1) (4.2) S0 = {v G S, J(v) > 0} then we immediately have the following proposition. Proposition 4.1.1 (i) J G 0 ° ( 5 , R) ; J G C 2 (5o,R). (ii) On So, one has: I'(X(v)v) = A(v) _ 1J(u); where X(v) is the unique positive solution of J(v) = I(Xv).. That is, for any u e So and cj) G TUS, the tangent space of S at u, one has: (I'(X(u)u),u) = 0 and {I'(X(u)u), (j)) — X(u)~l(J1 (u), 4>) (4.3) 4.2 Properties of the Energy Functional For u G S, we have that Xp Xq r r /(Au) = — - — / \u\q - A fu • " " Jn Jn dX d2 p q /(AM) = AP-X - A9"1 / \u\q - t fu Jn Jn /(AM) = (p-l)Xp-2-(q-l)Xq-2 [ Jn u (4.4) (4.5) (4.6) dX2 ' ' ' ' ' Jn Assume A = X(u) > 0 is the unique solution determined from the following conditions /(AM) > 0 dJ^ = (I'(Xu), «) = 0 d^ = I"(Xu)(u, u)<0 (4.7)' Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 46 and J(u) = I(\(u)u) = sup A e[ 0 o o) /(Aw), for u G 5". L e m m a 4.2.1 (i) For any sequence {un} C S, un 0 iff J(un) —^  co. ^MJ J satisfies the Palais-Smale condition and for any sequence {un} C S fl suc/i £/m£ 0 < C < J(u n) < C' and J'(uN)\EN = 0, one can extract a convergent subsequence, where EN is a n-dimensional subspace of HQ'p(Q) such that U ^ E ^ = Hl'P(Q). Proof: (i) For un —1 0, since for n sufficiently large, J(un) > 0 ( for I(un)> 0), we may assume that J(un) > 0 for all n. From (4.6) we get that: A'"7J«"|'>!FT (4'8» Since un —¥ 0 strongly in L9, Xn —>• co and that . A„ ~ k | T ^ (4.9) Now, 7(A n M n ) = A p ( J - i A « - p /a K l 9 - K~p la K ) ^ oo. Conversely since / ( « „ ) = I(\nun) —>• co, thus A„ —>• co because / is bounded on a bounded set. By (4.9) un —> 0 strongly in Lq. Since the weak topology on S is metrizable (HQ'P(Q) is separable) and / /o ' p (ft) ->• L 9 (0) is compact, so u„ ^ 0 in i/o'P(^)-(ii) We prove the second part, since the proof of the first part is the same. Set vn = X(un)un, then . ' I(vn) = I(\(un)un) = J(un) <C . / > n ) k = 0 Thus {vn} is.bounded (by the usual method to handle this) in i/o'p(fi), w e m a y assume that vn -¥ v0 weakly in HQ'p(Q). And also since {vn} is bounded in HQ'P(Q), we have (v0i I'(vn)) = {v0 - Pnv0, J ' K ) ) -> 0, (4.11) Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 47 where Pn is the partial sum projection to En. For T'(vn) is bounded because {vn} is bounded.. From this we get o(l) = (vn - v0, I'(vn)) = ( v n - v 0 , I'(vn) - I'(v0)) + o{iy > Cp [ \V(vn-v0)\p- f ' \ v n - v 0 \ q + o(l), (4.12) by Theorem 1.2.2. So u„'-> v0 strongly in #01 ,p(fi). v For any a 6 R we set Ja — {vE S, J(v) > a} (4.13) L e m m a 4.2.2 Let J satisfy the condition of Lemma 1, and a > 0, then there exist p(a) > d such that J^a) is contractible. in Ja. Furthermore, this contraction can be chosen so that the subspaces En are invariant under it ( provided En D ^$). Proof: In the proof in [3], replace the projection by the partial sum projection will do. L e m m a 4.2.3 Let v € S then, (i) there exist Ci > 0, C 2 > 0 and C3 > 0 and C € R such that ' C+ Cl{J{v))v < \{v) < C2(J(v))p + C 3 (4.14) (ii) there exist C[ and C'2 >0 such that • \J(v)-J(-v)\<C[(J(v)^+C2.- (4.15) Proof: (i) From (4.5) and (4.6), we get that Xq~p fn \v\q > Thus by applying the Sobolev embedding theorem, we conclude that there exists 5 > 0 such that X(v) > 5 for all v. Furthermore, we can conclude A"~" / \.v\" < 1 +A'-"| [ fv\ .hi Jn v < .•l + <J1.-p| I fv\<C - constant. (4.16) Jn Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 48 Since J(v) = I(Xv) = (J - J)AP + (J - 1)A / „ fv, and |A/./V < A(/' f\«')•:>([ \v\^ Jn Jn Jn < Ai(Xq [ \v\")< Jn < A2X* (4.17) £ 1 where Ax and A2 are constants, then J(v) > AXV — BXi and X(v) < C2J(v)p + C3. For the other half inequality of part (i), since Jnfv and Jn \v\Q are bounded and if X{u) -> 00, then Xfnfv + \Xq Jn \v\q = A p (A x - p . . / n fv + ±Xq~p Jn \v\q) > 0, so we conclude that A fn fv + ±Xq J n \v\q > C for some constant C. So ±AP = J(v) + Xfn fv + \Xq fn \v\q > J(v) + C. From this, the inequality in (i) follows. (ii) For any v G 5, with (i) and (4.16), |A/ fv\<CA(J(v))* +C 5 (4.18) Jn and by observation we have that J(-v) > J(y) + 2X(v) f fv (4.19) . . . Jn by symmetric property, we have \J(—v) — J(v)\ < Ce(J(v))^ + C7. L e m m a 4.2.4 Given any f G H0~1'p (ft), A > 0,e > 0, there exists ue G' i/d'p(ft) such that ||Apu£ + \u£\q~2u£ + /.llff-i.P'(n) < e, |K||Hi,P(n) > A. (4.20) Proof: We may assume / G Z>2(ft), we fix / in this set. Set b > max I(u) ll«llHi,P(n)<2A and 4>{x) — X — C4XP . Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 49 If J has no critical value in [b, +00), one can choose n G N, and c G R large enough such that (1) 0(c) > b, <j)'(x) > 0 if x > c. (2) The restriction Jn of J to En D S has no critical value in [0(c), c]. (3) Jc n Z?n is a contractible and compact ENR ( See Dold [18] p. 81 for the definition of ENII). Let LO. be a C°°-function from S Cl En into [0,1] equal to 1 if Jn(x) G [0(c), c] and 0 if Jn{xo) $L (0(c) — £ 1 , c + £ 1 ) where 6 R is such that J n has no critical value in (0(c) - e ^ c + ex). Consider the unique solution of the following problem dr](t,x) Vn S - = Jj[TVn) ( 4 2 1 ) 77(0,2;) —. X where Vn is the pseudo-gradient associated with Jn such that. I I K I I < 2 | | J ; I I , . ( J ; , K ) > I I J ; I I 2 ! and the norm is taken over the corresponding subspace. As one can easily see, J(rj(c,x)) is an increasing function of t and its derivative Lo(rj(t,x)) equals to 1 if J(rj(t,x)) G [0(c), c] for u G Jcr\En, define Z(u) = r/(sup(0, c — Jn(-u)), -u). • L e m m a 4.2.5 [3] If u G Jn H Z(u) is also in this set. By lemma 4.2.5, Z(u) is a continuous function from J c D En into itself. Z has a fixed point, by the Lefschetz-Hopf fixed point theorem. Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 50 Let u0 G J* n En, Z(uo) = u0, it is easy to see that ./«(-'«()) < c, Jn(u0) = c. The idea now is that r(t) = n(t, —uQ) is a path which leads from — UQ to u0, on the unit sphere (as n(t,x) acts by construction, from this sphere into itself), when t runs in the interval [0, c — «/„(—u0)], then . r c - J n ( « o ) ^ rc-Jn(-u0) 2dt Jn{uo) rim  KM*))" We then have for some £ G [0, c — «/„(—%)] nwo)i i< 2 ( c " J n ( "" o ) ) -Set n n (c) = A(u)u wi th v = r(£) = 77(cf, —tio), by relation of / and J , i i / : K ( c ) ) i i < 2 ( c "^ ( : " o ) ) . 7rA(u) Since Jn(v) > 0(c), we have X(v) > c(<j)(c))p.. Since J n ( « o ) = c, we have c— J n(—^o) < clC". Thus I ' | / ; K ( C ) ) | | <d^K. 0(c) P To complete the proof, we remark that IkWIIff^n) = A( u) < BCP + D . since J(i>) < c. Thus when n —>• 00, un(c) belongs to a compact set in Z7(Q). For any tp G £ n, (I'n(un(c)), (p) = ( A p u „ + \un\q-2un + / , <p). Since {un} is bounded, we may assume that um —> u weakly in ifg' p(f2), and therefore also strongly in Lr(£l) for al l r < p* by Rellich-Kondrakov theorem. . Now note that | | M n | | > 2A, as I[un) = - J ( r (£) ) > 4>(c) > b. We need to estimate Aga in by the weak convergence of {un}, we get ••(un-u,I'(un)) = {un-uj'\un)-l\u)) (4.22). Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 51 '•••••> Cp f | V ( « n - u)\p - f \un - u\qdx + o(l) . . ' = Cp f | V K - w ) | p + o(l) (since q < p* ). (4.23) From this relation, we get Cp\\un - u\\p = (un -u,I'(un)) + 6(1) • < \\un - u\\\\I'(un)\\ + o(l) Ci < | |u n -w| |c ' r + o ( l ) . (J)(C)P Case (1), if 5(c) = infn \\un — u\\ —> 0 as c —> oo, then the proof is done. Case (2), if 5(c) = infn \\un — u\\> 5 for some constant 5, then by (4.24) we have i ( | k - u\\)(Cv\\un - u\\p~l - C ' - ^ T ) . < o(l).. . . <f)(c)r From this we see that 5(c) —> 0 as c —» oo, this case will not occur. This completes the proof of Lemma 4.2.4. Since we are going to use the implicit function theorem,, we have to consider the following linear operator A : #o'P(fi) -» H~ l' p'(Q) v -> div(|VM|p-2V7j), for v G H0 hp(tt). (4.24) The functions in HQ'P(Q) with gradient everywhere non zero are dense, since in any neighborhood of a constant has some non-constant element and the trigonometric poly-nomials are dense in Co°(fi) (e.g. for the rectangle domain). L e m m a 4.2.6 Assume that fl C R n is a bounded domain, and u with Vu ^ 0 a.e. then A is surjective in the following sense, i. e. for any f G H-l'p'(ty, there exists vE # 0 1 , p ( f i ) , such that f (|Vu|p-2V?j • V0 - f4>)dx = 0, V0 G Hl' p(Q). (4.25) Jn Chapter 4., Multiple Solutions for Generic Linear Non-homogeneities 52 Proof: The functional J(„) = lf \Vur2\Wv\2 - f fv, 2 Jn Jn v G HQ'p(Q,) is differential, Hr)'p(£l) is reflexive and (1) J is coercive: Let ||u n | | —¥ oo, then there exists 8 such that . m(\Vvn\ > r l l ^ n l l ) > (2m(0))? We may choose some n 0 such that m(|Vit| < ^) < f. Thus we can conclude that J(vn) -> oo (2) J is weakly lower semi-continuous. Hence we can apply the abstract minimization result to J to find a minimum. Proof of (2): Since for any vn —^ v in HQP(Q,), Jn fvn —>• fn fv, we need only show that I{y) = f \Vu\p-2\Vv\2 Jn is weakly lower semi-continuous. For any a > 0, f3 > 0, a + ft = 1 and v\, v2 G HQ'p(Q) |V(OJTJI + / ? T J 2 ) | 2 = « 2 | V ? J I | 2 + /3 2 |Vu 2 | 2 + 2a/3Vui • V u 2 = a |V7Ji| 2 4- / ? | V T J 2 | 2 + (a2 - a)\VVl\2 + (f32 - / ? ) | V u 2 | 2 + ^ V v i • V u 2 = a | V u i | 2 + (5\Vv2\2 - a(3\V(vi - v2)\2 < o!Vi'i! 2 + ;^;V(.;2:2. So \Vv\2 is a convex function of v. Suppose vn -^v weakly in HQ'p(Q), then Vvn —^  Vw weakly in L2(Q) and I(vn) is bounded. We may assume that I(vn) is convergent. By theorem 3.13 of Rudin[43], for any m0 G JV, there exists a sequence {Pl}i of convex linear combinations i i P L = AZ °LVvm, 0<alm<l, arnl = 1, I >m0. m=mo m=mo Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 53 such that Pl —» Vv strongly in L 2(fi) and pointwise almost everywhere as I —> oo. By convexity, for any mo, any I > m0, and almost every i 6 0, i v « ( x ) r 2 | v p f < £ ^ | v u r 2 | V 7 j m ( x ) i 2 . Integrating over fi, and passing to the limit I —>• oo, from Fatou's Lemma, we obtain: / | V U | P - 2 | V T J | 2 < liminf / | V 7 j | P - 2 | V P ' | 2 Jn ' >oc 7n < sup / | V u | p _ 2 | V 7 j m | 2 . m>mo J n Since m 0 was arbitrary and I(vn) is convergent , this implies that [ \Wu\p~2\Vv\2 < limsup / | V M | P - 2 | V P ( | 2 Jn i-yoo Jn = liminf f \Vu\p-2\Vvm\2, i.e. J(r>) is weakly lower semi-continuous. 4.3 Proof of the M a i n Theorem Proof: Let On be the subset of # - 1 ' p ( f i ) consisting in functions / such that ( P / ) has at least n distinct solutions, u\, • • • un satisfying the following property:' • ' :.\.Ti(f) :Hl0'p(n) - > / f - ^ ' ( f i ) = ft - ) • (p-l)AUih + (q-l)\ui\''-2h ' (4.26) V is an isomorphism for i = 1, • • •, n. We are going to prove that On is an open subset of H~l'p' for any n. For this, we use induction. For n = 1: that 0\ is open is an immediate consequence of its definition, we have to prove that 0\ is dense in H~l'p'. We fix / G H~1,p'. By Lemma 4.2.4; for any prescribed Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 54. e > 0, we can find u£ G H Q ' P such that \\/\vu£ + \uE\q~2u£ +f\\H^pl[Q)<e . . (4.27) Let —fe — A p u e + \up£~2u£. Consider . Ti(f£) :"HQ'p(Q.) —>• / / „ ' ' " ' ( f t ) /i -4 (p-l)AUek -(q- l)|u,j" 2h (4.28) Ti( / £ ) may not be an isomorphism. In that case, one could find ho ^  0 such that: ( p - l ) A U £ / i o + ( g - l ) | ^ r _ 2 / i o = 0 (4.29) By Lemma 4.2.6, we may assume that AUc is invertible. Thus the operator * - ^ ^ ( k r 2 ^ ) ( 4 - 3 ° ) has an eigenvalue Ao = 1. . L e m m a 4.3.1 KUe is a compact operator. Proof of Lemma 4.3.1: Since 7, : HQ'P{Q) -> L"(ft), h -> h (4.31) is compact, and J 2 : L 9 ( f t ) ^ L ^ ( f t ) , ft -4 k l 9 - 2 ^ (4.32) is continuous, so J 2 ° h is compact. For any v G i/d'p(ft), by Sobolev embedding \{\u£\q-2h, v)\ = I / \ue\*-2hv\ Jn < ( / ^ k l ^ (-1-33) Chapter 4. Multiple Solutions for Generic Linear Non-honiogeneities 55 thus \u£\Q~2h e i / _ 1 ' p ' ( f i ) . So K is compact. by the above Lemma, K is a compact operator; hence A 0 cannot be a point of accu-mulation of its eigenvalues. Hence, for any n > 0, there exists A € R such that: | A - 1 | < v, (4.34) {p-l)AUeh+(q-l)\\ue\q-2h = 0 has no solution but h = 0. (4.35) Set v£ = Xu£; where, A satisfies (4.35). and -g£ = Apv£ + \v£\p~2v£. (4.36) By (4.28), g£ is in d as Ti(ge) :h^(p- l)AVch + (q- l)\v£\q-2h being injective is an isomorphism( it is a null index Fredholm operator). Furthermore, when choosing 77 sufficiently small, we can ensure: \\9e - / H t f - i * < £ . " • • . (4.37) as we already have j | / £ — f\\H-iP> < Thus, Oi is an open dense subset of H~l,p'(fi). • Assume now that we have proved that Ono is an open dense subset of H~1,p'. We wish to prove that same statement for Ono+\. It is clear that O n o + i is open. We have to show that it is dense. For this purpose, we are going to prove that O„ 0 +i is dense in Ono. Let / e O n o , and wi(/)> un0{f) be the n 0 solutions of (Pf) (4.38) and (4.27) holds. Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 56 Using the implicit function theorem in each Ui(f), one can find e\ >-0 such that: For any g E H~1'p' satisfying \g - f\ liP/ < ex • 0 problem (1) has n 0 solutions, U\(g) ••• uno(g). (4.39) Furthermore, \ui(g)\Hi,P < max \ui(f)\Hi,P + 1. Fix A > o satisfying A > max|uj(/)|Hi,p + 1. (4.40) i o We apply Lemma 4.2.4 with e < e\, A and / . Then, we get u € Hr),p(Q.) such that: | |u e | |Hx, > A, (4.41) | | A p u £ + \u£\p~2u£ + f\\H-iy <e. (4.42) Let / e = Ap«e+|«£r\. ' (4.43) The argument in the first inductive step remains valid for u£ and K(u£). Hence, if v£ = Xu£; • A satisfying (4.35) (4.44) and -g£ = Apv£ + \\v£\\p-2v£, (4.45) we derive for n sufficiently small, that: • \\ge - f\\H-^ < ei. (4.46) This follows from that fact that \\ge - f\\H-i,P> ->• 0 when 77 ->• 0 and the fact that ll'/s ~~ /lli?-i.p' < €i- Hence, problem (P/)-with #e has already n 0 distinct solutions, u-i(g), • • •, uno(g£), all of them satisfying (4.27) and also • hi(9e)\\ri* < A. (4-47) Chapter 4. Multiple Solutions for Generic Linear Non-homogeneities 57. But, ve is another solution of (Pf) with g£ by the definition of g£. Furthermore, would n be sufficiently small, then we can ensure: • > / l (4.48)' since ||we||ffi.p > A By (4.28): • Tno+i(v£) : HoP(fl) -> II ^(il) h -> (p-l)AVch+(q-l)\v£\p-2h (4.49) is an isomorphism. g£ is therefore in 0 „ o + i . The proof of the theorem is thus complete. Chapter 5 P D E s with Non-homogeneous Boundary Value Conditions 5.1 Introduction In order to improve the results of Ekeland, Ghoussoub and Tehrani [24] on the Bolza prob-lem, Bolle [9] introduced a method that proved to be useful for other non-homogeneous boundary value problems, see Bolle-Ghoussoub-Tehrani [10]. In this chapter, we adapt this method to study the non-homogeneous boundary value problems involving the p-Laplacian. For UQE C2(dQ, R), we will consider the following non-homogeneous boundary value problem { — A B M = \u\Q~2u + fix) in ft, u = u0 on dft Let u = v + UQ, then the above problem is equivalent to r -Ap(v + u0) = \v + uQ\q~2(v + u0) + f(x) in ft, < v = 0 on 9ft V Solutions of this, problem are critical points of the functional I(v) = [ ( - I V T J + Vu0\p --\v + u0\q - fv)dx. Jn p q Bolle's method deals with the functional / as the end-point of a continuous path of functionals {/^ }^ e[o,i] which starts at the symmetric functional Io which corresponds to / = u0 = 0. The main results of this chapter are the following: 58 Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 59 Theorem 5.1.1 Assume u0 E (7 2(£L R ) , 2 < p < q < p* and ^ < n^qq_p) - 1. then for any f E U'(Si) ( i + i = 1), the problem —Avu = \u\q~2u + f (x) in Q, • (Pf) u = 0 on Oil has infinitely many solutions. This theorem is due to Azorero and Peral [25]. In this chapter, we will show how this result can be easily derived by Bolle's method. The next result partially extends the result of Bolle-Ghoussoub-Tehrani [10] to the p-Laplacian case. Theorem 5.1.2 Assume u0 E C 2 ( 0 , R ) , 2 < p < q < g ^ p ( < P*), then for'any f£ C ( Q , R ) , the problem — A„u = \u\q~2u + f(x) • in 0, (Pf,u0) . II. — '»() on dfl has infinitely many solutions. . . . 5.2 Preservation of a Min-max Crit ical Level Along a Path, of Functionals In this section we recall Bolle's method for dealing with problems with broken symmetry and rewrite it in a Banach space setting. Let E be a Banach space and consider a C2 functional / = [0,1] x E -> R. We denote || • || the associate norm on E. For 9 E [0,1] we shall use the abbreviation Ig for 7(0, •). We make the following hypotheses: (HI) I satisfies the Palais-Smale condition, which means here that for every sequence {(9n,Xn)} (with 8n E [0,1], xn E E) such that | | / e n (£n) | | —> 0 as n -> +oo and-I(9n,xn) is bounded, there is a subsequence converging in [0,1] x E. Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 60 (H2) For all b > 0 there is a constant C\(b) such that: \Ie(x)\<b implies \^I(8, x)\ < Ci(6)(||4(:r)|| + i)(||x|| + 1). (H3) There exists two continuous function fi and f2: [0,1] x R —> R with / : < f2, that are Lipschitz-continuous relative to the second variable and such that, for all critical points x of Ig, h(o,ie(x)) < ^i(e,x) < f2(e,ie(x)). (H4) There are two closed subsets of E, A and B C A, such that: (i) T0 has an upper-bound on A and lim| x|_ > + O O ] X 6 y l(sup 6, 6[ 0 ) 1] Ig(x)) = —oo. (ii) CA,B > c B where c B = s u p s / 0 and C A , B = inf 9 e X ) / 1 b sup s ( A ) T 0 where T)A,B = {g £ C(E, E); g\B = MB and g(x) = x for x 6 A and ||a;|| > i?}. Denote by ^(z = 1, 2) the functions defined on [0,1] x R by ipi(0,s) = s { | ^ ( M = £ ( M i ( M ) Note that ipi and ip2 are continuous and that for all 9 € [0,1], ipi(6, •) and if2(6,-) are non-decreasing on R Moreover, since / i < f2, we have ipi < ip2- ' In the sequel, we set /2(s) = sup e e[ 0 1] |/2(0, s)|, i = 1,2. Here is the Banach space analogue of the result of Bolle [9]. • Theorem 5.2.1 Assume that I = [0,1] x E R 'is C2 and satisfies (HI), (H2), (H3) and (H4). If ip2(l,Cs) < ^ ( l ) c A , B ) , then I\ has a critical point at a level c such that IPI(1,CA,B) < c < "02(1, C A , B ) -Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 61 Assume now E = E- (B E+ and let {En}n be an increasing sequence of subspaces of E such that E0 = E_ and E n + i = En © Re„+i . If E-. is finite dimensional, set: Qk = {g G C(E, E); g is odd and g(x) = x for x G Ek and ||a;|| large}, and cfc = migegk s\xpg^Ek^ J 0 . The following result belongs to Bolle, Ghoussoub and Tehrani [10]. Theorem 5.2.2 Assume I satisfies hypothesis (HI), (H2) and (H3). In addition, we suppose (H4)' h is even and for any finite dimensional subspace W of E, we have sup I(6,y)-—> —oo as y € W and \\y\\ —» oo.' 0e[o,i] Then, there is C > 0 such that for every k: Either ~ (1) I\ has a critical level Ck with ip2(l,ck) < ipi(l,Ck+i) < c~k < -^ ( l jSup^ J 0). Or (2) ck+l - c k < C(h(ck+1) + f2{ck) + 1). Now, let's look at the pseudo-gradient vectors associated to a family of functionals. Let E be & real Banach space, for 9 £ [0.1], x e E, let Ig{x) = 1(9, x) be a family of functionals from E to R. Assume Ig(x) G C 1 ^ , 1] x E, R), for 9 e [0,1], u G E, we say v G E is a pseudo-gradient vector for / at (^, u), if the following hold: (0 ••Nl<2||/i(u)||, («) "<>sc~)»«> > In the following, pseudo-gradient will be denoted by p.g. for short. Let E = {(9, v) G [0,1] x E : I'g(v) 7^ 0}, then V : E -> E is called a p.g. vector field on if Vg is locally Lipschitz continuous and Vg(x) is a p.g. vector for I for all (9,x) in E. Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 62 L e m m a 5.2.1 If Ie(v) E C^QO,!] x E,K) for every 9 E [0,1], then there exists a p.g. vector field for I on E. . Proof: For (9,u) E E, we can find a vector tu E E, such that = 1 and (I'e(u),w) > ^\\I'e(u)\\, then z = ^\\I'e(u)\\w is a p.g. -vector for Ig at u with strictly inequality in (*). The continuity of Ig(u) then shows that z is a p.g. vector for all v E Ng>u, an open neighborhood of (9, u). Since {NgtU : (9,u) E E} is an open covering of E, it possesses a locally finite refinement which will be denoted by {Mj}. Let pj(9,x) denote the distance from (9,x) to the complement of M,-, then pj(9,x) is Lipschitz continuous and Pj(9,x) = 0 if (9,x) E,Mj- Set The denominator of ftj is only a finite sum since each (9,x) E [0,1] x E belongs to only finitely many sets Mk, each of the. sets lies in some Ngj>u.. Let Zj = 1 1 1 ^ , ( ^ ) 1 1 ^ , a p.g. vector for / in Mj and set Vg(x) = J2j Zj0j(9,x), since 0 < /3j(9,x) < 1 and J2j Pj(9,x) = 1 for each x E E, Vg(x) is a convex combination of p.g. vectors for Ig at x, so a p.g. vector field. Moreover Vg is locally Lipschitz continuous. The proof of the lemma is complete. With this generalized pseudo-gradient and the same methods used by Bolle [9] and Bolle-Ghoussoub-Tehrani [10], Theorem 5.2.1 and Theorem 5.2.2 follow immediately. 5.3 Mult ipl ic i ty of Solutions for Non-homogeneous Boundary Value Prob-lems Assume / E C(f i ,R) and u0 in C 2 ( f i ,R) . Consider the following problem —Anu = \u\q~2u + f in fi (Pf,uo) [u - M 0 ) Ian = 0, Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 63 where p < q < p*. Reformulate this problem by setting u = v + u0. then it is equivalent to ' • • •"• ' -Ap(v + u0) = \u0 + v\q~2(u0 + v) +• / in ft v = 0 on dfl. Let E-= H^'p(Q)be the Sobolev space with the norm ||u|| = JQ \Vv\pdx. We define on [0,1] x E the functional I by ' I(9\v) = J^\V(v + 9u0\p-^\v + 9uQ\q-9fv)dx. The solution of the problem (PfUo) coincide with the critical points of Ii — 1(1, •) in E. In order to apply Theorem 5.2.2, we must check the conditions there. For (HI), the (PS) condition, the proof is standard. Now we establish (H2). L e m m a 5.3.1 For all b > 0, there is a constant Cx(b) such that: |^/(MI < CiWdlWHI + 1)(IMI + 1) for all (9,v) with \Ie(v)\ < b. Proof: From the definition of the functionals, we have T^I(9,V) = [ \Vv + 9Vu0\p-2(Vv + 9Vu0)Vu0 - \v + 9u0\q-2(v + 9u0)u0 - fv, o9 Jn and (Fe(v),v) = f \Vv + '0Vuo\p- f \Wv + 9Vu0\p-2(Vv + 9Vu0)9Vu0 , Jn Jn - [ \v±9u0\q + I \v + 9u0\q-2(v + 9u0)9u0- [ 9fv Jn Jn Jn Remark: From this expression, we have {Fe(v),v) = r-9-^I(9,v) + 1(9, v) — /n fp, because of the 9 there, we can not get the required estimate from this expression directly. By assumption, I / (-\Vv + 9Vu0\p -~\v + 9u0\q - 9fv)dx\ < b. Jn v a Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 64 Thus / \v + 9u0\q>^-[ \Vv + eVuo\p -9q\ I fv\-qb. Jn p Jn Jn P And furthermore, we have \(I'e(v),v)\ > (--!)[ \Vv + 9Wu0\p-qb-9q\ f fv\-\[ 9fv\ p Jn Jn Jn -C1\\v + Ouo\\'>-1-C2\\v + 0uo\\q-1 > - + ou0\\p - dWv + ^ o i r 1 - C2\\v + euoW"'1 - C3\\v + 9uo\\g -d. Since | / n fv\ < C5\\v + 9uQ\\q + C 6 . Again by \I{9,v)\ < b, we have \\v + 9u0\\Qq < C7\\v + 9u0\\p + C 8 . Combine with the above inequalities, we get \(I'e(v),v)\>C9\\v + 9u0\\p-Cw. On the other hand, |^/(MI < Cn\\v + 9u0\\p-1+C12\\v + 9u0\\q-1+Cl:i\\v + 9Uo\\q + Cu < C15\\v + 9u0\\p + Cl6. So \p(9,v)\<C(\\I'e(v)\\ + l)(\\v\\ + l) for some positive constant C. ' L e m m a 5.3.2 There exists a constant C > 0 such that if v is a critical point for Ig, then .[ (-\Vu\p - \Wu\p-2&2) <C [ (\Vu\p+\u\(> + l)dx, Jdn p on, Jn where u = v + 9UQ. Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 65 Proof: By regularity theorem, we have only that u G C ^ O ) . In order to overcome this difficulty, we may use the uniform approximation as that used by Guedda and VerOn in [31]. Because of this, we assume we have the necessary regularity in the proof of this lemma and the proof of other results in this chapter. Since v is a critical point of Ig, one has -Ap(v + $uo) = \9u0 + v\c>-2(9uo + v) +finti ' v = 0 on <9Q. For x G O, let £(x) — d(x,dCl) be the distance to the boundary. Since. O is of C 2 , there is S > 0 such that £ is in C2 on Q D {£ < 6} and n(x) — V£[x) coincides on dVt with the inner normal. Let ip denote a smooth function R —» [0,1] such that <p = 1 on (—oo, 0] and = 0 on [S, +oo). Set g(x) = cp(£(x)). Multiply now the equation — Apu = \u\Q~2u + f in 0 u = 9UQ on dQ, by g(x)Vu • n(x) and integrate over 0. For the first term, we get: / -Apu • g{x)Vu • n(x) = - f | V u | p - 2 & 2 + [ \Vu\p-2Vu • V f p V u • nix)). Jn Jan on Jn The last term in this equality is equal to / zZ ^u^u^gu^n^^dx • = • / | V u | p ~ 2 ^ u X i u X j ( g n j ) X i Jni<i,j<n J n -. i,j • + iVuf^Y^iUxiXjQnjdx = 0 ( j T \ V u \ » ) + ZZJV\Vur-2(±\uxf)Xjgn3 = o(/jv»r) + / n E ( i | v < M n , Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 66 = o([\Vu\>)+[ £(£lv«lp)-sl Jn Jdn~~p = 0(f \Vu\p) + -f | V d p Jn p Jan V That is , du [ —Apu • g(x)Vu • n(x) = f . (-\Vu\p - | v u | p " 2 & ) + 0(f \Vu\p). Jn Jan v on Jn V Similarly, we have / \u\q-2ug(xWu • n(x)dx = 9q [ ^ - + 0([ \u\gdx), Jn Jan q Jn and f f-g-Vu-n(x)dx = 9 f fu0 + O(([\u\qdx)h. Jn Jan Jn Now, put everything together, we have •/ (-\Vu\p - | V u | p - 2 & 2 ) < C [ (\Vu\p + \u\q + l)dx. Jan p on Jn Lemma 5.3.3 There exists a constant C > 0 such that if v is a critical point of Ig, then • \^ei{e,v)\<c((i{o,v)f + i)^. Proof: Since v is a critical point of Ig. Then -Ap(v + u0) — \v + u0\q~2(v + u0) + f(x) in Q, v = 0 on dQ. Thus . -SLl(9, v) = I \Vv + 6Vu0\p-2(Vv + e\/uo)Vu0 -\v + 6u0\Q-2(v + 6u0)u0 - fv oa Jn = [\Vv.+ uVuo\p-2(Vv + 9Vuo)Vuo+[{{Ap(v + duo)+0f)uo-fv) Jn Jn = f \Vv + eVuo\p-2(Vv + 9Vu0)Vuo ' Jn Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 67 - j \V(v + 9u0)\p'2V(v + 9u0) • V u 0 + / Mv + O u 0 ) r 2 d - ^ ^ u 0 + e[fu0-[^ Jan on Jn Jn Jan on Jn Jn Let u = v + 9uo, we need only show that ,du r rill V>—1 / iv«r2—uo\ <c(((/(^W)) 2 + i ) V Van an . • • Since v = 0 on 90, from elementary calculus, Vu is proportional to the normal vector of the surface 90, we have Vv = | ^ • n on 90. / V 7 i p - — u ° ^ C i / V M . P - I — P - l P Jan on Jan on dn < C2( \Vv\ p-i F T ) V ./an dn = c2(f i v . n - i 2 ) 2 ^ . ./an 9ra i 9U l 9 s £=1 < G Van 9n Since Jao^lVur' - |Vu|^" 2 | |^j 2) .= JdnC-\D9nu0\2 - | V u j ^ 2 | | ^ | 2 ) , where Ddnu0 is the gradient of «o|an, and u0 G C 2 (0) , we get from lemma 5.3.2, / i v M r 2 i^i 2 < c 6 ( / ( i v 7 i i p +i« i 9 ) ^ + i ) . Jan on Jn Since v is a critical point, we have (I'0(v), v) = 0 and thus deduce that / \Vv + 9Vu0\p-~ / |?;+ •. 7n ./n and •/"•iVu + eVuor < C7(I(9,v) + l), [ \v + 9u0\q <C8{I{9,v) + l). Jn Jn The proof of the lemma is thus complete. Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 68 Proof of the Theorem 5.1.2: We show that 1(9, x) satisfies the hypothesis of Theorem 5.2.2. It is clear that IQ is even and that , in view of the above lemmas, it satisfies hypothesis (HI), (H2) and (H3). It is also clear that for every finite dimensional subspaces W of E, sup06[Oil] 1(9, y) —y —oo as \\y\\ —> oo, this implies (HA)'. To simplify the notation, we may assume that fl = (0, 1)", since any bounded domain can be bounded by a rectangle type domain and we can extend the functions on fl to the outside of fl by letting it to be zero in the outside of fl. Let Ek be the A;-dimensional subspace of E = Ho'p(fl), generated by the first k functions of the basis {(sink\ffxi, sinkn7rxn), ki G N , . i = 1, n}. For an increasing sequence of numbers R3, > 0 ( as big as we wish), such that Io < 0 for u G E.j n BCR.: Let Gk = {he C(Ej,H01,p(fl)) : h is odd, such that h(x) = x if x G E3- and > Rk}. Set cfc = inf maxI0(h(u)), •• heGj ueEj Theorem 5.2.2 applied with fi(9,s) = -a(s2 + 1 ) ^ and f2(9,s) = b(s2 + 1 ) ^ \ yields that if the set of critical levels of I\ has an upper bound then: \ck+1-ck\<C(c^ +c^i+l), which implies that ck < C'kP-, on the other hand, it is shown by Azorero and Peral in [25] that there is a positive constant'!/ > 0 such that ck > Lkn^-^ . This is a contradiction as long as n^p^ — 1 > P or q < . The proof of the theorem is complete. Proof of Theorem 5.1.1 Consider the functional 1(9,v) = [ (-\Vv\p - -\v\q - 9fv)dx, • / o n • • n Chapter 5. PDEs with Non-homogeneous Boundary Value Conditions 69 then the solutions of (Pf) are the critical points of 1(1, v). To apply Theorem 5.2.2, we must show the functional 1(9, v) satisfies all of the conditions there. Since problem (P/)'is a sub-critical problem, the (PS) condition is satisfied, (HI) is true. Since dI<$e'v^ = — / n fvdx, and | | / | | is a finite number, so for some constant C and all (9,v). (H2) is thus true. For (H3), we prove the following: Cla im: There exists a constant K > 0, such that if v is a critical point of Ig, then i « i < ^ m » ) + i ) A . Proof of the Cla im: If v is a critical point of /a,, then J^\Vv\p -\v\g -9fv = 0. ' ( * ) ' ' Thus, I(9,v)=(l-I)/n|w|9+ So . IJ(MI ^ ( i - ^ i i H i ^ - a - ^ i i / n i b i i , , -and \\vfrq<Cl(\I(d,v)\ + C2). So • l ^ f ^ l = \-Jnfv\< C3\\v\\q < C3(CJ(9,v) + C 2 )* < K(P(9,v) + 1)^ for some constant K. The proof the Claim is complete. Now, we have (HI), (H2) and (H3). Same as the proof of Theorem 5.1.2, we have (HA)' and we can get a sequence of critical values {ck}k such that i • i |c f c + i - ck\ < C(cqk +cqk+l + 1) which implies that ck < C'k^1, in the usual way, see Struwe [45]. On the other hand, we have ck > Lkn^-p — 1 [25]. This is a contradiction as long as < n(^lp^ — 1- The proof of Theorem 5.1.1 is thus complete. Chapter 6 V i r t u a l Solutions for Non-homogeneous P D E s Involving the p-Laplacian 6.1 Introduction In this chapter we continue our study of the following problem: where 2 < p < q < p*. Unlike the previous chapters, we will study the virtual solutions of this problem. In order to do that, we need to extend the corresponding results of Ekeland and Ghoussoub [23] from the Hilbert space to the Banach space setting. Since on a Banach space, the derivative of a functional is in the dual space, the method used for Hilbert space can not be applied here directly. Since our objective is to study the problems like (P/), we shall restrict our discussion to special Banach spaces. 6.2 Duality Maps and Uniformly Convex Banach Spaces In this section, we recall some properties about general Banach space which will be used in the sequel. Definition 6.2.1 A Banach space X is said to be uniformly convex if for any s > 0 ; there exists 0 < 6 = 6(e) < 1 such that for any x, y G X with \\x\\ = 1, ||y|| = 1 and \\x ~ y\\ > £, w e have Apu — \u\Q 2u + f(x) in fi, (Pf) < u = 0 on dfi x + y 2 | < 1 - 6. 70 Chapter 6. Virtual Solutions for Non-homogeneous PDEs 71 In nonlinear analysis, the dual map plays an important role in non-Hilbertian setting. Definition 6.2.2 Let X be a real normed linear space, X* be its dual. The set-valued map J : X —> 2X* defined by J(x) = {feX*: ( / , x) = \\xf = H/ll2} is called the normalized duality map on X. By the Hahn-Banach theorem, for each x £ X, J(x) is a non-empty subset of X*. That is J is defined on the whole space X. It has the following simple properties. (1) J is odd, i.e. «/(—x) — —J(x); (2) J is positive homogeneous, i.e. for any A > 0, J(Xx) = XJ(x); (3) J is bounded. We need the following properties of the Duality map Proposition 6.2.1 Let X be a Banach space with both X and X* uniformly convex. Then (1) the dual map .J : X -> X" is single valued and is an odd homeomorphism between X and X*.. (2) The inverse, J~l, is the dual map from X* to X** = X. (3) J is uniformly continuous on any bounded set, especially on the unit'sphere. We also need the following properties of a uniformly convex Banach space. Proposition 6.2.2 Let X be a uniformly convex Banach space, then Chapter 6. Virtual Solutions for Non-homogeneous PDEs 72 (1) If {xn}n C X is a sequence in X such that xn x and \\xn\\ —> \\x\\, then xn —» x. (2) For two sequences {xn}n; {yn}n inX such that \\xn\\ = \\yn\\ = 1 and\\xn+yn\\ —>• 2, then \\xn -yn\\ 0. • 6.3 V i r t u a l Crit ical Point Theory In this section, we always assume that X is a uniformly convex Banach space and the dual X* is also uniformly convex. L e m m a 6.3.1 Assume f is a Cl-functional on a uniformly convex Banach space X and let K be a symmetric compact subset of X that is disjoint from a symmetric closed subset B of X and 0 ^ K. Suppose K does not contain any critical point of f. and that there exists e (0 < e < 1), such that ' Jf'C-x) Jf'tx) „ n " 1 + F 7 7 7 7 ^ l l < 2 - £ - (6:1) "||J/'(-s)|| \\Jf'(x) Then there exists n > 0, an equivariant and continuous vector field V : X —>• X such that • . • (1) V{x) = 0 for every x G B; (2) \\V(x)\\ < 1 for every x € X and \\V{x)\\ = 1 on K*. (3) (f'(x), V(x)) < -l\\f'(x)\\ for every x e Proof: We can find rji such that Km nB = 0 and on Km (2) still satisfies, thus we have {m^ywx^j\\}<1-£-, (6-2) Let Wl(x) = - i j ^ j i + ij^ElJii, on then {\\f(x)\y 1 [ ) } - vim: www n / 'Wi r wm-xw < -1 + 1—e = -e. ; (6.3) Chapter 6. Virtual Solutions for Non-homogeneous PDEs 73 By the definition of wi(x), we know that e < ||iyi(x)|| < 2, thus Let ^(x) = n^)jgj | | , then we have' • (f(x), Vl(x)) < ~£\\f'(x)][ for every x E K7* ':. (6.4) Now let v2(x) be any continuous extension of V\ to the whole of X and let v3(x) = |(w2(x) — w2(—x)) be its corresponding equivariant field. Note that v3 = vi on Km and hence is a continuous equivariant extension of V\. We still need to make the vector field uniformly bounded everywhere and zero on B. For that, consider the set N = {x E X\v3(x) = 0}. It is a closed symmetric set which is disjoint from Km, and therefore choose some smaller 77, we may assume that w n IF - 0. Let now ' if x £ N v4(x) 0 otherwise and let J^tNL,, and d i S t ^ B ' dist(x, X \ iV") + dist(x, iV) v y dist(x, X \ B^) + dist(x, 5 ) ' It is clear that h and g are even and continuous functions and the vector field defined as V(x) = h(x)g(x)vi(x) is continuous, equivariant and satisfies assertions (1) and (2) while it coincides with V\(x) on Kv and hence (3) is also satisfied. L e m m a 6.3.2 Let f be a C1+-functional on a uniformly convex Banach space X and let B be a given closed symmetric set. Suppose K\ and K2 are two compact subsets of X that are disjoint from B and such that for some e ( 0< e < 1), \\f'{x)\\> € for every x E KxyjK2. (6.5) Chapter 6. Virtual Solutions for Non-homogeneous PDEs 74 Assume further that Ki'D (,—K2) = 0 while K\ is a symmetric set not containing 0 on which we have • j f ( - x \ Jf'(x) ll\\jf(-x)w+wm11 <2~£for every x € Ki- ^ Then, there exists 5 > 0 and a continuous and equivariant deformation a in C([0, 1] x X; X) such that for some to > 0, the following holds for every t G [0, to), (i) a(t; x) = x for every x G B. (ii) \\a(t, x)\\ < t for every x G X. (iii) f(a(t, x)) - f{x) < -^t for every x G K\ U K52. Proof: We start by showing that there exists n > 0 and an equivariant and continuous vector field V : X —> X such that (a) V(x) — 0 for every x G B. (b) \\V(x)\\ < 1 for every x G X and \\V(x)\\ = 1 on K\ U 4 (c) </'(*), V(x)) < f\\f'(x)\\ for every x G K\ u'#J. To do that, we first apply Lemma 6.3.1 to K\ to find r\\ > 0 and a vector field V\ that satisfies the conclusion of that lemma on Kf^ • In order to deal with K2, first note that since K2 fl (—K2) = 0, we can take -q2 small enough so that •K?n(-K2»)=Q, ... (6.7) and such that for X G K22, we still have \\f'(x)\\>e. (6.8) > Chapter 6. Virtual Solutions for Non-homogeneous PDEs 75 Define on Kf the vector field v2 = -Jf'(x) and note that (5.8) allows us to extend it equivariantly and unambiguously to Kf U {-K22 by letting v2{-x) — ~v2(x) for each x G\ K2 2 . It is clear that < m < ~ m m ( 6 ' 9 ) for every x G K22. In order to glue the two vector fields vx and v2, take S = min^x, n2} and consider an.even partition of unity li, l2 associated to the symmetric sets Kx and K2 U (—K2): h(x) = dist(x, K{ \ (K52 U (-K52))), l2{x) = dist(z, {K&2 U ( - # £ ) ) \ Iff). The vector field ii(x) + Z 2 W Ji(x) + Z2(z) is therefore equivariant and continuous on the ^-neighborhood of K\ U (K2 U (—K2)). Moreover, for any x in K{ U K2, we have </'(*), Wl(x)) < - mm{£-, l}\\f(x)\\. (6.11) Now let w2 be any continuous extension of w\ to the whole of X and let tu3 = \{w2(x) — w2{—x)) be its corresponding equivariant field. Note that w3 = W\ on the set Kf U. [K2 U (—X|)) and hence is a continuous equivariant extension of w\. We still need to make the vector field uniformly bounded everywhere and zero on B. For that, consider again the set N = {x G X; w3 = 0}. It is a closed symmetric set which is disjoint from Kf U (K2 U (—K2)) and we can clearly assume, by choosing a smaller 5, that the latter set is disjoint from N5. Let Now ' F(x) \\w3(x)\\ . y-0 otherwise Chapter 6. Virtual Solutions for Non-homogeneous PDEs 76 and let Q(X) = d i s t ( y ' N) a n d k ( x ) _ d i s t ( * > B) y { } distfo X \ Ns) + dist(x, N) 1 ] dist(x, X \ £ « ) + dist(x, B)' Again, h and g are even and continuous functions and the vector field defined by V{x) — h(x)g(x)F(x) is continuous, equivariant and satisfies assertions (a) and (b) while it co-incides with Wi(x) on Kf U K2 and hence (c) is also satisfied. Since / ' is locally Lipschitz and K = K\ U K2 is compact, we may assume that n0 and n Tj0 < rj < 5) is small enough and there exists finitely many xi, x2, xm G K such that K1* C VJf=lO{xh rj) and for any x, y G 0(^, 277), - /'(t/)| < M\\x - y\\ for some fixed M and 7 7 0 M < y-Now for any x £ X and r; G [0, 1] define a(t, x) = x + tV(x), then for any x G Km and rj < 77 /(<*(*, x)) - /(x) = {f'(x + t6V(X)), tV(x)) for some 0 < 6 < 1. Assume x G O(XJ 0 , 77), then x + £#V(a;) G O(xio, 77) also, thus f(a(t, x))-f(x) = (f'(x + teVix))-f'(x),tV(x)) + (f'(x),tV(x)) < t2M-te\\f'(x)\\ < t(tM-e2) e2 < t(r]0M-e2) < --t. Now, we have the deformation lemmas, the proof of the following theorem is the same as in [23] Theorem 6.3.1 Let f be a C l + -functional on a uniformly convex Banach space X and consider a Z2-homotopy stable family J7 in X with a closed symmetric boundary B. Suppose that • sup/(J5) < c •:= inf maxf(x) Chapter 6. Virtual Solutions for Non-homogeneous PDEs ' 77 and that /(O) ^ c. Then, there exists a sequence {xn}n in X such that lim n/(a; n) — c and which also satisfies either (a) l im n / '(:r n ) =0, or limn /.(-a:n) = c.'and lin^ l l ] ^ ^ , , - U^ ES^ II .^-0-Definition 6.3.1 Say that a C1-functional f on a Banach space X satisfies the sym-metrized Palais-Smale Condition at level c ((sP-S)c) if f satisfies (P-S)c and if a sequence {xn}n in X is relatively compact in X whenever it satisfies the following conditions: \imf(xn) = limf(-xn) = c and lim II f ' ^ X n ) f'(~xn) II = Q " "-ll/'WII ll/'(-x„)| |". . . With this definition and the above theorem, we have the, following Theorem 6.3.2 Let f be a C 1 + -functional on Banach space X and consider a Z2-homotopy stable family T in X with a closed symmetric boundary B. Suppose that sup f (B) < c := inf max f(x) and that /(0) ^ c and f satisfies (sP-S)c. Then there exists x G X with f(x) = c such that one of the following holds: (a) f'(x) = 0,or (b) f(x) = f(—x) = c and f'(x) = Xf'(-x) for some A > 0. As in Ekeland and Ghoussoub [23], we can also prove ' Theorem 6.3.3 Let f be a C1+-functional satisfies (sP-S)c on Banach space X — Y®Z with dim(Y) < oo. Assume /(0) = 0 as well as the following conditions: Chapter 6. Virtual Solutions for Non-homogeneous PDEs 78 (1) There is p > 0 and a > 0 such that inf f(Sp(Z)) > a. • (2) There exists an increasing sequence {En}n of finite dimensional subspaces of X, all containing Y such thai \imn dim(En) — oo and for each n, sup f(SRn(En)) < 0 for some Rn > p. Then f has an unbounded sequence of virtual critical values. 6.4 V i r t u a l Solutions for Non-homogeneous p-Laplacian Equations Now, we are in a position to study the virtual solutions of problem —Avu = \u\q~2u + fix) in fi-, - (Pf) u = 0 on dfi, where / € X* = H'1*'(Q) with X = H^(Q). We consider the sub-critical case, that is when p < q < p*. Weak solutions for this problem are the critical points of the functional Ij(u) = - [ \Vu\pdx - - [ \u\qdx - [ fudx p Jn q Jn Jn defined on X. The difficulty comes from the fact that this functional is not even. If / = 0, then this problem has infinitely many solutions. The same is true if < n(P<Lp) — 1- It is still an open problem whether this remains true for all p and q without those, assumptions. In this section, we consider the following problem — Apu = \u\q~2u + \xf in fi, . u = 0 on 9fi (Pfj.f) — 1 < A* < 1 and fn fudx — 0. The solution of this problem is called the virtual solution of the problem (Pf). Our main result is the following Chapter 6. Virtual Solutions for Non-homogeneous PDEs 79 Theorem 6.4.1 Assume that 2 < p < q < p*, then for every f G X*, either (Pj) or (P^f) has an unbounded sequence of solutions. Proof: We shall show that / / satisfies the hypothesis of Theorem 6.2.3. To simplify the notation, we assume that 0 = (0, l ) n . Let EK be the ^-dimensional subspace of X = HQ p(0), generated by the first k functions of the basis {(sinkiirxi, • • •,. sinkn7rxn), ki G N , i—1, n} Let ZK denote the complement of EK in X, that is, generated by the base vectors not in EK. For any u G EK_X, the topological complement of E k - i , . \\u\\p < C\\Vu\\p/kn ('[41] and [26] ). ' (6.12) C l a i m 1: For k sufficiently large, there exist p > 0 such that If(u) > 1 for all u G with \\u\\Hi,P(Q) = p. The following proof of this claim is adapted from ( [26] ). Let u e'dBp n E ^ . T , then / / ( « ) > ~ [ | V « | P - - [ \u\" - dIHIp • (6.13) p Jn q Jn where d = Ci(||/||p')- By using the Gagliardo-Nirenberg inequality ( [51] ). (/ < C2{[ \Vu\p)r([ \u\p)^ (6.14) Jn Jn Jn with a.= |(1 - J), we get If(u) > - [ \Vu\p-C2(f \Vu\p)?([ l u H 2 ^ - d ( / |uH'. (6.15). • p Jn Jn Jn Jn Combine with (6.12), we obtain '/(«> a V - « J ^ ) P < -- . ^ r ^ ^ - ^ ( 6 ' 1 6 ) Chapter 6. Virtual Solutions for Non-homogeneous PDEs 80 Now, choose j .g( l -a) /n , . , ^ 2pC 4 J ' therefore V If(u) > ±-(f-- %> > C 5 A ; ^ r a > l (6.17) 2p A: n for ft large enough. This completes the proof of Claim 1. Let Y = EK with the k choosing in Claim 1, we now show the following C l a i m 2: On finite dimensional subspace EK C HI'P(VL), there exist positive .constants C i , C 2 , C3 ( depending on EK ) such that •. ; sup If(u)<C1Rp-C2Rg + C3R. (6.18) uedBR(Ek) Indeed, for any fixed u E-HQ'p(Q) and any R > 0, we have : (^izt*) < ^ | | „ | | ^ -".--^iiiiiii + c? !^*!!^ . •" (6.19) Since £^ is a finite dimensional space, it is closed and the two norms || • | | 9 and || • \\Hhp are equivalent. This implies the claim. C l a i m 3: If satisfies (sP-S)c-It is well known that / / satisfies (P-S)c for any c. Now, Assume un E X such that / ; ( u n ) - > c a n d / / ( - u n ) - > c (6.20) and ,, I'f(un), I'f(^Un) „ ' ' „ 11 n> ^ —[S—"^n^o (6.21) "| | /}(u») |[ \\rs{-un) we need to prove that {un}n is relatively compact. By (6.20) (6.21) we have that [ fundx -)• 0 (6.22) Jn Chapter 6. Virtual Solutions for Non-homogeneous PDEs 81 and that for any v E X, | ( P } K ) i i + i i / x - t i n ) ^ ^ ^ ' . R } " ( i i^wif" wif(-un)^ L fw\dxl - ' (6.23) uniformly for « G l If | | l £ (u n ) | | = | |Jy-(-M n ) | | , the second term on the left hand side of the above inequality is understood to be 0. From this we get: ( p ^ + p ^ ' 1 1 ^ 1 1 - ^ ^ w^I ) l l / l l + 0 (1 )' (6:24) For any v E X, (I'f(un) - I'f(-un), v) = 2(I'0(un), v) and (I'f(un) + I'f{~un), v) = -2 J^fv. (6.25) By the second equality in (6.25), if ||/y-(un)|| —> oo, then ||I'f(—un)\\ —>• oo also. Combine with (6.23), we get that for any v £ X This means that (pr^-jjj + | 7 <( \ )\\)\\I'f(un)\\ 0: That is impossible, so we conclude that ||7y(u„)||, '||i}(—un)\\ and | | / o ( — « 7 i ) | | are all bounded. And combine with (6.22) and (6.23) we deduce that \{I'0(un), un)\ < o{l)\\un\\ •• ' (6.26) Therefore for n large enough, c + l + o ( l ) |k | | >I'0(un)-^(I'0(un), un) = (±-^)|K||p- (6.27) • which clearly implies that {un}n is bounded in X = HQ'P(Q,). Since X is reflexive, we may assume that un,—1 u in X and use the compactness of the Sobolev embedding to deduce that \\un — u\\q —> 0. By (6.22) and (6.23), we have ( / 0 ( O , u) ->0. Thus o ( l ) = (I'0(un)-I'0(u),un-u) '••> Cp\\un - u\\Hi,rm +0(1). • Chapter 6. Virtual Solutions for Non-homogeneous PDEs i.e. un.-+ u in HQ'p(Q). Let p be the cluster point of the sequence 1#K)I -Ur, 1 + > ( « n ) l l \\Tf{ since it is bounded. Then u will satisfy problem —Apu = \u\Q~2u + pf in ft, // = 0 - on <9ft — 1 < < 1 and fn fudx — 0. The theorem is thus proved. Chapter 7 Min-max Principles for Convex Sets and Applications to p-Laplacian 7.1 Introduction In this chapter we use yet another method to continue our study of the following problem: —A„7i = \u\q~2u + f (x) in Q, (Pf) u = 0 on <9Q where again 2 < p < q < p*. . In this chapter, we show that for any / in H~l>p'(n), there exists a constant a< 1. such that the following problem — A M — \u\q~2u + a f (x) in O, (Paf) u — 0 on Oil -has a solution. However, if Jnuf ^ 0 , then u is a solution of (Pf). For that, we first extend the minimax principle to closed convex subsets of Banach spaces. 7.2 A Minimax Principle on Closed Convex Subsets of Banach Spaces Suppose M is a closed, convex subset of a Banach space V, and suppose that E : M —> R possess an extension E G Cl(V; R) to V. For u E M define g(u) = sup (u — v,E'(u)) v€M,\\u-v\\<l as a measure for the slope of E in M. Clearly, if M = V we have g(u) = ||7i"(u)||. And if E G C 1 ( K ) , the function g is continuous in M. 83 Chapter 7. Min-max Principles for Convex Sets and Applications to p-Laplacian 84 L e m m a 7.2.1 Let B and C be two closed and disjoint subsets of M, suppose that C is compact and that: \/u0 G C, g(u0) > 3e, then there is adeformation a in C([0,1] x M , M) such that for some 8 > 0 and to > 0, the following conclusions hold: •• (i) a(t, x) = x for (t, x) G ([0,1] x B) U ({0} x M); (ii) \\a(t, x) - x\\ < t for (t,x) G [0,1] x M; (iii) E(a(t,x)) - E(x) < -2et for (t,x) G [0,i0) x C5, where ' Cs = {xe M,dist(ar,C) < 8}. Proof: First note that since C is.compact, we can find some So > 0 such that for any x;yeCSo; (Ci0 D B n M = 0) and ||x - y\\ < 50, then \\E\x) - E'(y)\\< e. For u0 G C, u0 corresponding to v0 G M, in the sense that {E'(u0),u0 - v0) > 3e; | | M 0 - ^o|| < L then UQ =fi VQ. SO there exists neighborhood UQ of Uo such that vo£U~o; U0 C Cs^f] M- for u G Uo, {E'(u),u-v0) > 3e. 2 and for any v G Uo, \\v — v0\\ < 1. The sets {Uo} cover C, let {Ui}ieJ be a finite sub-covering. Denote by Uj andwj the points corresponding to Ui in the same way as UQ and v0 corresponding to U0. Let Pi(x) = dist(x,M \ U(), i G J, p(x) = dist(rr,C) for x G M. Define Xi(x)= - • 2~LiejPi(x) + p(x) then \i(x) is Lipschitz continuous and \i(x) = 0 for x £ U and Aj(x) > 0 for x G C/j. Take 0 < 5 < dist(C, X \ UieJUi), then C 7 n ( M \ U < € J U I ) = 0 ; C 7 n B c C ^ n / 3 = 0. Chapter 7. Min-max Principles for Convex Sets and Applications to p-Laplacian 85 Let / : M —> [0,1] be a continuous function such that f 1 i f x G G l(x) = '<, - " • 0 itxe (M\UiejUi)uB. Define the mapping «(- , •) as: For (t,x) G [0,1] x M ; a(t,x) = x + tl(x)^2 \i(x)(vi — x) = .r(1 - t • /(x) ^ A,;(.r)) -i- / • l{x) Y, Az(-r) • a (7.1) It is immediately seen that: a(t,x) G C([0,1] x M) and a;(rj, x) = x for (rj, x) G ({0} x M) U ([0,1] x fi). and (ii) is satisfied. ' For x £ Cs and 0 < t < ^ , set a(t,x) = x by mean-value theorem, •* E(a(t,x)) = E(x+ tw) = E(x)+ t(E'(x),w)+ r(t) with \r(t)\ < t sup | | £ ' (x + sw) - E'(x)\\ < et, since < 1 0<s<t Now, by (1) a(t,x) G M and E(a{t,x)) < E(x)+tJ2\i(x)(E'(x),vi-x) + ei ie-.i ' ' < E(x)-3etY,\i(x)+et < E(x)-2et; i£.J - ' So (iii) is satisfied. With the above deformation, we have the following: . Theorem 7.2.1 Let E be a C1 functional on a B.anach space V and M be a closed convex subset of V and consider a homotopy stable family T of compact .subsets of M with a closed boundary B. Assume c = c(E,T) = inf^^ max x e ^ .E'(x) is finite and let F be a dual set to T which satisfies that inf E(F) > c. Then there exists a sequence {xn}n in M such that Chapter 7. Min-max Principles for Convex Sets and Applications to p-Laplacian 86 (i) lim„ .E(a;n) = c; (ii) \\mng(xn) = 0; (iii) lim„ dist(:rn, F) = 0; The proof of this theorem can be found in [28]. Definition 7.2.1 E satisfies the Palais-Smale condition, on M at level c (abbreviated as (P-S)M,c) if-' For any sequence {un} in M such that \E(un)\ < c uniformly while g{un) 0, is relatively compact. With this definition, it is clear that if E satisfies ( P - S ) M , c , then under the hypothesis of Theorem 7.2.1, there exists x G M n F such that E(x) = c and g(x) = 0. 7.3 Applications of the Minimax Principle to p-Laplacian The solutions of problem (Pf) are the critical points of functional over H0l'p(fl). Let M = {u G HQ,p(SI), fn fu < 0}. and consider the following homotopy stable family T = {jeC ([0,1]; M); 7(0) = 0 , 7 (1) ^ 0 and 7,(7(1)) < 0}. whose boundary is B = {u G M, If(u) < 0} and let c = c(If,T) = inf supIf(u). and F = {ueM, u^0, (Ff(u), u) = 0}. We have the following: Chapter 7. Min-max Principles for Convex Sets and Applications to p-Laplacian 87 Proposition 7.3.1 F is closed for p < q < p*. Proof: Since (Ff(u), u) = Jn \S7u\p — fn \u\Q — Jnfu, by Sobolev embedding, we have constant c such that < / W\Q <c([ \Vu\p)p. • Jn Jn Thus we have (Ff(u), u) > \\u\\p - c\\u\\q > \\u\\p(l-c\\u\\q-p) Choose 5 such that for any u e M with < 5, 1 - c||?i[|9 _ p > 0. This means that 0 is not the limit point of the set F, so F is closed. L e m m a 7.3.1 F is dual to the family T and i n f u 6 F If(u) = c. Proof: First note that F n B = 0, since for any u G F, fn\Vu\p - Jn \u\9 - Jn fu = 0 thus For the intersection property, fix 7 G T joining 0 to v, where « / 0 and If(v) < 0. By the proof above about the closedness of F , for t sufficiently close to 0, ( i ^ ( 7 ( £ ) ) , j(t)} > 0 and because ' (7J(7(1)),7(1)> = / | V 7 ( l ) r - / ' 17(1)1" - / / 7 ( 1 ) : 1 Jn Jn Jn < [ i v 7 ( i ) r - - / m w - p f M I ) v in c/ Jn Jn " ; . = Plf(v)<0 - ' (7.3) Chapter 7. Min-max Principles for Convex Sets and Applications to p-Laplacian 88 By intermediate value theorem, there exists 0' < t0 < 1 such that (Dlfdfa)), j(t0)) = 0 i.e. 7(*o) £ F- So F is dual to JF, and consequently c> M{If(u), ueF} To prove the reverse inequality, for any u G F, u ^ 0, consider the straight line j(t)-= tu. Since • *-1 iv«r - -p in q we have that lim^oo If(tu) — —oo and c < s u p 0 < t < o o Ij(tu). From di / ( fa) we get w«-) = ^ / 0 i v u r - ^ / n i « r - . / / « , - (7.4) = tp~1 [ i V M ^ - i 9 - 1 • • / " | u | 9 - [ fu = 0 (7.5) in in in / 0 | V « r = ^ / n M . + ^ T / n / U . (7.6) Then by using the fact that 2 < p < q and fnfu<0 we have / | V u | p <! Jn >IMq + Infu if * > 1 I < / n M 9 + J n / « if * < 1 Thus /; = 1 since u £ F and we conclude that c = in f u G F If[uY-L e m m a 7.3.2 Let M'. = {u G HQ'P{Q); Jn:fu < 0}' and Assume the sequence {un} C M satisfies: If(un) is bounded;,g(un) 0, (g{un) = supveM.\lUn_vll<l(u_n - v,rf(un)));,then {un} has a convergent subsequence in Hl'P(Q) converges to some u G M'. Proof: Since g(un) -> 0, let vn = 4|Kf^u™> t h e n u « e M a n d u™ ~ v™ = ~ 2 J K (21KH> J/K )> > L e- K ^ / W ) > IKI|o(l). Thus, 0(1) + ||un||o(l) > plf(un) - {un,I'f(un)) = ~ - / k n l 9 -PI fun + I Wn\q + / fu„, q Jn in in in = (1 - P-) [ \un\q + (1 - p) f fun > (1 - P-) [ \un\q. q Jn Jn q Jn U n so Chapter 7. Min-max Principles for Convex Sets and Applications to p-Laplacian 89 So Ikir = W / K ) + - / K\9 + P I fUn q Jn Jn p r < plf(un) + -Q J 0 177 q q Jn and {un} is a bounded sequence in HQ'P(SI). Since HQ'p(Q) is a reflexive Banach space, we may.assume that {un} weakly convergent to u G HQP(Q). By Sobolev's embedding, {un} strongly converges to u in LQ, since q < p*. It is clearly that u G M. Case 1: If ||un — T*|| < 1 for a subsequence of {un}, we may assume. ||t/n — u|| < 1 for all n. then g(un) > (un - Ti, /}(?/„)) = '(?/„ - w, /}(«/„) - //(«)) + o(l) > C p / | V K - u ) | p - / K - « | « + o(l),'' in in Thus C P JQ | V(u n — 7i)| p < g(un) + J N |un — Ti|9 + o(l) —» 0 and un^r u strongly. : Case 2: No subsequence of — Ti|| < 1, we may assume that \\un — u|| > 1 for all n. Let wn — (1 — ^ r t — - — u ) u n + TM~-—rrTi, then un E M and un — wn = n»n~u„ and g(u„) > (un - Wn, l'f(un)) -{un - //,/}(?/„)) 1 2| K - 7i|| 1 2| k -TiH I > r i r ; - ^ ( C P / | V ( n n - Ti ) | p - / > n - u|« + o(l)) l\\un — Ti|| in in (Cp\\un-u\\p + o(l)) . 2||un--«ii i.e. C p | | T i n - T i | | p < 2||wn - u\\g(un) + o(l), so T i n ->• Ti strongly. Chapter 7. Min-max Principles for Convex Sets and Applications tb p-Laplacian 90 Theorem 7.3.1 For any f G H 1'p> (fl) and 2 < p < q < p* there exists a constant a < 1 such that problem —Avu=\u\q~2u + af(x) in fl, < (Paf) u = 0 on dfl has a weak solution u in Hl'p(fl) and if fn fu ^ 0, then a = 1. Proof: With the above notation and lemmas, there exists u 6 F such that g(u) = 0, where g(u)= sup (u-v,I'f(u)) v£M;\\u-v\\<l and If(u) = ^ju\Vu\vdx ~~Jn WQdx - fudx for u e Hl'p(Q). Thus for any w G M with \\w\\ <-T, u + w E M. Let v = u + w, then {u — v, I'j(v)) = ( - iw , / } (« )> < 0, i.e. ^ ( A p w + | u | g - 2 M + /)u; < 0. ' (7.7) If Apu + \u\g~2u.<£ span{/}, then there exists w G HQP(fl) with \\w\\ < 1 and such that (/, w) = 0 and (Apu + | « | 9 _ 2 u , iw) > 0,' a contradiction. So there exists a constant a such that —Apu — \u\Q~2u = af in the weak sense. Now if we choose w with \\w\\ < 1 such that fn fw < 0 in (7.7), we get a < 1. Bibliography R. A. 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