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Oscillation theorems for elliptic differential equations Headley, Velmer Bentley 1968

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OSCILLATION THEOREMS FOR ELLIPTIC DIFFERENTIAL EQUATIONS by VELMER BENTLEY HEADLEY (Hons.), U n i v e r s i t y o f London, 1962  B.Sc.  M.A., U n i v e r s i t y o f 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 i n the Department ' of MATHEMATICS  We accept t h i s required  THE  t h e s i s as conforming t o the  standard  UNIVERSITY OF BRITISH COLUMBIA June, 1968  In p r e s e n t i n g  for  this  thesjs  in partial  an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y  that  the Library  Study.  thesis  shall  I further  make i t f r e e l y  agree that  publication  Department o f  of this  thesis  I agree  f o r r e f e r e n c e and  f o r extensive  copying of this  MATHEMATICS  PQth July.  1Q68  It i s understood  for financial  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada t e  Columbia,  f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my  w i t h o u t my w r i t t e n  Da  of British  available  permission  D e p a r t m e n t o r b y h iis r e p r e s e n t a t i v e s .  or  f u l f i l m e n t of the requirements  Columbia  gain  shall  that  copying  n o t be a l l o w e d  ii  ABSTRACT C r i t e r i a w i l l be obtained f o r a l i n e a r elliptic partial  differential  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 E u c l i d e a n space (i)  E  n  .  The  self-adjoint  R  of  n-dimensional  c r i t e r i a are o f two  main t y p e s :  those i n v o l v i n g i n t e g r a l s of s u i t a b l e majorants of the  coefficients,•and ( i i )  those i n v o l v i n g l i m i t s o f these major-  ants as the argument tends to i n f i n i t y . Our differential  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 to equations  partial  of well-known c r i t e r i a o f H i l l e ,  P o t t e r , Moore, and Wintner f o r o r d i n a r y d i f f e r e n t i a l  Leighton,  equations.  In g e n e r a l , our method p r o v i d e s the means f o r extending manner any o s c i l l a t i o n differential  equations.  criterion Our  for self-adjoint  equation i n  In the d e r i v a t i o n of the o s c i l l a t i o n R  E  n  .  criteria i t is  i s e i t h e r quasieonical ( i . e . contains  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 cylinder).  ordinary  r e s u l t s imply Glazman's theorems  i n the s p e c i a l case o f the Schrodinger  assumed t h a t  in this  ( i . e . c o n t a i n s an  In the d e r i v a t i o n o f the n o n o s c i l l a t i o n  an infinite  criteria  no s p e c i a l assumptions r e g a r d i n g the shape o f the domain are needed. Examples i l l u s t r a t i n g the t h e o r y are g i v e n . particular,  In  i t i s shown t h a t the l i m i t c r i t e r i a obtained i n  the second order case are the best p o s s i b l e o f t h e i r k i n d .  iii  TABLE OP CONTENTS Page INTRODUCTION  1  CHAPTER I Second Order S e l f - A d j o i n t  Equations  1.  D e f i n i t i o n s and n o t a t i o n s  4  2.  Auxiliary results  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.  C o n d i t i o n s of l i m i t type  19  5.  Nonoscillation  23  6.  Sharpness  theorems  of the r e s u l t s  ]  33  CHAPTER I I Equations of A r b i t r a r y Even Order 7.  Preliminaries  40  8.  D e f i n i t i o n s and n o t a t i o n s  41  9.  Auxiliary results  43  10.  Oscillation criteria  48  11.  Equations w i t h one v a r i a b l e  12.  F o u r t h order equations on l i m i t - c y l i n d r i c a l  separable  57  domains 13.  F o u r t h o r d e r equations on a l l of  14.  O s c i l l a t i o n theorems  BIBLIOGRAPHY  55  E  n  63 66 75  iv  ACKNOWLEDGEMENT I am g r e a t l y Indebted t o my a d v i s o r , Dr. C A . Swanson, f o r suggesting  the t o p i c , and f o r h i s help and  encouragement throughout the p r e p a r a t i o n  o f t h i s work.  I a l s o wish t o thank Dr. G.E. Hu'ige f o r h i s u s e f u l  comments  on the manuscript. I must a l s o express my g r a t i t u d e t o the- U n i v e r s i t y o f B r i t i s h Columbia and the N a t i o n a l Research C o u n c i l o f Canada f o r t h e i r generous f i n a n c i a l  support.  INTRODUCTION C o n d i t i o n s on the c o e f f i c i e n t s  of c e r t a i n  linear  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  f o r the equations t o be o s c i l l a t o r y i n  unbounded domains of n-dimensional E u c l i d e a n space The  c r i t e r i a are of two main types:  i n t e g r a l s o f s u i t a b l e majorants ( i i ) those i n v o l v i n g l i m i t s ment tends t o i n f i n i t y .  ( i ) those  n  involving  o f the c o e f f i c i e n t s ;  of these majorants  E .  and  as the argu-  C r i t e r i a o f type ( i ) were obtained  by Swanson [26] f o r second-order separable and the fundamental  equations with one v a r i a b l e  domain l i m i t - c y l i n d r i c a l ( i . e .  c o n t a i n i n g an i n f i n i t e c y l i n d e r ) ;  and c o n d i t i o n s o f type  ( i i ) by Glazman [9] f o r the Schrqdinger operator i n the case t h a t the domain i s a l l o f Our  one-dimensional [19],  3  n  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 t o n dimen-  sions ( i . e . p a r t i a l  [17]  E .  d i f f e r e n t i a l equations) o f well-known  results  [213, [283).  i n the l i t e r a t u r e  Our method p r o v i d e s the means f o r  g e n e r a l i z i n g t o n-dimensions  (i.e.partial  equations) any g i v e n one-dimensional The  results  ( c f . [9], [113 > [16  differential  o s c i l l a t i o n c r i t e r i o n . -.  we o b t a i n serve t o i l l u s t r a t e the power of the  method. There  i s an e x t e n s i v e l i t e r a t u r e  on o s c i l l a t i o n  .theorems f o r o r d i n a r y d i f f e r e n t i a l equations.. A. complete b i b l i o g r a p h y may be found i n the forthcoming book o f Swanson [273.  The corresponding theory f o r p a r t i a l  differential  -2-  (see, however, [ 9 ] ,  equations i s not as w e l l developed [13],  fl *], 1  [15], [26],  [27]),  l a c k of an n-dimensional  l a r g e l y because of the  analogue  [10], earlier  of Sturm's comparison  theorem.  In t h i s work we use the r e c e n t Clark-Swanson r e s u l t [ 4 ] , t o g e t h e r with a comparison  theorem of Swanson [24] f o r e i g e n -  v a l u e s , as the b a s i c t o o l s f o r d e r i v i n g our  oscillation  criteria. The  d e f i n i t i o n of an o s c i l l a t o r y e q u a t i o n given  below i s c l o s e l y r e l a t e d t o the n o t i o n of conjugacy used Kreith  [14],  An e q u a t i o n o s c i l l a t o r y i n our sense  s a i d t o have the nodal p r o p e r t y ( c f . [ 2 6 ] ) . i s s t r o n g e r than the requirement a zero i n every neighbourhood s e l f - a d j o i n t second-order  by  i s often  In g e n e r a l , t h i s  t h a t a s o l u t i o n e x i s t s with  of i n f i n i t y .  linear e l l i p t i c  In f a c t , i f a equation i s o s c i l l a -  t o r y i n our sense, every s o l u t i o n of the e q u a t i o n has a zero i n every neighbourhood  of i n f i n i t y .  T h i s w i l l be seen t o be  a consequence of the Clark-Swanson s e p a r a t i o n theorem In Chapter  I, equations of the second  [4].  order w i l l  be c o n s i d e r e d on q u a s i c o n i c a l " domains ( i . e . domains c o n t a i n i n g a cone). all  of  The E , n  Glazman [9]  criteria and w i l l  obtained are e a s i l y s p e c i a l i z e d t o c o n t a i n the c o r r e s p o n d i n g r e s u l t s of  f o r the Schrodinger o p e r a t o r .  Swanson comparison  theorem, we a l s o d e r i v e n o n o s c i l l a t i o n  theorems, but without making any  s p e c i a l assumptions  the shape of the domain ( i n c o n t r a s t t o the theorems, where we  Using the C l a r k -  regarding  oscillation  assume that the domain i s quasiconical.)'.  -3-  Our  results  in this  dimensional last  direction  theorems of H i l l e  s e c t i o n of Chapter  strating limit  the  theory.  criteria  obtained  Our  approach  [13]  and  Swanson [26]  i s an  be  devoted  are  Potter  [21],  order w i l l  be  results  The  (i.e. will  theorems  [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 (m  any  positive  integer),  fourth-order equations. son  I n the  Leighton-Nehari  absence  t h e o r e m , however, our methods w i l l  theorems f o r h i g h e r equations.  The  order  our  operators  of even order,  are  [18]).  valid  (m  o p e r a t o r we  form, but  use  and  methods w i l l  >_ 2)  we  know o f no  [17]  for compari-  nonoscillation  simple  self-adjoint  variational principles operators  one-dimensional,  o p e r a t o r s , we  ( c f . [ l ] and oscillation  f o r such  general  criteria  c o u l d be  g e n e r a l i z e d by u s i n g e x t e n s i o n s [24,  order  a relatively  criteria  Swanson c o m p a r i s o n . t h e o r e m  of  Glazman  differential  work f o r more g e n e r a l  f o r general e l l i p t i c  Although  partial  c o n s i d e r has  s i n c e the  of  of a s u i t a b l e not y i e l d  case,  extant.  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 o f r e s u l t s equations  funda-  contain  fewer i n t h i s  t h e r e a r e n o t many o n e - d i m e n s i p n a l  kind.  Kreith  equations.  are  the  s t u d i e d i n Chapter  Our  2m  illu-  show t h a t  e x t e n s i o n o f t h a t u s e d by  The  The  t o examples  shall  limit-cylindrical  cylinder).  one-  the b e s t p o s s i b l e o f t h e i r  f o r second-order  m e n t a l domain w i l l be  since  and  I will  of higher  II.  infinite  [11],  I n p a r t i c u l a r , we  Equations  an  g e n e r a l i z e well-known  p.517]'  remark t h a t of  such the  we  -4-  CHAPTER I SECOND ORDER SELF-ADJOINT EQUATIONS 9  1.  Definitions We  and  notations.  s h a l l o b t a i n 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  integral  erential  equation  (1.1)  type f o r the l i n e a r e l l i p t i c p a r t i a l  n £ D.(a. .D.u) i,j=l  Lu =  1  i n unbounded domains E .  Our  n  in  + bu = 0  J  n-dimensional E u c l i d e a n  theorems are, f o r the most p a r t , extensions  dimensional  o s c i l l a t i o n theorems of K n e s e r - H i l l e  type), Leighton [28]  R  1 J  [ l 6 ] , Moore [19], P o t t e r  ( i n t e g r a l type).  The  diff-  of  [11]  [21], and  remainder are the  space one-  (limit  Wintner  n-dimensional  second order analogues of one-dimensional 2m-th order  oscilla-  t i o n theorems of Glazman [9]. Points and  in  E  n  d i f f e r e n t i a t i o n with  i=l,2,...,n.  The  are denoted by  x = ( x ^ , x , . . . ,>')  respect to  i s denoted by  coefficients  x^  a. .  2  n  are supposed to be  1 J  r e a l and  of c l a s s  definite  in  and  continuous on  t o be C (R). 2  R.  C (R), The R.  and  the matrix  coefficient The  domain  b  (a..)  i s assumed to be D(L)  of  the set of a l l r e a l - v a l u e d f u n c t i o n s on The  is positive  L  real  i s defined R  of  class  c o n d i t i o n s on the c o e f f i c i e n t s , although not  the  - 5 -  weakest p o s s i b l e case  m = 1  treated  f o r example, [4.]), are  of those we  impose i n the  the  special  2m-th order case  i n Chapter I I .  Definition. Lu = 0  (see,  A function  if  u e D(L)  Shape of the o r i g i n and c o n t a i n the  R.  R  cone  w i l l be  and  domain  that  u u We  assume that  R  contains  enough i n the  = fx e E  a  > Ixl n —  :' x  n  x cos  a] '  1  0 < a < TT,  |x|  being the  boundary  dR  of  R  to  f o r some  Euclidean distance v  The  R.  the  direction  n  I  a,  of  s a t i s f i e s ( l . l ) everywhere i n  i s large C  called a solution  ^ 2\^ (Ex.) i=l  '  i s supposed to have a piecewise  continuous u n i t normal v e c t o r a t each p o i n t . We .  R  r  = R n  Definition.  s h a l l make use {x e E  n :  |x|  of the  > r};  A bounded domain  domain of a n o n t r i v i a l s o l u t i o n Definition.  The  oscillatory  in  S  following = (x 6 R  f  Nc  R u  notation:  u  : |x|  i s s a i d to be of  (l.l) iff  =  a nodal  u = 0  d i f f e r e n t i a l e q u a t i o n ( l . l ) i s s a i d to R  i f f there e x i s t s  of ( l . l ) w i t h a nodal domain i n  R  r  I t f o l l o w s from C l a r k and  rJ.  on be  a nontrivial solution for a l l  r >  0.  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 e q u a t i o n vanishes a t some R  Definition.  r  for a l l The  r >  u^  Swanson's n-dimensional  analogue of Sturm's s e p a r a t i o n theorem [4] that  point i n  dN.  0.  d i f f e r e n t i a l equation ( l . l ) Is  s a i d to  be  -6-  non-oscillatory  i f f there  in  R  nodal  2.  Auxiliary results.  have no  r  The will  be  basic  a recent  a d d i t i o n we  E  n  .  need two (Cf.  The  that  the  solutions  domains,  comparison theorem of  shall  such  t o o l f o r d e r i v i n g our  Minimum p r i n c i p l e . domain i n  r > 0  exists  oscillation  CA.  Swanson  well-known p r o p e r t i e s [6,  p.399])  function  Let  fi  2  u  e C (Q)  of  be  theorems  [24].  In  eigenvalues.  a bounded  which minimizes  the  functional (2.1)  J[u]  under the  condition  t o the  smallest  The  norm  Hull  a. .D.uD.u - bu  Hull = 1  eigenvalue -Lu  (2.2)  n E  T {  =  = Xu  i s the  }  dx  i s ,an e i g e n f u n c t i o n  of the  in  problem u = 0  fi;  usual  L  2  norm:'  on  Hull  dfi •  =  [ J  Proof.  In  [6]  i t i s shown t h a t  corresponding  i f the  p  2  ^~  |u'| d x ] ^  .  fi  minimizing  function  2  exists  and  i s of  class  c o r r e s p o n d i n g t o the Let  the  minimum v a l u e  C  ,  t h e n i t i s an  smallest of  eigenvalue  J[u]  be  X  .  eigenfunction  of the  problem  Then the  (2.2).  results  of  o [18, which  §11]  show t h a t  there  i s a weak s o l u t i o n o f <u , Q  exists a minimizing function (2.2)  (L-X )cp> = 0 , Q  i n the cp  e  following c"(Cl),  sense:  U  Q  -7-  2 < , > our  being  the u s u a l  c o n d i t i o n s on  [ l , §9]  of  imply  the  L  (fi)  inner product.  coefficients  U  that  a. . 3  is in fact  q  On  and  b,  account the  a classical  of  results  solution  of  (2.2). Monotonicity G(t)  be  G(t ) c  a bounded G(t ),  1  \ (t)  principle  of the  Q  -Lu is  domain i n  G(t ) y  2  1  lim t-0+  fi(t),  = Xu  where  X ( t ) = +oo °  Proof.  The  adapting  G(t ),  0^ < t-^ < t g < co  If  then  2  in  G(t); in  the  u = 0  t.  n(t)  first  let implies  eigenvalue  on  dG(t)  Moreover,  i f f o r some  e E  <  = {x  n  : r  o  |x|  r  Q  +t),  <-r  >  0,  then  .  monotonicity  the  R.  0 < t < oo  For  problem  monotone d e c r e a s i n g  G(t) c  f o r eigenvalues.  proof  given  ^ (t)  of  may  Q  i n [7,  be  e s t a b l i s h e d by  pp. 4-00-401 ] , f o r 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  smallest  ^- 3  eigenvalue  A (X)  of the  matrix  infimum  G(t),  since  G(t) c  in  ( a . .(x)) R  and  has  ( a . .) J.  definite  in  R.  elliptic  in  G ( t ) , i . e . there  that  Zn  I n o t h e r words, the  a. . ( x ) z . z . > u  exists  (t)|z|  non-zero is positive-  j  operator  L  i s uniformly  a number. H ( t )  > 0  such  x e G(t), z  e-E .  Q  for a l l  n  Since the f u n c t i o n  b  i s u n i f o r m l y continuous  there e x i s t s a number  on G ( t ) ,  k ( t ) > -oo such t h a t o ' v  2,  bu dx  (2.3)  > • k (t) u dx .— o G(t) v  G(t) for  all  u e C (R).  Let  1  {  J [u] t  £  G(t) '  a. .D.uD.u - tiu |dx .  i j J = 1  Then ( 2 . 3 ) and the u n i f o r m  ellipticity  of  L  imply  i n G(t)  that  J [u] t  > u (.t) Q  n 2, u dx 2 (D.u) dx + k ( t ) ° G(t) i = l G(t) p  But u d x = J*  u F(r,S)drdS  2  G(t) where  F(r,S)  2  ,  G(t)  i s the J a c o b i a n of the t r a n s f o r m a t i o n from  rectangular coordinates polar coordinates  (x-^Xg,...x ) n  ( r , Q ^ , . . .  Q^  3  to hyperspherical  ^) = ( r ^ S ) ,  relations  d e f i n e d by the  . x  1  n = r ~JT s i n 0 i=l  x  n  = r cos  8,, 1'  n-1 x  ±  We now extend  = r cos Q _ n  .u  i + 1  TT^ s i n 6j  ,  i=2,5,... n-l.  c o n t i n u o u s l y t o a l l of the annulus  a  Q(t)  by r e q u i r i n g i t t o be zero o u t s i d e i s zero on annulus  |x| = r , o  1  r  1  G ( t ) . In p a r t i c u l a r ,  u  b e i n g the i n n e r r a d i u s of the  o  Q ( t ) . Hence  u(x)  =  v (t,0 ,.•.,@ _ )dt , t  1  n  1  o where u(x) = v ( r , 8 , . • • , 8 _ ) , 1  By Schwarz's  n  = dv/dt  r +t o  <  dy o  o  D(t)  t  inequality,  u(x)r whenever  v  1  x e G(t) c  fi(t).  o Integrating this inequality  over  we o b t a i n u dx = J  u F(r,S)drdS  2  n(t)  2  n(t)  < t  2  J n(t)  < t'  F(r,S)drdS n 2 (D.u) dx , i=l 1  unce  2 where the  ( D . u )  hu  8  . |grad u| -= 2  ( f l f  +  are c e r t a i n (known) f u n c t i o n s of  r  and  S.  -10-  Since  u  i s d e f i n e d t o be zero o u t s i d e 2  u dx  (2.5)  <  t  n £ (D.u) dx  2  G(t)  G ( t ) , we have p  G(t)  i=l  1  Combining t h i s with i n e q u a l i t y ( 2 . 4 ) we get 2, u dx  2,P J [u] > [k (t) + H (t)/t ] c :  t  Q  Q  G(t) A c c o r d i n g t o Courant's \ (t)  where  Hull  ^  Q  =  i s the  L  (2.6) Since it  inf{J [u] t  : Hull  norm of  2  u  =  t  on  \ (t) > k (t) + n (t)/t Q  HQV^)  Q  cannot decrease as the domain  S i m i l a r l y , we may k (t) Q  =  1}  p.399], , •  G ( t ) . Thus  .  2  Q  may be chosen to be  remains p o s i t i v e as  [6,  minimum p r i n c i p l e  inf{/\ (x) : x e G ( t ) } , Q  G(t)  s h r i n k s , and t h e r e f o r e  t -* 0+.  choose  inf{-J  b u d x / f t u ^ : u € C(G(t)} , 2  G(t) that i s  k (t) Q  = inf|-J  bu dx/  Q(t) since' u  i s zero o u t s i d e  •  u dx : ue C(G(t) n(t)  G ( t ) . Now  b  and i s t h e r e f o r e a.continuous f u n c t i o n of continuous f u n c t i o n of  t  i s continuous i n |x|.  i n every i n t e r v a l  Thus  b  0 < t < 6,  R  is a  -11-  and k  i s t h u s bounded  (t)  0 <_ t <_ 6 '. f o r a l l  must r e m a i n bounded as  c o u l d have any  on  chosen  k Q  t -» 0+.  ( t ) independent  e v e n t , t h e i n e q u a l i t y (2.6) now  lim t-0+  X ( t ) = + oO, °-  Remark.  and the theorem " '  I n the a p p l i c a t i o n s below,  (This  of  t  implies  we p r o c e e d as f o l l o w s :  J+[u  z  o  J G(t)  ]  x  ( Ul  n- i i  J  r  by  defining  because  U  U  q  according the be and  last  U  Xu  in  depends  J.) t  21  - bu \ dx oJ  2i - •  Z a. .D.u D.u - bu^dx i ji o j o oJ  outside  G(t) ;  ,  G ( t ) . This  i s possible  of the problem u = 0  on  to hyperspherical  the l i m i t s  0  i n t h e domain o f  BG(t) It i s clear  above d e p e n d s c o n t i n u o u s l y  s e e n by t r a n s f o r m i n g considering  Q  ^ ("t)  1  t o C o u r a n t ' s minimum p r i n c i p l e . integral  In  (by Courant's p r i n c i p l e )  a. .D.u D .u i ji o j o  i s an e i g e n f u n c t i o n  -Lu.=  = 1}  ^ ('t)  n  t o be z e r o  Q  =  we  that  that  ( f o r some u o  £  •{ fi(t)  To p r o v e  : |u|  that  we a c t u a l l y need  continuously  X (t) = inf{J,[u]  implies  i s proved.  to  t,  Therefore  i n (2.3)).  to tend -continuously on  +<o.  6 > 0.  that  on  t , ' as may  polar  coordinates  of the r e s u l t i n g m u l t i p l e  integral,  -12-  Let- A ( X ) denote the l a r g e s t e i g e n v a l u e  Definition. matrix  (a . ( x ) ) , x e R .  A majorant o f  ly f e C (0,oo)  valued f u n c t i o n  ( ij)  function  g  s  positive-  a  such t h a t  ' f ( r ) _> max ( A ( x ) : x e S } , The  ^  a  o f the'  < r <oo) .  (0  d e f i n e d by  g ( r ) = min ( b ( x ) : x e S ) ,  (0  r  < r <oo)  i s c a l l e d a majorant of b ( x ) . Let equations The  A  and  B  be f u n c t i o n s i n G  A(x) = f ( | x | ) , B(x) = g ( | x | ) ,  comparison e q u a t i o n .  f o r equation  d e f i n e d by the  respectively.  We s h a l l o b t a i n o s c i l l a t i o n theorems  ( l . l ) by comparing i t w i t h t h e s e p a r a b l e  equation  n 2 D. (AD., v) + Bv = 0 . i=l  (2.7)  1  As b e f o r e , l e t  1  r , 6 ^ , 0 ^ . .. ,0 _-^ n  denote h y p e r s p h e r i c a l p o l a r  By w r i t i n g ( 2 . 7 ) i n terms o f these  coordinates.  coordinates,  we f i n d t h a t ( 2 . 7 ) has s o l u t i o n s ( i n p a r t i c u l a r ) o f t h e form (2.8)  v(x) = p(r)cp(0 ), 1  where (2.9)  p '  i ,„ \ •  (2.10) n  and  0 < r <oO,  0 <d  cp s a t i s f y the o r d i n a r y d i f f e r e n t i a l  < a ,  1  equations  ^ [ ^ - ^ ( r ) ^ ] + r -- [g(r) - X r" f(r)]p' = 0 n  1  2  a  d , ^  r  . n-2  [sin _  a  6  dcp  n  1  . n-2 .  ,.  g^J  + \  a  cp s i n  " _  9  1  = 0  ,  -Ir-  respectively.  We  X  choose  t o be the s m a l l e s t number f o r  itfhich (2.10) has a n o n t r i v i a l s o l u t i o n s a t i s f y i n g 'cp(a) = 0.  cp  0 <_ 0-^ <_ a  on  I t i s w e l l known [5]  that" X  exists  a  as the s m a l l e s t eigenvalue of a s i n g u l a r S t u r m - L i o u v i l l e problem. We  s h a l l suppose, f o r d e f i n i t e n e s s , t h a t the  corresponding  e i g e n f u n c t i o n has been normalized by the c o n d i t i o n  3.  =  1.  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. The  theorems i n t h i s s e c t i o n c o n s t i t u t e  Potter E  n  [21]  Theorem 1. a cone  C  [28].  and Wintner  They may  v = p, cp = 1, X  by t a k i n g  Equation ( l . l ) (a > 0),  extensions  theorems of L e i g h t o n [ l 6 ] ,  of the one-dimensional  of  cp(0)  and  Moore  [19],  be s p e c i a l i z e d to a l l  = 0.  a  is oscillatory (a..), ^  in  R  if  have majorants  R  contains  f,g  r e s p e c t i v e l y such that'  (3.1)  '-nIT:  =  + 0  °  J r  A N D  n  [g(r)-X r a  f ( r ) ] d r = +00.  1 Proof.  The hypotheses  Wintner theorem [l6]_, s o l u t i o n of (2.9) k  condition  imply t h a t the o r d i n a r y d i f f e r e n t i a l  i s o s c i l l a t o r y i n . 0 < r < o O by the L e i g h t o n -  equation (2.9)  & tco . . If  (3-1)  cp  [28],  Let  with zeros a t  p ( r ) • be a n o n - t r i v i a l r = 6^/6  . . . ,8^,-.  . . ,  where  i s an e i g e n f u n c t i o n of (2.10) 'with boundary  cp(a) = 0  corresponding t o the eigenvalue  X. , - the  ;  -14-  functlon  v  d e f i n e d by  IS c l s o l u t i o n of the  (2.8)  comparison  equation (2.7) with nodal domains i n the form of t r u n c a t e d cones  C  alc  =  ^  X  ^  e  ' n  ^  x  k=l,2,...,,  c  o  s  a  '  5  k  <  <  5  0 < a < TT ,  k-r-l^ '  with piecewise smooth boundaries.  nodal domain  C , c R ak p  g i v e n , • choose  k  f o r a l l p > 0;  1  implies that  x e R . P  Since  |x| > 6^ _> p  < A(x) |z|  E a. .(x)z.z i,j=l  v  for i f  p > .0  6^ _> p,-  and  x e C  c R,  has a is  5  l a r g e enough so t h a t  x e C ^  Thus  2  and  < f(r)|z|  a  clearly so t h a t  = A(x) | z | , z e. E  2  2  n  1  and  b(x) _> g(|x|) = B ( x ) ,  (2.7)  equation ( l . l ) majorizes  equation  i n the f o l l o w i n g sense: f  Pv  n  r  \  E  (A5 . . - a. .)D.uD .u + (b-B)u  dx _> 0 .  ak I t then f o l l o w s from a comparison theorem of C.A.-  Swanson  [24]  t h a t the s m a l l e s t eigenvalue of the problem -Lw satisfies  = w(a  n < 0.  6,  < t < 6, ' ,  of  the problem  in  C ,  Let  M^  and l e t  w = 0  a R  t  * {x e  m(t)  on  aC^ : 6  R  < |x| < t} ,  denote the s m a l l e s t eigenvalue  -15-  -Lw By  = (-i(t)w  the monotonicity  monot one  Since  in  in  I- ]_( ^+2.) £ °> LI, (T) = 0. 1 '  6^  < t _< 6^+1  T h i s means t h a t  v  M  of a n o n t r i v i a l s o l u t i o n u ^ Of ( l . l ) , for arbitrary  p > 0  equation  is oscillatory  (l.l)  provided  k in  T  and  =  in  and  (6^  [19],  ( C f . , e.g.  [21],  of t h i s f a c t The  [26],  m  <z  since  [27],  integral  conditions  In f a c t , i n the [28]) We  f ( r ) > 1,  the  the  literature  integral  A + b(x)  s h a l l have o c c a s i o n  to  Example  r e q u i r e s the  J  function  l  dr „ , <oo. r f(r)  In such cases  n _ 1  of Moore's o s c i l l a t i o n theorem  [19]  valid:  Theorem 2. contains  The  a cone  equation C (a  (l.l) is oscillatory  > 0),  4).  i n three dimensions, s i n c e i n  so t h a t  the f o l l o w i n g e x t e n s i o n  criteria  i n f a c t does not h o l d f o r the  P°° t h i s case  R^  the theorem i s proved.  f i r s t p a r t of c o n d i t i o n ( 3 . 1 )  operator  c  large,  i n the a p p l i c a t i o n s (Cf. § 6 ,  to grow q u i t e s l o w l y , and  Schrodinger  k+l-'  i f the i n t e g r a l c o n d i t i o n s h o l d when 1  are o f t e n s t a t e d i n t h i s manner. make use  6  +<o.  i s a nodal domain  akT  i  if  lim  is sufficiently R  i s r e p l a c e d by a p o s i t i v e number.  is  u-^t)  the number 1 as the lower l i m i t of i n t e g r a t i o n , but  theorems remain v a l i d  f  SM^.  and  M  I t i s convenient to s t a t e the with  on  there e x i s t s a number  6  such t h a t  w = 0  a k t >  p r i n c i p l e f o r eigenvalues,  nonincreasing  1  M  and  (a. . ) , b  in  R  if  R.  have majorants  -16-  f,  g,  r e s p e c t i v e l y , such  , J  (3.2)  dr  0 0  1  r  n _ 1  f o r some number  Proof.  f(r)  that  < co  m > 1,  and  h™(r)[g(r)-X r- f(r)]dr 2  a  1  where  oO  h (r) =  dt  r t n-1 f ( t )  A c c o r d i n g t o 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 of the p r o o f f o l l o w s  (3.2).  The  remainder  t h a t 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  slightly  A(X)  (a. .)  of  Theorem 3. —  Let  i n the case t h a t the l a r g e s t eigenvalue  i s bounded i n Po.  R  contain  the cone  and l e t A ( X ) be bounded i n majorants tory  in  and f o r  f,g, R  for  n > 3  respectively. n = 2  holds with  a  Suppose  f o r some  a > 0, '  (a..) _ and  Then.equation  b  have  (l.l) is oscilla-  .  i f there e x i s t s a number  6 > 0  such  that  f g ( r ) - X r ~ f ( r ) , ] d r = + oo .  (3.4)  In the case  if  R.  C  2  n = 1/ 6=1  equation ( l . l ) i s o s c i l l a t o r y (the Leighton-Wintner.theorem).  i f (3.4)  -17-  Froof.  A(X)  If  we'can choose conditions  i s bounded i n  are f u l f i l l e d  A(X) < A  say  0 < r • < eo.  f(r) = A ,  (3-1)  R,  x e  1 ?  Then, f o r  R,  n = 2,  and hence the f i r s t  the  statement  of the theorem f o l l o w s from Theorem 1. For and  n  ( )  =  r  n  6 > 0  n _> 3,  the f i r s t p a r t of (3-2)  /(n-2) f\- .  r  such that (3.4)  By hypothesis  holds.  Let  is f u l f i l l e d ,  there  exists  m = 1 + 6/(n-2).  d i r e c t computation shows t h a t c o n d i t i o n (3.4) p a r t of c o n d i t i o n (3.2), and.therefore  Then  i m p l i e s the  second  the second statement of  the theorem f o l l o w s from Theorem 2. It to  n  i s c l e a r t h a t our method enables us  dimensions any  s u f f i c i e n t condition for a self-adjoint  o r d i n a r y l i n e a r d i f f e r e n t i a l equation be o s c i l l a t o r y .  to g e n e r a l i z e .  In what f o l l o w s we  of the second order  s h a l l therefore  only a r e p r e s e n t a t i v e number of the e x i s t i n g  to  generalize  one-dimensional-  o s c i l l a t i o n theorems. Our  next theorem g e n e r a l i z e s P o t t e r ' s  [21] of Leighton's introduce  theorem.  We  refinement  s h a l l f i n d i t convenient to  the f o l l o w i n g n o t a t i o n .  Let  h  be a p o s i t i v e  2 C  f u n c t i o n d e f i n e d by  (3.5)  .  m  [h(r)]"  2  = g(r) - A - J ^ + ( n - l ) ( n - 3 ) / 4 ] r " , 2  Let the f u n c t i o n s H  l ^  r  ;  ~  and hfFT.  H rr) 2^ - hJTJ 1  H  r j  -  "  H  be d e f i n e d  2  4h(r)  ^ ( r ) ] ~ 4h(r)  >  +  2  by  h'(r)  >  0 < r < oo  -18-  where primes denote d i f f e r e n t i a t i o n with r e s p e c t t o r . Theorem 4.  Let R  c o n t a i n the cone  and l e t A ( X ) be bounded Then equation  (l.l)  positive  function  C  i n R,  say  i s oscillatory h  C  a  a > 0,  f o r some  A ( X ) £ A^, x e R .  in R  i f there e x i s t s a  s a t i s f y i n g (3-5) f o r large  r  and  either  <-> 3  ' J^.ErlT"'J"" i  6  H  ( t 1 d t= + c 0  or i  (3.7)  J ^ H ( t ) d t = + oo . 2  Proof.  In equation  of t h i s equation,  '  (3-8)  dr  to  1  + ( n - l ) ( n - 3 ) / 4 ] r - } a = 0. . 2  n a  (3.6) (or (3-7))  i m p l i e s t h a t the equation  i s o s c i l l a t o r y by the theorem of P o t t e r mentioned i n the  remark above. (3.6)  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  +'{g(r) - AA\  The hypothesis  The normal form  v  A-, 1  f ( r ) = A^.  (l-n)/2 p = r • " a , is  " ... transformation  (3.8)  ( 2 . 9 ) choose  Thus the equation  or ( 3 . 7 ) h o l d s .  ( 2.9) i s a l s o o s c i l l a t o r y i f  The remainder of the p r o o f is. s i m i l a r  t h a t of Theorem 1 and w i l l be omitted. Because of the p o s i t i v i t y c o n d i t i o n on  h,  c l e a r t h a t Theorem 4 i s i n some r e s p e c t s l e s s g e n e r a l Theorem 3 .  i t is than-  However, i n s e c t i o n 6 we s h a l l e x h i b i t an example  -19-  f o r which Theorem 4 gives, i n f o r m a t i o n  not. o b t a i n a b l e  i n any-  obvious way from Theorem  4.  Conditions The  of l i m i t first  type.  theorem i n t h i s s e c t i o n i s a g e n e r a l i z a t i o n  of the c l a s s i c a l K n e s e r - H i l l e contains  theorem [ 1 1 ] .  Glazman's g e n e r a l i z a t i o n [ 9 , T h . 7  Schrodinger operator a new proof  -v  2  of h i s r e s u l t .  p . 1 5 8 ] f o r the  3  n - b(x), x e E ,  Our theorem a l s o  and i n f a c t  provides  I t w i l l a l s o be seen t h a t our  c o n d i t i o n i s sharp. j  Theorem 5»  Suppose that  R  contains  the. cone  C  f o r some  Cfc  1  a > 0,  A ( X ) i s bounded i n R . f o r some s  and t h a t  A(X) < A., , x e R .  say  Let  b  have majorant  g.  s > 0,  Then  equation ( l . l ) i s o s c i l l a t o r y i n R i f Lim i n f r g ( r ) > A-, [\ r-* oo  (4.1)  Proof. r  o  + (n-2) /4].-  2  2  The hypothesis ( 4 . 1 ) i m p l i e s t h a t there e x i s t  and  v  such t h a t r  for a l l  2  r > r .  .  g(r) >  > A  Y  1  [ \  + (n-2) /4] 2  A  We then compare ( 2 . 9 ) with the E u l e r  I  ^ with s o l u t i o n s  V " p =  constants  ,  1  ^  +  where  ( Y  " V l ) ' " " " " - °. 3  1  ,•  •  equation  -20-  (4.3)  Since  3  + (n-2)p + ( /A ) - \  2  Y  y > A - , [X  + (n-2) / 4 ] ,  1  a  = 0.  the q u a d r a t i c e q u a t i o n ( 4 . 3 )  has  complex r o o t s , and t h e r e f o r e e q u a t i o n ( 4 . 2 )  in  ( r , cd).  comparison  f(r). = A-^  We may choose  theorem  on  [ 5 ,p.208]  i s oscillatory  and apply Sturm's  (r-^oo), (^  to deduce t h a t e q u a t i o n ( 2 . 9 ) i s o s c i l l a t o r y  = max{r ,s}) o  on account  of the  hypothesis r  n _ 1  [g(r)  The remainder  - \  a  r  -  2  f(r)] > r  of the p r o o f proceeds  n  _  5  (  Y  - X ^ ) ,  r > max{r ,s}. o  as i n Theorem 1 and w i l l  be omitted. Our next theorem i s an e x t e n s i o n o f a well-known theorem of H i l l e paper j u s t  [ll,  cited,  Th.5]»  As H i l l e p o i n t s , out i n the  the c o n d i t i o n s are a refinement o f those  ?  i n Theorem 5 , s i n c e i n t e g r a t i o n smooths out i r r e g u l a r i t i e s i n the f u n c t i o n Theorem 6 .  g. Let  and l e t A(x)  R  c o n t a i n the cone  be bounded i n  R,  f o r some  say  Then e q u a t i o n ( l . l ) i s o s c i l l a t o r y  in  A ( X ) _< A - ^ ,  a > 0, xeR.  R i f  2  r  g(r) >  (4.4)  A-,  [X  + (n-1) ( n - 3 ) / 4 J  lim inf r r - oo r J  f o r large  g ( t ) dt > A , [\ . x a n  r  + {n-2)-  and e i t h e r /k]  -21-  or  (4.5)  l i m sup r-» oO  where  g ( r ) = min  •Proof.  rf°° g ( t ) dt > A , [X r  + (n -4n+7)/4], 2  (b(x) : x e S }.  In e q u a t i o n ( 2 . 9 ) choose  f ( r ) = A-^.  We-recall that  the normal form of t h i s e q u a t i o n , 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  (3.8)  '  A  N  dr  The hypothesis oscillatory  in  + (g(r) -  p = r ^  -  1  1  ^  2  a,  is  '[X„ + ( n - l ) ( n - 3 ) / 4 ] r - } a = 0. 2  A 1  a  ( 4 . 4 ) i m p l i e s t h a t the e q u a t i o n ( 3 . 8 ) i s 0 < r < oo  by a theorem o f H i l l e  [ 1 1 , Th.5],  since  lim i n f r-» oo '=  -1 1  > \  r f " {g(t)/A, - [\ r 0  lim inf r-oo  r  .00  g ( t ) dt - [\  r  + ( n - 2 ) / 4 - [\ 2  a  + (n-l) (n-3)/4]t~ }dt 2  n  Q  + (n-l)(n-3)/4] = £ .  Thus 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 hypothesis  + (n-l)(n-3)/4] • •  in  0 < r < oo i f  (4.'4) h o l d s .  ; I f the hypothesis (4.5') h o l d s , then e q u a t i o n ( 3 . 8 ) is oscillatory above, s i n c e  in  0 < r < oo  by the theorem of H i l l e  cited  -22-  l i m sup r-> oO  A,  -1  1  >  [\  l i m sup r-» oo  f  0  - • [\  (2.9)  g(t)dt  i s also  permits  - [\  Q  Q  +  dimensional o f Glazman possible, we s h a l l  p.102].  i t i s useful demonstrate  Theorem 7.  Let  R  oscillatory  in  0 < r < cO omitted,  change.  show, by e x h i b i t i n g  (4.1) i s sharp.  a  Our n e x t  T h i s g i v e s the second  Although  this  countertheorem  impose  o r d e r , nresult  i s not the sharpest  o f t h e t h e o r y , as  i n s e c t i o n 6.  c o n t a i n t h e cone  C  f o r some  a > 0,  QT  A ( X ) be bounded  Then e q u a t i o n  (l.l)  for  and  large  result  i n the a p p l i c a t i o n s  !_  let  }dt  + ( n - l ) ( n - 3 ) / 4 ] = 1.  a n a l o g u e o f a 2m-th q r d e r , o n e - d i m e n s i o n a l [9,  _ 2  (n-1)(n-3)/4]  u s t o r e l a x t h e c o n d i t i o n ( 4 . 1 ) p r o v i d e d we  an a d d i t i o n a l h y p o t h e s i s .  Proof.  (n-l)(n-3)/4]t  i t f o l l o w s t h a t o f Theorem 1 w i t h o u t  example, t h a t t h e i n e q u a l i t y  (4.6)  +  The r e m a i n d e r o f t h e p r o o f w i l l be  I n s e c t i o n 6 we s h a l l  and  a  0  2  (4.5) holds.  since  r  1  +(n -4n+7)/4] - [ X  a  Thus t h e e q u a t i o n if  (g(t)/A  r  R,  i s oscillatory  say  A ( X ) < A-^J x e R .  2  2-  i f r g ( r ) > A-, [\ +(n-2) / 4 ] °  l i m sup ( l o g r ) r-» oo In equation  in  r  t { g ( t ) - A [\ +(rx-2) / ^ t ) d t = + co .  ( 2 . 9 ) choose  n  n  f ( r ) = A-^.  We t h e n  reduce  -23-  (2.9)  t o normal  (3.8)  f o r m as  A .  where  p(r) =  +  r  ^  1  {  -  g  ^  n  t h e e q u a t i o n (3.8) hypothesis  (4.6)  (  r  _  )  a.  2  6:  i n t h e p r o o f o f Theorem  r,  . ( _i)( _ )/4] - }a = 0 , 2  +  n  n  By a t h e o r e m  is oscillatory  5  r  0 < r < co,  in  [9,  o f Glazman  p.  since  102],  the  implies  = + °0 . Thus t h e e q u a t i o n (2.9) formation  p = r^'  in  0 < r<oO.  as  i n Theorem 1.  5.  by  ^  2  a  p r o o f of the theorem  We  therefore  may  R  quasicylindrical no l o s s  and  this  trans-  behaviour  now  be  completed  omit the. d e t a i l s . . "  gives  t h e shape  t o be  theorems  rise  will  no  special  that  i s c o n n e c t e d and unbounded,  R.  ( a s i n §§3  or- q u a s i b o u n d e d  i n assuming  be-proved  t o an i n t e r e s t i n g  o f t h e domain  quasiconical  ( a s i n [26])  of g e n e r a l i t y R  the  theorems.  regarding  necessary .for  since  preserves o s c i l l a t o r y  n o d a l domains a r e n o t c o n s t r u c t e d ,  a r e needed  Since  n  dscillatory,  of our n o n o s c i l l a t i o n  contradiction,  since  -  The  Nonoscillation All  L  i s also  R S  feature:  assumptions  I t i s not and  [9].  contains  4), There i s the o r i g i n . •  i s n o n v o i d f o r each  r >  -2k-  Theorem 8. s > 0, E  Let  L  be u n i f o r m l y e l l i p t i c i n R  i . e . there e x i s t s a number a. . ( x ) z . z . > A |z|  fora l l  > 0  A  such t h a t  x e R . z e E . Let  g ( r ) = max (b(x) : x e S ) , 0 < r <oo. Q  f o r some  Then e q u a t i o n ( l . l )  r  is non-oscillatory i n R i f (5.1)  l i m sup r r-» co  Proof. that  2  solution  i n R.  r  r  (5»l) implies- t h a t t h e r e , e x i s t  But the h y p o t h e s i s  r , v such t h a t o ' o r g (r)  < y  2  Q  (5.2)  Q  Then'there e x i s t s a n o n t r i v i a l i of ( l . l ) with a nodal domain N c R for a l l  u  constants  for.all  2  Suppose the c o n c l u s i o n of the theorem i s f a l s e , i . e .  ( l . l ) i s oscillatory  r > 0.  (n-2) s' /k.  g (r) <  r > r .  I  < (n-2)  o  We compare  A /k, Q  ( 1 . l ) . with.the  v = 0,  A D^v + Y | x | Q  2  Q  equation  x e R.  Because o f the hypotheses  E a i,j=l 1  > A |z| , .  (x)z.z J  y ~ > S ( ) >. ( ) ^ r  o  2  r  b  x  Q  x e R , z e E S  xeS , r> r r  r  equation  (5-2) majorizes equation  ( l . l ) i n the f o l l o w i n g sense:  -25-  N  { S (a i,3=1  - A 6  )D uD  u +  (  Y o  (5.2)  has  the  a zero  solution  in  equation  for  the  o  for a l l  r  analogue  |]  1  V  +  0  (5.. 2)  of a  r  - 3  n  0  solution  (  r  the  0  (5-2)  of  a  {r ,s}.  )  shall  But  Q  equation  , 0 (5«3)  being  quadratic  the  radial  equation  Y  ( ~2)  <  O  A  n  i s non-zero  Q /  exhibit  a  A,  in  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  I n s e c t i o n 6 we  of  = 0.  r o o t s because  v = r  This  p  (equation  satisfies  real  r > max  [h]  solution  ordinary d i f f e r e n t i a l  + (n-2)a + Y A  2  has  r > 0.  a l l  u dN  r  of the  a  ( 5 . 2 ) ) , and a  hence  N  p = r  also a solution  This  max{r ,s})  Clark-Swanson n-dimensional  [A  form of  >  comparison theorem i m p l i e s t h a t every  (5.3) is  2  J  t h e r e f o r e the  o f Sturm's  > 0,  - b)u }dx  2  L r  (r and  |x|-  N  and ij dN  r  r  theorem.  counterexample  to  2 constant  (n-2)  A /4  i . e . there  exist  oscillatory  show t h a t t h e be  improved,  l i m sup r  2  g (r) = (n-2)  2  A  Q  i n c o n d i t i o n (5-1) equations  cannot  f o r which  A .  r-* co The  theorem  just  proved,  together  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,  Hille's  [ll].  result  results and  the  proof  In the are  new.  cases  where  with Th.7, 1,  5  Theorem p.158] both  of  the  -26-  Th eo rem 9-  L  A  s > 0,  t  e  L  be u n i f o r m l y e l l i p t i c  b e i n g the e l l i p t i c i t y  q  in  R  constant.  f o r some  Let  g^  be the  f u n c t i o n defined, by  g l  (t)  = g (t) - A (n-l)(n-3)/4t Q  (b(x) : x e S ),  with  S ( ) =  (l.l)  i s nonoscillatory  (5.4)  l i m sup r - co  r  m  a  in  r J  r  g ^ ( ) = max r  Proof.  0 < r <oo.  r  where  that  x  Q  ,  2  G  R  Then equation  if  gT(t)  dt <  {g (.r) , 1  A  A,  °  1  0},  0 < r < co.  Suppose the c o n c l u s i o n of the theorem i s f a l s e , i . e .  ( l . l ) is oscillatory in  solution,  u  R. "Then there e x i s t s  of ( l . l ) with a nodal domain  N  c R  r  r  a nontrivial for a l l  r > 0. We now  (5.5)  n E AD i=l °  compare  ( l . l ) with the e q u a t i o n  p  n  1  v• + g (|x|)y_= 0, °  x e R.  Because of the hypotheses n  E a i,j=l  1 J  (x) z z 1  J  g (r) > b(x), Q  ? > A |z| , •  x e S , r  x e R , s  0 < r <  oo  n z e E , n  -27-  equation  (5.5) majorizes  equation  n. { E i,j=l  - A 6  J N  J  (a  L r  t h e r e f o r e the  [4]  implies that every  for  all  of  we  we  no  shall  zeros in  also (radial)  s o l u t i o n s of  (5.7)  A  ^  2  w,  S^( )  [11, Th.7  r  r  A  1  % (  r  )  p > 0  and  To  see  (r)p The  U  /  =  =  equation  of  trans-  0.  .  equation  0  0  A  of hypothesis  o  =  g  o( ^ r  / A  o  "  (5.4).  r  this,  - 0 normal form  5N  solution  a well-known theorem of H i l l e  + gt(r)y/A_  1  g o  2  n o n o s c i l l a t o r y on a c c o u n t  st( )/ o  1  (5.5).  C o r . l ] i m p l i e s t h a t the  •^k dr  a  oscillation-preserving  nonnegative,  i s  N  exists  r > p.  -'(n-l)(n-3)A/4r ]w o  n  d r  Since  in  is  ^g+-[g (r) o ^ o n  a zero  above  ordinary d i f f e r e n t i a l  r -  +  making the n  has  f o r some  for a l l  1  p = r ^ ~  (5«5)  R  r  rr"" ^]  ( 5 . 6 ) , o b t a i n e d by  is  in R  Swanson c i t e d  show t h a t t h e r e  *&  formation  of  s o l u t i o n s of the  0  (r>s)  J  solution  zeros  note t h a t the  (5-6) are  But  no  2  t h e o r e m o f C l a r k and  ( 5 - 5 ) w h i c h has  t h e r e f o r e has  u + (g - b ) u } d x > 0 , °  )D uD  1 J  and  r > s.  (l.l):  Moreover,  (n-D(n-3)/4r , 2  -28-  so that Sturm's comparison theorem [ 5 , p . 2 0 8 ] equation ( 5 - 7 ) i s n o n o s c i l l a t o r y . tion some  Hence 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 P therefore  that  Thus there e x i s t s a s o l u -  of ( 5 - 7 ) which has no zeros i n  w = w(r) p > 0.  implies  for  (p.,00)  v = (l~ )/ n  2  r  p > 0.  f o r some  w  We have  a r r i v e d a t a c o n t r a d i c t i o n , and the theorem i s  proved. Remark.  If  n = 1,  b(x)  0  and  a ^ ( x ) = 1,  reduces t o the c l a s s i c a l theorem of H i l l e  case  p =0  a  s  the s p e c i a l  of the f o l l o w i n g n-dimensional analogue of a  theorem of H i l l e Theorem 9 A .  theorem  c i t e d i n the p r o o f .  t o regard Theorem 9  It i s possible  this  [11, p.250,  Th.12]:  The e q u a t i o n ( l . l ) i s n o n o s c i l l a t o r y  there e x i s t s a p o s i t i v e i n t e g e r .oO ,00  ,  .  p  such  in R i f  that  „ CoO 0  p for s u f f i c i e n t l y large  S (r)=  r,  I [L (r)]- , 2  k  where  L (x) = L  ( x ) l o g x,  p=l,2,3,..,  with L  ( ) X  Q  = > x  l o g x = l o g l o g x, 2  l o g x = l o g l o g ^x. p  Proof. • The p r o o f i s s i m i l a r t o t h a t o f Theorem 9 . t o H i l l e ' s Theorem 12 i n s t e a d  We appeal  of the C o r o l l a r y 1 t o h i s  -29-  Theorem 7 a t the a p p r o p r i a t e p l a c e s .  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 o f Glazman [ 9 , p . 9 9 , Th.10].  As w i l l be noted below, our r e s u l t  c o n t a i n s 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  first  proved by H i l l e [ l i ] . Theorem 10.  Equation ( l . l ) i s n o n - o s c i l l a t o r y  in  R  i f the  inequality  rg!T(r')dr < co o I  holds f o r some  M  and  6 > 0  Proof.  6 > 0,  where  = (r : r g ( r ) / A 2  6  6]V  o  A G  Q  - 6),  have the meanings a s s i g n e d i n Theorem 9«  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 t h a t  (5.7) i s nonoscillatory.  [9> V'99> Th.10] are s a t i s f i e d , r [g (r)A 2  0  for a l l  >^=^-  The c o n d i t i o n s  of  since  - ( n - l ) ( n - 3 ) / 4 r ] > * - 6' 2  o  r e M . . Thus,(5.7) i s n o n o s c i l l a t o r y .  The remaining  &  d e t a i l s are as i n Theorem 9« " C o r o l l a r y 1. 0 < x < oO ,  ( C f . [11, p . 2 3 7 , T h . 2 ] ) . the o r d i n a r y  (a(x)y')' + b(x)y = 0  differential  If  b  i s bounded on  equation  i s non-oscillatory  i f  x b (x) dx <oo, J  1  -30-  where  b (x) +  i s the p o s i t i v e p a r t of  i s bounded below on Proof.  b ( x ) , provided  a(x)  0 < x < oo .  The c o n d i t i o n s o f Theorem 10 are f u l f i l l e d ,  c u l a r , i f f o r some  i n parti  m _> 1,  (5.8)  r  2  m  "  1  g"|( ) r  d  <  r  •  0 0  o I f we s e t m = 1,  n = 1  we o b t a i n  x b (x) dx < <=o , and . o  t h i s i s e q u i v a l e n t t o the h y p o t h e s i s o f the c o r o l l a r y , s i n c e b  i s bounded on Since  (0,co). g (r) = 6 ( ) 0  1  Theorem 10 g i v e s a simple  C o r o l l a r y 2. if  L  when  r  or n = 3,  n = 1  criterion for non-oscillation i n  The e q u a t i o n ( l . l ) i s n o n - o s c i l l a t o r y i n  "E?  i s u n i f o r m l y e l l i p t i c and  r g ( r ) dr < oo , J  where Proof.  o  g ^ ( ) = max [ g ( r ) , 0} r  Q  Set  m = 1,  true f o r g e n e r a l Co r o l l a r y 3.  Let  Then the e q u a t i o n  n = 3  and  & ( ) =  i n (5.8).  r  Q  m  a  x  {^(x) : jx|  A similar result i s  n: r  2  g ( ) >. A ( n - l ) ( n - 3 ) r  G  Q  f o r large  ('l.l) i s n o n - o s c i l l a t o r y i n E  n  i f  r.  -31-  (r) - A ( n - l ) ( n - 3 ) / 4 r ] d r  r[g  < oo .  2  n  o Proof.  Set  Remark.  i n (5-8).  m = 1  Each of the  corollaries  "been d e d u c e d f r o m the of H i l l e It  should  in  the  Swans on  [11,  p . 2 3 7 , Th.2]  be  n o t e d t h a t we  the  of the  fact  of equation  method o f Theorem  case,  since  the  b(x)  by  use  result  o f the  t o remove t h e  eventually  the  even  require-  p o s i t i v e , on  > b(x),  x e S , r  hypotheses  imply  »0 < r <  oo ,  the n o n - o s c i l l a t i o n  (5.7). next  t h e o r e m , a g e n e r a l i z a t i o n o f a remark  I n f a c t , we  s e c t i o n 4 are. the  P o t t e r , we  9.  Clark-  [ 2 1 , p.468], i s u s e f u l i n . t h e a p p l i c a t i o n s o f  of Potter  in  the  theorem  inequality  that  Our  theory.  using  have i m p r o v e d H i l l e ' s  coefficient  o  the  hy  m i g h t have  one-dimensional  c o m p a r i s o n t h e o r e m e n a b l e s us  g (r) and  corresponding  one-dimensional  ment t h a t account  t o Theorem 10  introduce  s h a l l use best the  i t t o show t h a t  possible  of t h e i r  the  kind.  following notation.  Let  the  estimates Following r\  be  a  2 positive ' C  (5.9)  Let  function defined  Mr).]"  the- f u n c t i o n s  2  by  = A ; g ( r ) * - (n-l) (n-3)/4r , 1  2  1  G.  and  G~  be  defined  by  0 < r < oO .  -32-  G,(r)  1  [V(r)]  1  h ' ( r ) ]  2  4n(r)  r|(r)  "  +  +  V(r) 2  T)'(r)  2  where primes denote d i f f e r e n t i a t i o n with r e s p e c t Theorem 11. s > 0,  A  Let q  L  be u n i f o r m l y  elliptic  b e i n g the e l l i p t i c i t y  in  constant.  the maximum of the p o s i t i v e part- of  b(x)  Then equation ( l . l ) i s n o n - o s c i l l a t o r y i n  to  R  f o r some  Let  R  g-^(r)  S ,  on  r.  be  0 < r < oo.  i f there  exists  2 a positive and  C  either-  (5.10)  function  s a t i s f y i n g (5-9)  r\  for large  r  (r) < 0  G,  or G (r)  (5.11)  2  holds f o r l a r g e Proof.  We  use  <  0  r. the argument of Theorem 9,  except t h a t  appeal to the remark of P o t t e r mentioned above and comparison theorem [5, is non-oscillatory. ordinary  To see  t h i s , we  together:  (5.12)  w" + [ri(r)] -2  (5.13)  h  ( r ) v '  the  p.208] to show that the equation note that the  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  2  we  ]'  + v  w = 0, =  0,  Sturm (5.7)  following  or n o n o s c i l l a t o r y  -33-  (5.1 0  [ri(r)z' ]' + G ( r ) z =  (5.15)  h ( r ) y ' ]' + G ( r ) y = 0,  J  0,'  1  2  f o r the d e r i v a t i v e of a s o l u t i o n of ( 5 - 1 2 ) (5.13),  tion  (5.14) i s obtained from  equation  z =  [ri(r)]  w,  - l !  the s u b s t i t u t i o n  (5.15)  and  y =  [r|(r)]  v.  K  i s a s o l u t i o n of  (5.12)  by  the  (5.13)  i s obtained from I f the h y p o t h e s i s  (or ( 5 . I I ) ) h o l d s , the e q u a t i o n ( 5 - l 4 ) (or ( 5 . 1 5 ) ) o s c i l l a t o r y by'the Sturm comparison theorem [ 5 ] « equation  (5.12)  also non-oscillatory.  The  remaining  t h i s theorem are s i m i l a r to those  by  (5-10) i s non-  Thus the  therefore (5-7)  i s n o n - o s c i l l a t o r y , and  subs-titu-  is  d e t a i l s of the p r o o f of  of Theorem 9 and w i l l  be  omitted.  6.  Sharpness of the  results.  In t h i s s e c t i o n we the theory.  s h a l l give examples to  In p a r t i c u l a r , we  s h a l l show t h a t the  c o n d i t i o n s of K n e s e r - H i l l e type b e s t p o s s i b l e of t h e i r k i n d . estimates  (4.1)  and  (4.4)  i n s e c t i o n s 4 and  Our  first  illustrate limit  5 are  the  example shows t h a t the  are sharp f o r each p o s i t i v e i n t e g e r  n, i . e . there e x i s t s a n o n - o s c i l l a t o r y e q u a t i o n f o r which e q u a l i t y holds i n ( 4 . 1 ) Example 1.  and  (4.4).  For each p o s i t i v e  integer  n  there i s a non-  o s c i l l a t o r y e q u a t i o n f o r which lim  inf  r-*  00  r  2  g(r)  =  A^X  +  (n-2) /4] 2  -34-  and  lim  inf  r °° g ( t ) d t = r  r-» oo  f  \ [\ 1  +  a  (n-2) /4] 2  r 2  Construction. ,x e E ,  Let  be t h e S c h r o d i n g e r  operator  v  + b(x),  with  n  (6.1) Then  L  b(x) = ( n - 2 ) / ( 4 r ) , 2  A  =1,  X  JL  = 0.  -|x| = r .  2  Simple  computations  show t h a t  vX  •n(r) = 2r,  G - ( r ) = 0.  Thus e q u a t i o n  L  by Theorem  i s non-oscillatory  11. On t h e o t h e r h a n d , s i n c e  have  (l.l)  lim i n f • r - co  r  lim i n f r-» oo  r  2  g(r) =  b  i s g i v e n by  ( n - 2 ) / 4 = A , [X  +  2  ( 6 . 1 ) , we  (n-2) /4] 2  a  Also, (r\  f°° J  g(t)dt  =  sharp  Example  liminf r — co  (n-2) /4 = 2  example shows t h a t t h e e s t i m a t e s  f o r every  n = 1  2  r =  Our n e x t  P ^  J>  K  this 2.  positive  reduces  integer  t o t h e example  F o r each p o s i t i v e  equation f o r which  n.  l i m sup r-* oo  2  A [\ +(n-2) /4]. 2  1  a  (5.1) and  (5.4) a r e  In the s p e c i a l  g i v e n by H i l l e  integer r  "r  n  there  [11].  i s an  2 g ( r ) = (n-2) A /4. ° 0  case  oscillatory  -35-  Construction. x € E ,  Let  b(x)  (6.2)  since  |x| = r  Schrodinger  A  o  of  = (n-2) /4r 2  and  operator  is  l i m sup r-» oo  r  = 1  clear g (r)  2  in this  +  2  Y l  /(.r  v  2  +  b(x),  r) , 2  y^ ^  that = (n-2) /4  =_(n-2) A /4, 2  2  case.  solutions  On t h e  (viz.  q  o t h e r h a n d , our  solutions  (2-n)/2 u = r^ (log  the form  log  i s . a number s a t i s f y i n g  y^  Then i t  has p a r t i c u l a r alone)  be t h e  and t a k e  n  where  L  ; /  r)  equation  d e p e n d i n g on s  ,  where  r  s  is  a  2 ' r o o t of the q u a d r a t i c e q u a t i o n Y-j the  > -^p  this  (l.l)  Theorem 1 t h e r e f o r e  by  is  is  oscillatory.  shows t h a t  if  and  thus  The a r g u m e n t o f  equation ( l . l )  also clear that  is  oscillatory.  the f u n c t i o n  b  is  defined  (6.2), we h a v e l i m sup r-» co  r  H o w e v e r , we h a v e Remark.  00  i  g£(t)  dt  r  •lim sup r-*oO  for  = 0.' S i n c e  - 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 ,  r a d i a l form of  It  s  r  c  0 0  [1/41  shown t h a t  P  (l.l)  Examples 1 and 2 have  the Schrodinger  operator  + y-,/(t  in  is  shown, E  l o g t)  p  ]dt  oscillatory  in this  in particular, the  constant  case.  that"  (n-2)  /4  -36-  i s c r i t i c a l f o r each p o s i t i v e i n t e g e r both o s c i l l a t o r y and  n,  n o n - o s c i l l a t o r y equations f o r which  e q u a l i t y holds i n the estimates (4.1), We  s h a l l now  of r e l a x i n g one  or both of the  <oo  x  1 r  conditions  and  f (r)  There a l s o e x i s t s a n o n o s c i l l a t o r y oo  dr  1 r  n  Construction. and  _  Then we  may  take  (since  R =  E )  the  (3.1)  i n the  effect  n-  L  be  [°° r " 1 " 1 1  r- f(r)]dr 2  e q u a t i o n f o r which  and  the  [g(r) - X  1  r  n _ 1  [g(r)-X r- f(r)dr]  < oo  2  a  Schrodinger operator  v  +  b(x),  = e"  1  f(r) = 1  2 „ ,, 2+ ( n - 2 ) V ( 4 r ), (since  x  a. . = 6. .)  = r. and  X  = 0  n  n >_ 3,  [°° r " [ g ( r ) 1 On  (5.4).  take b(x)  If  = co  f(r)  1  Let-  (6.3)  1  and  Leighton-Wintner theorem.  d r  n  (5-1)  There e x i s t s a n o n o s c i l l a t o r y ^ e q u a t i o n f o r which  n  x e E-,n ,  (4.4),  give an example to i l l u s t r a t e the  dimensional form of the Example 3.  i . e . there e x i s t  then  CO  dr . n-1 ~,\ 1 r f(r)  < oo  - X r-. f ( r ) ] d r = T°° r"- " [- - 1 2  other hand, a s t r a i g h t f o r w a r d  n ±  11  e  r  and  +  (n-2) /(4r )]dr 2  2  computation shows t h a t  -37-  ( i n the n o t a t i o n  i)  of Theorem 11)  <  G r 1  0  that equation ( l . l ) i s n o n o s c i l l a t o r y by  If  n < 2,  Cg(r)  r ^  a  However, s i n c e shows t h a t (l.l)  = e~  < 0  This  [e~  •  if  )'  b  + x [e~ s  continue our  r,  and  .>  s o  and  therefore  i s g i v e n by  x  < «D  computation equation  ( 6 . 3 ) and on the  a... - 6. .. oscillatory  [ a ( x ) u ' ] ' + b(x)u  + (s~l) /(4x )]u = 2  = 0.  0,  2  on  0 < x <  i n s p e c t i o n of the hypotheses of  theorems i n t h i s chapter with the  ) ~ °°  (Potter's  oQ the  following  Example 1 p r o v i d e s us with a n o n o s c i l l a t o r y  f o r which . '°°  r  routine  a nonnegative i n t e g e r ) , i s n o n o s c i l l a t o r y  Remark.  e  equation  (xV  We  S  d  a  one-dimensional e q u a t i o n  In p a r t i c u l a r , the  r  + (n-2) /(hr)]dr  r  example a l s o throws some l i g h t  behaviour of the  •  n _ 1  + l/(4r )  for large  is nonoscillatory  Remark.  r  a  J  [r\(r)]~  G^(r)  °o  f(r)  1  - X r~^f(r)]dr = | 1  1  (s  l  l  Theorem 11.  dr  then J  r  f o r  condition).  This  equation illustrates  1. the if  e f f e c t of r e l a x i n g one b  i s . g i v e n by  However. •,  of* the  ( 6 . 1 ) , then  H ( r ) s 0,  conditions  h(r) = 2 r ,  so t h a t 1  (3.6). so that  H, ( t ) dt < oO,  In  fact,  r  dt F["t)  00  =  °°  -38-  In the remark f o l l o w i n g Theorem 4 (the n-dimensional form of a r e s u l t  of P o t t e r [ 2 1 ] )  that the f u n c t i o n  h  we  noted that the  be p o s i t i v e was  requirement  i n g e n e r a l more r e s t r i c t i v e  than the corresponding c o n d i t i o n s i n Theorem 3 (the n-dimensional form of the Leighton-Moore-Wintner  Theorem).  However, our  next example shows t h a t there are equations f o r which Theorem 4 o b t a i n a b l e from Theorem 3-  g i v e s i n f o r m a t i o n not immediately Example 4 . (6.4)  The d i f f e r e n t i a l e q u a t i o n v u 2  is oscillatory  + [(n-2) /(4r ) 2  on  2  E^ ,  + l/(4r  condition (3-4)  even though  1  log r)]u = 0  2  does not  hold. To see t h i s , we note that  (with the n o t a t i o n of  Theorem. .4) [h(r)]" A  r,  2  = l/(4r ) + l/(4r 2  r o u t i n e computation and t h e r e f o r e  shows that  2  log r ) .  Hg(r) = 0 ( r  H„(r)dr = oo  _ 1  (s > l ) .  )  f o r large  Thus e q u a t i o n  s  (6.4)  on account of Theorem 4 and the remark  is. o s c i l l a t o r y  f o l l o w i n g Theorem 1 r e g a r d i n g the l i m i t s n = 2,  Except i n the case obvious way  °° r s  1  -  6  this  cannot be concluded i n an  from Theorem 3 , s i n c e  [g(r) - X r " '  a  2  of i n t e g r a t i o n .  (for  f(r)]dr = C  s > l)  r " [-^ 1  - s for  each  6  +  4r 6 > 0.  ]dr < oo 4r log r  -39-  E q u a t i o n (6.4) a l s o s a t i s f i e s the c o n d i t i o n s o f Theorem 7.  In f a c t ,  A-^ = 1 ,  i n t h i s example  = 0,  X.  so  .that  + (n-2) /4]t" = l/(4t  g ( t ) - A [\ 1  2  a  2  2  log t ) .  Thus J  t[  g (  t)  -  A  ]  >  (n-2)V4]t- dt 2  a  +  }  = j  4  t  t  =  cO  T h i s shows t h a t we might have used Theorem 7 i n s t e a d of Theorem 4 to show t h a t e q u a t i o n (6.4) i s o s c i l l a t o r y .  - 40 -  CHAPTER I I  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 e q u a t i o n g i v e n  here reduces t o Glazman's i n the case of o r d i n a r y equations (m = 1, 2, ...)(of.  of order 2m  o b t a i n e d below f o r 2m-th equations are d i r e c t [9]  s u l t s of Glazman  (-l)  m  d  ,2m - ^ dx  L  [9, p. 4 0 ] ) .  The  criteria  order p a r t i a l d i f f e r e n t i a l  g e n e r a l i z a t i o n s of c o r r e s p o n d i n g r e f o r 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  - b ( x ) u = 0 , 0 < x < oo .  We a l s o o b t a i n f o r f o u r t h - o r d e r p a r t i a l  differen-  t i a l equations c r i t e r i a extending those obtained, by L e i g h ton  and Nehari  [17] f o r 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  2 2 ^ [a(x) dx" dx"  - b(x)u = 0 ,  a(x) > 0 .  Most of our theorems are proved by a p p e a l i n g to t h e i r one-dimensional  forms. 'An e x c e p t i o n i s Theorem  20,  which i s proved by i n v e s t i g a t i n g d i r e c t l y the s o l u t i o n s of  -  the  41 -  comparison e q u a t i o n .  8.  Definitions  and n o t a t i o n .  We s h a l l c o n s i d e r the l i n e a r e l l i p t i c t i a l operator  (8.1)  L  Lu = ( - l )  d e f i n e d by  m  S_ D i ) j—1  on unbounded domains E  n  .  R  m i  (a  a  D™u) - bu  i j  ±  = a  J  J  ±  ,  i n n-dimensional E u c l i d e a n space  We s h a l l use the n o t a t i o n s of Chapter I , except where  otherwise i n d i c a t e d .  The c o e f f i c i e n t s  be r e a l and of c l a s s  C  is positive  m  definite i n  in  R U dR  R .  of  L  R U SR  Definition.  where i n  are assumed t o  and the matrix ( i j ) a  b  R U 3R .  is The domain  i s d e f i n e d t o be the s e t of a l l r e a l - v a l u e d  f u n c t i o n s on  a function  a ^  The c o e f f i c i e n t  assumed t o be r e a l and continuous on D(L)  differen-  of c l a s s  A solution u e D(L)  C  (R) .  of the e q u a t i o n  Lu = 0  which s a t i s f i e s the e q u a t i o n e v e r y -  R . We assume that  R  contains a c y l i n d e r  form  .  G x (x  n  is  : 0 < x  n  < oo}  ,  of the  - 42 -  where  G  i s a bounded  following  (n-1)-dimensional domain.  n o t a t i o n w i l l be  R  r  = R n (x e E  Definition.  used:  : |x[ > r} .  n  A bounded domain  N c R  a nodal domain of a n o n t r i v i a l s o l u t i o n u  The  and i t s p a r t i a l d e r i v a t i v e s  u  i s s a i d to be of  Lu = 0 i f f  of order • <_ m-l  v a n i s h on  dN .  Definition.  The d i f f e r e n t i a l e q u a t i o n  s a i d to be o s c i l l a t o r y i n solution all  u  of  r  Lu = 0  R  Lu = 0  is  i f f there e x i s t s a n o n t r i v i a l  w i t h a nodal domain i n  R^  for  r > 0 .  Definition. tory i n  A function  R  iff'u  u  has a zero i n  i s s a i d to be R  p  oscilla-  f o r a l l .r > 0 .  We s h a l l a l s o use the standard n o t a t i o n CC  D u = D^ a  1  D  CL  CL  D  2 2  n n  u ,  a = (c^,  n  a ) , |a| =  2  n  a . .  Our o s c i l l a t i o n c r i t e r i a i n §10 w i l l be proved under the assumption bounded: A ( X )  <.  that  the l a r g e s t  F o r some  A ,  s > 0  there e x i s t s  , for a l l x e R  x  eigenvalue  A ( X ) of such  ( j_j) a  i  s  that  . s  Let  n  be the s m a l l e s t e i g e n v a l u e of the problem  - 43 -  r  (8.2)  D. (A  S  (-1)  m  i=l  J I D cp = 0  on  a  Let the f u n c t i o n  1  D,  n  1  g e C(0,co )  G^ c R o  cp) = ucp  dG , |a[ = 0, 1,  on each bounded subdomain of bounded domain  m  Auxiliary  G  ra-1.  R .  .  sC^)  be such t h a t  <. ° ( ) 1  x  (For example, on a  we might s e t to  g ( t ) = min (b(x) : x e G  9.  in  1  Q  and  x  n  = t} .)  results.  As i n Chapter I , we s h a l l make use of a monotonic 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 , which we s h a l l deduce from a form of P o i n c a r e ' s i n e q u a l i t y . ^  Definition. bounded width  We s h a l l say t h a t a domain < d  line p a r a l l e l to  I  i s no g r e a t e r than  iff  there i s a l i n e  I  0 has  such t h a t each  i n t e r s e c t s fi- i n a s e t whose diameter d .  F o r example, the t r u n c a t e d cone 3 has bounded width  " section  < t sec a , and the (open) c y l i n d e r  G,(t) of s e c t i o n 10 has bounded width  Lemma (Poincare's i n e q u a l i t y ) . ed width <. d , then  or  <_ t .  I f a domain  0  has bound-  -  m-j  (9.1)  |cp|  for a l l  j j Q  < d  m  Mm a = J :  u  J  and  **]  2  |a|=m  0  Y  i s a constant  n , and  ,? ^  [  ,  m  , 0 <_ j <_ ra-1 , where  cp € C™(fi)  m  |cp| ^  Y  depending o n l y on  Proof.  2J4 -  1 / 2  T h i s i s g i v e n on pp. 7 3 - 7 5 of  [1].  We s h a l l s t a t e our next r e s u l t f o r the s i m p l e s t boundary value problem, but the method c l e a r l y works f o r g e n e r a l boundary c o n d i t i o n s of the k i n d g i v e n i n [ 2 4 ] . account  of the form of the Poincare  the r e s u l t  (a m o n o t o n i c i t y  inequality  cited,  p r i n c i p l e f o r eigenvalues)  will  (8.1).  be o b t a i n e d f o r a more g e n e r a l o p e r a t o r than Let  here  On  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 o p e r a t o r  M  be d e f i n e d by  (9.2)  Mu a ( - i )  m  2  D^)  |p|=|q|=m .,  where  p = ( p , •.., P ) 1  n  -  B  >  u  ^  , q = (q^ • q  n  )  are m u l t i -  i n d i c e s w i t h i n t e g r a l nonnegative components. |pj =  n Ep, 1=1  .  F o r each  1  The c o e f f i c i e n t s  z e E  n  o we w r i t e  As u s u a l , -£U i \J z. i=l p  z  x  =  .  1  A^^(x)  are supposed to be o f c l a s s  C" (R)  - 45 -  and symmetric i n the i n d i c e s . M elliptic  F o l l o w i n g Browder [ 3 ] , we  i f the f o l l o w i n g two (a)  c o n d i t i o n s are  2  The form  A  |ph|q|=m  d e f i n i t e a t each p o i n t  (x) z P  for  lP\i D u  A  q  a l l u e C (R) m  +  fulfilled: i s positive  q  q  x e R .  (b) F o r each bounded domain e x i s t s a number U (G) > 0 such t h a t o 2  p  call  2  dx > \x(G)\  G  G c R  with  (L^u)  there  dx  2  .  We  note t h a t c o n d i t i o n  If  the o p e r a t o r  (a) i s the u s u a l  ellipticity  condition.  condition  (b) i s redundant,  condition-(a). form  = \ p  q  where  i s ' o f the form  (8.1), then  and i n f a c t i s a consequence of  To see t h i s , we note t h a t i f  M  has  the  (8.1), then  a. . A  M  0  (me^)  if  p =  (me.,)  and  q = (me.)  ,  otherwise,  i s the v e c t o r i n  p l a c e and zeros elsewhere. there e x i s t s a number  E  n  with  If condition  m  (a) h o l d s , then  1^ (G) > 0 "such t h a t 0  i n the i - t h  - he -  inf xeG  since  n inf Z |z|=l i , j = l  G  a. ( x ) z m z m  J  i s compact and t h e c o e f f i c i e n t s  tinuous .  a^^  a r e con-  Hence  m  n  This  m= u (G) ,  m J  m  v  ..  i2m  i  „n.  , z e E  implies  5  a.. z , i,j=l ^  z.  m  m  S  > u (G) ° • i=l  2  , z e  m  Z  E  n  1  T h i s may be v / r i t t e n i n t h e f o r m  (*)  S i,j=l  where  §  a  S  §  ±  > u (G)  1  i s the vector  2 § i=l  2  ,  , the signs being  (±ZJ_ ) M  t h e same  as t h o s e o f t h e c o r r e s p o n d i n g c o m p o n e n t s i n t h e v e c t o r Every vector therefore  § e E  condition  S a i,j=l  1 J  and  this  § 1  J  g  n  may be w r i t t e n i n t h e a b o v e (*)  form;.  implies  > u (G) . 2 i=l 0  z .  S  2  § e E  n  ,  1  implies condition (b).  V.  - 47 -  Monotonicity  Principle.  domain contained If  implies  eigenvalue  Mu  ^ (t)  = Xu i n  il,  G  c  p  G,  ;  Du a  400-401].  i s uniformly k. > -co o  be  of bounded width  G  , G  g  r  in  on  ^ G  t , and  5G,  l i m -X  g  , then  continuous on  G  J  Then, c o n d i t i o n  Extend  the  'G  (t) = + oo  .  may  adapt the argument i n note t h a t s i n c e  G , there e x i s t s a  Bu dx  > k  u dx  for a l l  2  G  J [u] P u  u  G  (  2  |p| = |q|=m  (b) i m p l i e s  > n_(G)  A  P  L^u  d u q  - Bu }  M  u e be  .  q  that  f 2 . (D u) dx + k • G-' | p | =m  continuously  dx  2  p  to a l l of  2  fi  B  constant  by  J [u]  (9-3)  <_ t .  l,..,m-l  Let the E u l e r - J a c o b i f u n c t i o n a l corresponding to defined  a  , |a| = 0,  t-o+ °  For the second p a r t we  such t h a t  G^  fi  = 0  For the f i r s t p a r t we  PP*  let  of the problem  Q  i s monotone d e c r e a s i n g  Proof.  0 < t < oo  w i t h i n a domain  0 < r < s < o o  first  For  f u dx G 2  0 J n  by s e t t i n g  u = 0  C (R) m  -  outside  G .  48  -  Apply P c i n c a ^ ' s i n e q u a l i t y with  j = 0  in  ( 9 . 1 ) to o b t a i n  u^dx  < V  [  m  Y  J  fi  2  \Jpu\ dx]  .  d  |pj=m  Hence  J  f G  u dx < 2  2 Y  t  2  [f G  m  E |p|=m  J  (DP )  2  U  dx]  Combining t h i s with i n e q u a l i t y (9*3) we get  J [u] Q  > (k  The remainder that  10.  Q  +  u (G)/ o  2 Y  t  2 m  )  u  dx  of the p r o o f i s as i n s e c t i o n 2.  Y >' 0 , s i n c e  Oscillation  ||u||  ^ 0  Q  (We  note  .)  criteria.  In t h i s s e c t i o n we o b t a i n o s c i l l a t i o n theorems under the h y p o t h e s i s t h a t the l a r g e s t e i g e n v a l u e of  (a. .) J  i s bounded.  Our theorems g e n e r a l i z e r e s u l t s of Glazman  [ 9 ] f o r one dimension to the n-dimensional case. be s p e c i a l i z e d to a l l of  E  n  by t a k i n g  They  may  cp 3 1 , |i = 0 .  - 49 -  THEOREM 1 0 .  (10,1)  The d i f f e r e n t i a l  Lu = ( - l )  is oscillatory number  in R  A-. > 0  r  co  s > 0  u  there e x i s t s a  A ( X ) <. A. , f o r a l l x e R_ , S  and  [ g ( t ) + u] d t = + oo ,  are d e f i n e d above i n s e c t i o n 8 .  g  ( 8 . 1 ) w i t h the separable e q u a t i o n  We compare  (-l)  S  m  D, (A  m  1  '  1  - b*v = 0  D, v)  M  i=l  where  m  i f f o r some  such t h a t  o  (10.3)  (a, . D, u) - bu = 0  m  JL  (10.2)  Proof.  D,  J.  and i f  where  2  m  equation  1  The h y p o t h e s i s ( 1 0 . 2 )  b*(x) = g ( x ) . n  the o r d i n a r y d i f f e r e n t i a l  implies that  equation  o (10.4)  (-l) A  is oscillatory  m  X  D  2 m n  on account  w - [ g ( x ) + n]w = 0 n  o f Glazman's g e n e r a l i z a t i o n [ 9 ,  p. 1 0 4 , Th. 1 3 ] of the theorem o f L e i g h t o n [ 1 6 ] and  - 50 -  Wintner [ 2 8 ] . solution x  n  =  5  w  1' 2 6  Let  r > 0  be g i v e n .  Then there e x i s t s a  of (10.4) w i t h zeros of order where  J  6  2  > 6.^ >_ max  e i g e n f u n c t i o n of ( 8 . 2 ) then the f u n c t i o n x = (x^, x ,  v  (r,s] .  m  If  at cp  i s an  corresponding to the e i g e n v a l u e d e f i n e d by  .n ,  v ( x ) = w(x )cp(x) , where n  ..., _ ] _ ) * i s a s o l u t i o n of (10.3) by d i r e c t x  2  n  c a l c u l a t i o n , w i t h a nodal domain  G  = G x (x  ±  In f a c t , i f  6. 1  domain r  hence  n  for  a n  n  Thus  v  In f a c t ,  f G  r  .  = (a,a ) n  a  v  |a| = 0, 1,  implies  ( E (A i,j=l  holds whenever  m-l, s i n c e and  has a nodal domain  x € G^  G  n  1  i  c  6 J  R s  , then  — _ ®(x) . •'  cp has nodal .  N  c R for a l l r r |x| _> |x | > 6^ >_ r , n  Thus (10.3) i s o s c i l l a t o r y .  1  1  2  w(x ) D . n'  a theorem of Swanson [ 2 4 , Th. 4 ] .  J  6}  v  0  x e R  bG^  <  a^)  a D v = D  n  n  are m-fold zeros of w(x ) • • n'  2  G .  > 0 .  a  on  a  6  < x  l  2  D v =0  and  6  a = (a^, a ,  D v = D  Hence  :  n  The  We now appl;  inequality  - a, ,)D, v D, v + (b - b * ) v } dx > 0 10 ^ i J m  >  o  n  account  m  of the  2  hypotheses  - 51 -  S  a. .(x)z,z . <_ A  i,j=l  1  J  1  J  £  1=1  1  2  z,  n , z e E  , .x e R_  1  s  b*(x) = g ( x ) < b(x) , x e G n  .  1  Hence the e i g e n v a l u e problem  Lu = Xu  in  ;  D u = 0 a  on  3G  1  , |a| = 0_, 1,  m-l  has a t l e a s t one ( i n p a r t i c u l a r , the s m a l l e s t ) e i g e n v a l u e l e s s than or equal t o z e r o .  G(t) = G x [ x  and l e t  :  n  Let  < x  < t) , 6  n  < t < 6  1  2  ,.  ^ ( t ) denote the s m a l l e s t eigenvalue of the p r o 0  blem  Lu = X ( t ) u  in  G(t) ; D u = 0 a  on  3G(t)  j a | = 0,1,...,m  By the m o n o t o n i c i t y p r i n c i p l e i n s e c t i o n 9, tone n o n i n c r e a s i n g i n  Since  ^ ( 2^ — Q  such t h a t  6  0  *  t  h  e  6  r  X (T) = 0 . Q  e  n  < t <_ 6  exists  0  and  a number  T h i s means t h a t  ^ (t) 0  i s mono  l i m X ( t ) = + oo  T  in  G(T)  (6^,62]  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 s i n c e c G  G(T)  1  c  f o r arbitrary  oscillatory i n R .  r > 0 , e q u a t i o n (10.1) i s  T h i s completes the p r o o f of the  theorem. T h i s theorem 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, p. 104] of the Leighton-Wintner theorem. set  n = 1  and  we may take  a^(x)  To see t h i s ,  = 1 , and r e c a l l t h a t f o r R = E  n  cp s 1 , n = 0 .  Our  next theorem extends t o  n  dimensions  Glazman's g e n e r a l i z a t i o n [9, p» 100] of a r e s u l t o f H i l l e [11, Th. 5 ] .  Glazman's r e s u l t i s the s p e c i a l case  n = 1 , a-^x) =1  of our theorem.  THEOREM 11.  Let  s > 0 If  s(  there x n  )  exists  + M >. 0  A^ > 0  v* ~ m  x  n  , »oo J [g(t)+u]dt  x  m  - l m  _ J2^1 " (m-l)J  ? k=l  t  A ( X ) <_  , x e R  and  where  A  n  be bounded, i . e . f o r some  such t h a t  f o r large  9  l i m sup  A(X)  R = E  i\k-l/m-l\ (k-l> • 2m -,k  then the e q u a t i o n (10.1) i s o s c i l l a t o r y .  2 > A .A 1  m  ,  g  - 53  As i n the proof of Theorem 1 0 , we compare  Proof.  (10.3),  with  [11,  oscillatory.  n  > A  a  X  sufficiently  (10.1)  (a  2m  n  large  00  Proof.  x  2 m _ 1  1g(t)+ii-A..a x  i s oscillatory  (10.4)  i s oscillatory  of Glazman [ 9 , p. 1 0 2 ] . l a r to that  all x e R R  2  t"  2  m  g  R  g  and some  |  dt = 0 0  m  R .  the o r d i n a r y " d i f f e r e n on account  of a theorem  The remainder, of the proof i s s i m i omitted.  L e t the l a r g e s t e i g e n v a l u e  be bounded i n  tory i n  in  of.Theorem 10 and w i l l be  Corollary.  = (2m-l)U/2 )m  m  and  The hypotheses imply t h a t  t i a l equation  for  x~  l i m sup ( l o g r ) [ t r r  then e q u a t i o n  i  2  00  (10.6)  t h a t of  I f the i n e q u a l i t y  g(* )-Hi  holds f o r  is  change.  THEOREM 1 2 .  (10.5)  (10A)  Th. 5 ] t o show t h a t  The remainder o f the proof f o l l o w s  Theorem 10 without  (10.1)  [ 9 , P« 1 0 0 ]  q u o t i n g Glazman's g e n e r a l i z a t i o n  of a theorem of H i l l e  (a ^)  -  f o r some  s > 0 :  A^' > 0 .  i f for sufficiently  large  A ( x ) <_ A ^ ,  Then ( 1 0 . 1 ) x  n  A ( X ) of  is oscilla  and some  5 > 0  - 54  the  inequality  x  is  n  2 m  Cs( n> x  + M ]  >  A  l^ m a  2 + 6  )  satisfied.  Proof.  The f i r s t hypothesis  clearly f u l f i l l e d . > A 6t~ 1  Remark.  ( 1 0 . 3 ) ' o f Theorem 12 i s  Moreover, s i n c e  f o r large  2 m  Theorem 12 i s a l s o  g(t-) + |i -  A  i  a  2 m  t "  2  m  ( 1 0 . 6 ) of  t , the second h y p o t h e s i s  satisfied.  T h i s r e s u l t g e n e r a l i z e s the c l a s s i c a l  H i l l e theorem [11] of  -  i n four directions:  Kneser-  ( i ) to equations  a r b i t r a r y even o r d e r , ( i i ) to. equations w i t h v a r i a b l e  leading.coefficients,  ( i i i ) to  equations not d e f i n e d on a l l of cylindrical  n  dimensions, E  n  ( i . e . on  ( i v ) to  limit-  domains).  It  i s p o s s i b l e 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 ) , [ 9 , Th. 9 , p. 96]  c i t i n g Glazman's g e n e r a l i z a t i o n  of the K n e s e r - H i l l e theorem [11]  to show  t h a t 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 ( 1 0 . 4 ) i s o s c i l l a tory. It just c i t e d  should be noted .that the r e s u l t of Glazman  [ 9 , Th. 9 , p. 9 6 ]  i s the s p e c i a l case  R = E  n  ,  - 55 -  n = 1 , n ( a  11.  x  )  ~  1  o  f  o  u  corollary.  r  Equations w i t h one v a r i a b l e  separable.  In t h i s s e c t i o n we s h a l l c o n s i d e r the e q u a t i o n  (11.1) D  u+ V D [a (x)D. u] i j j—1 m  ±  - (-l) b(x )u = 0 ,  m  m  i:s  n  x = (x-,, Xg , •.• x _ ) . F o l l o w i n g Swanson [26] we  where let  2 m n  }  |i*  be the s m a l l e s t e i g e n v a l u e of the problem  (-l) *" 1  (11.2) ^  where  n  1  D cp = 0  G  V  D [ a ( x ) D c p ] = n*cp m  i  on  m  i j  j  dG , |a| = 0 , 1,  ...  in G } m-l ,  i s as i n s e c t i o n 8. . Each o f the theorems of the p r e c e d i n g  has an analogue i n t h i s case, but without that  A ( X ) i s bounded.  section  the assumption  As an example ire s t a t e and prove  the f o l l o w i n g analogue of Theorem 10:  THEOREM 10A.  (11.3)  The d i f f e r e n t i a l  (-l) [D m  2 1  % +  D ™[ i,j=l  a i 1  equation  (x)DAi]}  - b(xju = 0 n  - 56 -  is oscillatory i n  R  if  f.00  (11.4)  J  Proof.  The hypothesis  differential  (11.5)  [b(t)+|i*] dt = + oo  o  (11.4) i m p l i e s t h a t the o r d i n a r y  equation  (-D D m  n  2 m  v  - [b(x )-Hi*] v = 0  of Glazman's g e n e r a l i z a t i o n  p. 104] of the theorem of L e i g h t o n  Let  r > 0  be g i v e n .  2  >  5  1 —  r  *  I  f  ^  i  ponding t o e i g e n v a l u e by  u(x) = v(x )cp(x) n  s  a  n  m  e i  S  at  e n : £ > u n c  x  1  = [x : x e G , 6  nodal domain implies  1  G.^ c R^  = -j_, 5  n  tion  s 2  v  of  > where  of (11.2) c o r r e s -  u* , then the f u n c t i o n  u  defined  i s a s o l u t i o n of (11.3) b y . d i r e c t c a l -  c u l a t i o n , w i t h a nodal domain G  [16] and Wintner [ 2 8 ] .  Then there e x i s t s a s o l u t i o n  (11.5) w i t h zeros of order 6  ' .  n  i s o s c i l l a t o r y on account [9,  .  < x  n  , .  < 6 ] 2  .  Thus there e x i s t s u w i t h a r > 0 , s i n c e . x e G-j^  for arbitrary  |x| >_ |x | > 6^ >. r , so t h a t n  x e R^  .  Hence  •equation (11..3) i s o s c i l l a t o r y . The analogues of Theorems (11.3) are proved  similarly.  11 and 12 f o r e q u a t i o n  We s t a t e them without  proof.  - 57 -  THEOREM 11A.-  If  b ( x ) + u* > 0  2m-l m  x  l i m sup v* ~  «J  r -• oo  f o r large  n  oo  [b(t)+n*] dt > A  R  x  and  ,  2  r  in R .  The e q u a t i o n (11.3) i s o s c i l l a t o r y i n  i f the i n e q u a l i t y  large  n  III  then the e q u a t i o n (11.3) i s o s c i l l a t o r y  THEOREM 12A.  x  b ( x ) + \i* >  x "  R  holds f o r  2 m  n  and  n  00  l i m sup ( l o g r ) f t ^ ' ^ b f t ^ * ^ r -» oo •r  Remark.  2  t~  2 m  | - dt = oo  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 [ a ( x ) D u ] + _ 2 D [a .(x)D u], + b ( x ) u = 0 . i f J I n  n  n  i  lj  j  n  =  Lack of s u i t a b l e one-dimensional  o s c i l l a t i o n theorems has  restricted  a(x )s 1  us t o equations with  n  i n the g e n e r a l _  even order case.  12.  F o u r t h - o r d e r equations on l i m i t - c y l i n d r i c a l domains. In t h i s s e a t i o n we s h a l l d e r i v e o s c i l l a t i o n c r i -  - 58 -  t e r i a f o r the e q u a t i o n  (12.1) D ^ [ a ( x ) D u ] + V D ^ a ^ U j D ^ u ] - b(x )u = 0 , i j j l 2  n  n  n  n  =  a , a^^ , b  where the c o e f f i c i e n t s i n section 8.  s a t i s f y the c o n d i t i o n s  We a l s o suppose t h a t  are p o s i t i v e f o r l a r g e  x  n  .  a(x )  and  b(x )+u* n  Our theorems c o n s t i t u t e  t e n s i o n s of well-known r e s u l t s of L e i g h t o n and Nehari  THEOREM 13. and l e t x  n  .  Let  a. be an a r b i t r a r y r e a l  a(x ) > 0 , b ( x ) - H i * > 0 n  Then e q u a t i o n  ex[17] •  constant,  f o r s u f f i c i e n t l y large  (12.1) i s o s c i l l a t o r y i n  R  if  /-lim sup t " " a ( t ) < 1 t -» oo 2  a  (12.2)< .lim i n f t " [b(t)+u*] t -• CO 2  where  Proof.  | i * i s g i v e n by  >:• I :?' 1  ,.2x2  ,  (11.2).  The hypotheses (12.2) imply t h a t the o r d i n a r y  differential  (12.3)  a  D  equation  2 n  [a(x )D n  2 n  v ] - [b(x )..+ u*] v = 0 n  - 59 -  has an o s c i l l a t o r y s o l u t i o n , i . e . has a s o l u t i o n w i t h i n f i n i t e l y many z e r o s , on account [17, Th. 6 . 2 ] .  and Nehari  Let r > 0  theorem i n the paper j u s t c i t e d a solution where  v  6^ >  >_ r .  The remainder  zeros a t  x  n  = 6^, 6^ ,  of the proof f o l l o w s  t o n-dimensions p a r t of the  theorem of L e i g h t o n and Nehari  [ 1 7 , Th. 6 . 2 ]  cited. In the case t h a t  [17,  By another  change and w i l l be omitted.  T h i s r e s u l t extends  just  be g i v e n .  [17, Th. 3 « 6 ] there e x i s t s  of ( 1 2 . 3 ) with double  that of Theorem 10A without  one-dimensional  of a theorem of L e i g h t o n  a(x ) s 1 , another p a r t of  Theorem 6 . 2 ] shows t h a t the c o n c l u s i o n of Theorem 13  holds i f the hypotheses  ( 1 2 . 2 ) are r e p l a c e d by  lim inf t [ b ( t ) t -* oo H  THEOREM 14.  The  R  a > 0  i f there e x i s t s co  Proof.  + u * ] > 9/16 .  equation  (12.1) i s o s c i l l a t o r y i n  such t h a t oo  The hypotheses  (12.4)  imply t h a t the e q u a t i o n  ( 1 2 . 3 ) has an o s c i l l a t o r y s o l u t i o n , on account  o f a theorem  - 60 -  of L e i g h t o n and Nehari  [17, Th. 6.11].  d e t a i l s of the proof f o l l o w f a m i l i a r  The remaining  l i n e s and w i l l be  omitted. In the case  a(x ) =• 1 , the above r e s u l t n'  takes  the f o l l o w i n g form:  THEOREM 15.  The e q u a t i o n  is oscillatory i n R  holds f o r . some  Proof.  i f there e x i s t s  a > 0  such t h a t  r < 3  Since we s h a l l use the argument of Theorem 13, we  need o n l y show t h a t  (12.3) has an o s c i l l a t o r y  solution.  T h i s f a c t i s a consequence of a r e s u l t of L e i g h t o n and Nehari  [17, Cor. 6.10].  We omit the remaining d e t a i l s of  the p r o o f . Our next two theorems give o s c i l l a t i o n c r i t e r i a for a special  case of (12.1), namely  - 61  (12.5)  D  2 n  [a(x )D n  2 n  u]+ V D [a .(x)D u]+c(x )u = 0 , 1, j — 1 2  1  where the c o e f f i c i e n t and  the f o l l o w i n g  (12.6)  c  n  Let  oscillatory i n  x  s  R  x  n  R ,  :  .  be an a r b i t r a r y r e a l  the i n e q u a l i t i e s n  n  c(x ) - u* > 0  n  c i e n t l y large  J  i s r e a l and continuous on  a(x ) > 0 ,  suppose that  2  i j  i n e q u a l i t i e s hold f o r large  THEOREM 1 6 . and  -  constant,  ( 1 2 . 6 ) hold f o r s u f f i ( 1 2 . 5 ) has a  . . Then e q u a t i o n  solution  If  l i m sup t -» 00  t" " 2  s  a(t) < 1  and  lim i n f t t - 00  where the  u*  2  "  s  [c(t) - n*] > s /4 ,  i s g i v e n by ( 1 1 . 2 ) .  2  I n the case  a  (  x n  c o n c l u s i o n remains v a l i d i f ( 1 2 . 6 ) holds and  lim i n f t^[c(t) t -> 00  > 1  .  ) - 1  ,  - 62  Proof. tial  The  -  hypotheses imply t h a t the o r d i n a r y d i f f e r e n -  equation  (12.7)  D  2 n  [a(x )D n  2 n  v]  + [ c ( x ) - u*] v = 0 n  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 L e i g h t o n and Nehari  [17,  Th. 11.1].  Then there e x i s t s a s o l u t i o n .  If  cp  of  i s an e i g e n f u n c t i o n of  the eigenvalue n  be  given.  (11.2) corresponding u  R  .  p  Since  is oscillatory in  r  to  d e f i n e d by  i s a s o l u t i o n of .(12.5) by d i r e c t  t a t i o n , with a zero i n u  r > 0  ( 1 2 . 7 ) with a zero i n  u* , then the f u n c t i o n  u(x) = v(x )cp(x)  implies that  v  Let  of  compu-  i s arbitrary,  this  R , and the theorem i s  proved.  THEOREM 17. large  x  n  (12.6)  Let and suppose  a  (  x n  )  has a s o l u t i o n o s c i l l a t o r y i n  s  hold f o r s u f f i c i e n t l y •  1  R  Then e q u a t i o n i f there e x i s t s  (12.5) a > 0  such t h a t CO  a  Proof.  The proof i s s i m i l a r to t h a t of Theorem 1 6 ,  appeals  to a c r i t e r i o n of L e i g h t o n and N e h a r i - [ 1 7 , Th.  and 11.4]  - 63  -  to show t h a t ( 1 2 . 7 ) has an o s c i l l a t o r y s o l u t i o n . the  omit  details.  13.  F o u r t h order equations on a l l of The  (13.1)  E  n  n p £ D.*(a,, D u) - bu = 0* i,j=l 1  1 J  t h a t the domain  R  .  J  g e n e r a l c o n d i t i o n s on  L  are as i n s e c t i o n 8 ,  w i l l be a l l of  E  n  .  (x : r ^ < |x| < r )  form  , hence we need a s l i g h t e x t e n s i o n of  2  the m o n o t o n i c i t y p r i n c i p l e proved say t h a t an annulus t .  except  The nodal domains  of the comparison equation w i l l be a n n u l i of the  thickness  case  namely  p  Lu s  .  e q u a t i o n to be c o n s i d e r e d i s the s p e c i a l  m = 2 of ( 1 0 . 1 ) ,  The  We  of the form  i n Chapter  I.  We  shall  [x : r ^ < |x| < r^+t)  Even though t h i s annulus  has  has bounded width,  the l a t t e r does not approach zero as - t -» 0+ , so t h a t the form of the m o n o t o n i c i t y p r i n c i p l e i n §9 i s i n a p p l i c a b l e here.  Lemma (Poincare's i n e q u a l i t y f o r a n n u l i ) n  has t h i c k n e s s  (13.2)  t , then  iui  0 j f l  < t  2 f o r a l l u e C (fi) , where  2  |u| ; 2  n  I f an  annulus  - 64 -  | u |  Proof.  ^ • 'o , L [  ( D a u ) 2  a  3 x 1 1 / 2  In the course of p r o v i n g the m o n o t o n i c i t y  c i p l e f o r e i g e n v a l u e s i n the second  prin-  order case f o r annular  domains, we showed ( c f . ( 2 . 5 ) ) t h a t  ,  (13.3)  |u|  4-21 |2  /  2  A p p l y i n g t h i s i n e q u a l i t y to the f i r s t p a r t i a l D^u  derivatives  , we o b t a i n  =  =  t'  2 (D D.u) |a|=l a  n  t J  2  dx (by d e f i n i t i o n )  1  n 2 (D.D.u)" dx , i = l , 2 , . . . , n . f i .1=1 J  1  Hence n = f 2 (D.u)" dx fi i = l J  n  1  2 ^(D^u) fi i , j = l  2  dx = t J ^ S . (D u) fi |a|=2 2  a  (  dx  A/  - 65 -  Combining t h i s w i t h  from which  (13.2)  (13.3)  follows  we o b t a i n  immediately.  We now s t a t e the r e q u i r e d form o f the m o n o t o n i c i t y p r i n c i p l e f o r eigenvalues.  I n view o f the form of Poincare's  i n e q u a l i t y proved here, we s t a t e the r e s u l t f o r the more general operator  (13-4)  Mu  s  2  iP  |p|-|q|-2  (A  D u) q  - bu  i  *  The p r i n c i p l e w i l l theji be. true f o r the o p e r a t o r L  , since  L  i s a s p e c i a l case of M ( c f . s e c t i o n 9 ) .  We note t h a t the o p e r a t o r i n case  m =  conditions  2  of  (9.2).  (13.4)  i s the s p e c i a l  We s h a l l a c c o r d i n g l y suppose t h a t  (a) and (b) o f s e c t i o n 9 are s a t i s f i e d .  As noted  i n s e c t i o n 9 , however, when we a p p l y the m o n o t o n i c i t y c i p l e f o r the o p e r a t o r dition  L , I t s e l l i p t i c i t y alone  ( a ) ) i s enough t o guarantee  an annulus  of thickness  ( i . e . con-  the t r u t h o f the p r i n c i p l e  s i n c e c o n d i t i o n (a) i m p l i e s c o n d i t i o n (b) i n t h i s  Monotonicity P r i n c i p l e  prin-  (Annular Domains). t . Then the f i r s t  case.  L e t fi(t) be eigenvalue  X (t)  of the problem  Q  Mu = Xu  i n fi(t) ;  Du a  = 0  i s monotone n o n i n c r e a s i n g ( f o r  Proof.  Sfi(t),  |a| = 0 ,  1  t > 0) and l i m X ( t ) = +oo t-0+ °  The proof i s s i m i l a r to t h a t of the  r e s u l t i n s e c t i o n 9 and w i l l be  14.  on  .  corresponding  omitted.  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 s e c t i o n i s a theorem of  the K n e s e r - H i l l e type f o r e q u a t i o n  (13.1).  I t contains  the corresponding r e s u l t of L e i g h t o n and Nehari f o u r t h order 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  u^  [17] f o r the v  and extends the analogous theorem of Glazman [ 9 ] o p e r a t o r w i t h harmonic l e a d i n g term to one w i t h cular)  biharmonic  - bu = 0 f o r an (in parti-  l e a d i n g term.  F i r s t we need a few t e c h n i c a l lemmas of an elementary c h a r a c t e r . the s e p a r a b l e  (14.1)  where  s h a l l compare e q u a t i o n  (13.1) w i t h  equation  A  A.,  We  A y 2  1  - Bv = 0  ',  i s an upper bound on the l a r g e s t e i g e n v a l u e A ( X )  - 67  of the m a t r i x  ( a ^ j ( x ) ) ; i . e . there  A ( x ) < L A^  such t h a t  such t h a t there  B(x)  Notation. s  -  .  The  continuous f u n c t i o n  exists a function  = g ( | x | ) < b(x)  ,  0  Let  F(s,n)  be  e x i s t s a number  g  n  •  P(s,n) = s ( s - 2 ) (s+n-2) (s+n-4)  coordinates  i n Chapter I , we r , 8^  ,  ,  form  ( i n p a r t i c u l a r ) of the  A  P r o p o s i t i o n 18. maximum a t  - g (r)p  Lp 2  1  s = 2-n/2  ordinary  Q  The .  =  0  (14.1)  (14.1) has  so-  ,  differential  polynomial '  writing  f i n d that  = p(r) , 0 <_ r. < co  s a t i s f i e s the  (14.2)  By  n  lutions  p  spherical polar  ^ _]_? we  v(x)  .  introduce  i n terms of these c o o r d i n a t e s ,  where  .  the p o l y n o m i a l of degree f o u r i n  d e f i n e d by  As  is  satisfying  Q  x e E  B  equation  .  F(s,n)  has  a relative  -  Proof.  If  s = 2-n/2  68  -  i s a zero of  must be a repeated z e r o , s i n c e about  s = 2-n/2  .  F(s,n)  F(s,n)  Moreover, s i n c e  nomial of degree f o u r with p o s i t i v e  is  , then i t  symmetrical  F(s,n)  i s a poly-  leading coefficient  and at l e a s t two d i s t i n c t r e a l z e r o s , a c o n s i d e r a t i o n of the shape of i t s graph shows t h a t the repeated zero at s = 2-n/2  i s a maximum p o i n t . If  s = 2-n/2  i s not a zero of  P(s,n) , we  use l o g a r i t h m i c d i f f e r e n t i a t i o n ' to show t h a t  Mi, ^ u  o) o  ;  P'(s,n) P(s,n) ~ s  + +  s-2  +  1 s+n-2  . 1, s+n-4  +  where the prime denotes d i f f e r e n t i a t i o n with r e s p e c t to and the formula  (14.3) holds except at zeros of  F(s,n) .  Thus F'(2-n/2,n) = 0  Differentiation  of  (14.3)  yields  F"(s,n) = -P(s,n)[^p + — 3 — + s^ (s-2T  I t f o l l o w s t h a t when  .  F'(s,n) = 0  L  — p  +  (a+n-2)  D  1  (s+n-4r  ( i n p a r t i c u l a r , when  -  s = 2-n/2) , F"(s,n)  69 -  has s i g n o p p o s i t e t o t h a t of  F(s,n).  But P(2-n/2,n) = ( 2 - n / 2 ) ( n / 2 ) 2  s i n c e by h y p o t h e s i s  2  > 0 ,  F(2-n/2,n) ^ 0 .  Hence  P"(2-n/2,n)<0  and the p r o p o s i t i o n i s proved.  P r o p o s i t i o n 19*  (14.4)  I f the i n e q u a l i t y  «i) > A  n (n - 4) /l6 2  x  2  h o l d s , then the e q u a t i o n A  (14.5)  1  F ( s , n ) - cu ='0  has a t l e a s t one p a i r of complex r o o t s .  Proof.  By P r o p o s i t i o n 18, the polynomial' F ( s , n )  nonnegative  r e l a t i v e maximum a t  polynomial  F ( s , n ) - ou / A^  s = 2-n/2 .  s = 2-n/2 .  Hence the  has a l o c a l maximum a t  C o n d i t i o n (14.4) i m p l i e s t h a t  F(2-n/2,n) - «a/A = n ( n - 4 ) / l 6 - u)/A < 0. . 2  2  1  1  l a t i v e maximum of the p o l y n o m i a l s = 2-n/2  has a  i s negative.  Thus the r e -  F ( s , n ) - uu/A-^ a t  Hence e q u a t i o n  (14.5) has a t l e a s t  - 70 -  one p a i r of complex r o o t s . s t a t e and prove  We are now i n a p o s i t i o n t o  the main r e s u l t o f t h i s s e c t i o n .  THEOREM 2 0 .  Suppose t h a t the l a r g e s t  A(x)  i s bounded i n E  of  (a^j(x))  s i d e some hypersphere), (13.1)  i s oscillatory i n  (14.6)  l i m i n f r^g(r) r -> 00  E  (or a t l e a s t out-  n  A ( X ) <_ A ^ .  say  eigenvalue  Then e q u a t i o n  if  n  > A  n (n-4) /l6 2  2  ,  where g(r). = min (b(x) : |x| = r )  Proof. constants  The hypothesis r  and  o  (14.6) i m p l i e s  ID such  r ^ g ( r ) > ID > A  for a l l r > r  (14.7)  A  x  Q  A  .  2 V  that  n (n-4) /l6 2  1  2  We then compare  -  uur'V =  which i s the s p e c i a l case r a d i a l form of e q u a t i o n ( 1 4 . 7 ) "has s o l u t i o n s  t h a t these e x i s t  ( 1 3 - 1 ) w i t h the e q u a t i o n  0  B(x) = u>|x|"^  (l4.7)  of (14.1).  The  i s o f E u l e r type and thus  ( i n p a r t i c u l a r ) o f the form  v ( x ) = |x| , s  - 71  where  s  satisfies  (14.5).  -  T h i s i s e a s i l y seen by  noting  that  Ar  s  =  s(s+n-2)r  s  = s(s+n-2)Ar ~  £  and A r 2  s  2  = s(s+n-2) (s-2) ( s + n - 4 ) r ~ s  Since  u) > A  n (n-4) /l6 2  1  2  t i o n 19 i s f u l f i l l e d , at l e a s t p a i r of one  and  , hypothesis  (14.4) of P r o p o s i -  therefore equation  complex r o o t s .  i|  (l4-5)  has  This implies that  there e x i s t s an o s c i l l a t o r y s o l u t i o n of the r a d i a l form of  (14.7),  Let  a > 0  Nehari p  i . e . a s o l u t i o n with I n f i n i t e l y many z e r o s . be g i v e n .  [17, Th.  f  Then a theorem of L e i g h t o n  3.6]  i m p l i e s t h a t there e x i s t s a s o l u t i o n  of the o r d i n a r y d i f f e r e n t i a l  A_ A p  - tor" p  w i t h double zeros a t  r = 6 ,  (14.8)  2  4  ]  n  1  (v ' p )"  since  A p(r) =  may  transformed  be  2  (Note t h a t  n 1  +  u  equation  =0  6  where  6_  d  2  [(l-n)r " p>]' n  3  i n t o the form c o n s i d e r e d  (l-n)r ~ n  and  3  <. 0  > 6.  > max  1  , so t h a t  O  (l4.8)  i n [17,Th.  f o r a l l positive integers  [r  3-6] n  - 72 -  and  see the remark f o l l o w i n g  that the f u n c t i o n  v  [17, Th. 12.1]).  d e f i n e d by  I t follows  v(x) = p(x)  i s a solu-  t i o n of (14.7) with a nodal domain  N = • (x : &  < |x| < 6 }  1  2  .  In f a c t ,  D v = U  and  1  Y  ^ dr a x  ±  the r i g h t s i d e i s zero on  f o r any  a > 0  there  exists a solution  domain i n the r e g i o n |x| >  plies  r = 6.^ , r = 6 v  [x : |x| > a) , s i n c e  > a .  Hence  g  .  Thus  w i t h a nodal x e N  im-  (14.7) i s o s c i l l a t o r y .  We how a p p l y a theorem [24, Th. 4] of C. A. Swanson.  Because of the hypotheses  E a. .(x)z.z . <. A ( X ) | Z | i,j=l 1  J  1  < A |z| N  3  , x suff. large, z  x  u>r" < g ( r ) < b(x)  |x| > r  4  ,  we have  ^ i,ii C  ( A i 5  ^  D  " i j a  )  D  i  2  u  A  +  (B  - H*rV)  dx >  eE  - 73 -  Hence the eigenvalue  Lu = Xu  in  N ;  problem  u = L\j_u = 0  has a t l e a s t one eigenvalue l e s s than z e r o .  2,  ..., n  ( i n p a r t i c u l a r , the s m a l l e s t )  Let  N(t) = (x : 6  and l e t  5N , i = 1 ,  on  1  6  < |x| < t ) ,  <.t <  1  6 , 2  X ( t ) denote t h e . s m a l l e s t eigenvalue Q  of the p r o -  blem  Lu = Xu  in  N(t) ; u = D u = 0  By the m o n o t o n i c i t y  SN(t), 1 = 1 ,  on  ±  p r i n c i p l e of s e c t i o n 1 3 ,  monotone n o n i n c r e a s i n g i n  6  N  < t <_ 6 ~ and  2 , ..., n.  • X ( t ) Is l i m X ( t ) = +oo  t - 6 ^ °• . Since  ^ ( 2^ — 0  S  ' there e x i s t s  0  such t h a t . X (T) = 0 nontrivial solution  .  Corollary  (13-1)  and  ordinary d i f f e r e n t i a l equation  i v  - b(x)u = 0  T  in  ( ^ i * ^  i s a nodal domain of a  Moreover, a  i s oscillatory i n  (Leighton and Nehari  u  N(T)  of ( 1 3 . 1 ) .  N(T) C N C { X : |x| > a} equation  Thus  a number  i s arbitrary, E  n  therefore  .  [ 1 7 , p a r t o f Th. 6 . 2 ] ) .  The  - Ik  -  is oscillatory i f  (14.9)  l i m i n f x\>(x) > 9/16 .  X -> 00  Proof. n = 1  This c o r o l l a r y of Theorem 20.  i s the s p e c i a l  case  a. . =.6  - 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, P a r t i a l D i f f e r e n t i a l Equations, Proceedings of the Summer Seminar, Boulder, Colorado, 1 9 5 7 . F.E. Browder,  The D i r i c h l e t Problem for Linear E l l i p t i c  Equations of A r b i t r a r y Even Order with Variable C o e f f i cients, Proc. Nat. Acad. S c i . U.S.A., 38 ( 1 9 5 2 ) , pp. 230235.  C. Clark and CA.  Swanson,  Comparison Theorems for  • E l l i p t i c D i f f e r e n t i a l Equations, Proc. Amer. Math. Soc. 16 ( 1 9 6 5 ) , pp.  886-390.  E.A. Coddington and N. Levihson,  Theory of Ordinary  D i f f e r e n t i a l Equations, McGraw-Hill, New  York,  1955.  R. Courant and D. H i l b e r t , Methods of Mathematical Physics, Vol. I, Interscience, New York, 1966.P.R. Garabedian, P a r t i a l D i f f e r e n t i a l Equations, Wiley, New York, 1 9 6 4 . I.M. Glazman, On the Negative Part of the Spectrum of~ One-Dimensional and Multi-Dimensional D i f f e r e n t i a l 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 D i f f e r e n t i a l Operators,  Israel  Program  f o r S c i e n t i f i c Translations, Daniel Davey and  Co., New  York, 1 9 6 5 .  - 76 -  [10]  V.B. Headley  and C A .  Swanson,  Oscillation  Criteria  f o r E l l i p t i c Equations, P a c i f i c J . Math, ( t o appear). [11]  E. H i l l e ,  N o n - o s c i l l a t i o n Theorems,  Trans. Amer. Math.  Soc. 64 (1948), pp. 234-252. [12]  K. K r e i t h ,  A New Proof o f a Comparison Theorem f o r  E l l i p t i c Equations, Proc. Amer. Math. Soc. 14 (1963), pp. [13]  33-35.  K. K r e i t h ,  O s c i l l a t i o n Theorems f o r E l l i p t i c  Equations,  Proc. Amer. Math. Soc. 15 (1964), pp. 341-344. [14]  K. K r e i t h ,  Disconjugacy i n  E  n  ,  SIAM J . A p p l . Math.  15 (1967), pp. 767-770. [15]  K. K r e i t h ,  An A b s t r a c t O s c i l l a t i o n Theorem, Proc. Amer.  Math. Soc. 19 (1968), pp. 17-20. [16]  W. L e i g h t o n ,  On S e l f - A d j o i n t 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. L e i g h t o n and Z. Nehari,. On the O s c i l l a t i o n o f S o l u t i o n s of S e l f - A d j o i n t L i n e a r D i f f e r e n t i a l Equations o f the F o u r t h Order, Trans. Amer. Math. Soc. 89(1958), pp. 325-  377. [18]  S.G. M i k h l i n ,  The Problem of the Minimum o f a Quadratic  F u n c t i o n a l , Holden-Day, San F r a n c i s c o , 1965. [19]  R.A. Moore,  The Behaviour  o f S o l u t i o n s of a L i n e a r  D i f f e r e n t i a l E q u a t i o n o f Second Order, P a c i f i c J . Math. • 5 (1955), PP. 125-145.  - 77 -  [20]  C. B. Morrey, Lecture Notes on the Theory of P a r t i a l D i f f e r e n t i a l Equations, Department of Mathematics, University of Chicago, Summer i 9 6 0 .  [21]  Ruth Lind Potter, of Second Order,  [ 2 2 ] S.L. Sobolev,  On Self-Adjoint D i f f e r e n t i a l Equations P a c i f i c J . Math.3  ( 1 9 5 3 ) ,  pp.  4 6 7 - 4 9 1 .  Some Applications of Functional Analysis  i n Mathematical Physics, American Mathematical Society, Providence, 1 9 6 3 . [23]  C A . Swanson,  A Comparison- Theorem f o r E l l i p t i c  D i f f e r e n t i a l Equations, Proc. Amer. Math. Soc. 1 7 pp.  [24]  6 l l - 6 l 6 .  C A . Swanson,  A Generalization of Sturm's Comparison  Theorem, J . Math. Anal. Appl. [25]  ( 1 9 6 6 ) ,  C A . Swanson,  1 5  ( 1 9 6 6 ) ,  pp.  5 1 2 - 5 1 9 .  Comparison Theorems f o r 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]  C A . Swanson, An Identity f o r E l l i p t i c Equations with Applications, Trans. Amer. Math. Soc. (to appear).  [27]  C A . Swanson, Comparison and O s c i l l a t i o n Theory of Linear D i f f e r e n t i a l Equations, Academic Press, New York, 1 9 6 8 .  [28]  A. Wintner,  A C r i t e r i o n of O s c i l l a t o r y  Quart. Appl. Math.  7  ( 1 9 4 9 ) ,  1 1 5 - 1 1 7 -  Stability,  

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