"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Headley, Velmer Bentley"@en . "2011-07-19T21:58:51Z"@en . "1968"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "Criteria will be obtained for a linear self-adjoint elliptic partial differential equation to be oscillatory or nonoscillatory in unbounded domains R of n-dimensional\r\nEuclidean space E\u00E2\u0081\u00BF. The criteria are of two main types: (i) those involving integrals of suitable majorants of the coefficients, and (ii) those involving limits of these majorants\r\nas the argument tends to infinity.\r\nOur theorems constitute generalizations to partial differential equations of well-known criteria of Hille, Leighton, Potter, Moore, and Wintner for ordinary differential equations. In general, our method provides the means for extending in this manner any oscillation criterion for self-adjoint ordinary differential equations. Our results imply Glazman's theorems in the special case of the Schrodinger equation in E\u00E2\u0081\u00BF.\r\nIn the derivation of the oscillation criteria it is assumed that R is either quasiconical (i.e. contains an infinite cone) or limit-cylindrical (i.e. contains an infinite cylinder). In the derivation of the nonoscillation criteria no special assumptions regarding the shape of the domain are needed.\r\nExamples illustrating the theory are given. In particular, it is shown that the limit criteria obtained in the second order case are the best possible of their kind."@en . "https://circle.library.ubc.ca/rest/handle/2429/36119?expand=metadata"@en . "OSCILLATION THEOREMS FOR ELLIPTIC DIFFERENTIAL EQUATIONS by VELMER BENTLEY HEADLEY B.Sc. (Hons.), University of London, 1962 M.A., University of B r i t i s h Columbia, 1966 A THESIS SUBMITTED.IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department ' of MATHEMATICS We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1968 In p r e s e n t i n g t h i s t h e s j s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I agree t h a t t he L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department or by h iis r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f MATHEMATICS The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada D a t e P Q t h J u l y . 1Q68 i i ABSTRACT C r i t e r i a w i l l be obtained for a l i n e a r s e l f - a d j o i n t e l l i p t i c p a r t i a l d i f f e r e n t i a l equation to be o s c i l l a t o r y or n o n o s d i l a t o r y i n unbounded domains R of n-dimensional Euclidean space E n . The c r i t e r i a are of two main types: ( i ) those involving integrals of suitable majorants of the coefficients,\u00E2\u0080\u00A2and ( i i ) those involving l i m i t s of these major-ants as the argument tends to i n f i n i t y . Our theorems constitute generalizations to p a r t i a l d i f f e r e n t i a l equations of well-known c r i t e r i a of H i l l e , Leighton, Potter, Moore, and Wintner for ordinary d i f f e r e n t i a l equations. In general, our method provides the means for extending i n this manner any o s c i l l a t i o n c r i t e r i o n f o r s e l f - a d j o i n t ordinary d i f f e r e n t i a l equations. Our r e s u l t s imply Glazman's theorems i n the s p e c i a l case of the Schrodinger equation i n E n . In the derivation of the o s c i l l a t i o n c r i t e r i a i t i s assumed that R i s either quasieonical ( i . e . contains an i n f i n i t e cone) or l i m i t - c y l i n d r i c a l ( i . e . contains an i n f i n i t e c y l i n d e r ) . In the derivation of the n o n o s c i l l a t i o n c r i t e r i a no s p e c i a l assumptions regarding the shape of the domain are needed. Examples i l l u s t r a t i n g the theory are given. In p a r t i c u l a r , i t i s shown that the l i m i t c r i t e r i a obtained i n the second order case are the best possible of t h e i r kind. i i i TABLE OP CONTENTS Page INTRODUCTION 1 CHAPTER I Second Order Self-Adjoint Equations 1. De f i n i t i o n s and notations 4 2. A u x i l i a r y r e s u l t s 6 3. O s c i l l a t i o n c r i t e r i a of i n t e g r a l type 13 4. Conditions of l i m i t type 19 5. Nonoscillation theorems 23 6. Sharpness of the re s u l t s ] 33 CHAPTER II Equations of Arbitra r y Even Order 7. Preliminaries 40 8. Defin i t i o n s and notations 41 9. A u x i l i a r y r e s u l t s 43 10. O s c i l l a t i o n c r i t e r i a 48 11. Equations with one variable separable 5 5 12. Fourth order equations on l i m i t - c y l i n d r i c a l domains 5 7 13. Fourth order equations on a l l of E n 63 14. O s c i l l a t i o n theorems 66 BIBLIOGRAPHY 7 5 iv ACKNOWLEDGEMENT I am greatly Indebted to my advisor, Dr. CA. Swanson, for suggesting the topic, and for his help and encouragement throughout the preparation of t h i s work. I also wish to thank Dr. G.E. Hu'ige for his useful comments on the manuscript. I must also express my gratitude to the- University of B r i t i s h Columbia and the National Research Council of Canada for t h e i r generous f i n a n c i a l support. INTRODUCTION Conditions on the c o e f f i c i e n t s of ce r t a i n l i n e a r e l l i p t i c p a r t i a l d i f f e r e n t i a l equations w i l l he obtained which are s u f f i c i e n t for the equations to be o s c i l l a t o r y i n unbounded domains of n-dimensional Euclidean space E n . The c r i t e r i a are of two main types: ( i ) those involving integrals of suitable majorants of the c o e f f i c i e n t s ; and ( i i ) those involving l i m i t s of these majorants as the argu-ment tends to i n f i n i t y . C r i t e r i a of type ( i ) were obtained by Swanson [26] for second-order equations with one variable separable and the fundamental domain l i m i t - c y l i n d r i c a l ( i . e . containing an i n f i n i t e c y l i n d e r ) ; and conditions of type ( i i ) by Glazman [9] for the Schrqdinger operator i n the case that the domain i s a l l of E n . Our theorems constitute generalizations to n dimen-sions ( i . e . p a r t i a l d i f f e r e n t i a l equations) of well-known one-dimensional results i n the l i t e r a t u r e (cf. [9], [113 > [16 [17]3 [19], [213, [283). Our method provides the means for generalizing to n-dimensions ( i . e . p a r t i a l d i f f e r e n t i a l equations) any given one-dimensional o s c i l l a t i o n c r i t e r i o n . -. The results we obtain serve to i l l u s t r a t e the power of the method. There i s an extensive l i t e r a t u r e on o s c i l l a t i o n .theorems for ordinary d i f f e r e n t i a l equations.. A. complete bibliography may be found i n the forthcoming book of Swanson [273. The corresponding theory f o r p a r t i a l d i f f e r e n t i a l - 2 -equations i s not as well developed (see, however, [ 9 ] , [ 1 0 ] , [ 1 3 ] , f l 1 * ] , [ 1 5 ] , [ 2 6 ] , [ 2 7 ] ) , l a r g e l y because of the e a r l i e r lack of an n-dimensional analogue of Sturm's comparison theorem. In t h i s work we use the recent Clark-Swanson r e s u l t [4], together with a comparison theorem of Swanson [24] for eigen-values, as the basic tools for deriving our o s c i l l a t i o n c r i t e r i a . The d e f i n i t i o n of an o s c i l l a t o r y equation given below i s closely related to the notion of conjugacy used by Kreith [14], An equation o s c i l l a t o r y i n our sense i s often said to have the nodal property (cf. [ 2 6 ] ) . In general, t h i s i s stronger than the requirement that a solution exists with a zero i n every neighbourhood of i n f i n i t y . In f a c t , i f a s e l f - a d j o i n t second-order l i n e a r e l l i p t i c equation i s o s c i l l a -tory i n our sense, every solution of the equation has a zero i n every neighbourhood of i n f i n i t y . This w i l l be seen t o be a consequence of the Clark-Swanson separation theorem [ 4 ] . In Chapter I, equations of the second order w i l l be considered on quasiconical\" domains ( i . e . domains containing a cone). The c r i t e r i a obtained are e a s i l y specialized t o a l l of E n , and w i l l contain the corresponding results of Glazman [9] for the Schrodinger operator. Using the Clark-Swanson comparison theorem, we also derive n o n o s c i l l a t i o n theorems, but without making any special assumptions regarding the shape of the domain ( i n contrast to the o s c i l l a t i o n theorems, where we assume that the domain i s quasiconical.)'. - 3 -Our r e s u l t s i n t h i s d i r e c t i o n g e n e r a l i z e well-known one-dimensional theorems of H i l l e [ 1 1 ] , and P o t t e r [ 2 1 ] , The l a s t s e c t i o n of Chapter I w i l l be devoted to examples i l l u -s t r a t i n g the theory. In p a r t i c u l a r , we s h a l l show t h a t the l i m i t c r i t e r i a o b t ained are the b e s t p o s s i b l e of t h e i r k i n d . Equations of h i g h e r order w i l l be s t u d i e d i n Chapter I I . Our approach i s an e x t e n s i o n of t h a t used by K r e i t h [13] and Swanson [26] f o r second-order equations. The funda-mental domain w i l l be l i m i t - c y l i n d r i c a l ( i . e . w i l l c o n t a i n an i n f i n i t e c y l i n d e r ) . The r e s u l t s are fewer i n t h i s case, s i n c e there are not many one-dimensipnal theorems extant. Our theorems c o n s t i t u t e g e n e r a l i z a t i o n s of r e s u l t s of Glazman [9] f o r two-term o r d i n a r y d i f f e r e n t i a l equations of order 2m (m any p o s i t i v e i n t e g e r ) , and Leighton-Nehari [17] f o r f o u r t h - o r d e r equations. In the absence of a s u i t a b l e compari-son theorem, however, our methods w i l l not y i e l d n o n o s c i l l a t i o n theorems f o r h i g h e r order (m >_ 2) p a r t i a l d i f f e r e n t i a l e quations. The o p e r a t o r we c o n s i d e r has a r e l a t i v e l y simple form, but our methods w i l l work f o r more g e n e r a l s e l f - a d j o i n t o p e r a t o r s of even order, s i n c e the v a r i a t i o n a l p r i n c i p l e s we use are v a l i d f o r g e n e r a l e l l i p t i c o perators ( c f . [ l ] and [ 1 8 ] ) . Although we know of no one-dimensional, o s c i l l a t i o n c r i t e r i a f o r such g e n e r a l o p e r a t o r s , we remark t h a t such c r i t e r i a c o u l d be g e n e r a l i z e d by u s i n g extensions of the Swanson comparison.theorem [ 2 4 , p.5 1 7 ] ' - 4 -CHAPTER I SECOND ORDER SELF-ADJOINT EQUATIONS 9 1. Definitions and notations. We s h a l l obtain o s c i l l a t i o n c r i t e r i a of l i m i t type and i n t e g r a l type for the l i n e a r e l l i p t i c p a r t i a l d i f f -e r e n t i a l equation n (1.1) Lu = \u00C2\u00A3 D.(a. .D.u) + bu = 0 i , j = l 1 1 J J i n unbounded domains R i n n-dimensional Euclidean space E n . Our theorems are, for the most part, extensions of one-dimensional o s c i l l a t i o n theorems of Kneser-Hille [11] ( l i m i t type), Leighton [ l 6 ] , Moore [19], Potter [21], and Wintner [28] ( i n t e g r a l type). The remainder are the n-dimensional second order analogues of one-dimensional 2m-th order o s c i l l a -t i o n theorems of Glazman [9]. Points i n E n are denoted by x = (x^,x 2,... ,>'n) and d i f f e r e n t i a t i o n with respect to x^ i s denoted by i=l,2,...,n. The c o e f f i c i e n t s a. . are supposed to be 1 J r e a l and of class C (R), and the matrix (a..) i s pos i t i v e d e f i n i t e i n R. The c o e f f i c i e n t b i s assumed to be r e a l and continuous on R. The domain D(L) of L i s defined to be the set of a l l real-valued functions on R of class C 2(R). The conditions on the c o e f f i c i e n t s , although not the - 5 -weakest possible (see, for example, [4.]), are the special case m = 1 of those we impose i n the 2m-th order case treated i n Chapter II. D e f i n i t i o n . A function u w i l l be c a l l e d a s o l u t i o n of Lu = 0 i f u e D(L) and u s a t i s f i e s ( l . l ) everywhere i n R. Shape of the domain R. We assume that R contains the o r i g i n and that R i s large enough i n the x n d i r e c t i o n to contain the cone C = f x e E n :' x > I x l cos a] for some a n \u00E2\u0080\u0094 1 ' I ^ 2\^ a , 0 < a < TT, | x | being the Euclidean distance ( E x . ) v i = l ' The boundary dR of R i s supposed to have a piecewise continuous unit normal vector at each point. We s h a l l make use of the following notation: . R r = R n {x e E n : |x| > r}; S f = (x 6 R u : |x| = r J . D e f i n i t i o n . A bounded domain N c R i s said to be a nodal domain of a n o n t r i v i a l solution u of ( l . l ) i f f u = 0 on dN. D e f i n i t i o n . The d i f f e r e n t i a l equation ( l . l ) i s said to be o s c i l l a t o r y i n R i f f there exists a n o n t r i v i a l solution u^ of ( l . l ) with a nodal domain i n R r for a l l r > 0. It follows from Clark and Swanson's n-dimensional analogue of Sturm's separation theorem [4] that every solution of. an o s c i l l a t o r y d i f f e r e n t i a l equation vanishes at some point i n R r for a l l r > 0. D e f i n i t i o n . The d i f f e r e n t i a l equation ( l . l ) Is said to be -6-n o n - o s c i l l a t o r y i f f there e x i s t s r > 0 such t h a t the s o l u t i o n s i n R have no nodal domains, r 2. A u x i l i a r y r e s u l t s . The b a s i c t o o l f o r d e r i v i n g our o s c i l l a t i o n theorems w i l l be a r e c e n t comparison theorem of C A . Swanson [ 2 4 ] . In a d d i t i o n we s h a l l need two well-known p r o p e r t i e s of eigenvalues. Minimum p r i n c i p l e . (Cf. [6, p.399]) L e t fi be a bounded n 2 domain i n E . The f u n c t i o n u e C (Q) which minimizes the f u n c t i o n a l n ( 2 . 1 ) J[u] = T { E a. .D.uD.u - bu } dx under the c o n d i t i o n Hull = 1 i s ,an e i g e n f u n c t i o n corresponding to the s m a l l e s t eigenvalue of the problem ( 2 . 2 ) -Lu = Xu i n fi; u = 0 on dfi \u00E2\u0080\u00A2 2 p 2 ~^ The norm H u l l i s the u s u a l L norm:' H u l l = [ |u'| dx]^ . J fi Proof. In [6] i t i s shown t h a t i f the m i n i m i z i n g f u n c t i o n 2 e x i s t s and i s of c l a s s C , then i t i s an e i g e n f u n c t i o n c o r r e s p o n d i n g t o the s m a l l e s t eigenvalue of the problem ( 2 . 2 ) . Let the minimum value of J [ u ] be X . Then the r e s u l t s of o [18, \u00C2\u00A7 1 1 ] show t h a t there e x i s t s a m i n i m i z i n g f u n c t i o n U Q which i s a weak s o l u t i o n of ( 2 . 2 ) i n the f o l l o w i n g sense: = 0 , cp e c\"(Cl), - 7 -2 < , > b e i n g the u s u a l L (fi) i n n e r product. On account of our c o n d i t i o n s on the c o e f f i c i e n t s a. . and b, the r e s u l t s 3 of [ l , \u00C2\u00A79 ] imply t h a t U q i s i n f a c t a c l a s s i c a l s o l u t i o n of (2.2). M o n o t o n i c i t y p r i n c i p l e f o r e i g e n v a l u e s . For 0 < t < oo l e t G(t) be a bounded domain i n R. I f 0^ < t-^ < tg < co i m p l i e s G ( t 1 ) c G ( t 2 ) , G ( t 1 ) y G ( t 2 ) , then the f i r s t e i genvalue \ Q ( t ) of the problem -Lu = Xu i n G ( t ) ; u = 0 on dG(t) i s monotone d e c r e a s i n g i n t. Moreover, i f f o r some r Q > 0, G(t) c fi(t), where n(t) = {x e E n : r < |x| <-r +t), then o l i m X ( t ) = +oo . t-0+ \u00C2\u00B0 Proof. The monotonicity of ^Q(t) may be e s t a b l i s h e d by a d a p t i n g the proof g i v e n i n [ 7 , pp. 4-00-401 ] ,for the Laplace operator. The c o n t i n u i t y of a. . i m p l i e s t h a t the s m a l l e s t ^- 3 eigenvalue A ( X ) of the matrix (a. .(x)) has non-zero infimum i n G ( t ) , s i n c e G(t) c R and (a. .) i s p o s i t i v e -J . j d e f i n i t e i n R. In other words, the operator L i s u n i f o r m l y e l l i p t i c i n G ( t ) , i . e . there e x i s t s a number. H Q ( t ) > 0 such n t h a t Z a. . ( x ) z . z . > u ( t ) | z | f o r a l l x e G ( t ) , z e-E n. Since the function b i s uniformly continuous on G(t), there exists a number k (t) > -oo such that o v ' ( 2 . 3 ) G(t) bu dx > \u00E2\u0080\u00A2 k (t) . \u00E2\u0080\u0094 o v 2, u dx G(t) for a l l u e C 1(R). Let J t [ u ] { \u00C2\u00A3 a. .D.uD.u - tiu |dx . G(t) ' i j J = 1 Then ( 2 . 3 ) and the uniform e l l i p t i c i t y of L i n G(t) that imply But J t [ u ] > u Q(.t) G(t) n p 2 (D.u) dx + k (t) i=l \u00C2\u00B0 2 , u dx G(t) u 2dx = J* u 2F(r,S)drdS , G(t) G(t) where F(r,S) i s the Jacobian of the transformation from rectangular coordinates (x-^Xg,...x n) to hyperspherical polar coordinates ( r , Q ^ , . . . 3Q^ ^) = (r^S ) , defined by the rela t i o n s . n x = r ~JT s i n 0 1 i = l x = r cos 8,, n 1' n-1 x\u00C2\u00B1 = r cos Q n_ i + 1 TT^ s i n 6j , i = 2 , 5 , . . . a n - l . We now extend . u continuously to a l l of the annulus Q(t) by requiring i t to be zero outside G(t). In p a r t i c u l a r , u i s zero on |x| = r , r being the inner radius of the 1 1 o o annulus Q(t). Hence u(x) = v t ( t , 0 1 , . \u00E2\u0080\u00A2 . , @ n _ 1 ) d t , o where u(x) = v ( r , 8 1 , . \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , 8 n _ 1 ) , v t = dv/dt By Schwarz's inequality, u(x)r < dy o o r +t o o whenever x e G(t) c fi(t). Integrating t h i s inequality over D(t) we obtain u 2dx = J u 2F(r,S)drdS n(t) n(t) < t 2 J F(r,S)drdS n(t) -< t' n 2 (D.u) dx , i=l 1 unce 2 ( D . u ) 8 . |grad u|2-= ( f l f + where the hu are certa i n (known) functions of r and S. -10-Since u i s defined to be zero outside G(t), we have ( 2 . 5 ) 2 2 u dx < t n p \u00C2\u00A3 (D.u) dx i = l 1 G(t) G(t) Combining th i s with inequality ( 2 . 4 ) we get 2,P J t [ u ] > [ k Q ( t ) + H Q ( t ) / t c : ] 2, u dx G(t) According to Courant's minimum p r i n c i p l e [6, p.399], \ Q ( t ) = i n f { J t [ u ] : H u l l t = 1} , \u00E2\u0080\u00A2 where Hull ^ i s the L 2 norm of u on G ( t ) . Thus ( 2 . 6 ) \ Q ( t ) > k Q ( t ) + n Q ( t ) / t 2 . Since HQV^) may be chosen to be inf{/\ Q(x) : x e G(t)} , i t cannot decrease as the domain G(t) shrinks, and therefore remains posi t i v e as t -* 0+. S i m i l a r l y , we may choose k Q ( t ) = inf{-J bu 2dx/ftu^ : u \u00E2\u0082\u00AC C(G(t)} , G(t) that i s k Q ( t ) = i n f | - J bu dx/ Q(t) \u00E2\u0080\u00A2 n(t) u dx : ue C(G(t) since' u i s zero outside G(t). Now b i s continuous i n R and i s therefore a.continuous function of |x|. Thus b i s a continuous function of t i n every i n t e r v a l 0 < t < 6, -11-and i s thus bounded on 0 <_ t <_ 6 '. f o r a l l 6 > 0. Therefore k ( t ) must remain bounded as t -\u00C2\u00BB 0+. ( T h i s i m p l i e s t h a t we could have chosen k Q ( t ) independent of t i n (2.3)). In any event, the i n e q u a l i t y (2.6) now i m p l i e s t h a t l i m X ( t ) = + oO, and the theorem i s proved. t-0+ \u00C2\u00B0- \" ' Remark. In the a p p l i c a t i o n s below, we a c t u a l l y need ^ 0 ( \" t ) to tend -continuously t o + max (A(x) : x e S } , ( 0 < r 0), and (a..), ^ have majorants f,g respectively such that' (3.1) ' - n I T : = + 0 \u00C2\u00B0 A N D J r n [ g ( r ) - X a r f ( r ) ] d r = +00. 1 Proof. The hypotheses (3-1) imply that the ordinary d i f f e r e n t i a l equation (2.9) i s o s c i l l a t o r y in. 0 < r < oO by the Leighton-Wintner theorem [l6]_, [28], Let p(r) \u00E2\u0080\u00A2 be a n o n - t r i v i a l solution of (2.9) with zeros at r = 6^/6 . . . ,8^,-. . . , where & ktco . . If cp i s an eigenfunction of (2.10) 'with boundary condition cp(a) = 0 corresponding to the eigenvalue X. , - the ; -14-functlon v defined by (2.8) IS c l s olution of the comparison equation (2.7) with nodal domains i n the form of truncated cones Calc = ^X e ^ ' xn ^ c o s a ' 5 k < < 5k - r-l^ ' 0 < a < TT , k=l,2,...,, with piecewise smooth boundaries. Thus v has a nodal domain C , c R for a l l p > 0;1 for i f p > .0 i s ak p 5 given, \u00E2\u0080\u00A2 choose k large enough so that 6^ _> p,- and c l e a r l y x e C ^ implies that |x| > 6^ _> p and x e C a c R, so that x e R . Since P E a. .(x)z.z < A ( x ) | z | 2 < f ( r ) | z | 2 = A(x) | z | 2 , z e. E n i , j = l 1 and b(x) _> g(|x|) = B(x), equation ( l . l ) majorizes equation (2.7) i n the following sense: f r n Pv \ E (A5 . . - a. .)D.uD .u + (b-B)u dx _> 0 . ak It then follows from a comparison theorem of C.A.- Swanson [24] that the smallest eigenvalue of the problem -Lw = (aw- i n C a R, w = 0 on aC^ s a t i s f i e s n < 0. Let M^ t * {x e : 6 R < |x| < t} , 6, < t < 6, ' , and l e t m(t) denote the smallest eigenvalue of the problem - 1 5 --Lw = (-i(t)w i n M a k t > w = 0 on S M ^ . By the monotonicity p r i n c i p l e for eigenvalues, u-^t) if monot one nonincreasing i n 6^ < t _< 6^+1 and lim = + there exists a number T i n ( 6 ^ 6k+l-' such that LI, (T) = 0. This means that M i m i s a nodal domain M 1 v ' akT of a n o n t r i v i a l solution u^ Of ( l . l ) , and since 0 provided k i s s u f f i c i e n t l y large, equation ( l . l ) i s o s c i l l a t o r y i n R and the theorem i s proved. It i s convenient to state the integral conditions with the number 1 as the lower l i m i t of integration, but the theorems remain v a l i d i f the i n t e g r a l conditions hold when 1 i s replaced by a p o s i t i v e number. In f a c t , i n the l i t e r a t u r e (Cf., e.g. [ 1 9 ] , [ 2 1 ] , [ 2 6 ] , [ 2 7 ] , [ 2 8 ] ) the i n t e g r a l c r i t e r i a are often stated i n t h i s manner. We s h a l l have occasion to make use of this fact i n the applications (Cf. \u00C2\u00A7 6 , Example 4 ) . The f i r s t part of condition ( 3 . 1 ) requires the function f to grow quite slowly, and i n f a c t does not hold for the Schrodinger operator A + b(x) i n three dimensions, since i n P\u00C2\u00B0\u00C2\u00B0 dr th i s case f ( r ) > 1 , so that \u00E2\u0080\u009E , 0), and (a. .), b have majorants -16-f, g, respectively, such that (3.2) , J 0 0 dr 1 r n _ 1 f ( r ) < co and 1 h \u00E2\u0084\u00A2 ( r ) [ g ( r ) - X a r - 2 f ( r ) ] d r oO dt fo r some number m > 1, where h (r) = n-1 f ( t ) r t Proof. According to Moore's o s c i l l a t i o n theorem [ 1 9 ] , the ordinary d i f f e r e n t i a l equation (2.9) i s o s c i l l a t o r y i n 0 < r < oO on account of the hypotheses ( 3 . 2 ) . The remainder of the proof follows that of Theorem 1 without change and w i l l be omitted. The c r i t e r i a obtained i n Theorems 1 and 2 may be sharpened s l i g h t l y i n the case that the largest eigenvalue A ( X ) of (a. .) i s bounded i n Po. Theorem 3. Let R contain the cone C for some a > 0, \u00E2\u0080\u0094 a ' and l e t A ( X ) be bounded i n R. Suppose (a..) _ and b have majorants f,g, respectively. Then.equation ( l . l ) i s o s c i l l a -tory i n R for n = 2 i f . and for n > 3 i f there exists a number 6 > 0 such that In the case n = 1/ equation ( l . l ) i s o s c i l l a t o r y i f (3.4) holds with 6 = 1 (the Leighton-Wintner.theorem). (3.4) fg(r) - X r ~ 2 f ( r ) , ] d r = + oo . -17-Froof. If A (X ) i s bounded i n R, say A (X ) < A 1 ? x e R, we'can choose f ( r ) = A , 0 < r \u00E2\u0080\u00A2 < eo. Then, for n = 2, the conditions (3-1) are f u l f i l l e d and hence the f i r s t statement of the theorem follows from Theorem 1. For n _> 3, the f i r s t part of (3-2) i s f u l f i l l e d , and n n ( r ) = r /(n-2) f\- . By hypothesis there exists 6 > 0 such that (3.4) holds. Let m = 1 + 6/(n-2). Then d i r e c t computation shows that condition (3.4) implies the second part of condition (3.2), and.therefore the second statement of the theorem follows from Theorem 2. It i s clear that our method enables us to generalize . to n dimensions any s u f f i c i e n t condition for a s e l f - a d j o i n t ordinary l i n e a r d i f f e r e n t i a l equation of the second order to be o s c i l l a t o r y . In what follows we s h a l l therefore generalize only a representative number of the e x i s t i n g one-dimensional-o s c i l l a t i o n theorems. Our next theorem generalizes Potter's refinement [21] of Leighton's theorem. We s h a l l f i n d i t convenient to introduce the following notation. Let h be a p o s i t i v e 2 C function defined by -(3.5) . m [ h ( r ) ] \" 2 = g(r) - A - J ^ + ( n - l ) ( n - 3 ) / 4 ] r \" 2 , 0 < r < oo Let the functions and H 2 be defined by H l ^ r ; ~ h f F T . \" 4h(r) + > H rr) - 1 ^ ( r ) ] 2 h'(r) H 2 ^ r j - hJTJ ~ 4h(r) > -18-where primes denote d i f f e r e n t i a t i o n with respect to r. Theorem 4. Let R contain the cone C for some a > 0 , a and l e t A ( X ) be bounded i n R, say A ( X ) \u00C2\u00A3 A^, x e R . Then equation ( l . l ) i s o s c i l l a t o r y i n R i f there exists a pos i t i v e C function h s a t i s f y i n g (3-5) for large r and either < 3 - 6 > ' J ^ . E r l T \" ' J \" \" H i ( t 1 d t = + c 0 or i (3 .7) J ^ H 2 ( t ) d t = + oo . Proof. In equation (2 .9) choose f ( r ) = A^. The normal form of t h i s equation, obtained by making the o s c i l l a t i o n - p r e s e r v i n g \" ... (l-n ) / 2 transformation p = r v \u00E2\u0080\u00A2 \" a , i s ( 3 . 8 ) ' A-, +'{g(r) - AA\n + (n - l ) ( n - 3 ) / 4 ] r - 2 } a = 0. 1 dr 1 a . The hypothesis ( 3 . 6 ) (or (3-7)) implies that the equation (3-8) i s o s c i l l a t o r y by the theorem of Potter mentioned i n the remark above. Thus the equation ( 2.9) i s also o s c i l l a t o r y i f ( 3 . 6 ) or (3 .7) holds. The remainder of the proof is. s i m i l a r to that of Theorem 1 and w i l l be omitted. Because of the p o s i t i v i t y condition on h, i t i s clear that Theorem 4 i s i n some respects less general than-Theorem 3 . However, i n section 6 we s h a l l exhibit an example -19-for which Theorem 4 gives, information not. obtainable i n any-obvious way from Theorem 4. Conditions of l i m i t type. The f i r s t theorem i n thi s section i s a generalization of the c l a s s i c a l Kneser-Hille theorem [ 1 1 ] . Our theorem also contains Glazman's generalization [ 9 , Th.7 3 p.158] for the 2 n Schrodinger operator -v - b(x), x e E , and i n fac t provides a new proof of his re s u l t . It w i l l also be seen that our condition i s sharp. j Theorem 5\u00C2\u00BB Suppose that R contains the. cone C for some 1 Cfc a > 0, and that A(X) i s bounded i n R . for some s > 0, s say A(X) < A., , x e R . Let b have majorant g. Then equation ( l . l ) i s o s c i l l a t o r y i n R i f (4.1) Lim i n f r 2 g ( r ) > A-, [\ + (n-2) 2/4].-r-* oo -Proof. The hypothesis (4.1) implies that there ex i s t constants r and v such that o . r 2 g(r) > Y > A 1 [ \ A + ( n - 2 ) 2 / 4 ] for a l l r > r . We then compare (2.9) with the Euler equation I ^ V \" 1 ^ + ( Y \" V l ) ' \" \" 3 \" \" - \u00C2\u00B0. with solutions p = , where 1 ,\u00E2\u0080\u00A2 \u00E2\u0080\u00A2 - 2 0 -( 4 . 3 ) 3 2 + (n-2)p + ( Y/A 1) - \ a = 0. Since y > A - , [X + (n-2) / 4 ] , the quadratic equation ( 4 . 3 ) has complex roots, and therefore equation ( 4 . 2 ) i s o s c i l l a t o r y i n (r , cd). We may choose f ( r ) . = A - ^ and apply Sturm's comparison theorem [ 5 , p . 2 0 8 ] on (r-^oo), (^ = max{r o,s}) to deduce that equation ( 2 . 9 ) i s o s c i l l a t o r y on account of the hypothesis r n _ 1 [ g ( r ) - \ a r - 2 f ( r ) ] > r n _ 5 ( Y - X ^ ) , r > max{r o,s}. The remainder of the proof proceeds as i n Theorem 1 and w i l l be omitted. Our next theorem i s an extension of a well-known theorem of H i l l e [ l l , T h . 5 ] \u00C2\u00BB As H i l l e points, out i n the paper just c i t e d , the conditions are a refinement of those ? in Theorem 5 , since integration smooths out i r r e g u l a r i t i e s i n the function g. Theorem 6 . Let R contain the cone for some a > 0 , and l e t A(x) be bounded i n R, say A ( X ) _< A - ^ , x e R . Then equation ( l . l ) i s o s c i l l a t o r y i n R i f 2 r g(r) > A - , [X + (n-1) ( n - 3 ) / 4 J for large r and either ( 4 . 4 ) lim i n f r g(t) dt > A , [\n + {n-2)- /k] r- oo J r . x a -21-or (4.5) lim sup rf\u00C2\u00B0\u00C2\u00B0 g(t) dt > A , [X + (n 2-4n+7)/4], r-\u00C2\u00BB oO r where g(r) = min (b(x) : x e S }. \u00E2\u0080\u00A2Proof. In equation (2.9) choose f ( r ) = A-^ . We-recall that the normal form of th i s equation, obtained by making the o s c i l l a t i o n - p r e s e r v i n g transformation p = r ^ - 1 1 ^ 2 a, i s (3.8) ' A N + (g(r) - A '[X\u00E2\u0080\u009E + ( n - l ) ( n - 3 ) / 4 ] r - 2 } a = 0. dr 1 a The hypothesis (4.4) implies that the equation (3.8) i s o s c i l l a t o r y i n 0 < r < oo by a theorem of H i l l e [ 1 1 , Th.5], since lim i n f r f \" 0 {g(t)/A, - [\n + (n-l) ( n - 3 ) / 4 ] t ~ 2 } d t r-\u00C2\u00BB oo -1 r .00 ' = l i m i n f r 1 r-oo g(t) dt - [\ + ( n - l ) ( n - 3 ) / 4 ] r \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 > \ a + (n-2) 2/4 - [\ Q + ( n - l ) ( n - 3 ) / 4 ] = \u00C2\u00A3 . Thus equation (2.9) i s also o s c i l l a t o r y i n 0 < r < oo i f hypothesis (4.'4) holds. ; If the hypothesis (4.5') holds, then equation (3.8) i s o s c i l l a t o r y i n 0 < r < oo by the theorem of H i l l e c i t ed above, since -22-l i m sup ( g ( t ) / A 1 - \u00E2\u0080\u00A2 [\ a + ( n - l ) ( n - 3 ) / 4 ] t _ 2 } d t r-> oO -1 f 0 0 1 r-\u00C2\u00BB oo A , l i m sup r g ( t ) d t - [\ Q + ( n - 1 ) ( n - 3 ) / 4 ] > [\ a +(n 2-4n+7)/4] - [ X Q + ( n - l ) ( n - 3 ) / 4 ] = 1. Thus the e q u a t i o n (2.9) i s a l s o o s c i l l a t o r y i n 0 < r < cO i f (4.5) h o l d s . The remainder of the p r o o f w i l l be omitted, s i n c e i t f o l l o w s t h a t of Theorem 1 without change. In s e c t i o n 6 we s h a l l show, by e x h i b i t i n g a counter-example, t h a t the i n e q u a l i t y (4.1) i s sharp. Our next theorem permits us t o r e l a x the c o n d i t i o n (4.1) p r o v i d e d we impose an a d d i t i o n a l h y p o t h e s i s . T h i s g i v e s the second order, n-dimensional analogue of a 2m-th qrder, one-dimensional r e s u l t of Glazman [9, p.102]. Although t h i s r e s u l t i s not the sharpest p o s s i b l e , i t i s u s e f u l i n the a p p l i c a t i o n s of the theory, as we s h a l l demonstrate i n s e c t i o n 6. Theorem 7. L e t R c o n t a i n the cone C f o r some a > 0, ! _ QT and l e t A ( X ) be bounded i n R, say A ( X ) < A -^J x e R . 2 2-Then e q u a t i o n ( l . l ) i s o s c i l l a t o r y i f r g ( r ) > A-, [\ +(n-2) /4] f o r l a r g e r and \u00C2\u00B0 (4.6) l i m sup ( l o g r ) t { g ( t ) - A n [\n+(rx-2) / ^ t ) d t = + co . r-\u00C2\u00BB oo r Proof. In equ a t i o n (2.9) choose f ( r ) = A-^ . We then reduce -23-(2.9) to normal form as i n the proof of Theorem 6: (3.8) A . + { g ( r ) _ r, +. ( n _ i ) ( n _ 5 ) / 4 ] r - 2 } a = 0 , where p ( r ) = r ^ 1 - n ^ 2 a. By a theorem of Glazman [9, p. 102], the e q u a t i o n (3.8) i s o s c i l l a t o r y i n 0 < r < co, s i n c e the hypothesis (4.6) i m p l i e s = + \u00C2\u00B00 . Thus the equation (2.9) i s a l s o d s c i l l a t o r y , s i n c e the t r a n s -f o r m a t i o n p = r ^ ' L - n ^ 2 a preserves o s c i l l a t o r y b e h a v i o u r i n 0 < r -2k-Theorem 8. Let L be uniformly e l l i p t i c i n R for some s > 0 , i . e . there exists a number A > 0 such that E a. .(x)z.z. > A |z| f o r a l l x e R . z e E . Let g Q ( r ) = max (b(x) : x e S r ) , 0 < r 0. But the hypothesis (5\u00C2\u00BBl) implies- that there,exist constants r , v such that o ' o r 2 g Q ( r ) < y o < ( n - 2 ) 2 AQ/k, f o r . a l l r > r . We compare (1 . l ) . with.the equation ( 5 . 2 ) I AQD^v +Y Q|x| v = 0 , x e R. Because of the hypotheses E a (x)z.z > A |z| , x e R , z e E i , j = l 1 J . - S yor~2 > S Q ( r ) >. b ( x ) ^ x e S r, r > r r equation (5-2) majorizes equation ( l . l ) i n the following sense: -25-{ S (a - A 6 )D uD u + ( Y o | x | - 2 - b)u 2}dx > 0 , N r L i , 3 = 1 J ( r > max{r o,s}) and t h e r e f o r e the Clark-Swanson n-dimensional analogue [h] of Sturm's comparison theorem i m p l i e s t h a t every s o l u t i o n of (5.2) has a zero i n N r u dN r f o r a l l r > max { r Q , s } . But the s o l u t i o n p = r a of the o r d i n a r y d i f f e r e n t i a l e q uation (5.3) [ A 1 | ] + V 0 r n - 3 p ( r ) , 0 i s a l s o a s o l u t i o n of (5.. 2) ( e q u a t i o n (5\u00C2\u00AB3) b e i n g the r a d i a l form of (5.2)), and a s a t i s f i e s the q u a d r a t i c e q u a t i o n a 2 + (n -2)a + Y 0 A 0 = 0. T h i s e q u a t i o n has r e a l r o o t s because Y O < (n~2) A Q / A , and hence the s o l u t i o n v = r a of (5-2) i s non-zero i n N r ij dN r f o r a l l r > 0. T h i s c o n t r a d i c t i o n e s t a b l i s h e s the theorem. In s e c t i o n 6 we s h a l l e x h i b i t a counterexample to 2 show t h a t the constant (n-2) A Q / 4 i n c o n d i t i o n (5-1) cannot be improved, i . e . there e x i s t o s c i l l a t o r y equations f o r which l i m sup r 2 g ( r ) = ( n - 2 ) 2 A A . r-* co The theorem j u s t proved, together with Theorem 5 above, c o n t a i n s Glazman's g e n e r a l i z a t i o n [9, Th . 7 , p.158] of H i l l e ' s r e s u l t s [ l l ] . In the cases where 1, both the r e s u l t and the proof are new. -26-Th eo rem 9- L e t L be uniformly e l l i p t i c i n R f o r some s > 0, A q being the e l l i p t i c i t y constant. Let g^ be the function defined, by g l ( t ) = g Q ( t ) - A G ( n - l ) ( n - 3 ) / 4 t 2 , with S Q ( r ) = m a x (b(x) : x e Sr), 0 < r 0. We now compare ( l . l ) with the equation n p (5.5) E A nD v\u00E2\u0080\u00A2 + g (|x|)y_= 0, x e R. i=l \u00C2\u00B0 1 \u00C2\u00B0 Because of the hypotheses n ? n E a (x) z z > A |z| , x e R , z e E n , i , j = l 1 J 1 J \u00E2\u0080\u00A2 s g Q ( r ) > b(x), x e S r, 0 < r < oo -27-equation (5.5) majorizes e q u a t i o n ( l . l ) : n. J { E (a - A 6 )D uD u + (g -b)u 2}dx>0, (r>s) J N r L i , j = l 1 J \u00C2\u00B0 J and t h e r e f o r e the theorem of C l a r k and Swanson c i t e d above [4] i m p l i e s t h a t every s o l u t i o n of (5\u00C2\u00AB5) has a zero i n N U 5N r f o r a l l r > s. But we s h a l l show t h a t there e x i s t s a s o l u t i o n of (5-5) which has no zeros i n R f o r some p > 0 and t h e r e f o r e has no zeros i n R r f o r a l l r > p. To see t h i s , we note t h a t the s o l u t i o n s of the o r d i n a r y d i f f e r e n t i a l e q u a t i o n (5-6) *0& r r \" \" 1 ^ ] + r - 1 g o ( r ) p - 0 are a l s o ( r a d i a l ) s o l u t i o n s of ( 5 . 5 ) . The normal form of (5.6) , obtained by making the o s c i l l a t i o n - p r e s e r v i n g t r a n s -f o r m a t i o n p = r ^ ~ n ^ 2 w, i s (5.7) A n ^ g + - [ g n ( r ) - ' ( n - l ) ( n - 3 ) A / 4 r 2 ] w = 0. o d r ^ o o Since S ^ ( r ) i s nonnegative, a well-known theorem of H i l l e . [11, Th.7 C o r . l ] i m p l i e s t h a t the e q u a t i o n \u00E2\u0080\u00A2^k + g t ( r ) y / A _ = 0 dr 1 0 i s n o n o s c i l l a t o r y on account of h y p o t h e s i s ( 5 . 4 ) . Moreover, s t ( r ) / A o 1 % ( r ) / A o = g o ( r ^ / A o \" ( n - D ( n - 3 ) / 4 r 2 , - 2 8 -so that Sturm's comparison theorem [ 5 , p . 2 0 8 ] implies that equation ( 5 - 7 ) i s nonoscillatory. Thus there exists a solu-t i o n w = w(r) of ( 5 - 7 ) which has no zeros i n ( p . , 0 0 ) for some p > 0 . Hence there exists a solution v = r ( l ~ n ) / 2 w of ( 5 - 5 ) which has no zeros i n R f o r some p > 0 . We have P therefore arrived at a contradiction, and the theorem i s proved. Remark. If n = 1 , b(x) 0 and a ^ ( x ) = 1 , this theorem reduces to the c l a s s i c a l theorem of H i l l e c i t e d i n the proof. It i s possible to regard Theorem 9 a s the spe c i a l case p = 0 of the following n-dimensional analogue of a theorem of H i l l e [ 1 1 , p . 2 5 0 , Th . 1 2 ] : Theorem 9A. The equation ( l . l ) i s nonoscillatory i n R i f there exists a posit i v e integer p such that , 0 0 , \u00E2\u0080\u009E C 0 p .oO . oOfor s u f f i c i e n t l y large r, where S (r) = I [ L k ( r ) ] - 2 , L (x) = L (x)log x, p = l , 2 , 3 , . . , with LQ ( X ) = x> l o g 2 x = log log x, l o g p x = log log ^x. Proof. \u00E2\u0080\u00A2 The proof i s si m i l a r to that of Theorem 9 . We appeal to H i l l e ' s Theorem 12 instead of the Corollary 1 to his -29-Theorem 7 at the appropriate places. We omit the d e t a i l s . Our next theorem i s the n-dimensional analogue of a 2m-th order one-dimensional o s c i l l a t i o n theorem of Glazman [9, p.99, Th.10]. As w i l l be noted below, our r e s u l t contains a well-known c r i t e r i o n f o r n o n - o s c i l l a t i o n f i r s t proved by H i l l e [ l i ] . Theorem 10. Equation ( l . l ) i s non-oscillatory i n R i f the inequality rg!T(r')dr < co o I holds for some 6 > 0, where M6 = (r : r 2 g o ( r ) / A Q > ^ = ^ - - 6), and 6 0> 6]V A G have the meanings assigned i n Theorem 9\u00C2\u00AB Proof. Since we s h a l l use the argument' of Theorem 9> i t s u f f i c e s to show that (5.7) i s nonoscillatory. The conditions of [9> V'99> Th.10] are s a t i s f i e d , since r 2 [ g 0 ( r ) A o - ( n - l ) ( n - 3 ) / 4 r 2 ] > * - 6' for a l l r e M&. . Thus,(5.7) i s nonoscillatory. The remaining d e t a i l s are as i n Theorem 9\u00C2\u00AB \" Corollary 1. (Cf. [11, p.237, Th.2]). If b i s bounded on 0 < x < oO , the ordinary d i f f e r e n t i a l equation (a(x)y')' + b(x)y = 0 i s non-oscillatory i f x b (x) dx 1, (5.8) r 2 m \" 1 g\"|(r) d r < 0 0 \u00E2\u0080\u00A2 o If we set m = 1, n = 1 we obtain x b (x) dx < <=o , and . o this i s equivalent to the hypothesis of the co r o l l a r y , since b i s bounded on (0,co). Since g 1 ( r ) = 6 0 ( r ) when n = 1 or n = 3, Theorem 10 gives a simple c r i t e r i o n for n o n - o s c i l l a t i o n i n Corollary 2. The equation ( l . l ) i s non-oscillatory i n \"E? i f L i s uniformly e l l i p t i c and r g (r) dr < oo , J o where g^( r) = max [ g Q ( r ) , 0} and & Q( r) = m a x {^(x) : jx| Proof. Set m = 1, n = 3 i n ( 5 . 8 ) . A sim i l a r r e s u l t i s true for general n: 2 C or o l l a r y 3. Let r g G ( r ) >. A Q ( n - l ) ( n - 3 ) for large r. Then the equation ('l.l) i s non-oscillatory i n E n i f -31-r [ g ( r ) - A n ( n - l ) ( n - 3 ) / 4 r 2 ] d r < oo . o Proof. Set m = 1 i n (5-8). Remark. Each of the c o r o l l a r i e s t o Theorem 10 might have \"been deduced from the c o r r e s p o n d i n g one-dimensional theorem of H i l l e [11, p.237, Th.2] hy u s i n g the method of Theorem 9. I t should be noted t h a t we have improved H i l l e ' s r e s u l t even i n the one-dimensional case, s i n c e the use of the C l a r k -Swans on comparison theorem enables us to remove the r e q u i r e -ment t h a t the c o e f f i c i e n t b(x) by e v e n t u a l l y p o s i t i v e , on account of the i n e q u a l i t y g o ( r ) > b ( x ) , x e S r, \u00C2\u00BB0 < r < oo , and the f a c t t h a t the hypotheses imply the n o n - o s c i l l a t i o n of e q u a t i o n (5.7). Our next theorem, a g e n e r a l i z a t i o n of a remark of P o t t e r [21, p.468], i s u s e f u l in. the a p p l i c a t i o n s of the theory. In f a c t , we s h a l l use i t t o show t h a t the estimates i n s e c t i o n 4 are. the b e s t p o s s i b l e of t h e i r k i n d . F o l l o w i n g P o t t e r , we i n t r o d u c e the f o l l o w i n g n o t a t i o n . Let r\ be a 2 p o s i t i v e ' C f u n c t i o n d e f i n e d by (5.9) M r ) . ] \" 2 = A ; 1 g 1 ( r ) * - ( n - l ) ( n - 3 ) / 4 r 2 , 0 < r < oO . Let the- f u n c t i o n s G. and G~ be d e f i n e d by -32-G , ( r ) r | ( r ) \" 1 1 [ V ( r ) ] 2 + V ( r ) 4n(r) + 2 h ' ( r ) ] 2 T)'(r) where primes denote d i f f e r e n t i a t i o n with respect to r. Theorem 11. Let L be uniformly e l l i p t i c i n R for some s > 0, A q being the e l l i p t i c i t y constant. Let g-^(r) be the maximum of the p o s i t i v e part- of b(x) on S , 0 < r < oo. Then equation ( l . l ) i s non-oscillatory i n R i f there exists 2 a p o s i t i v e C function r\ s a t i s f y i n g (5-9) for large r and either-holds f o r large r. Proof. We use the argument of Theorem 9, except that we appeal to the remark of Potter mentioned above and the Sturm comparison theorem [5, p.208] to show that the equation (5.7) i s non-oscillatory. To see t h i s , we note that the following ordinary d i f f e r e n t i a l equations are o s c i l l a t o r y or nonoscillatory together: (5.10) G, (r) < 0 or (5.11) G 2 ( r ) < 0 (5.12) w\" + [ri(r)] -2 w = 0, (5.13) h 2 ( r ) v ' ] ' + v = 0, -33-( 5 . 1 J 0 [ri(r)z' ]' + G 1 ( r ) z = 0,' ( 5 . 1 5 ) h ( r ) y ' ]' + G 2(r)y = 0, for the derivative of a solution of ( 5-12 ) is a solution of ( 5 . 1 3 ) , equation (5.14) i s obtained from ( 5 . 1 2 ) by the subs-titu-t i o n z = [ r i ( r ) ] - l ! w, and ( 5 . 1 5 ) i s obtained from ( 5 . 1 3 ) by the su b s t i t u t i o n y = [r|(r)] K v. If the hypothesis ( 5-10 ) (or ( 5.II)) holds, the equation ( 5-l4) (or ( 5 . 1 5 ) ) i s non-o s c i l l a t o r y by'the Sturm comparison theorem [ 5 ] \u00C2\u00AB Thus the equation ( 5 . 1 2 ) i s non-oscillatory, and therefore (5-7 ) i s also non-oscillatory. The remaining d e t a i l s of the proof of this theorem are similar to those of Theorem 9 and w i l l be omitted. 6. Sharpness of the r e s u l t s . In t h i s section we s h a l l give examples to i l l u s t r a t e the theory. In p a r t i c u l a r , we s h a l l show that the l i m i t conditions of Kneser-Hille type i n sections 4 and 5 are the best possible of t h e i r kind. Our f i r s t example shows that the estimates (4 . 1 ) and (4.4) are sharp for each posi t i v e integer n, i . e . there exists a non-oscillatory equation for which equality holds i n (4 .1 ) and (4.4). Example 1. For each posi t i v e integer n there i s a non-o s c i l l a t o r y equation for which l i m i n f r 2 g(r) = A ^ X + ( n - 2 ) 2 / 4 ] r-* 00 -34-and l i m i n f r r\u00C2\u00B0\u00C2\u00B0 g ( t ) dt = f \ 1 [ \ a + (n-2) 2/4] r-\u00C2\u00BB oo r 2 C o n s t r u c t i o n . Let L be the Schrodinger operator v + b ( x ) , ,x e E n , with (6.1) b(x) = ( n - 2 ) 2 / ( 4 r 2 ) , -|x| = r . Then A =1, X = 0. Simple computations show that JL vX \u00E2\u0080\u00A2n(r) = 2r, G- L(r) = 0. Thus eq u a t i o n ( l . l ) i s n o n - o s c i l l a t o r y by Theorem 11. On the other hand, s i n c e b i s g i v e n by (6.1), we have l i m i n f r 2 g ( r ) = (n-2) 2/4 = A , [X + (n-2) 2/4] \u00E2\u0080\u00A2 r - co a A l s o , f\u00C2\u00B0\u00C2\u00B0 (r\ P ^ 2 l i m i n f r g ( t ) d t = K J> l i m i n f r-\u00C2\u00BB oo J r r \u00E2\u0080\u0094 co \" r = (n-2) 2/4 = A 1[\ a+(n-2) 2/4]. Our next example shows t h a t the estimates (5.1) and (5.4) are sharp f o r every p o s i t i v e i n t e g e r n. In the s p e c i a l case n = 1 t h i s reduces to the example g i v e n by H i l l e [11]. Example 2. For each p o s i t i v e i n t e g e r n there i s an o s c i l l a t o r y 2 2 equation f o r which l i m sup r g ( r ) = (n-2) A /4. r-* oo 0 \u00C2\u00B0 -35-2 C o n s t r u c t i o n . L e t L be the S c h r o d i n g e r o p e r a t o r v + b ( x ) , x \u00E2\u0082\u00AC E n , and t a k e (6.2) b ( x ) = ( n - 2 ) 2 / 4 r 2 + Y l / ( . r l o g r ) 2 , where |x| = r and y^ i s . a number s a t i s f y i n g y^ ^ Then i t i s c l e a r t h a t l i m sup r 2 g ( r ) = (n-2) 2/4 = _ ( n-2) 2 A q/4, r-\u00C2\u00BB oo s i n c e A = 1 i n t h i s c a s e . On the o t h e r h a n d , our e q u a t i o n o has p a r t i c u l a r s o l u t i o n s ( v i z . s o l u t i o n s d e p e n d i n g on r (2 - n )/2 s a l o n e ) o f the f o r m u = r^ ; / ( l o g r ) , where s i s a 2 ' r o o t o f the q u a d r a t i c e q u a t i o n s - s + = 0.' S i n c e Y-j > -^ p t h i s q u a d r a t i c e q u a t i o n has complex r o o t s , and t h u s t h e r a d i a l f o r m o f ( l . l ) i s o s c i l l a t o r y . The argument of Theorem 1 t h e r e f o r e shows t h a t e q u a t i o n ( l . l ) i s o s c i l l a t o r y . I t i s a l s o c l e a r t h a t i f t h e f u n c t i o n b i s d e f i n e d by (6.2), we have l i m sup r r-\u00C2\u00BB co 0 0 i g \u00C2\u00A3 ( t ) d t r c 0 0 P p \u00E2\u0080\u00A2lim sup r [1/41 + y-,/(t l o g t ) ] d t r-*oO However , we have shown t h a t ( l . l ) i s o s c i l l a t o r y i n t h i s c a s e . Remark . Examples 1 and 2 have shown, i n p a r t i c u l a r , t h a t \" f o r t h e S c h r o d i n g e r o p e r a t o r i n E t h e c o n s t a n t (n-2) /4 -36-is c r i t i c a l for each positive integer n, i . e . there exist both o s c i l l a t o r y and non-oscillatory equations for which equality holds i n the estimates (4.1), (4.4), (5-1) and (5.4). We s h a l l now give an example to i l l u s t r a t e the e f f e c t of relaxing one or both of the conditions (3.1) i n the n-dimensional form of the Leighton-Wintner theorem. Example 3. There exists a nonoscillatory^ equation for which n xd r _ 3, then CO dr . n-1 ~ , \ 1 r f ( r ) < oo and [\u00C2\u00B0\u00C2\u00B0 r n \" 1 [ g ( r ) - X r-. 2 f ( r ) ] d r = T\u00C2\u00B0\u00C2\u00B0 - n \" 1 - - r 1 1 r\"-\u00C2\u00B1[e-1 + ( n - 2 ) 2 / ( 4 r 2 ) ] d r On the other hand, a straightforward computation shows that -37-( i n the notation of Theorem 11) G1i r) < 0 f o r l a r S e r .> s o that equation ( l . l ) i s nonoscillatory by Theorem 11. If n < 2, then dr J l r ^ 1 f ( r ) \u00C2\u00B0o and r Cg(r) - X r ~ ^ f ( r ) ] d r = | r n _ 1 [ e ~ r + ( n - 2 ) d / ( h r ) ] d r < \u00C2\u00ABD 1 a J 1 However, since [r\(r)]~ = e~ + l / ( 4 r ) a routine computation shows that G^(r) < 0 for large r, and therefore equation ( l . l ) i s nonoscillatory i f b i s given by (6.3) and a... - 6. .. Remark. This example also throws some l i g h t on the o s c i l l a t o r y behaviour of the one-dimensional equation [a(x)u']' + b(x)u = 0. In p a r t i c u l a r , the equation \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ( x V ) ' + x s [ e ~ x + ( s ~ l ) 2 / ( 4 x 2 ) ] u = 0, (s a nonnegative integer), i s nonoscillatory on 0 < x < oQ We continue our inspection of the hypotheses of the theorems i n this chapter with the following Remark. Example 1 provides us with a nonoscillatory equation for which . '\u00C2\u00B0\u00C2\u00B0 ) ~ \u00C2\u00B0\u00C2\u00B0 (Potter's condition). This i l l u s t r a t e s 1. the e f f e c t of relaxing one of* the conditions ( 3 . 6 ) . In f a c t , r00 dt i f b is . given by ( 6 . 1 ) , then h(r) = 2 r , so that F[\"t) = \u00C2\u00B0\u00C2\u00B0 However. \u00E2\u0080\u00A2, H (r) s 0, so that H, (t) dt < oO, 1 -38-In the remark following Theorem 4 (the n-dimensional form of a r e s u l t of Potter [21 ] ) we noted that the requirement that the function h be posi t i v e was i n general more r e s t r i c t i v e than the corresponding conditions i n Theorem 3 (the n-dimensional form of the Leighton-Moore-Wintner Theorem). However, our next example shows that there are equations for which Theorem 4 gives information not immediately obtainable from Theorem 3-Example 4 . The d i f f e r e n t i a l equation ( 6 . 4 ) v 2u + [ ( n - 2 ) 2 / ( 4 r 2 ) + l / ( 4 r 2 log r)]u = 0 i s o s c i l l a t o r y on E^1, even though condition (3-4) does not hold. To see t h i s , we note that (with the notation of Theorem. .4) [ h ( r ) ] \" 2 = l / ( 4 r 2 ) + l / ( 4 r 2 log r ) . routine computation shows that Hg(r) = 0 ( r _ 1 ) for large A r, and therefore H\u00E2\u0080\u009E(r)dr = oo (s > l ) . Thus equation s ( 6 . 4 ) is. o s c i l l a t o r y on account of Theorem 4 and the remark following Theorem 1 regarding the l i m i t s of integration. Except i n the case n = 2 , this cannot be concluded i n an obvious way from Theorem 3 , since (for s > l ) \u00C2\u00B0\u00C2\u00B0 r 1 - 6 [g(r) - X r \" 2 f ( r ) ] d r = C r 1 \" 6 [-^ + ]dr < oo s ' a - s 4 r 4 r log r for each 6 > 0. -39-Equation (6.4) also s a t i s f i e s the conditions of Theorem 7. In f a c t , i n this example A-^ = 1 , X. = 0 , so .that g(t) - A1[\a + ( n - 2 ) 2 / 4 ] t \" 2 = l / ( 4 t 2 log t ) . Thus J t [ g ( t ) - A ] > a + ( n - 2 ) V 4 ] t - 2 } d t = j 4 t t = cO This shows that we might have used Theorem 7 instead of Theorem 4 to show that equation (6.4) i s o s c i l l a t o r y . - 40 -CHAPTER II EQUATIONS OP ARBITRARY EVEN ORDER 7. Preliminaries. The d e f i n i t i o n of an o s c i l l a t o r y equation given here reduces to Glazman's i n the case of ordinary equations of order 2m (m = 1, 2, ...)(of. [9, p. 40]). The c r i t e r i a obtained below f o r 2m-th order p a r t i a l d i f f e r e n t i a l equations are di r e c t generalizations of corresponding r e -sults of Glazman [9] f o r the ordinary d i f f e r e n t i a l equation ,2m ( - l ) m d - ^ L - b(x)u = 0 , 0 < x < oo . dx We also obtain f o r fourth-order p a r t i a l d i f f e r e n -t i a l equations c r i t e r i a extending those obtained, by Leigh-ton and Nehari [17] f o r the ordinary d i f f e r e n t i a l equation 2 2 ^ [a(x) - b(x)u = 0 , a(x) > 0 . dx\" dx\" Most of our theorems are proved by appealing to t h e i r one-dimensional forms. 'An exception i s Theorem 20, which i s proved by inv e s t i g a t i n g d i r e c t l y the solutions of - 41 -the comparison equation. 8. Definitions and notation. We s h a l l consider the l i n e a r e l l i p t i c differen-t i a l operator L defined by (8.1) Lu = ( - l ) m S_ D i m ( a i j D\u00E2\u0084\u00A2u) - bu a \u00C2\u00B1 J = a J \u00C2\u00B1 , i ) j\u00E2\u0080\u00941 on unbounded domains R i n n-dimensional Euclidean space E n . We s h a l l use the notations of Chapter I, except where otherwise indicated. The c o e f f i c i e n t s a ^ are assumed to be r e a l and of class C m i n R U dR and the matrix ( a i j ) i s p o s i t i v e d e f i n i t e i n R . The c o e f f i c i e n t b i s assumed to be r e a l and continuous on R U 3R . The domain D(L) of L i s defined to be the set of a l l real-valued functions on R U SR of class C (R) . D e f i n i t i o n . A solution of the equation Lu = 0 i s a function u e D(L) which s a t i s f i e s the equation every-where i n R . We assume that R contains a cylinder of the form . G x ( x n : 0 < x n < oo} , - 42 -where G i s a bounded (n-1)-dimensional domain. The following notation w i l l be used: R r = R n (x e E n : |x[ > r} . D e f i n i t i o n . A bounded domain N c R i s said to be a nodal domain of a n o n t r i v i a l s o l u t i o n u of Lu = 0 i f f u and i t s p a r t i a l derivatives of order \u00E2\u0080\u00A2 <_ m-l vanish on dN . D e f i n i t i o n . The d i f f e r e n t i a l equation Lu = 0 i s said to be o s c i l l a t o r y i n R i f f there exists a n o n t r i v i a l s o l u t i o n u r of Lu = 0 with a nodal domain i n R^ f o r a l l r > 0 . D e f i n i t i o n . A function u i s said to be o s c i l l a -tory i n R i f f ' u has a zero i n R p f o r a l l .r > 0 . We s h a l l also use the standard notation CC CL CL D au = D^1 D 2 2 D n n u , a = (c^, a n ) , |a| Our o s c i l l a t i o n c r i t e r i a i n \u00C2\u00A710 w i l l be proved under the assumption that the largest eigenvalue A ( X ) of ( aj_j) i s bounded: For some s > 0 there exists such that A ( X ) < . A , , f o r a l l x e R . x s Let n be the smallest eigenvalue of the problem n = 2 a . . - 43 -r (-1) S D . m ( A n D,m cp) = ucp i n G i = l 1 1 1 (8.2) J I Dacp = 0 on dG , |a[ = 0, 1, ra-1. . Let the function g e C(0,co ) be such that s C ^ ) <. 1 \u00C2\u00B0 ( x ) on each bounded subdomain of R . (For example, on a bounded domain G^ c R we might set o to g(t) = min (b(x) : x e G Q and x n = t} .) 9 . A u x i l i a r y r e s u l t s . As i n Chapter I, we s h a l l make use of a monotoni-c i t y p r i n c i p l e f o r eigenvalues, which we s h a l l deduce from a form of Poincare's inequality.^ D e f i n i t i o n . We s h a l l say that a domain 0 has bounded width < d i f f there i s a l i n e I such that each l i n e p a r a l l e l to I intersects fi- i n a set whose diameter i s no greater than d . For example, the truncated cone o r\" section 3 has bounded width < t sec a , and the (open) cylinder G,(t) of section 10 has bounded width <_ t . Lemma (Poincare's i n e q u a l i t y ) . If a domain 0 has bound-ed width <. d , then - 2J4 -m-j ( 9 . 1 ) | c p | j j Q < Y d |cp| m^ f o r a l l cp \u00E2\u0082\u00AC C\u00E2\u0084\u00A2(fi) , 0 <_ j <_ ra-1 , where Y i s a constant depending only on m and n , and M m a = [ J , ? ^ 2 * * ] 1 / 2 m , u J 0 |a|=m Proof. : This i s given on pp. 73-75 of [ 1 ] . We s h a l l state our next r e s u l t f o r the simplest boundary value problem, but the method c l e a r l y works f o r general boundary conditions of the kind given i n [ 2 4 ] . On account of the form of the Poincare inequality here c i t e d , the r e s u l t (a monotonicity p r i n c i p l e f o r eigenvalues) w i l l be obtained f o r a more general operator than ( 8 . 1 ) . Let the l i n e a r e l l i p t i c d i f f e r e n t i a l operator M be defined by ( 9 . 2 ) Mu a ( - i ) m 2 D^) - B u > |p|=|q|=m ., ^ where p = (p 1, \u00E2\u0080\u00A2.., P n) , q = ( q ^ \u00E2\u0080\u00A2 q n ) are multi-indices with i n t e g r a l nonnegative components. As usual, n n o -\u00C2\u00A3U p i |pj = E p , . For each z e E we write zx = \J z. . 1=1 1 i = l 1 The c o e f f i c i e n t s A^^(x) are supposed to be of class C\" (R) - 45 -and symmetric i n the in d i c e s . Following Browder [3], we c a l l M e l l i p t i c i f the following two conditions are f u l f i l l e d : (a) The form 2 A (x) z p + q i s posi t i v e |ph|q|=m P q d e f i n i t e at each point x e R . (b) For each bounded domain G with G c R there exists a number U (G) > 0 such that o 2 A lP\i D qu dx > \x(G)\ 2 (L^u) 2 dx f o r a l l u e C m(R) . We note that condition (a) i s the usual e l l i p t i c i t y condition. If the operator M i s ' o f the form (8.1), then condition (b) i s redundant, and i n f a c t i s a consequence of condition-(a). To see t h i s , we note that i f M has the form (8.1), then a. . i f p = (me.,) and q = (me.) , A = \ p q 0 otherwise, where (me^) i s the vector i n E n with m i n the i - t h place and zeros elsewhere. If condition (a) holds, then there exists a number 1^0(G) > 0 \"such that - he -n m m i n f i n f Z a. (x) z m z m = u (G) , xeG |z|=l i , j = l J J s i n c e G i s compact and the c o e f f i c i e n t s a^^ are con-t i n u o u s . Hence n m m v . . i i2m \u00E2\u0080\u009E n . , z e E T h i s i m p l i e s 5 a . . z , m z . m > u (G) S Z 2 m , z e E n i , j = l ^ \u00C2\u00B0 \u00E2\u0080\u00A2 i = l 1 T h i s may be v / r i t t e n i n t h e f o r m (*) S a S \u00C2\u00B1 \u00C2\u00A7 1 > u (G) 2 \u00C2\u00A7 2 , i , j = l i = l where \u00C2\u00A7 i s the v e c t o r (\u00C2\u00B1ZJ_ M) , the s i g n s b e i n g the same as t h o s e of the c o r r e s p o n d i n g components i n the v e c t o r z . E v e r y v e c t o r \u00C2\u00A7 e E n may be w r i t t e n i n the above form;. t h e r e f o r e c o n d i t i o n (*) i m p l i e s S a \u00C2\u00A7 g > u (G) . 2 S 2 \u00C2\u00A7 e E n , i , j = l 1 J 1 J 0 i = l 1 and t h i s i m p l i e s c o n d i t i o n ( b ) . V. - 47 -Monotonicity P r i n c i p l e . For 0 < t < oo l e t G^ be a domain contained within a domain fi of bounded width <_ t. If 0 < r < s < o o implies G p c G g , G r ^ G g , then the f i r s t eigenvalue ^ Q ( t ) of the problem Mu = Xu i n G, ; D au = 0 on 5G, , |a| = 0, l,..,m-l i s monotone decreasing i n t , and lim -X (t) = + oo . t - o + \u00C2\u00B0 Proof. For the f i r s t part we may adapt the argument i n il, PP* 400-401]. For the second part we note that since B i s uniformly continuous on G , there exists a constant k. > -co such that o Bu dx > k 'G u 2dx f o r a l l u e C m(R) G Let the Euler-Jacobi f u n c t i o n a l corresponding to M be defined by J G [ u ] ( 2 A L^u d qu - Bu2} dx . J G |p| = |q|=m P q Then, condition (b) implies that ( 9 - 3 ) J P[u] > n _ ( G ) f 2 . (D pu) 2dx + k f u 2dx u \u00E2\u0080\u00A2 G-' | p | =m 0 J n G Extend u continuously to a l l of fi by s e t t i n g u = 0 - 48 -outside G . Apply P c i n c a ^ ' s inequality with j = 0 i n (9.1) to obtain u^ dx < Y V m [ 2 \Jpu\d dx] . J f i |pj=m Hence f u 2dx < Y 2 t 2 m [f E (DP U) 2 dx] J G J G |p|=m Combining t h i s with inequality (9*3) we get JQ[u] > (kQ + u o ( G ) / Y 2 t 2 m ) u dx The remainder of the proof i s as i n section 2. (We note that Y >' 0 , since ||u|| Q ^ 0 .) 10. O s c i l l a t i o n c r i t e r i a . In t h i s section we obtain o s c i l l a t i o n theorems under the hypothesis that the largest eigenvalue of (a. .) J i s bounded. Our theorems generalize r e s u l t s of Glazman [ 9 ] f o r one dimension to the n-dimensional case. They may be s p e c i a l i z e d to a l l of E n by taking cp 3 1 , |i = 0 . - 49 -THEOREM 1 0 . The d i f f e r e n t i a l equation ( 1 0 , 1 ) Lu = ( - l ) m 2 D,m (a, . D,mu) - bu = 0 i s o s c i l l a t o r y i n R i f f o r some s > 0 there exists a number A-. > 0 such that A ( X ) <. A. , f o r a l l x e R_ , J. JL S and i f rco ( 1 0 . 2 ) [g(t) + u] dt = + oo , o where u and g are defined above i n section 8 . Proof. We compare ( 8 . 1 ) with the separable equation ( 1 0 . 3 ) ( - l ) m S D , M ( A 1 D, mv) - b*v = 0 i = l 1 ' 1 where b*(x) = g(x n) . The hypothesis ( 1 0 . 2 ) implies that the ordinary d i f f e r e n t i a l equation o ( 1 0 . 4 ) ( - l ) m A X D n 2 mw - [g(x n) + n]w = 0 i s o s c i l l a t o r y on account of Glazman's generalization [ 9 , p. 1 0 4 , Th. 13] of the theorem of Leighton [16] and - 50 -Wintner [ 2 8 ] . Let r > 0 be given. Then there exists a soluti o n w of (10.4) with zeros of order m at x n = 5 1 ' 62 J where 6 2 > 6.^ >_ max (r,s] . If cp i s an eigenfunction of (8 .2) corresponding to the eigenvalue . n , then the function v defined by v(x) = w(xn)cp(x) , where x = (x^, x 2 , ..., x n_]_) * i s a solution of (10.3) by d i r e c t c a l c u l a t i o n , with a nodal domain G \u00C2\u00B1 = G x ( x n : 6 l < x n < 62} In f a c t , i f a = (a^, a 2 , a^) = (a,a n) , then a a a \u00E2\u0080\u0094 _ D v = D n D v = D n w(x )D a \u00C2\u00AE(x) . n n . n' v \u00E2\u0080\u00A2 ' Hence D av = 0 on bG^ f o r |a| = 0, 1, m-l, since 6. and 6 0 are m-fold zeros of w(x ) and cp has nodal 1 2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 v n' . domain G . Thus v has a nodal domain N c R f o r a l l r r r > 0 . In f a c t , x \u00E2\u0082\u00AC G^ implies |x| _> |x n| > 6^ >_ r , hence x e R r . Thus (10.3) i s o s c i l l a t o r y . We now appl; a theorem of Swanson [24, Th. 4 ] . The inequality f ( E (An 6 - a, ,)D, mv D, mv + (b - b*)v 2} dx > 0 JG1 i , j = l 1 1 J 10 ^ i J holds whenever G i c R s > o n account of the hypotheses - 51 -2 n S a. .(x)z,z . <_ A \u00C2\u00A3 z, , .x e R_ , z e E i , j = l 1 J 1 J 1 1=1 1 s b*(x) = g(x n) < b(x) , x e G1 . Hence the eigenvalue problem Lu = Xu i n ; D au = 0 on 3G 1 , |a| = 0_, 1, m-l has at least one (in p a r t i c u l a r , the smallest) eigenvalue less than or equal to zero. Let G(t) = G x [ x n : < x n < t) , 6 1 < t < 6 2 ,. and l e t ^ 0 ( t ) denote the smallest eigenvalue of the pro-blem Lu = X(t)u i n G(t) ; D au = 0 on 3G(t) ja| = 0,1,...,m By the monotonicity p r i n c i p l e i n section 9, ^0(t) i s mono tone nonincreasing i n 6n < t <_ 6 0 and lim X (t) = + oo Since ^ Q ( 6 2 ^ \u00E2\u0080\u0094 0 * t h e r e exists a number T i n (6^,62] such that X Q(T) = 0 . This means that G(T) i s a nodal - 52 -domain of a n o n t r i v i a l s o l u t i o n of (10.1), and since G(T) c G 1 c f o r a r b i t r a r y r > 0 , equation (10.1) i s o s c i l l a t o r y i n R . This completes the proof of the theorem. This theorem contains Glazman's generalization [9, p. 104] of the Leighton-Wintner theorem. To see t h i s , set n = 1 and a^(x) = 1 , and r e c a l l that f o r R = E n we may take cp s 1 , n = 0 . Our next theorem extends to n dimensions Glazman's generalization [9, p\u00C2\u00BB 100] of a r e s u l t of H i l l e [11, Th. 5] . Glazman's r e s u l t i s the sp e c i a l case R = E n n = 1 , a-^x) = 1 of our theorem. THEOREM 11. Let A ( X ) be bounded, i . e . f o r some s > 0 there exists A^ > 0 such that A ( X ) <_ , x e R g I f s ( x n ) + M >. 0 f o r large x n and 9 , \u00C2\u00BBoo 2 l i m sup v*m~x J [g(t)+u]dt > A 1 . A m , where m t i\k-l/m-l\ A - l _ J2^1 ? (k-l> m \" (m-l)J k=l \u00E2\u0080\u00A2 2m -,k then the equation (10.1) i s o s c i l l a t o r y . - 53 -Proof. As i n the proof of Theorem 1 0 , we compare ( 1 0 . 1 ) with ( 1 0 . 3 ) , quoting Glazman's generalization [ 9 , P\u00C2\u00AB 100] of a theorem of H i l l e [ 1 1 , Th. 5] to show that (10A) i s o s c i l l a t o r y . The remainder of the proof follows that of Theorem 10 without change. THEOREM 1 2 . I f the ine q u a l i t y ( 1 0 . 5 ) g(* n)-Hi > A X a 2 xn~2m ( a m = ( 2 m-l ) U / 2 m ) -ho lds f o r s u f f i c i e n t l y l a r g e x and 0 ( 1 0 . 6 ) l i m sup (log r ) [ t 2 m _ 1 1 g ( t ) + i i - A . . a 2 t \" 2 m | dt = 0 0 r - 0 r x m then equation ( 1 0 . 1 ) i s o s c i l l a t o r y i n R . Proof. The hypotheses imply that the ordinary\"differen-t i a l equation ( 1 0 . 4 ) i s o s c i l l a t o r y on account of a theorem of Glazman [ 9 , p. 1 0 2 ] . The remainder, of the proof i s simi l a r to that of.Theorem 10 and w i l l be omitted. Corollary. Let the largest eigenvalue A ( X ) of (a i^) be bounded i n R g f o r some s > 0 : A(x) <_ A^ , f o r a l l x e R g and some A^' > 0 . Then ( 1 0 . 1 ) i s o s c i l l a tory i n R i f f o r s u f f i c i e n t l y large x n and some 5 > 0 - 54 -the i n e q u a l i t y x n 2 m C s( xn> + M ] > A l ^ a m 2 + 6 ) i s s a t i s f i e d . Proof. The f i r s t hypothesis ( 1 0 . 3 )'of Theorem 12 i s c l e a r l y f u l f i l l e d . Moreover, since g(t-) + |i - A i a m 2 t \" 2 m > A 1 6 t ~ 2 m f o r large t , the second hypothesis ( 1 0 . 6 ) of Theorem 12 i s also s a t i s f i e d . Remark. This r e s u l t generalizes the c l a s s i c a l Kneser-H i l l e theorem [11] i n four d i r e c t i o n s : ( i ) to equations of a r b i t r a r y even order, ( i i ) to. equations with variable l e a d i n g . c o e f f i c i e n t s , ( i i i ) to n dimensions, (iv) to equations not defined on a l l of E n ( i . e . on l i m i t -c y l i n d r i c a l domains). It i s possible to prove t h i s c o r o l l a r y by com-paring ( 1 0 . l ) with ( 1 0 . 3 ) , c i t i n g Glazman's generalization [ 9 , Th. 9 , p. 96] of the Kneser-Hille theorem [11] to show that the ordinary d i f f e r e n t i a l equation ( 1 0 . 4 ) i s o s c i l l a -tory. I t should be noted .that the r e s u l t of Glazman just c i t e d [ 9 , Th. 9 , p. 96] i s the s p e c i a l case R = E n , - 55 -n = 1 , a n ( x ) ~ 1 o f o u r c o r o l l a r y . 11. Equations with one variable separable. In t h i s section we s h a l l consider the equation (11.1) D n 2 mu+ V D\u00C2\u00B1m[ai:s(x)D.mu] - ( - l ) m b ( x n ) u = 0 , i j j\u00E2\u0080\u00941 where x = (x-,, Xg , \u00E2\u0080\u00A2.\u00E2\u0080\u00A2 } x _ n ) . Following Swanson [26] we l e t |i* be the smallest eigenvalue of the problem ( - l ) 1 * \" 1 V D i m [ a i j ( x ) D j m c p ] = n*cp i n G (11.2) ^ -D cp = 0 on dG , |a| =0, 1, ...} m-l , where G i s as i n section 8. . Each of the theorems of the preceding section has an analogue i n t h i s case, but without the assumption that A ( X ) i s bounded. As an example ire state and prove the following analogue of Theorem 10: THEOREM 10A. The d i f f e r e n t i a l equation (11.3) (-l) m[D 2 % + D \u00E2\u0084\u00A2 [ a i 1 ( x ) D A i ] } - b ( x j u = 0 1 i , j = l n - 56 -i s o s c i l l a t o r y i n R i f f.00 (11.4) [b(t)+|i*] dt = + oo . J o Proof. The hypothesis (11.4) implies that the ordinary d i f f e r e n t i a l equation (11.5) ( - D m D n 2 m v - [b(x n)-Hi*] v = 0 ' . i s o s c i l l a t o r y on account of Glazman's generalization [ 9 , p. 104] of the theorem of Leighton [16] and Wintner [ 2 8 ] . Let r > 0 be given. Then there exists a s o l u t i o n v of (11.5) with zeros of order m at x n = 5 - j _ , s 2 > where 6 2 > 51 \u00E2\u0080\u0094 r * I f ^ i s a n e i S e n : \u00C2\u00A3 > u n c t i o n of (11.2) corres-ponding to eigenvalue u* , then the function u defined by u(x) = v(x n)cp(x) i s a so l u t i o n of (11.3) by.direct c a l -c u l a t i o n , with a nodal domain , . G 1 = [x : x e G , 6 1 < x n < 6 2] . Thus there exists u with a nodal domain G.^ c R^ f o r a r b i t r a r y r > 0 , since . x e G-j^ implies |x| >_ |x n| > 6^ >. r , so that x e R^ . Hence \u00E2\u0080\u00A2equation (11..3) i s o s c i l l a t o r y . The analogues of Theorems 11 and 12 f o r equation (11.3) are proved s i m i l a r l y . We state them without proof. - 57 -THEOREM 11A.- I f b(x n) + u* > 0 f o r large x n and 2m-l oo lim sup v*m~x [b(t)+n*] dt > A 2 , \u00C2\u00ABJ III r -\u00E2\u0080\u00A2 oo r then the equation (11.3) i s o s c i l l a t o r y i n R . THEOREM 12A. The equation (11.3) i s o s c i l l a t o r y i n R i f the inequality b(x R) + \i* > x n \" 2 m holds f o r large x n and 00 l i m sup (log r)f t ^ ' ^ b f t ^ * ^ 2 t ~ 2 m | - dt = oo r -\u00C2\u00BB oo \u00E2\u0080\u00A2 r Remark. Swanson [23] has obtained o s c i l l a t i o n c r i t e r i a f o r the second order separable equation n-1 . _ D n[a(x n)D nu] + _ 2 D i[a l j.(x)D ju], + b ( x n ) u = 0 . i f J = I Lack of suitable one-dimensional o s c i l l a t i o n theorems has r e s t r i c t e d us to equations with a(x n) s 1 i n the general _ even order case. 12. Fourth-order equations on l i m i t - c y l i n d r i c a l domains. In t h i s seation we s h a l l derive o s c i l l a t i o n c r i -- 58 -t e r i a f o r the equation (12.1) D n^[a(x n)D n 2u]+ V D ^ a ^ U j D ^ u ] - b(x n)u = 0 , i j j = l where the c o e f f i c i e n t s a , a^^ , b s a t i s f y the conditions i n section 8 . We also suppose that a(x ) and b(x n)+u* are p o s i t i v e f o r large x n . Our theorems constitute ex-tensions of well-known r e s u l t s of Leighton and Nehari [17] \u00E2\u0080\u00A2 THEOREM 13. Let a. be an a r b i t r a r y r e a l constant, and l e t a(x ) > 0 , b(x n)-Hi* > 0 f o r s u f f i c i e n t l y large x n . Then equation (12.1) i s o s c i l l a t o r y i n R i f /-lim sup t \" 2 \" a a ( t ) < 1 (12.2)< t -\u00C2\u00BB oo ,.2x2 .lim i n f t 2 \" a [ b ( t ) + u * ] >:\u00E2\u0080\u00A2 I1:?' , t -\u00E2\u0080\u00A2 CO where | i * i s given by (11.2). Proof. The hypotheses (12.2) imply that the ordinary d i f f e r e n t i a l equation (12.3) D n 2 [ a ( x n ) D n 2 v ] - [b(xn)..+ u*] v = 0 - 59 -has an o s c i l l a t o r y solution, i . e . has a soluti o n with i n -f i n i t e l y many zeros, on account of a theorem of Leighton and Nehari [17, Th. 6.2]. Let r > 0 be given. By another theorem i n the paper just c i t e d [17, Th. 3\u00C2\u00AB6] there exists a solution v of (12 .3) with double zeros at x n = 6^, 6^ , where 6^ > >_ r . The remainder of the proof follows that of Theorem 10A without change and w i l l be omitted. one-dimensional theorem of Leighton and Nehari [ 1 7 , Th. 6 . 2 ] just c i t e d . [ 1 7 , Theorem 6 . 2 ] shows that the conclusion of Theorem 13 holds i f the hypotheses ( 1 2 . 2 ) are replaced by This r e s u l t extends to n-dimensions part of the In the case that a(x ) s 1 , another part of lim i n f t H [ b ( t ) + u * ] > 9/16 . t -* oo THEOREM 14. The equation (12.1) i s o s c i l l a t o r y i n R i f there exists a > 0 such that co oo Proof. The hypotheses ( 1 2 . 4 ) imply that the equation ( 1 2 . 3 ) has an o s c i l l a t o r y solution, on account of a theorem - 60 -of Leighton and Nehari [17, Th. 6.11]. The remaining d e t a i l s of the proof follow f a m i l i a r l i n e s and w i l l be omitted. In the case a(x ) =\u00E2\u0080\u00A2 1 , the above r e s u l t takes n' the following form: THEOREM 15. The equation i s o s c i l l a t o r y i n R i f there exists a > 0 such that need only show that (12.3) has an o s c i l l a t o r y solution. This f a c t i s a consequence of a r e s u l t of Leighton and Nehari [17, Cor. 6.10]. We omit the remaining d e t a i l s of the proof. Our next two theorems give o s c i l l a t i o n c r i t e r i a f o r a s p e c i a l case of (12.1), namely holds for. some r < 3 Proof. Since we s h a l l use the argument of Theorem 13, we - 61 -( 1 2 . 5 ) D n 2 [ a ( x n ) D n 2 u ] + V D 1 2[a i j.(x)D J 2u]+c(x n)u = 0 , 1, j\u00E2\u0080\u0094 1 where the c o e f f i c i e n t c i s r e a l and continuous on R , and the following i n e q u a l i t i e s hold f o r large x n : ( 1 2 . 6 ) a(x n) > 0 , c ( x n ) - u * > 0 . THEOREM 1 6 . Let s be an a r b i t r a r y r e a l constant, and suppose that the i n e q u a l i t i e s ( 1 2 . 6 ) hold f o r s u f f i -c i e n t l y large x n . . Then equation ( 1 2 . 5 ) has a solution o s c i l l a t o r y i n R If lim sup t \" 2 \" s a(t) < 1 t -\u00C2\u00BB 00 and lim i n f t 2 \" s [c(t) - n*] > s 2/4 , t - 00 where u* i s given by ( 1 1 . 2 ) . In the case a ( x n ) - 1 , the conclusion remains v a l i d i f ( 1 2 . 6 ) holds and lim i n f t ^ [ c ( t ) > 1 . t -> 00 - 62 -Proof. The hypotheses imply that the ordinary d i f f e r e n -t i a l equation has an o s c i l l a t o r y s o l u t i o n , on account of a r e s u l t of Leighton and Nehari [ 1 7 , Th. 11.1]. Let r > 0 be given. Then there exists a sol u t i o n v of (12 .7) with a zero i n . I f cp i s an eigenfunction of (11.2) corresponding to the eigenvalue u* , then the function u defined by u(x) = v(x n)cp(x) i s a solution of .(12.5) by d i r e c t compu-ta t i o n , with a zero i n R p . Since r i s a r b i t r a r y , t h i s implies that u i s o s c i l l a t o r y i n R , and the theorem i s proved. THEOREM 17. Let (12 .6) hold f o r s u f f i c i e n t l y large x n and suppose a ( x n ) s 1 \u00E2\u0080\u00A2 Then equation (12.5) has a solu t i o n o s c i l l a t o r y i n R i f there exists a > 0 such that Proof. The proof i s s i m i l a r to that of Theorem 1 6 , and appeals to a c r i t e r i o n of Leighton and Nehari- [ 1 7 , Th. 1 1 . 4 ] ( 1 2 . 7 ) D n 2 [ a ( x n ) D n 2 v ] + [c(x n) - u*] v = 0 C O a - 63 -to show that ( 1 2 . 7 ) has an o s c i l l a t o r y s o l u t i o n . We omit the d e t a i l s . 1 3 . Fourth order equations on a l l of E n . The equation to be considered i s the s p e c i a l case m = 2 of ( 1 0 . 1 ) , namely n p p ( 1 3 . 1 ) Lu s \u00C2\u00A3 D.*(a,, D u) - bu = 0* . i , j = l 1 1 J J The general conditions on L are as i n section 8 , except that the domain R w i l l be a l l of E n . The nodal domains of the comparison equation w i l l be annuli of the form (x : r ^ < |x| < r 2 ) , hence we need a s l i g h t extension of the monotonicity p r i n c i p l e proved i n Chapter I. We s h a l l say that an annulus of the form [x : r ^ < |x| < r^+t) has thickness t . Even though th i s annulus has bounded width, the l a t t e r does not approach zero as -t -\u00C2\u00BB 0+ , so that the form of the monotonicity p r i n c i p l e i n \u00C2\u00A79 i s inapplicable here. If an annulus Lemma (Poincare's inequality f o r annuli) n has thickness t , then ( 1 3 . 2 ) i u i 0 j f l < t 2 |u| 2; n 2 f o r a l l u e C (fi) , where - 64 -| u | ^ \u00E2\u0080\u00A2 ['o , a L ( D a u ) 2 3 x 1 1 / 2 Proof. In the course of proving the monotonicity p r i n -c i p l e f o r eigenvalues i n the second order case f o r annular domains, we showed (cf. ( 2 . 5 ) ) that (13.3) |u| , 2 / 4-21 |2 Applying this i n e q u a l i t y to the f i r s t p a r t i a l derivatives D^u , we obtain = t' 2 (D aD.u) 2 dx (by d e f i n i t i o n ) n |a|=l 1 = t n 2 (D.D.u)\" dx , i=l,2,...,n . J f i .1=1 J 1 Hence n = f 2 (D.u)\" dx J fi i = l 1 n 2 ^ ( D ^ u ) 2 dx = t 2 J ^ ( S . (D au) fi i , j = l fi |a|=2 dx A / - 65 -Combining t h i s with ( 1 3 . 3 ) we obtain from which ( 1 3 . 2 ) follows immediately. We now state the required form of the monotonicity p r i n c i p l e f o r eigenvalues. In view of the form of Poincare's in e q u a l i t y proved here, we state the r e s u l t f o r the more general operator ( 1 3 - 4 ) Mu s 2 iP (A D qu) - bu i | p | - | q | - 2 * The p r i n c i p l e w i l l theji be. true f o r the operator L , since L i s a sp e c i a l case of M (cf. section 9 ) . We note that the operator i n ( 1 3 . 4 ) i s the s p e c i a l case m = 2 of ( 9 . 2 ) . We s h a l l accordingly suppose that conditions (a) and (b) of section 9 are s a t i s f i e d . As noted i n section 9 , however, when we apply the monotonicity p r i n -c i p l e f o r the operator L , Its e l l i p t i c i t y alone ( i . e . con-d i t i o n (a)) i s enough to guarantee the truth of the p r i n c i p l e since condition (a) implies condition (b) i n t h i s case. Monotonicity P r i n c i p l e (Annular Domains). Let fi(t) be an annulus of thickness t . Then the f i r s t eigenvalue X Q ( t ) of the problem Mu = Xu i n fi(t) ; D au = 0 on Sfi(t), |a| =0, 1 i s monotone nonincreasing (for t > 0) and lim X (t) = +oo . t-0+ \u00C2\u00B0 Proof. The proof i s s i m i l a r to that of the corresponding r e s u l t i n section 9 and w i l l be omitted. 14. O s c i l l a t i o n theorems. The main r e s u l t of t h i s section i s a theorem of the Kneser-Hille type f o r equation (13.1). I t contains the corresponding r e s u l t of Leighton and Nehari [17] f o r the fourth order ordinary d i f f e r e n t i a l equation u ^ v - bu = 0 and extends the analogous theorem of Glazman [9] f o r an operator with harmonic leading term to one with (in p a r t i -cular) biharmonic leading term. F i r s t we need a few technical lemmas of an elementary character. We s h a l l compare equation (13.1) with the separable equation (14.1) A 1 A 2y - Bv = 0 ', where A., i s an upper bound on the largest eigenvalue A ( X ) - 67 -of the matrix (a^j(x)) ; i . e . there exists a number such that A(x) < L A^ . The continuous function B i s such that there exists a function g Q s a t i s f y i n g B(x) = g 0(|x|) < b(x) , x e E n . Notation. Let F(s,n) be the polynomial of degree four i n s defined by \u00E2\u0080\u00A2 P(s,n) = s(s-2) (s+n-2) (s+n-4) . As i n Chapter I, we introduce spherical polar coordinates r , 8^ , , ^ n _ ] _ ? By writing (14.1) i n terms of these coordinates, we f i n d that (14.1) has so-lutions (in p a r t i c u l a r ) of the form v(x) = p(r) , 0 <_ r. < co , where p s a t i s f i e s the ordinary d i f f e r e n t i a l equation (14.2) A 1 L2p - g Q ( r ) p = 0 . Proposition 18. The polynomial F(s,n) has a r e l a t i v e maximum at s = 2-n/2 . ' - 6 8 -Proof. If s = 2-n/2 i s a zero of F(s,n) , then i t must be a repeated zero, since F(s,n) i s symmetrical about s = 2-n/2 . Moreover, since F(s,n) i s a poly-nomial of degree four with posi t i v e leading c o e f f i c i e n t and at least two d i s t i n c t r e a l zeros, a consideration of the shape of i t s graph shows that the repeated zero at s = 2-n/2 i s a maximum point. If s = 2-n/2 i s not a zero of P(s,n) , we use logarithmic d i f f e r e n t i a t i o n ' to show that M i , o) P'(s,n) + 1 . 1 , u ^ o ; P(s,n) ~ s + s-2 + s+n-2 + s+n-4 where the prime denotes d i f f e r e n t i a t i o n with respect to and the formula (14.3) holds except at zeros of F(s,n) . Thus F'(2-n/2,n) = 0 . D i f f e r e n t i a t i o n of ( 1 4 . 3 ) y i e l d s F\"(s,n) = -P(s,n)[^p + \u00E2\u0080\u0094 3 \u00E2\u0080\u0094 + L \u00E2\u0080\u0094 p + 1 s^ (s-2T (a+n-2)D (s+n-4r I t follows that when F'(s,n) = 0 ( i n p a r t i c u l a r , when - 6 9 -s = 2-n/2) , F\"(s,n) has sign opposite to that of F(s,n). But P(2-n/2,n) = (2-n/2) 2(n/2) 2 > 0 , since by hypothesis F(2-n/2,n) ^ 0 . Hence P\"(2-n/2,n)<0 and the proposition i s proved. Proposition 19* I f the ineq u a l i t y (14.4) \u00C2\u00ABi) > A x n 2 ( n - 4 ) 2 / l 6 holds, then the equation (14.5) A 1 F(s,n) - cu ='0 has at lea s t one p a i r of complex roots. Proof. By Proposition 18, the polynomial' F(s,n) has a nonnegative r e l a t i v e maximum at s = 2-n/2 . Hence the polynomial F(s,n) - ou / A^ has a l o c a l maximum at s = 2-n/2 . Condition (14.4) implies that F(2-n/2,n) - \u00C2\u00ABa/A1 = n 2 ( n - 4 ) 2 / l 6 - u)/A1 < 0. . Thus the r e -l a t i v e maximum of the polynomial F(s,n) - uu/A-^ at s = 2-n/2 i s negative. Hence equation (14.5) has at least - 70 -one pa i r of complex roots. We are now i n a p o s i t i o n to state and prove the main r e s u l t of t h i s section. THEOREM 2 0 . Suppose that the largest eigenvalue A(x) of (a^j(x)) i s bounded i n E n (or at lea s t out-side some hypersphere), say A ( X ) <_ A^ . Then equation ( 1 3 . 1 ) i s o s c i l l a t o r y i n E n i f (14.6) l i m i n f r^g(r) > A n 2 ( n - 4 ) 2 / l 6 , r -> 00 where g(r). = min (b(x) : |x| = r) Proof. The hypothesis (14.6) implies that these e x i s t constants r and ID such that o r ^ g(r) > ID > A 1 n 2 ( n - 4 ) 2 / l 6 f o r a l l r > r Q . We then compare ( 13-1) with the equation (14.7) A x A 2 V - uur'V = 0 which i s the sp e c i a l case B(x) = u>|x|\"^ of (14.1). The r a d i a l form of equation ( l 4 . 7 ) i s of Euler type and thus (14 . 7 ) \"has solutions ( i n p a r t i c u l a r ) of the form v(x) = |x| s, - 71 -where s s a t i s f i e s (14.5). This i s e a s i l y seen by noting that and A r s = s(s+n-2)r \u00C2\u00A3 A 2 r s = s(s+n-2)Ar s~ 2 = s(s+n-2) (s-2) (s+n-4)r s~ i | Since u) > A 1 n 2 ( n - 4 ) 2 / l 6 , hypothesis (14.4) of Proposi-t i o n 19 i s f u l f i l l e d , and therefore equation (l4-5) has at least p a i r of one complex roots. This implies that there exists an o s c i l l a t o r y solution of the r a d i a l form of (14.7), i . e . a solution with I n f i n i t e l y many zeros. Let a > 0 be given. Then a theorem of Leighton and Nehari [17, Th. f 3.6] implies that there exists a s o l u t i o n p of the ordinary d i f f e r e n t i a l equation (14.8) A]_ A 2 p - tor \" 4 p = 0 with double zeros at r = 6 n , 6 where 6_ > 6. > max [r 1 d 2 1 O since A 2p(r) = (vn'1pu)\" + [ ( l - n ) r n \" 3 p > ] ' , so that (l4.8) may be transformed into the form considered i n [17,Th. 3-6] (Note that ( l - n ) r n ~ 3 <. 0 f o r a l l pos i t i v e integers n - 72 -and see the remark following [17, Th. 12.1]). I t follows that the function v defined by v(x) = p(x) i s a solu-t i o n of (14.7) with a nodal domain N = \u00E2\u0080\u00A2 (x : &1 < |x| < 62} . In f a c t , D v = ^ U 1 Y dr a x \u00C2\u00B1 and the r i g h t side i s zero on r = 6.^ , r = 6 g . Thus f o r any a > 0 there exists a solu t i o n v with a nodal domain i n the region [x : |x| > a) , since x e N im-p l i e s |x| > > a . Hence (14.7) i s o s c i l l a t o r y . We how apply a theorem [24, Th. 4] of C. A. Swanson. Because of the hypotheses E a. .(x)z.z . <. A ( X ) | Z | < A N |z| , x s u f f . large, z e E i , j = l 1 J 1 3 x u>r\"4 < g(r) < b(x) |x| > r , we have ^ C i , i i ( A i 5 ^ \" a i j ) D i 2 u DA + ( B - H*rV) dx > - 73 -Hence the eigenvalue problem Lu = Xu i n N ; u = L\j_u = 0 on 5N , i = 1 , 2, ..., n has at least one eigenvalue ( i n p a r t i c u l a r , the smallest) less than zero. Let N(t) = (x : 6 1 < |x| < t) , 6 1 <.t < 6 2 , and l e t X Q ( t ) denote the.smallest eigenvalue of the pro-blem Lu = Xu i n N(t) ; u = D \u00C2\u00B1u = 0 on SN(t), 1 = 1 , 2, ..., n. By the monotonicity p r i n c i p l e of section 1 3 , \u00E2\u0080\u00A2 X (t) Is monotone nonincreasing i n 6 N < t <_ 6 ~ and lim X (t) = +oo t - 6 ^ \u00C2\u00B0 \u00E2\u0080\u00A2 . Since ^ 0 ( S 2 ^ \u00E2\u0080\u0094 0 ' there exists a number T i n ( ^ i * ^ such that . X (T) = 0 . Thus N(T) i s a nodal domain of a n o n t r i v i a l s o l u t i o n of ( 1 3 . 1 ) . Moreover, N(T) C N C { X : |x| > a} and a i s a r b i t r a r y , therefore equation ( 13-1) i s o s c i l l a t o r y i n E n . Corollary (Leighton and Nehari [ 1 7 , part of Th. 6 . 2 ] ) . The ordinary d i f f e r e n t i a l equation u i v - b(x)u = 0 - Ik -i s o s c i l l a t o r y i f (14.9) l i m i n f x\>(x) > 9/16 . X -> 00 Proof. This c o r o l l a r y i s the sp e c i a l case a. . =.6 n = 1 of Theorem 20. - 75 -BIBLIOGRAPHY S. Agmon, Lectures on E l l i p t i c Boundary Value Problems, Nostrand, Princeton, 1 9 6 5 . L. Bers, P. John, and M. Schechter, Partial Differential Equations, Proceedings of the Summer Seminar, Boulder, Colorado, 1957 . F.E. Browder, The Dirichlet Problem for Linear E l l i p t i c Equations of Arbitrary Even Order with Variable Coeffi-cients, Proc. Nat. Acad. Sci. U.S.A., 38 ( 1 9 5 2 ) , pp. 230-2 3 5 . C. Clark and CA. Swanson, Comparison Theorems for \u00E2\u0080\u00A2Elliptic Differential Equations, Proc. Amer. Math. Soc. 16 ( 1 9 6 5 ) , pp. 886-390 . E.A. Coddington and N. Levihson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Interscience, New York, 1966.-P.R. Garabedian, Partial Differential Equations, Wiley, New York, 1964 . I.M. Glazman, On the Negative Part of the Spectrum of~ One-Dimensional and Multi-Dimensional Differential Operators on Vector-Functions, Do'kl. Akad. Nauk SSSR(N.S. 119 ( 1 9 5 8 ) , pp. 421-424/ I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Israel Program for Scientific Translations, Daniel Davey and Co., New York, 1 9 6 5 . - 76 -[10] V.B. Headley and CA. Swanson, O s c i l l a t i o n C r i t e r i a for E l l i p t i c Equations, P a c i f i c J. Math, (to appear). [11] E. H i l l e , Non-oscillation Theorems, Trans. Amer. Math. Soc. 64 (1948), pp. 234-252. [12] K. Kreith, A New Proof of a Comparison Theorem for E l l i p t i c Equations, Proc. Amer. Math. Soc. 14 (1963), pp. 33-35. [13] K. Kreith, O s c i l l a t i o n Theorems for E l l i p t i c Equations, Proc. Amer. Math. Soc. 15 (1964), pp. 341-344. [14] K. Kreith, Disconjugacy i n E n , SIAM J. Appl. Math. 15 (1967), pp. 767-770. [15] K. Kreith, An Abstract O s c i l l a t i o n Theorem, Proc. Amer. Math. Soc. 19 (1968), pp. 17-20. [16] W. Leighton, On Self-Adjoint D i f f e r e n t i a l Equations of , Second Order, J. London Math. Soc. 27 (1952), pp. 37-47. [17] W. Leighton and Z. Nehari,. On the O s c i l l a t i o n of Solutions of Self-Adjoint Linear D i f f e r e n t i a l Equations of the Fourth Order, Trans. Amer. Math. Soc. 89(1958), pp. 325-377. [18] S.G. Mikhlin, The Problem of the Minimum of a Quadratic Functional, Holden-Day, San Francisco, 1965. [ 1 9 ] R.A. Moore, The Behaviour of Solutions of a Linear D i f f e r e n t i a l Equation of Second Order, P a c i f i c J. Math. \u00E2\u0080\u00A2 5 (1955), PP. 125-145. - 77 -[ 2 0 ] C. B. Morrey, Lecture Notes on the Theory of Partial Differential Equations, Department of Mathematics, University of Chicago, Summer i 9 6 0 . [ 2 1 ] Ruth Lind Potter, On Self-Adjoint Differential Equations of Second Order, Pacific J. Math.3 ( 1 9 5 3 ) , pp. 4 6 7 - 4 9 1 . [ 2 2 ] S.L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, American Mathematical Society, Providence, 1 9 6 3 . [23] CA. Swanson, A Comparison- Theorem for E l l i p t i c Differential Equations, Proc. Amer. Math. Soc. 1 7 ( 1 9 6 6 ) , pp. 6 l l - 6 l 6 . [24] CA. Swanson, A Generalization of Sturm's Comparison Theorem, J. Math. Anal. Appl. 1 5 ( 1 9 6 6 ) , pp. 5 1 2 - 5 1 9 . [ 2 5 ] CA. Swanson, Comparison Theorems for E l l i p t i c Equations on Unbounded Domains, Trans. Amer. Math. Soc. 1 2 6 ( 1 9 6 7 ) , 2 7 8 - 2 8 5 . [26] CA. Swanson, An Identity for E l l i p t i c Equations with Applications, Trans. Amer. Math. Soc. (to appear). [27] CA. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1 9 6 8 . [ 2 8 ] A. Wintner, A Criterion of Oscillatory Stability, Quart. Appl. Math. 7 ( 1 9 4 9 ) , 1 1 5 - 1 1 7 -"@en . "Thesis/Dissertation"@en . "10.14288/1.0080554"@en . "eng"@en . "Mathematics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Oscillation theorems for elliptic differential equations"@en . "Text"@en . "http://hdl.handle.net/2429/36119"@en .