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Nonlinear eigenvalue problems in ellipticvariational inequalities:a local studyFrancis Conrad a , Francoise Issard-Roch b , Claude-Michel Brauner c & BasilNicolaenko da Département Informatique, Saint-Etienne, 42023, FranceU.A. du C.N.R.S. 740b Département Informatique, Saint-Etienne, 42023, FranceU.A. du C.N.R.S. 740c Département M.I.S., Ecully, 69131, FranceU.A. du C.N.R.S. 740d N-M 87545, U.S.AC.N.L.S. M.S. B-258Published online: 14 May 2007.

To cite this article: Francis Conrad , Francoise Issard-Roch , Claude-Michel Brauner & Basil Nicolaenko (1985)Nonlinear eigenvalue problems in elliptic variational inequalities:a local study, Communications in PartialDifferential Equations, 10:2, 151-190

To link to this article: http://dx.doi.org/10.1080/03605308508820375

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COMM. IN PARTIAL DIFFERENTIAL EQUATIONS, 1 0 ( 2 ) , 151-190 (1985)

NONLINEAR EIGENVALUE PROBLEMS IN ELLIPTIC VARIATIONAL INEQUALITIES :

A LOCAL STUDY

Francis Conrad Fran~oise Issard-Roch

Dkpartement Informatique U.A. du C.N.R.S. 740

Ecole des Mines 158 cours Fauriel

42023 Saint-Etienne, France

Dgpartement Informatique U.A. du C.N.R.S. 740

Ecole des Mines 158 cours Fauriel

42023 Saint-Etienne, France

Claude-Michel Brauner Basi 1 Nicol aenko

Dgpartement M. I .S. U.A. du C.N.R.S. 740 Ecole Centrale de Lvon

B.P. 163 6 9 1 3 1 Ecul ly, France

C.N.L.S. M.S. 8-258 Los A1 amos National Laboratory

Los Alarnos N-M 87545 U.S.A.

ABSTRACT

We consider a class of Nonlinear Eigenvalue Problems (N.L. E.P.) associated with Elliptic Variational Inequalities (E.V.I.):

I where a is a bilinear coercive form on Ho(B) and K a closed convex . .

set. First we introduce the main tools for a local study of

branches of solutions ; we extend the linearization process required in the case of equations.

Next we prove the existence of arcs of solutions close to regular vs singular points, and determine their local behaviour up to the first order.

Finally, we discuss the connection between our reqularity condition and some stability concept.

copyright O 1985 by Marcel Dekker, Inc.

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1. INTRODUCTION

Nonl inear Eigenvalue Problems (N.L.E.P.) o f t h e form :

have been t h e sub jec t o f ex tens i ve g loba l as w e l l as l o c a l s t u d i e s (e.g.

see [ 151 [ 161 [ 171 [ 351 and f o r a survey [ 301, where t h e b i f u r c a t i o n d i a -

grams assoc ia ted w i t h va r i ous n o n l i n e a r i t i e s F are g i v e n ) .

I n (1 .0 ) n i s a r e g u l a r domain i n I#, A As a u n i f o r m l y e l l i p t i c ,

second o rde r ope ra to r and F i s a non l i nea r , smooth f u n c t i o n o f iR i n t o i t s e l f

which may a l s o depend on t h e space va r i ab le .

Problems w i t h F o n l y de f i ned i n some open se t o f IR such as I F(u) = ---- , k 2 0, have a l s o been i n v e s t i g a t e d by two o f t h e authors

( I - u ) [ 3 - 91.

The case o f a non-smooth f u n c t i o n F has a l s o been cons idered by some

authors, i n c l u d i n g K.C. CHANG [ 1 2 ] who i n t roduced t h e concept o f D i s c o n t i -

nuous Nonl inear D i f f e r e n t i a l Equat ion (D.N.D.E. 1. It corresponds i n f a c t

t o t he case where u i s a s o l u t i o n o f some f r e e boundary problem. Although,

g loba l ex i s tence r e s u l t s o f m u l t i p l e s o l u t i o n s have been es tab l i shed , no

l o c a l s tudy has been undertaken, t o t h e bes t o f our knowledge.

It has been shown i n [ 61 [ 8 ] [13] that D.N.D.E. may appear as s i n g u l a r

l i m i t s o f N.L.E.P. Examples o f such a s i t u a t i o n may occur i n chemical c a t a l y s t

o r enzyme k i n e t i c s model l ing , where m u l t i p l e s o l u t i o n s have been po in ted

out f o r t he N.L.E.P. as we l l as f o r t h e l i m i t Free Boundary Problem. Le t us

mention two t y p i c a l cases :

The f i r s t example w i t h m = 1, k = 0, has l e d t o a sys tema t i c bounded pena-

l i z a t i o n approx imat ion o f E . V . I . v i a t he homographic approx imat ion [6-91 .

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EIGENVALUE PROBLEMS

I n some cases, t h e Free Boundary Problem may be cha rac te r i zed by

Nonl inear E l l i p t i c V a r i a t i o n a l I n e q u a l i t i e s [E.V. I.) o f t h e form :

a(u,v-u) 2 A (F(u) ,v -u) (1 .1)

P v e K ; u e K

K be in? a c e r t a i n c l osed convex s e t i n H;(n), e.g.

K = i v e H;(a) / v 5 Y a.e.on n) i n t h e obs tac le problems.

C lea r l y ( 1 . 1 ) appears t o be a n a t u r a l ex tens ion o f Problem ( 1 . 0 ) . C lass i ca l r e s u l t s o f RABINOWITZ [ 3 5 ] can be app l i ed t o Problem (1 .1) i n

o rde r t o h i g h l i g h t t h e ex i s tence o f a connected component c w i t h s o l u t i o n s

(x ,u) i n IR, x H;(a) ; c i s unbounded i n IR, x H;(n) and i n f a c t (1.1) admi ts

a s o l u t i o n u whenever A 2 0.

Suppose Y , t h e so -ca l l ed "obstac le" , i s > 0 and r e g u l a r i n 5 . Then

i t i s s t r a i g h t f o r w a r d t h a t , f o r smal l A , any s o l u t i o n u o f t he equat ion :

i s a l s o a s o l u t i o n o f (1 .1) a t l e a s t i f F i s bounded.

On t h e o the r hand, i f F 3 c > 0 on R+ , any s o l u t i o n u o f (1.1)

has a nonvoid co inc idence se t i x / u ( x ) = Y ( x ) 1 w i t h A s u f f i c i e n t l y

l a rge , f o r , i f not , u would be a s o l u t i o n o f (1 .3) [25, 28 ] t end ing t o

i n f i n i t y as A t - , which c o n t r a d i c t s t h e c o n s t r a i n t u e K.

Therefore we can d e p i c t t h e general f e a t u r e o f a b i f u r c a t i o n diagram

cor responding t o a component c o f s o l u t i o n s ( i , u ) o f (1 .1) (see F igu re 1 ) :

c con ta ins an equat ion branch i . e . a subset o f s o l u t i o n s (x,u)

o f (1.3) w i t h u Y and an E.V.1.-branch i . e . a subset o f s o l u t i o n s

(x,u) o f (1 .1) w i t h i u = Y ) # 0 (see [25], [ 3 1 ] and sec t i on 2 f o r

a more accurate d e f i n i t i o n o f t h e co inc idence s e t i u =%'I). A p o i n t

( h , ~ ) be long ing t o t h e E.V.1.-branch does n o t g e n e r a l l y s a t i s f y ( 1 .3 )

( f o r i ns tance when A r # A F ( Y ) ) .

The two branches are connected through a t r a n s i t i o n p o i n t (A, ,U~) .

The d e f i n i t i o n o f a t r a n s i t i o n p o i n t w i l l be g i ven more p r e c i s e l y

below. For e x p l i c i t diagrams, we r e f e r t o [ 6 - 81 [ 1 3 ] .

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CONRAD ET AL.

The aim o f t h e paper i s t o s tudy , f r o m a l o c a l p o i n t o f v iew, an

a r c o f s o l u t i o n s t o ( 1 . 1 ) . For an e q u a t i o n ( b r a n c h ) t h e t h e o r y i s we l l -known

and i s based on t h e i m p l i c i t f u n c t i o n theorem, c o n v e n t i o n a l o r adapted t o

t u r n i n g p o i n t s [ 15,161 . The key e q u a t i o n i n t h i s s t u d y i s t h e l i n e a r i z e d

e q u a t i o n o f ( 1 . 3 ) .

T h i s work i s an a t t e m p t t o g e n e r a l i z e t h e method t o t h e E . V . I . branch,

w i t h s p e c i a l a t t e n t i o n t o t r a n s i t i o n and t u r n i n g p o i n t s . As f a r as a l o c a l

s tudy i s concerned, t h e b a s i c p rob lem i s t h a t no sharp i m p l i c i t f u n c t i o n theorem

i s a v a i l a b l e f o r o p e r a t o r s wh ich a r e n o t F r e c h e t d i f f e r e n t i a b l e .

The paper i s o r g a n i z e d as f o l l o w s :

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EIGENVALUE PROBLEMS

In Section 2, we define the conical linearization of (1.1) which

is the main tool for the local study of (1.1) and extends to

(1 .l) the linearization process used for equations, The points

(h,u) of a branch are then classified as regular vs singular.

Section 3 is devoted to the inversion lemma, a basic result for

the local study in the next two sections. We also give existence

results and properties of one-sided derivatives of u with respect

to A, at a generic E.V.I. point and at a transition point, when a

condition of coerciveness for the "tangent operator" is satisfied.

When this operator is a linear one, the coerciveness may be

relaxed.

In Sections 4 and 5 we obtain the main results concerninq the lo-

cal behaviour of branches of solutions (h,u) in the reqular vs

singular case, using a fixed point formulation of (1.1) and de-

gree theory. This gives a desi.ription in terms of components ema-

nating from a solution (h,u) and strongly differentiable at

(h,u). It is also possible, by use of Schauder's theorem, to give

a local expansion of solutions near (h,u) in terms of multi-

valued mappings. We mention these results, but do not develop

them (for the latter, we refer to [ 1 4 ] and [13,23] for further

details).

Finally, in Section 6, we discuss condition (S) used in the

inversion lemma and in Section 4, for the standard regular case ; (S) is an attempt of extension to E.V.I.'s of an invertibility

condition for equations. It turns out that (S) is closely connec-

ted to some stability condition, as shown by numerical experi-

ment.

2. THE MAIN TOOLS FOR A LOCAL STUDY.

2.1. Basic assumptions

We recall the problem under study :

within the following framework :

Let n be a regular ( * ) bounded domain of Wn, n>l, a the bili-

near continuous coercive form on H~(Q) :

( * ) that is, aQ is c2 and Q is locally on one side of an

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C O W ET AL.

av a(u ,v ) = a x Ei dx , a = ~ , , e ~ z ( i i ) , asso-

1 a 1 J J l

- c i a t e d w i t h t h e o p e r a t o r A = - fJ a a x . r a i 1 ,

1 J

L e t Y e H 1 ( a ) n e( a) ( the " o b s t a c l e " ) be p o s i t i v e , and d e f i n e

K = i v e H;(n) / v 5 Y a.e.on a1 . We suppose subsequent ly Y i s

q u a s i - c o n t i n u o u s ( q . ~ . ) w i t h r e s p e c t t o t h e f o r m a [ 1 ] ; t h e r e f o r e ,

c o n s i d e r i n g t h a t any i n e q u a l i t y quas i -everywhere (q .e . ) [ 1 ]

between elements o f H;(n) i n v o l v e s t h e i r q.c. r e p r e s e n t a t i v e s ,

K = { v e H;(a) / v 5 Y q.e.on n l .

L e f f : ( t , x ) e [ 0 , Y , ] x a + f ( t , x ) E IR be a bounded p o s i t i v e ,

Caratheodory t y p e f u n c t i o n , i n c r e a s i n g and t w i c e d i f f e r e n t i a b l e i n

t, it and ftt b e i n g a l s o Cara theodory and bounded. We suppose

admi ts an .?xtension f on lR x a h a v i n g t h e same p r o p e r t i e s as -? and

we deno ie h f F, F ' , T " t h ? Nemytskii o p e r a t o r s on L 2 ( n ) a s s o c i a t e d

w i t h f , ft, ftt. A t y p i c a l case i s when f i s a c2 f u n c t i o n on

[ O , Y / _ ] , independent o f x.

I n t h e f o l l o w i n g , ( , ) denotes t h e usua l i n n e r p r o d u c t i n LZ (a),

1 1 t h e a s s o c i a t e d norm and 1 / t h e usua l norm on H;(n).

I f G : L z ( n ) + H;(a) i s t h e Green 's (compact) o p e r a t o r a s s o c i a t e d

w i t h t h e second o r d e r o p e r a t o r A w i t h homogenous D i r i c h l e t boundary

c o n d i t i o n s and PK i s t h e a - p r o j e c t i o n on t h e c l o s e d convex s e t

K c H;(a) [ 3 1 ] d e f i n e d by :

t h e n i t i s easy t o see t h a t ( 1 . 1 ) admi ts t h e e q u i v a l e n t f i x e d p o i n t

f o r m u l a t i o n

( 1 . 2 ) u = PK [ A G F ( u ) ] .

Since F i s bounded and G r e g u l a r , t h e n o n l i n e a r mapping

T ( L , u ) = P [ A G F ( u ) ] fromIR, x H;(a) t o H;(n) i s compact. There- K

f o r e a p p l y i n g a s tandard g l o b a l r e s u l t o f RABINOWITZ [351, we

o b t a i n t h e e x i s t e n c e o f a component C o f s o l u t i o n s ( i , u ) o f ( 1 . 1 )

i n IR, x H;(n), c o n t a i n i n g (0,O) ; c i s unbounded i n l R + x H;(n).

Moreover, c b e i n g c o n s i d e r e d as a subset o f IR, x L 2 ( n ) , we o b t a i n

t h a t t h e p r o j e c t i o n on IR i s unbounded (see [ I 3 1 f o r d e t a i l s ) .

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EIGENVALUE PROBLEMS

I n f a c t ( 1 .1 ) admi ts a s o l u t i o n u f o r any p o s i t i v e A . Morecver, i f

A Y i s a measure such t h a t ( A Y ) - e ~ ' i a ) , then u E ~ ' ' ~ ( a ) by a

s tandard r e g u l a r i t y r e s u l t [ l o ] .

l o p rove t h a t ( 1 . 1 ) has a s o l u t i o n f o r any 1, 0, we cons ide r t h e

i t e r a t i v e scheme :

i a(uPtl , v - up+') 2 A(F (uP) , v - up+')

V v e K ; u P t l E K

w i t h u0 = 0 ( r esp . u0 = Y ! ; a l i m i t process a: p t + - g i ves t h e

ex i s t ence of a min imal s o l u t i o n !(A) ( r esp . maximal s o l u t i o n U ( A ) )

o f (1.11, i n c r e a s i n g and l e f t ( r esp . r i g h t ) cont inuous f r om

IR, t o H;(a) [13. 231.

The f o l l o w i n g s e t s o f f u n c t i o n s a re t o be used i n t h e n e x t s e c t i o n s :

L e t ( A , u ) elR, x K be a s o l u t i o n o f (1 .1) and C(K,u) =pya M ~ K - u l ;

t h e co inc i dence s e t ( u = Y i = ix E a / u ( x ) = ~ ( x ) q . e . l i s d e f i n e d

up t o a se t of a c a p a c i t y zero [ 1 I [ 313. Then, f o r t h e s t r ong

c l osu re , C(K,u)= i v f H;(n) / v 5 0 q.e.on i u = ~ ) [ 3 1 ] .

The a-orthogonal o f u - A G F ( u ) i s d e f i n e d as :

i u - 1 G ~ ( u ) l ' = i v E H;(n) I a(u ,v ) = i ( F ( u ) , v ) 1 . F i n a l l y , we se t Su = m) fl i u - x G F(U)I' wh ich i s a c l osed

convex cone i n H ; ( n ) .

2.2 The c o n i c a l spectrum

We d e f i n e t h e homogeneous c o n i f i e d E.V.I . o f (1.1) w . r . t . u, a t a

p o i n t ( i , u ) , as

D e f i n i t i o n 2.1. - If w = 0 i s t h e o n l y s o l u t i o n o f (2.11, we say t h a t ( 1 , ~ ) i s a

r e g u l a r s o l u t i o n o f ( 1 . I ) . Otherwise, we say (h,u) i s a s i n g u l a r one.

2.3 Con ica l d i r e c t i o n s

We d e f i n e t h e right (resp. l e f t ) c o n i c a l d i r e c t i o n s a t (h ,u) as t h e

s o l u t i o n s z, ( r esp . z - ) o f t h e non homogeneous E.V. I .

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CONRAD ET AL.

( r e s p ( 2 . 2 ) - where z, i s r e p l a c e d by z- w i t h h e - Su ; z- s - Su).

2.4 A geomet r ic i n t e r p r e t a t i o n o f t h e above concepts

We assume i n t h i s s u b s e c t i o n n 5 5 .

Theorem 2.1

L e t G ( A ) be t h e maximal s o l u t i o n o f ( I . ] ) , uo = i i ( k 0 ) and

( i ) e i t h e r zA i s bounded i n H;(d as A i io and e v e r y weak l i m i t

z, o f z, i s a s o l u t i o n o f (2.2), i .e. a r i g h t c o n i c a l

d i r e c t i o n a t (h0.u0)

( i i ) o r (aO,uO) i s s i n g u l a r .

P r o o f : ( i ) r e c a l l t h a t ? ( A ) i s r i g h t c o n t i n u o u s i n A ; Since - U ( A ) = u0 t ( A - A ) zA 6 K we have z A e CIK,uo) ; so as a weak

l i m i t p o i n t of z, when A + h0, Z, be longs t o t h e weak c l o s u r e

o f C(K,uo) = -i7(K,uo) ( s i n c e C(K,uo) i s convex) .

1 Next we prove z, e i u o - A, G F ( u o ) 1 :

S i n c e (A,, u0) i s a s o l u t i o n o f ( 1 . I ) we g e t :

a ( u o , v - u0) 2 " ( F ( u o ) , v - u0) V v e K.

Take v = G ( A ) and d i v i d e by A - xo > 0 :

a(uo, z,) 2 A ~ ( F ( u ~ ) , z A ) wh ich i m p l i e s , a t t h e l i m i t A - Ao,

t h a t a (uo , z,) 2 h 0 ( F ( u o ) , z,).

and a(uo , z,) 5 A ( F ( u o ) , 2,) s i n c e z, - z, weakly i n HA(n)

and ii ( A ) - uO s t r o n g l y i n H;(a).

Thereforea(uo,z,) = ~ ~ ( F ( u ~ ) , z , ) andz , E Sue.

F i n a l l y , l e t us p rove t h a t z.+ s a t i s f i e s (2.21, ; l e t E > 0 and

h 6 C(K,uO) ll i uo - A. G F ( u o ) } l ; add t h e E . v . 1 . ' ~ g i v i n g

u a n d i i ( i ) :

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EIGENVALUE PROBLEMS

i n which w i s chosen equal r e s p e c t i v e l y t o G ( A ) and uo + E h ,

F i r s t , l e t us t ake E = 0 i n ( 2 .3 ) and d i v i d e by (A - ~ 0 ) ~ :

Since zA i s bounded i n H;(a), hence i n L 3 ( ( a ) f o r n 5 5 by Sobolev

embeddings, we get :

Then a (z+ , 2,) 2 ( F ( u o ) , z,) + AO(F4 (uO) Z + , z + ) s i nce t he

form a i s weakly 1.s.c. Next, f o r E > 0, we d i v i d e (2 .3) by X - X o ,

and make use of t h e equat ion a(uo , E h ) = A (F (uO) , E h ) :

which g ives as A + A :

o r e l se , by combinat ion w i t h t h e p rev ious r e s u l t f o r c = 0 :

f o r any h e C ( K , uo) fl cu0 - h0 G ~ ( u , ) l ~ a n d by d e n s i t y f o r any

h 6 Su0 [311.

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- ( i i ) suppose t h e r e i s a t l e a s t a sequence ip > A, , up = u ( i p ) such

I u p - uo l l t h a t - - + - .

A ~ - A0

Without l o s s o f g e n e r a l i t y , we may suppose i+, # uo V p and se t

- Uo W,= -

Let w be a l i m i t p o i n t o f wp , i n Hh(i?) weak. We proceed as i n case ( i )

t o show t h a t w i s i n S u o Furthermore, we w r i t e t h e E.V. I . ' s s a t i s -

f i e d by uo and u p :

a(uo , v - u0) 1. h0 (F (uO) , v - uo )

a ( u p , v - up) 1. a p ( F ( u p ) , v - u P )

i n which we choose r e s p e c t i v e l y v = u and v = uo. D i v i d i n g by

J / u p - u ~ ( ( ~ , and adding i n e q u a l i t i e s , i t comes

where a i s a c o e r c i v i t y cons tan t f o r a . The l i m i t as p r - y i e l d s :

0 < a _i A ~ ( F ' ( U , ) w , w)

(aga in , i n T a y l o r ' s fo rmula , we use t h e f a c t t h a t f o r n 5 5, wp i s

bounded i n L 3 ( a ) w i t h a s i m i l a r argument as above i n T a y l o r ' s

fo rmul a ) .

The above i n e q u a l i t y i m p l i e s t h a t w # 0. F i n a l l y , we prove w i s a

s o l u t i o n o f (2 .1) e x a c t l y i n t h e same way as i n case ( i ) (see 1131

f o r d e t a i l s ) . ~

Remark 2.1

I f we cons ider t h e minimal branch o f (1 .1) , we would s i m i l a r l y o b t a i n

a r e s u l t about t h e l e f t con i ca l d i r e c t i o n s ( o r s i n g u l a r i t y ) . .

re mar^ 2.2

Theorem 2.1 i s an ex is tence r e s u l t f o r r e g u l a r p o i n t s and can be

viewed, as f a r as extremal s o l u t i o n s arecons idered, as a k i n d o f

" c o n i c a l u Fredholm a l t e r n a t i v e .

Remark 2.3 - U ( A ) - Uo I f z, denotes a weak l i m i t p o i n t o f --- , z+ 6 Su hence

A - ho

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z+ 5 0 q . e . on {uo = u ? . But , s i nce t h e m a x i m a l s o l u t i o n i s

nondecreasing i n A , z+z 0 a.e. on a hence q.e. and i n f a c t

z+ = 0 q.e. on i uo = Y } . Such a r e s u l t i s n o t t r u e f o r t h e l e f t

d i r e c t i o n s z - , see s e c t i o n 3.

2.5. R e l a t i o n s h i p w i t h c o n i c a l d i f f e r e n t i a t i o n o f p r o j e c t o r s

Reca l l (see e.g. [ 3 1 ] ) t h a t f o r a convex s e t o f t h e f o rm

K = i v e H1(n ) / v 2 Y q.e.1 t h e a - p r o j e c t o r PK admits a c o n i c a l d e r i v a t i v e ,

t h a t i s , f o r t 2 0, w e H;(n) , h E H;(a),one has t h e f o l l o w i n g f o rmu la :

l i m ~ ( t , h ) = 0 u n i f o r m l y i n h on compact subsets o f H;(Q) t r 0 1 wherte SV = m f l v)w - v l , v = PK [w] .

We suppose i n t h i s subsect ion t h a t n 4 which i m p l i e s Hh(a) c o n t i -

nuous ly embedded i n L b ( n ) .

Then, f o r u, h e H;(n), one has t h e f o l l o w i n g equa t i on i n Hh(a) :

[ u ( x ) + t h ( x ) , u ( x ) ] i f u ( x ) + t h ( x ) 2 U ( X ) w i t h 6 ( x ) e

[ u ( x ) , U ( X ) + t h ( x ) l e lsewhere

Now l e t ( A , u ) be a s o l u t i o n o f ( 1 .1 ) and h E H;(n). We combine t h e

prev ious e q u a l i t y w i t h t h e d e f i n i t i o n o f t h e c o n i c a l d e r i v a t i v e o f

PK w i t h

Then t a k i n g i n t o account t h e L i s p s c h i t z c o n t i n u i t y o f p r o j e c t o r s ,

we ge t t h e f o l l o w i n g f o rmu la i n H;(a) :

l i m E ( t , h ) = 0 ,un i f o rm ly f o r bounded h ( by t h e compactness o f G). t + o

I n t h e same way (see 113. 231 f o r d e t a i l s ) , we g e t a l s o :

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f o r a s o l u t i o n ( i , u ) o f ( 1 . 1 ) , h E H;(R), and l i m E ( t , h ) = 0

u n i f o r m l y f o r h bounded. t - 0

As i n t h e c l a s s i c a l e q u a t i o n case, t h e s p e c t r a l p rob lem i s expec ted

t o be o b t a i n e d by ( c o n i c a l ) d i f f e r e n t i a t i o n o f t h e Equat ion ( 1 . 1 ) f o r

( A , u + t W ) :

which g i v e s : w = P [ A G F ' ( u ) w] , a f i x e d p o i n t f o r m u l a t i o n e q u i v a l e n t t o Su

( 2 . 1 ) . T h i s f o r m u l a t i o n would suggest an e x t e n s i o n o f some c l a s s i c a l s p e c t r a l

t h e o r y s i n c e t h e o p e r a t o r w + PSu[ G F ' ( u ) w I i s compact, p o s i t i v e l y

homogenous, p o s i t i v e [ 13 1 b u t g e n e r a l l y n o n l i n e a r .

To o b t a i n t h e r i g h t - O r l e f t - c o n i c a l d i r e c t i o n s a t ( h , u ) , we d i f f e -

r e n t i a t e t h e E q u a t i o n (1 .1 ) w r i t t e n f o r ( h t t, u t t z ) :

u t t z = P K [ ( i + t ) G F ( u t t z ) ]

a t t = 0, , which y i e l d s :

a f i x e d p o i n t f o r m u i a t i o n e q u i v a l e n t t o ( 2 . 2 ) +

Thus we see how t h e t o o l s i n t r o d u c e d i n t h i s s e c t i o n a r e n a t u r a l

e x t e n s i o n s o f those o b t a i n e d by t h e l i n e a r i z a t i o n process i n t h e uncons-

t r a i n e d case.

3. THE BASIC INVERSION LEMMAFIRST PROPERTIES OF CONICAL DIRECTIONS

I n t h i s s e c t i o n t h e d imens ion n i s a r b i t r a r y . L e t ( i , u ) be a

s o l u t i o n o f ( 1 . 1 ) .

3.1 The genera l case

D e f i n i t i o n 3.1

L e t us say t h a t ( A , u ) s a t i s f i e s C o n d i t i o n (s) i f

i n f i a(w ,w) - a ( F ' ( u ) w , w ) / w s Su - Su , l lw l l = 1) > 0.

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Remark 3.1

C o n d i t i o n (s ) i m p l i e s t h a t h , u ) i s a r e g u l a r p o i n t : f o r i f w f 0

i s a s o l u t i o n o f :

t h e n a(w,w) - i ( F 1 ( u ) w, w) = 0, which c o n t r a d i c t s ( S ) .

Remark 3 .2

(5') i s o f course more r e s t r i c t i v e t h a n r e g u l a r i t y , see f o r i n s t a n c e

t h e case o f an e q u a t i o n , o r s e c t i o n 7 o f t h i s paper.

Remark 3 .3

For o t h e r f o r m u l a t i o n s e q u i v a l e n t t o (S) a t l e a s t when

F ' ( u ) > 0, see [131.

Lemma 3.1

L e t ( i , ~ ) s a t i s f y c o n d i t i o n (S). Then

( i ) f o r any h e Hh(a) , t h e n o n l i n e a r o p e r a t o r

w - w - Pa[ " F F ' ( u ) w + h i e H;l(a) i s one t o one h o m H;(a)

i n t o i t s e l f , and i t ' s i n v e r s e Rh i s L i p s c h i t z cont inuous,

t h e L i p s c h i t z c o n s t a n t b e i n g independent o f h .

( i i ) a s i m i l a r r e s u l t h o l d s when Su i s r e p l a c e d by - Su

P r o o f : l e t g e H;(a); we c o n s i d e r t h e e q u a t i o n :

( 3 . 1 ) w - P&[A G F ' ( u ) w t h ] = g .

The f i x e d p o i n t f o r m ( 3 . 1 ) i s c l e a r l y e q u i v a l e n t t o

o r e l s e , w = g + z, where z = Th(g) i s s o l u t i o n o f

S e t t i n g b (v,,v,) = a ( v , , v,) - A ( F 1 ( u ) v , , v , ) and

< l , v z = a (h ,v ) t x ( F 1 ( u ) g , v ) , b i s a b i l i n e a r c o n t i n u o u s

f o r m on H;(a), 1 a l i n e a r c o n t i n u o u s f o r m on H;(n), and (3 .3 ) i s

e a u i v a l e n t t o

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C o n d i t i o n ( S ) on Su i m p l i e s t h a t t h e f u n c t i o n a l

J : w e Su + J (w) = b(w,w) i s 1.s.c. , c o e r c i v e

and s t r i c t l y convex , t h e r e f o r e ( 3 . 4 ) admi ts a un ique s o l u t i o n z

which min imizes J ; t h e n ( 3 . 1 ) has a un ique s o l u t i o n w = R h ( g ) .

F i n a l l y , i t i s enough t o p rove t h e L i p s c h i t z c o n t i n u i t y o f Th.

L e t z, = T h ( g , ) , z, = Thlg,) , z, i z, , which s a t i s f y :

Choosing r e s p e c t i v e l y v = z, and v = z, i n t h e E.V. I . ' s and

adding, i t comes :

C o n d i t i o n ( 3 ) i m p l i e s t h e r e e x i s t s a > 0 such t h a t

b ( z , - , Z, - Z Z ) 2 a z , - z,1I2 s i n c e z, - z, 6 SU - SU

Then

i z I - z 2 1 1 2 z L l g l - g 2 / 12,- z 2 / 5 L , I ! g , - g 2 I H z 1 - z 2 1 L L

hence 1 2 , - z, l l 5 2 119,- g, 1 1 , and 2 does n o t depend on h. o a

Remark 3.4

I f ( S ) i s r e p l a c e d by t h e l e s s r e s t r i c t i v e c o n d i t i o n :

we s t i l l g e t an e x i s t e n c e r e s u l t f o r ( 3 . 2 ) o r ( 3 . 4 ) ( b y use o f

a m i n i m i z i n g sequence r e l a t i v e l y t o J ) b u t uniqueness i s n o t

guaranteed any more.

Remark 3.5

I f Su i s a l i n e a r space, Lemma 3.1 i s t r u e s i m p l y i f (A,u) i s

r e g u l a r .

Coro l 1 a r y 3.1

L e t ( A , u ) s a t i s f y c o n d i t i o n ( $ . There e x i s t s a un ique r i g h t ( r e s p .

l e f t ) c o n i c a l d i r e c t i o n z, ( r e s p z - ) . We have z z 0 a.e. on a , t -

and i n p a r t i c u l a r z+ = 0 q.e. on E = i u = Y } .

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Moreover, i f (A'?)- e i P ( a ) p 1 2 , t h e n z- 2 z, a.e. on n and

i n p a r t i c u l a r z- 2 0 a.e. on n.

Proo f : We a p p l y Lemma 3 . l ( i ) w i t h h = G F ( u ) ang g = 0 t o g e t

e x i s t e n c e and uniqueness o f z, s a t i s f y i n g :

z, = PSu [ i G F ' ( u ) z+ + G F l u ) ]

wh ich i s t h e f i x e d p o i n t f o r m u l a t i o n e q u i v a l e n t t o (2.2), ( s i m i l a r l y

w i t h z - u s i n g Lemma ( 3 . 1 ) ( i i ) ) .

For t h e p o s i t i v i t y o f z,, l e t z: = sup (O,z,) ; z: 2 0 a.e. on a ,

hence q.e. on E = { u = '?) ; t h e r e f o r e z: e - c(K,u) ; b u t t ---

z, s c X ) ->z, s C(K,u) [311 -> z:a C(K,u) n - c ? ) -> z: = 0 q.e. on E,hence z: e Su.

We choose h = z: E SU i n (2.21, :

a(z, , z;) - i ( F 0 ( u ) z+ , z;) 2 ( F ( u ) , z;)

- t w i t h z, = z, - z - e SU - SU -7

-> z; = 0 a.e. -7 z,, 0 a.e. o n a - > E , 2 0 q.e. on a ,

hence on E and z, 6 Su i m p l i e s t h a t z, = 0 q.e. on E.

To p r o v e t h a t z - 2 z, a.e., we r e w r i t e ( 2 . 2 ) + as :

a ( z t , - h ) - x ( F ' ( u ) z, , - h ) 2 ( F ( u ) , - h ) V h e - SU

a ( z - , h ) - i ( F 1 ( u ) z- , h) 2 ( F ( u ) , h ) 'V h e - SU

and add t h e i n e q u a l i t i e s :

Consider k = ( z - - 2,)- ; k i s 2 0 a.e.on a , h e n c e q.e . on a and

k e - m. Moreover, s i n c e u 6 ~ ~ ' ~ ( n ) , one has :

a ( u , k ) - A ( F ( u ) , k ) = (Au - A F ( u ) ) k d x + (Au - i F ( u ) ) k d x

The f i r s t i n t e g r a l i s z e r o ( [ 2 8 ] , see a l s o p a r t 1 ) o f p r o o f o f

Theorem 3 . 1 ) ; and t h e second t o o , s i n c e k = 0 a.e. on E . Conse-

q u e n t l y , k e - S u and choos ing h = k i n (3 .5 ) we g e t

a ( k , k ) - A ( F 1 ( u ) k , k ) 5 0 , so k = 0

by c o n d i t i o n ( S ) , and z - 2 z, o Dow

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3.2 The case o f a t r a n s i t i o n p o i n t

L e t ( 1 , ~ ) be a s o l u t i o n o f ( l . l ) . R o u g h l y speak ing , ( i , u ) i s a t r a n -

s i t i o n p o i n t i f t h e r e i s a sequence o f e q u a t i o n s o l u t i o n s , and a sequence

o f "pure" E . V . I . s o l u t i o n s , c o n v e r g i n g t o ( i , u ) (see f i g u r e 1 i n s e c t i o n 1 ) .

We s t a t e t h i s d e f i n i t i o n i n a more p r e c i s e way :

D e f i n i t i o n 3.2

L e t ( i , u ) be a s o l u t i o n o f ( 1 . 1 ) . We say ( A , u ) i s a t r a n s i t i o n p o i n t

i f , i n e v e r y ne ighbourhood o f (A,u) i n IR, x Hk(Q) , t h e r e e x i s t

( A , , u , ) s a t i s f y i n g ( 1 . 1 ) and ( 1 . 3 ) ( t h a t i s , an e q u a t i o n s o l u t i o n )

and (A,,u,) s a t i s f y i n g (1 .1 ) b u t n o t ( 1 . 3 ) ( t h u s cap tu, = 81 > 0 ) .

T h i s d e f i n i t i o n i s somewhat r e s t r i c t i v e i n o r d e r t o e x c l u d e t r i v i a l

s i t u a t i o n s (see [ 1 3 ] ) . However, t h e o n l y p o i n t needed be low i s t h a t ,

s i n c e ( A , u ) i s a s o l u t i o n o f ( 1 . 3 ) ( t a k e y + u , A , + i ) we have

i u - i G F ( u ) l 1 = H;l(a), hence S u = m) and S u - S u = H;(n).

C o n d i t i o n ( s ) a t a t r a n s i t i o n p o i n t ( A , u ) t a k e s t h e f o r m ( S ) t :

i n f ja(w,w) - A ( F ' ( u ) w, w) / l lw l l = 11 > 0

and i s o b v i o u s l y connected t o a l i n e a r i z e d s t a b i l i t y c o n d i t i o n ,

r e l a t i v e l y t o t h e e v o l u t i o n problem a s s o c i a t e d w i t h Equat ion (1.3).

Coro l 1 a r y 3.2

L e t (A ,u) be a t r a n s i t i o n p o i n t s a t i s f y i n g ( s ) ~ .

( i ) t h e r e e x i s t s a u n i q u e r i g h t ( r e s p , l e f t . ) c o n i c a l d i r e c t i o n

z, ( r e s p . z - )

( i i ) z+ 0 a.e. on a and i n p a r t i c u l a r z, = 0 q.e. on E = t u = r } ( i i i ) z- , z, a.e. on a, t h u s 2 - 2 0 a.e. on a and i n f a c t

z- > 0 a.e, on n

( i Q ) z - i s t h e u n i q u e s o l u t i o n o f t h e l i n e a r i z e d e q u a t i o n :

Az- - i F ' ( u ) z - = F ( u ) i n D ' ( a ) ; z- E H;(a) n H Z ( n )

P r o o f : thanks t o C o r o l l a r y 3.1, we have o n l y t o p rove ( i i i )

and ( i i j ) .

a) we prove z - ~ z , a.e, on a : l e t E = i u = Yl ( r e c a l l t h a t u

and u a r e quas i -cont inuous i n t h e d e f i n i t i o n o f E). Again, ( 2 . 2 ) + i m p l y :

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embe

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We add bo th i n e q u a l i t i e s and choose h = ( z - - 2,)- 2 0 a.e. on n , hence q.e. on E, hence h E - C(K,U) = - Su :

b ) l e t 2 be t h e unique s o l u t i o n o f t h e l i n e a r i z e d equat ion :

A z - A F 1 ( u ) z = F ( u ) i n D ' ( n ) ; z e HA(n)

Since A F ' ( u ) z + F i u ) e L ~ ( R ) , f o r some q = q i n ) > 2, z e ~ ~ ' ~ ( a )

and, by boo t - s t rap , z G ~ ~ ' ~ ( 0 ) V q e N, hence z i s i n c l t a ( % )

V cx E [ 0 ,1 [ . But a t a t r a n s i t i o n p o i n t , A u = h F ( u ) -> u e ~ ' " ~ ( 3 ;

thus i F 1 ( u ) z + F ( u ) e ca(.) and z e c2"(.), A z - i F 1 ( u ) z > 0

on n. Cond i t i on ( s ) ~ i m p l i e s t h a t t h e s t rong maximum p r i n c i p l e

ho lds [ 1 9 ] thus z > 0 a.e. on a and z 6 -Su

Then f o r any w e H;(a), and i n p a r t i c u l a r f o r w e - Su :

a (z , w - z ) - A ( F ' ( u ) z, w - z ) = (F (u ) , w - z )

which i m p l i e s z s a t i s f i e s ( 2 . 2 ) - -> z = z - by uniqueness ; t h i s

achieves t h e proof . o

3.3 The l i n e a r case

Suppose Su i s a l i n e a r space ; then

Su = Su n - Su = i w e H;(n) / w = 0 q.e, on E l

and ( h , ~ ) i s s i n g u l a r ( A # 0) i f f i s an e igenva lue o f t h e l i n e a r

ope ra to r w + PSu [ G F ' ( u ) w ]

I n e q u a l i t i e s (2.21, and (2 .2) - a re equ i va len t and reduce t o t h e

f o l l o w i n g equat ion :

C l e a r l y i n t h a t case Lemma (3 .1) and i t s c o r o l l a r y can be s t a t e d more

p r e c i s e l y s i nce w e H;(a) + w - PSu [ A G F ' ( u ) w + h ] e H;(n) i s

one t o one as soon as (A ,u ) i s r e g u l a r . I n t h a t case we have a

unique con i ca l d i r e c t i o n z = z, = z - s o l u t i o n o f

bu t z 2 0 does n o t n e c e s s a r i l y ho ld .

Remark 3.6

The l i n e a r case i s n o t t h e genera l one, s i nce a t a t r a n s i t i o n p o i n t

where E = { xo) and n = 1 we have z+(xo) = 0 b u t z - ( x 0 ) , 0 hence

z, # z - and Su i s n o t a l i n e a r space.

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Remark 3.7

On t h e o the r hand cons ider a t r a n s i t i o n p o i n t where cap E = 0

( f o r i ns tance E = { x o } and n > 1 ). Then

Su = = i w 6 H;(a) / w 5 0 q.e. on E l = H;(a) i s l i n e a r .

We are go ing t o show t h a t Su may be l i n e a r i n a l e s s simple case,

on t he E.V.1, branch.

Theorem 3.1

Le t ( 1 , ~ ) be a s o l u t i o n o f (1 .1) and E = i u = Y I ( * ) . We make t h e f o l -

lowing assumptions :

( i ) Y a w2 'p(n) , P, 2 and meas i A Y = & F ( Y ) ~ = 0 V 0

( i i ) E = E , E # 0 and {w = 0 q.e. on E -> w = 0 q.e. on E l

Then Su i s t h e l i n e a r space i w e H;(n)/ w = 0 q.e. on E} which can

be i d e n t i f i e d t o H;(axE) and I n e q u a l i t i e s (2 .1) and (2 .2 ) * may

be r e w r i t t e n as D i r i c h l e t problems i n n \ E :

Proof :

1 ) We f i r s t show t h a t S u c t w e HA(n)/w = 0 q.e. on E j .

Since ( A Y ) - s L Z ( n ) , u e H 2 ( n ) . Now l e t w 6 SU ;

a(u,w) = i / D F l u ) w dx, t h e r e f o r e [Au - i F i u ) ] w dx = 0 ( 3 .6 ) L Define I'(u) = i x / u ( x ) ~ ( x ) i n t h e sense o f H;(n)} as i n [ 2 8 ] ;

t hen I ( u ) and E a re equal up t o a s e t o f measure zero, and

Au - i F ( u ) = P , where P i s a (non p o s i t i v e ) measure whose sup-

p o r t i s i nc l uded i n I ( u ) , so - 1 ~ ( u ) ] w dx = 0 (3.6)bis

By Stampacchia's Lemma, Au = A Y a.e. on E. Therefore , (3.6) and

(3.6)bis imp l y : LIA Y - F( ' f ) ] w dx = 0 ( 3 . 7 )

( * ) Reca l l t h a t u and Y a re chosen quas i -cont inuous and E i s de f i ned up

t o a se t o f capac i t y zero.

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EIGENVALUE PROBLEMS

But w 5 0 q.e. on E and A Y - A F ( Y ) = Au - A F ( u ) a.e. on E.

As Au - A F ( u ) 5 0 a.e. on a, one has A Y - A F ( Y ) < 0 a.e. on E

by t h e f i r s t assumption ; f rom (3 .7 ) , we i n f e r w = 0 a.e. on E,

which i s e q u i v a l e n t t o w = 0 q.e. on E. From ou r second assumption,

we ge t w = 0 q.e. on E.

2 ) Next, we prove t h e converse i n c l u s i o n : tw E H;(a)/ w = 0 q.e. on E l

c Su.

L e t w e H;(n), w = 0 q.e. on E ; then w E C(K,lr) and

aiu,w) - A IFIUI,WI = ~ I A U - A F(U)I w dx = IE ~ A U - A F ( u ) l w dx = o s i nce w = 0 a.e. on E.

3 ) F i n a l l y , t h e f a c t t h a t Su can be i d e n t i f i e d t o H;(a\E) i s a r e s u l t

which i s con ta i ned i n t h e li t t e r a t u r e . More p r e c i s e l y , H;(o\E) c Su

f o l l o w s f r o m [ 2 2 , Theorem0.11 a n d S u c H ; ( a \ E I f r o m

[ 21 , Lemma 41 ; t h i s i d e n t i f i c a t i o n ho lds f o r any c l o s e d

s e t E o f non zero capac i t y , when Su i s a l i n e a r space (hence neces-

s a r i l y equal t o tw e H i ( a ) / w = 0 q.e. on E l ) .

T a Not i ce t h a t t h e c o n d i t i o n E = E may be r e l a x e d : i n f a c t , i f E = E

up t o a s e t o f c a p a c i t y zero , t hen c l e a r l y w = 0 q.e. on E <->

w = 0 q.e. on E and Theorem 3.1 i s t r u e , p rov i ded Su i s i d e n t i f i e d

Remark 3.8

We c l o s e t h i s s e c t i o n w i t h a d i scuss ion on t h e t e c h n i c a l assumption

( i i ) i n Theorem 3.1 : V 0

E = E , E # 0 and ( w = 0 q.e. on E -> w = 0 q.e. on E l . Th i s assump-

t i o n i s c l o s e l y r e l a t e d t o t h e problem o f app rox ima t i on o f f u n c t i o n s

o f Hb(a) t h a t van ish i n E by f u n c t i o n s i n D (n \E ) [ 2 1 , 221

A c t u a l l y , t h i s problem i s e q u i v a l e n t ( f o r i ns tance i n lR3 t o t h e

f o l l o w i n g : any a n a l y t i c f u n c t i o n i n E' be l ong ing t o LZ(E ' ) can

be approximated by r a t i o n a l f u n c t i o n s w i t h po les i n E.

V

F i r s t , i f n = 1 and E = E # 0, assumption ( i i ) i s always s a t i s f i e d :

i n f a c t , w E H;(a) has a cont inuous r e p r e s e n t a t i v e 2, t hus 2 = 0 V

i n E i m p l i e s 3 = 0 i n E = E by c o n t i n u i t y , hence w = 0 q.e. i n E.

We n o t i c e t h a t , i f (,I,u) i s a t r a n s i t i o n p o i n t w i t h E = ( x o l , t hen V

Su = tw e H 1 ( n ) / w(x ) 5 Ol i s n o t l i n e a r , b u t i n t h a t case E = 0. Dow

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CONRAD ET AL . a-

. For n = 2, [ 22 , E2ample 6.61 shows a case i n wh ich E = E,

w e Hb(iR2), w = 0 on E b u t w > 0 on a subset o f p o s i t i v e c a p a c i t y

o f aE.

a- . When n , 2, i f E = E, our assumption i s j u s t t h e s o - c a l l e d A ' -

s t a b i l i t y f o r E' as d e f i n e d i n [ 22 , f i n a l remarks ] and i s

e q u i v a l e n t t o Saak 's c o n d i t i o n [ 22 I cap ( B n E) = cap (B n E) f o r a l l open b a l l s B (see [ 22 I The-

orems 6.3 and 6 .51 for o t h e r s u f f i c i e n t A ' - s t a b i l i t y c o n d i t i o n s ) .

By [ 21 , Theorem 71 a p p l i e d t o E t h e S a a k ' s o c o n d i t i o n i s e q u i -

v a l e n t t o t h e p r o p e r t y t h a t t h e p a r t o f E where E i s " t h i n "

[ 22 , formula ( 1 . 1 4 ) ] has c a p a c i t y zero .

But [ 21 , Theorem 41 E i s t h i n a t x e E i f f

For x e E, B Y ( & ) , t h e open b a l l o f r a d i u s 6 and c e n t e r x i s i n E

f o r smal l 6 ; t h u s

t h e r e f o r e t h e i n t e g r a l i s i n f i n i t e f o r any n 2 2 and o u r c o n d i t i o n

reduces t o t h e f a c t t h a t t h e p a r t o f aE where E i s t h i n must be

o f zero c a p a c i t y .

N o t i c e t h a t f o r n = 2, our assumption i s always t r u e as soon as T

E = E, E connected [ 21 , C o r o l l a r i e s 1 and 31 o r a t l e a s t i f

E i s a f i n i t e u n i o n o f such d i s j o i n t s e t s .

-m . F i n a l l y we have t o ment ion t h a t i f E = E and aE i s r e g u l a r , t h e n our assumption i s t r u e .

T To prove t h i s f a c t , l e t w 6 H;(a), w = 0 a.e. on E . S i n c e E = E,

aE = aE i s r e g u l a r ( t h a t i s , c1 and E l o c a l l y on one s i d e o f aE) ; t h u s t h e t r a c e o p e r a t o r y on aE i s w e l l d e f i n e d .

S ince w e HI(:), we have v w s H t ( a i ) and

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EIGENVALUE PROBLEMS 171

But on E, w = 0 a.e. and by Stampacch ia 's Lemma ow = 0 a.e, on E, hence Y w = 0. Now f o r w = 0 a.e. on E,Y w = 0 we d e f i n e

0 on E v = l w O n Q ' E

S ince Y w = 0, v e H;(a) and I v - wll = 0 ; t h u s v and w have t h e

same q .c . r e p r e s e n t a t i v e up t o a s e t o f c a p a c i t y zero, so w = 0 q.e . on E (see 1281, Append ix ) .

A c o n d i t i o n i m p l y i n g C' smoothness o f aE i s g i v e n i n [ll], a t l e a s t

i f Y E c 3 ( E ) , A Y + A F ( u ) > 0 : meas { E n B X ( 6 ) 1

V x e E , - > a > O a s 6 + 0,. meas { B X ( 6 ) 1

It i s known t h a t aE has measure zero , t h e r e f o r e E may be r e p l a c e d

by :. Then t h e p r e v i o u s c o n d i t i o n i m p l i e s measi; ll B x ( 6 ) i ~ a 6 n

as 6 - Ot, hence i s n o t t h i n a t x 1 1 8 , Lemma 1 p. 1701.

Thus t h e c o n d i t i o n g i v e n i n t h e preceed ing remark i s weaker t h a n

t h e r e g u l a r i t y c o n d i t i o n .

4. LOCAL STUDY. THE REGULAR CASE.

I n t h i s s e c t i o n , we assume n 5 4. Suppose (h0,u0) i s a r e g u l a r s o l u t i o n

of ( 1 . 1 ) . We a r e go ing t o p rove t h a t z, ( resp .2- ) i s , i n some sense, t h e

f i r s t o r d e r t e r m o f an expans ion o f s o l u t i o n s o f ( 1 . 1 ) near (h0 ,u0) . S ince

PK i s n o t Frechet d i f f e r e n t i a b l e , t h e c l a s s i c a l i m p l i c i t f u n c t i o n theorem

does n o t h o l d . F i r s t , l e t us r e c a l l t h e f o l l o w i n g r e s u l t p roved i n [13 , 231.

Theorem 4.1 : L e t (h0,u0) be a s o l u t i o n o f ( 1 . 1 ) s a t i s f y i n g ( s ) , and l e t z+

( r e s p . 2-1 denote t h e un ique c o n i c a l d i r e c t i o n s a t ( i 0 , u o ) .

Then t h e r e e x i s t n z 0 and a m u l t i - v a l u e d mapping o+ : ( 0 , n) + H;(a)

( r e s p . o- : ( - q , o ) - H;(a)) such t h a t uo t tz, + to,( t ) ( r e s p .

uo + t z - + t o - ( t ) ) i s a s o l u t i o n o f ( 1 . 1 ) f o r A = ho + t . Moreover

I o , ( t ) / ( r e s p . 1 Q - ( t ) ) - 0 as t - 0, ( r e s p , t - 0 - ) .

The p r o o f i s based on t h e i n v e r s i o n lemma. Theorem 4.1 i s t r u e i n

t h e l i n e a r case under t h e weaker assumption o f r e g u l a r i t y (see Remark 3 . 5 )

and o f course we have z, = z- and e x i s t e n c e o f a m u l t i v a l u e d mapping

o : (-!l,Il) - H;(n).

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172 CONRAD ET AL.

I n t h i s s e c t i o n , we s h a l l deve lop another p o i n t o f v iew, namely t h e

t o p o l o g i c a l one, wh ich g i v e s r i s e t o t h e e x i s t e n c e o f connected components

i n ( t , o ) s t a r t i n g f r o m (0,O). A s i m i l a r f i x e d p o i n t f o r m u l a t i o n r e l y i n g on

t o p o l o g i c a l degree i n s t e a d o f c l a s s i c a l f i x e d p o i n t theorem was s e t up by

one o f t h e a u t h o r s i n a plasma b i f u r c a t i o n prob lem [ 3 3 ] .

4.1. The f i x e d p o i n t f o r m u l a t i o n

L e t (h,,u,) be a s o l u t i o n o f (1 .1 ) .

Set (4 .1 ) u = uo + sz, h = A, + s, s e IR, z E H & ( Q )

and A(s,z) = u - PK[A G F ( U ) ] = uo + sz - pK[ (Ao+s) G F ( u ~ + s z ) ] .

The aim o f t h i s s e c t i o n i s t o p rove e x i s t e n c e o f s o l u t i o n s o f ( 1 . I )

o f t h e f o r m ( 4 . 1 ) t h a t i s , t o So lve :

(4 .2 ) a ( s , z ) = 0 w i t h n (O,z ) = uo - P [ h o G F ( u o ) ] = 0 K

T h e r e f o r e (4 .2 ) f o r s f 0 i s e q u i v a l e n t t o :

S ince n 5 4 , Formula ( 2 . 5 ) o f S e c t i o n 2.4 shows t h a t A (s ,z ) i s

r i g h t d i f f e r e n t i a b l e a t s = 0 and

t h e d i f f e r e n t i a t i o n b e i n g u n i f o r m when z remains bounded i n H;(a). A s i m i l a r

r e s u l t h o l d s f o r s < 0 b u t , i n t h e sequel , we o n l y c o n s i d e r s > 0.

Whith these n o t a t i o n s , ( 4 . 3 ) w i t h s > 0 i s e q u i v a l e n t t o :

From now on, we suppose c o n d i t i o n (5') i s s a t i s f i e d a t (ho,u0) ( g e n e r a l

case) o r (hO,uO) i s s i m p l y r e g u l a r ( l i n e a r c a s e ) .

By lemma 3.1 (see a l s o remark 3 .5 ) t h e mapping

w - w - P [ i o G F 1 ( u o ) w + G F ( u 0 ) ]

i s i n v e r t i b l e , l e t T be i t s i n v e r s e . Then (4 .3 ) <->

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EIGENVALUE PROBLEMS 173

At t he l i m i t s + 0+, t h e r i g h t hand s i de o f 14.4) i s T (0 ) = z+ by

t h e d i f f e r e n t i a t i o n f o rmu la and d e f i n i t i o n o f T. Therefore , we set z = z, + $ , Q E H;(a) and, f o r s > 0, (4.4) i s equ i va len t t o

- T(0) . F i n a l l y we d e f i n e Q(O,Q) = 0 and +(s ,$) = t h e R.H.S. o f ( 4 . 5 ) f o r

s >0, and r e w r i t e ( 4 . 2 ) under t h e equ i va len t form, whenever s , 0 :

Lemma 4.1

Le t (ho,u0) be a s o l u t i o n o f ( 1 . l ) s a t i s f y i n g c o n d i t i o n ( s ) . Then + : R+ x H;(a) - H;(a) i s comple te ly cont inuous.

Proof : l e t r be a bounded set i n IR+ x H;l(a). Then, f o r a t l e a s t a

sequence isn,@,) E T ,sn + s i n R,, $n * $ i n L Z ( n ) ->

sn z, + sn $n + s Z, + s 0 i n L z ( n ) ; Since f ( t , x ) i s Caratheodory

and bounded, F i s cont inuous from L 2 ( n ) t o L 2 ( a ) and

F(sn z, + sn qn) - F ( s z, t s $ 1 i n L 2 ( a ) . Since F 1 ( u o ) E ~ ~ ( n ) ,

F 1 ( u 0 ) $, - F 1 ( u o ) 4 i n L 2 ( a ) .

The a p p l i c a t i o n o f t he Green ' s ope ra to r G y i e l d s convergences i n H;(n)

Suppose f i r s t s # 0 ; then, s i nce P K, PSuo and T a re cont inuous,

we o b t a i n l i m ~ ( s , , $n ) = + ( s , $ ) as n t -. , i n ~ ' , ( f i ) .

I f sn + s = 0, we r e c a l l t h a t t h e con i ca l d i f f e r e n t i a t i o n o f P i n K

t h e d i r e c t i o n io G F 1 ( u o ) z+ + " G FF'(u0) o + G F (uo ) i s u n i f o r m

w . r . t . en bounded i n H;(n) ; then + ( s n , @ " ) + 0 = Q ( 0 , q ) . I n any

case, we see t h a t Q ( r ) i s r e l a t i v e l y s e q u e n t i a l l y compact, thus Q

i s compact.

I n o rde r t o Drove t h a t + i s cont inuous, we cons ider d i r e c t l y a

sequence (s,, @,) converg ing i n lR+ x H;(a), t hus i n IR, x L Z ( a ) and

we app ly t h e same method o

Lemma 4.1 a l l ows t h e use o f c l a s s i c a l degree t heo ry [ Z 7 , 34, 361

4.2. Ex is tence O f components o f s o l u t i o n s o f (1.1 ) which a re s t r o n g l y r i g h t -

and l e f t - d i f f e r e n t i a b l e a t ( i0 ,u0) Dow

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174 CONRAD ET a.

Lemma 4.2 :

E i t h e r ( 1 . 1 ) admi ts l o c a l l y an a f f i n e r i g h t s o l u t i o n a r c s t a r t i n g

f r o m ( h o , u 0 ) , o r f o r any s* > 0, t h e r e e x i s t s s E ] 0, S * ] such t h a t

@(S,O) # 0.

P r o o f : suppose t h e r e e x i s t s s* > 0 such t h a t @ ( s , 0 ) = 0 V S e [ O,s*]

Then, f o r s e [ 0, s* :

The a p p l i c a t i o n o f T-I and t h e d e f i n i t i o n o f z+ l e a d t o :

uo + s z L = P ! ( A ~ + S ) G F(uo + s z + ) 1 K

t h e r e f o r e , f o r S e [O, S* I we have an a r c o f s o l u t i o n s o f t h e f o r m :

A = A o + 5 , U ' U o t S Z + 0

Remark 4.1 :

The case o f a l o c a l l y a f f i n e branch o f s o l u t i o n s i s indeed p o s s i b l e .

For i n s t a n c e , i f f ( t , x ) = 1, t h e e q u a t i o n branch i s a f f i n e and a l l

i t s p o i n t s ( i , ~ G ( 1 ) ) s a t i s f y c o n d i t i o n ( S ) . A more i n t e r e s t i n g ex-

ample o f an a f f i n e E . V . I . b ranch can be g i v e n : t a k e n = ] - l , l [ , A = - a , f ( t , x ) ! 1 and 't'(x) = 1 t 1x1. The e q u a t i o n branch i s

( A , u ( x ) = ( I - x ' ) ) f o r A ( 2. The t r a n s i t i o n p o i n t i s A. = 2,

u ( x ) = 1 - x Z .

Then ( A = 2 t s, u ( x ) = u O ( x ) + 3 (1x1 - x 2 ) = u ( x ) + s z + ( x ) ) f o r

O ( s ( 2 i s an a f f i n e branch o f s o l u t i o n s o f ( 1 . 1 ) which, f o r s > 0,

a r e n o t e q u a t i o n s o l u t i o n s ; t h e c o i n c i d e n c e s e t o f a l l t h e s e s o l u t i o n s

i s 101 and A U ! ~ ) + F ( u ( s ) ) = s 6 ( 0 ) f o r 0 < 5 < 2 ; a s t r a i g h t -

f o r w a r d c a l c u l a t i o n shows t h a t Su = 1 w e H;(a)/w(O) ( 0 1 f o r s = 0,

b u t Su = i w e H;l(a)/w(O) = 0 1 f o r s e ]0 ,2 [ i s l i n e a r t h o u g i ~ The-

orem 3.1 i s n o t a p p l i c a b l e . For A > 4, t h e c o i n c i d e n c e s e t o f t h e

s o l u t i o n o f ( 1 . 1 ) i s [ - r , r ] , r = 1 - and t h e E . V . I . b ranch i s no 6

l o n g e r a f f i n e (see F i g u r e 2 t h e graphs o f t h e s o l u t i o n s , w i t h r e s p e c t

t o A ) .

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EIGENVALUE PROBLEMS

- 1 - I I r 1

F i g u r e 2

Remark 4.2. :

A s t r a i g h t f o r w a r d a p p l i c a t i o n o f Krasnosel ' s k i i and Rab inowi tz t y p e

arguments based on t o p o l o g i c a l degree does n o t exc lude a l o c a l l y

a f f i n e component. I n t h e f o l l o w i n g , we w i l l e s t a b l i s h s t r onge r

r e s u l t s d i s t i n g u i s h i n g a f f i n e and non a f f i n e cases.

Lemma 4.3. :

Suppose t h a t t h e r e i s no l o c a l r i g h t a f f i n e s o l u t i o n branch s t a r t i n g

(0,O) i s an accumula t ion p o i n t o f n o n t r i v i a l s o l u t i o n s ( s , @ ) s # 0,

Proof : Suppose t h e c o n t r a r y . Then3 c o > 0, n o > 0 such t h a t

8 ( s , $ ) e (I:~ x Brio) n F. I n p a r t i c u l a r , any s o l u t i o n

( s , $ ) 6 I:~ x B \ o f (4.5) w i t h s # 0 i s n e c e s s a r i l y o f t h e f o rm

(s#O, QfO).

Consequent ly, i f s E 10, g ] and @ e a Bn0/2, t hen $ - + ( s ,$ ) # 0 ;

t h e t o p o l o g i c a l degree deg ( I - O ( s , . ) , B r b / 2 , 0) i s t h u s d e f i n e d f o r

s E [ 0 , E ~ / Z ] and i s cons tan t by homotopy.

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CONRAD ET AL.

By Lemma 4.2, l e t s, e ]0,c0/2] such t h a t +(s,,O) # 0. Then

+ - + (s ,,+) # 0 V $ E Bn0/2 and the re fo re , by homotopy,

deg ( I - ~ ( s , . ) ,Bno/2,0)=deg ( I - + ( s t , . ) , Bno/2, 0) = 0.

But we have a l s o deg ( I - +(s , . ) , Bvo/2, 0 ) - deg ( I - +(O,. 1, Bn0/2, 0 ) =

deg ( I , 811~12, 0 ) = 1, a c o n t r a d i c t i o n Q

Theorem 4.2 : Let (ho,uo) be a s o l u t i o n o f (1.1) which s a t i s f i e s c o n d i t i o n (S).

There e x i s t s a connected component Ct ( i n R+XH;(Q)) o f s o l u t i o n s

(s,$) o f ( 4 .5 ) , c o n t a i n i n g (0,0), unbounded i n R+xH;(Q), such t h a t

( h = ho + s, u = uo t sz+ + S Q ) i s s o l u t i o n o f ( 1 .1 ) .

I f (si ,$;) 6 e, s i+O then - - z+ s t r o n g l y , where z+ i s h i - Lo

t h e r i g h t c o n i c a l d i r e c t i o n at (Ao,uo), t h a t is, the unique s o l u t i o n

o f (2.2)+.

Moreover, e i t h e r t h e r e i s a l o c a l r i g h t a f f i n e s o l u t i o n branch o f (1.1)

s t a r t i n g from (hO,uO) o r (0,O) i s an accumulat ion p o i n t o f n o n t r i v i a l

s o l u t i o n s o f (4.5).

Proof o f Theorem 4.2 : l e t ct be t h e connected component o f s o l u t i o n s (s,$)

o f ( 4 .5 ) w i t h s 2 0 , c o n t a i n i n g ( 0 , 0 ) . I n t h e non a f f i n e case, Lem-

ma 4.3 i m p l i e s t h a t (0,O) i s a l i m i t p o i n t o f s o l u t i o n s ( ~ ~ $ 4 ~ ) o f ( 4 . 5 )

w i t h si > 0, mi # 0 and thus c t i s non empty, an obvious f a c t i n t h e

a f f i n e case. Since @ i s comple te ly cont inuous, t h e unboundeness o f C'

i s then a c l a s s i c a l r e s u l t [ 35 ] . I f si + 0,, t hen qi - 0, hence

Remarks

4.3- Theorem 4.2 j u s t i f i e s t h e geometr ic i n t e r p r e t a t i o n g i ven i n

Theorem 2.1 and i s more p rec i se s i nce we g e t s t r o n g convergence o f

4.4- A s i m i l a r r e s u l t o f course ho lds f o r h<h0 ; we g e t ' e x i s t e n c e o f

a com~onent C unbounded i n R - X H ~ ( O ) , c o n t a i n i n g (0,0), such t h a t if

4.5-The c r u c i a l f a c t i n t h e p roo f of Theorem 4.2 i s t h e i n v e r t i b i l i t y

o f t h e ope ra to r w - PSu [ ho G F' ( uo ) w + G F (uo ) 1 guaranteed by

c o n d i t i o n (S). ~ h e r e f o r g , i n t he l i n e a r case, Theorem 4.1 i s v a l i d

a t every r e g u l a r p o i n t ( x ~ , u ~ ) . Dow

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5. LOCAL STUDY. A SINGULAR CASE

5.1. P r e l i m i n a r i e s

We aga in suppose t h a t n 5 4. L e t (ho ,u0) be a s o l u t i o n o f (1 .1)

such t h a t Suo i s a l i n e a r space. We suppose ( io ,uo) i s s i n g u l a r . More spec i -

f i c a l l y , t h e ke rne l ,v o f t h e l i n e a r ope ra to r : w 6 H;(n) + w - PSu [xoGF' (uo)w]

e Hd(a) i s supposed t o be one d imens iona l . L e t z N be a genera tor o f O t h i s

k e r n e l .

Lemma 5.1 :

Le t R d e n o t e t h e range o f I - PSu [ " G F F ' ( u o ) . ] i n Hi("). Then .F:

i s a c l osed subspace o f codimensi8n 1 and, i n f a c t

R = i w e H;(n) / ( F 1 ( u 0 ) z!~, w ) = O i

Proof : l e t H = I - PSu [ " G F F ' ( u o ) ] . H i s a compact p e r t u r -

b a t i o n o f t h e identify.'^^ a p p l i c a t i o n o f Fredho lm's A1 t e r n a t i v e ,

R = Range(H) i s c losed, codim R = dim N = 1.

1 L e t us show t h a t R = i G F 1 ( u 0 ) zit) = i w E H ; ( ~ ) ~ ( F ' ( u ~ ) z ~ ~ , w) = 0 1

C l e a r l y , t h e l a t t e r space i s o f codimension 1, t h e r e f o r e we have

o n l y t o prove t h a t R c i w 6 H;(d I ( F 1 ( u O ) z , W ) = O i . N

L e t w c R : 3 y e HL(n) : w = y - PSu [ho G F ' ( u o ) y ] .

By t h e d e f i n i t i o n Ef t h e l i n e a r oper&or Psu :

V h E SuO : a ( y - w,h) = A O ( F 8 ( u O ) y, h); we ?hoose h = z N 6 SuO :

S ince z = P [ AO G F 6 ( u O ) z ] : m Suo N

V h s Suo : a ( z , h ) = h o ( F ' ( u o ) z h);we choose h = y - w : N N '

Since a i s symmetric, s u b t r a c t i n g ( 5 .2 ) f r om (5 .1 ) l eads t o

A ~ ( F ' ( U , ) zN, W) = 0 ( n o t e t h a t ho cannot be ze ro because ( \ , uo )

i s s i n g u l a r ! ) o

We suppose i n t h e f o l l o w i n g t h a t t h e a d d i t i o n a l c o n d i t i o n (t?) i s

f u l l f i l l e d :

(6 (F (uo ) , ziy) # 0

Remark 5.1 :

We n o t e t h a t a(G F 1 ( u 0 ) z , P [G F ( u o ) ] ) = a(PSu [ G F 1 ( u o ) z 1,G F ( u o ) ) N SUo 0 N D

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178 CONRAD ET AL .

i s s e l f - a d j o i n t f o r t h e a - s c a l a r p r o d u c t ) = a('; , G F ( u o ) ) = 0

1 - (Z F ( u O ) ) . There fore , s i n c e io # 0 and thanks t o Lemma 5.1,

1'

(9) i s e q u i v a l e n t t o t h e c o n d i t i o n : PSu [G F ( u o ) ] 6 R, a n a t u r a l 0

e x t e n s i o n o f t h e s o - c a l l e d t r a n s v e r s a l i t y c o n d i t i o n i n t h e e q u a t i o n

case (see [ 1 6 ] ) .

We denote by W a supplementary space o f N i n H;(n), f o r i n s t a n c e t h e

a-or thogona l supplementary o f N .

As i n t h e r e g u l a r case, by use o f Schauder 's theorem, i t i s p o s s i b l e ,

a t l e a s t f o r n 5 3 , t o g i v e an expans ion theorem i n v o l v i n g m u l t i v a l u e d

mappings, see [13, 14, 231.

Theorem 5.1 :

Under t h e assumptions o f t h i s s u b s e c t i o n and n 5 4, t h e r e e x i s t

n > 0 and a m u l t i v a l u e d mapping :

s e to,,,) - i ~ ( s ) , W ( S ) > E IR x w such t h a t

(A, + s r ( s ) , uo + s z + s w ( s ) ) i s a s o l u t i o n o f ( 1 . 1 ) ; moreover N

( ~ ( s ) , w ( s ) ) + 0 as s + O+. A s i m i l a r r e s u l t h o l d s f o r s G ( - r i , O ) .

Theorem 5.1 i s i n some weak sense an e x t e n s i o n o f t h e c l a s s i c a l

r e s u l t s o f C r a n d a l l and Rab inowi tz on t u r n i n g p o i n t s [ 16,171 . We

w i l l deve lop t h e p o i n t of v iew o f components by means o f t o p o l o g i c a l

degree t h e o r y ; we f o l l o w t h e same method, as i n s e c t i o n 4,and r e c a l l

t h a t , s i n c e PK i s n o t Fr'echet d i f f e r e n t i a b l e , an a p p l i c a t i o n o f t h e

i m p l i c i t f u n c t i o n theorem adapted t o t h i s s i n g u l a r case ( a s i n [ 1 6 ] )

i s n o t p o s s i b l e .

5.2. F i x e d p o i n t f o r m u l a t i o n

+ s ( z + w ) , I . c N ; w f b We s e t N N

t s 6 , 6 e R ; s ' 0

and l o o k f o r s o l u t i o n s of (1 .1) o f t h a t t y p e , i . e , we want t o s o l v e (5 .3 ) :

u - P [ A G F ( U ) ] = U ~ + S ( Z + w ) - P [ ( A o + ~ 6 ) G F ( ~ O t S Z + S W ) ] = O K N K N

o r e l s e :

Us ing aga in t h e second c o n i c a l d i f f e r e n t i a t i o n f o r m u l a ( 2 . 5 ) w i t h

t = s 6 , W e g e t :

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EIGENVALUE PROBLEMS

( h e r e PSu i s l i n e a r , and f o r s = 0-, we o b t a i n t h e same e x p r e s s i o n f o r t h e

l e f t d e r i v a t i v e ) .

S i n c e A(O,s,w) = P [ h O G F i l l o ) ] - P [ h G F (uO) I . I 0 , ( 5 . 3 ) K K 0

f o r s # 0 i s e q u i v a l e n t t o :

S i n c e z 6 fl ( 5 . 4 ) i s e q u i v a l e n t t o : N

( 5 . 5 ) w - A Psuo I G F 1 ( u 0 l w 1 - 6 P

We want t o s o l v e ( 5 . 5 ) i n te rms o f (w,s) ; t h e f i r s t s t e p c o n s i s t s i n

g i v i n g a f i x e d p o i n t f o r m u l a t i o n e q u i v a l e n t t o ( 5 . 5 ) .

Lemma 5.1 :

The l i n e a r o p e r a t o r (a , * ) E YI x w - w - i0 PSuo[G F 1 ( u 0 ) w ]

- 6 PSuo [ G F ( u o ) ] 6 H;(Q) i s one- to -one , w i t h a c o n t i n u o u s i n v e r s e .

P r o o f : l e t us s o l v e t h e e q u a t i o n

w - A Psuo L G F t ( u O ) w 1 - 6 PSUO I G F ( u O ) l = f .

T h i s e q u a t i o n a d m i t s a s o l u t i o n w iff

f + 6 PSu [ G F ( u o ) ] E R <-> 0

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CONRAD ET AL.

( F ' ( u O ) z N , f ) <-> 6 = - A ( see Remark 5 .1 )

O ( F ( u o ) , 7. ) A'

which i s d e f i n e d and c o n t i n u o u s i n f thanks t o t h e t r a n s v e r s a l i t y

c o n d i t i o n ("e. T h i s p a r t o f t h e p r o o f r e s u l t s f r o m t h e p r o j e c t i o n

o f t h e e q u a t i o n on t h e L2 ( 0 ) - o r t h o g o n a l o f R , which i s t h e one

d imens iona l space spanned by F ' ( u o ) z (Lemma 5.1 1 . N

Now we p r o j e c t t h e e q u a t i o n on R .

Since w E w + w - A PSuo [ G F 1 ( u O ) W ] admi ts a pseudo- inverse

f r o m R i n t o W, we o b t a i n a un ique s o l u t i o n w e w t o t h e e q u a t i o n

w - A PSuo [ G F ' ( u o ) w ] = f t a PSu [ G F(u,)j,and w i s cont 0

nuous w . r . t . f . o

L e t T denote t h e c o n t i n u o u s i n v e r s e d e f i n e d by Lemma 5.1. Then

f o r s # 0 i s o b v i o u s l y e q u i v a l e n t t o :

As s + 0, t h e l i m i t o f t h e r i g h t hand s i d e i s z e r o by t h e d i f f e r e n -

t i a t i o n f o r m u l a . T h e r e f o r e l e t m (s,6,w) be t h e R.H.S. o f ( 5 . 6 ) f o r

s # 0,and z e r o f o r s = 0. The f i n a l f i x e d p o i n t f o r m u l a t i o n f o r our

s p e c i a l s o l u t i o n s o f (1.1 ) i s

Subsequent ly, we suppose s 3 0.

Lemma 5 . 2 :

o : IR, x IR x H;(n) + IR x H;l(n) i s a n o n l i n e a r c o m p l e t e l y c o n t i n u o u s

mapping.

Proo f : As i n Lemma 4.1, we a g a i n have t o d i s c u s s , when

(sn,6,,wn) + (s,6,w),the cases s # o o r s = o o

Now we a r e a b l e t o a p p l y degree t h e o r y t o t h e e q u a t i o n ( 5 . 7 ) .

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EIGENVALUE PROBLEMS 181

5.3 E x i s t e n c e o f components o f s o l u t i o n s o f (1 .1 ) wh ich a r e s t o n g l y

r i g h t and l e f t d i f f e r e n t i a b l e a t (A0,u0)

Lemma 5.3 :

E i t h e r ( 1 . 1 ) admi ts l o c a l l y a " v e r t i c a l " b ranch o f s o l u t i o n s c o n t a i -

n i n g ( A ~ . u ~ ) o r , f o r any s * > 0 t h e r e e x i s t s s G 10, s*] such

t h a t o(s,O,O) # (0 ,o ) .

iUA : .lo+ s , i s a s o l u t i o n V s E [O,s*I

and we have a v e r t i c a l b ranch o f s o l u t i o n s ( i .e , a b ranch o f d i s t i n c t

s o l u t i o n s w i t h A c o n s t a n t ) o

Remark 5 .2 :

We do n o t know i f , f o r o u r c l a s s o f problems, a l o c a l l y v e r t i c a l b ranch

o f s o l u t i o n s i s p o s s i b l e . The c l a s s i c a l Rabinowi t z t o p o l o g i c a l degree

t e c h n i q u e does n o t e x c l u d e a l o c a l l y v e r t i c a l b ranch.

Lemma 5.4 :

Suppose t h e r e i s no l o c a l v e r t i c a l s o l u t i o n branch c o n t a i n i n g ( A0,u0)

Then : V E > 0, V Y > 0 , V n > 0, 3 (s,a,w) e 1: x Jy x BQ n G

where 1: x J r x Bq = I s , s , w / O ( s 2 E; 1 6 ( y; / I W ~ / ( > and

G = j s , s , w l ( s , w ) = m(s,a,w) ; s # 0 and (s,w) # (0 ,O) ) . I n

p a r t i c u l a r , (0,0,0) i s an accumula t ion p o i n t o f s o l u t i o n s (s,s,w)

w i t h s # 0 and ( 6 , ~ ) # (0 ,O) .

P r o o f : Suppose t h e c o n t r a r y . Then 3 c 0 >O, yo > 0, n o > 0 such

t h a t $ ( s , s , w ) c I:~ x Jyo x Bqo n G. I n p a r t i c u l a r any s o l u t i o n

( s , & , w ) 6 I:~ x Jyo x Bn0 w i t h s > O i s n e c e s s a r i l y o f t h e f o r m :

( 5 > 0, 6 = 0, W = 0 ) .

Consequent ly, if s E 10, E0/2] and ( 6 , ~ ) E a r w h e r e T = JuO/2 x Bq0/2,

t h e n ( 6 , ~ ) - @ ( s , & , w ) # 0.

The t o p o l o g i c a l degree deg (I - m(s,. , . ) , r , 0 ) i s t h u s d e f i n e d f o r

s E [O, c0/21 and i s c o n s t a n t by homotopy. Dow

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By Lemma 5.3, l e t s, e ] 0, ~ , /2 ] be such t h a t m(s,,0,0) # (0,O)

Then ( 6 , ~ ) - m(s,,a,w) f (0,O) V ( 6 , ~ ) c r and t h e r e f o r e by homotopy

deg ( I - m(s ,.,. ) , r , 0 ) = deg ( I - m(s ,,.,. ) , r , 0 ) = 0

But, on t h e o t h e r hand, deg ( I - m(s ,.,. ) , r , 0 ) = deg ( I - m(O ,.,. ) , r , 0 )

= deg ( ( I , r , O ) = 1, a c o n t r a d i c t i o n o

Remark 5.3 :

A s i m i l a r r e s u l t o b v i o u s l y h o l d s f o r s < 0 .

Theorem 5.2 :

Le t (Xo,uo) be a s i n q u l a r s o l u t i o n o f (1.1) which s a t i s f i e s a l l t h e

assumptions o f subsec t i on 5.1. There e x i s t two connected components

@ and C- o f s o l u t i o n s (s,b,w) o f (5 .7) , unbounded i n R ~ R ~ H ~ ( Q ) ,

such t h a t C' I7 C- = (0,0,0) and ( X = ho t sh, u = u o t s z ~ + sw)

i s s o l u t i o n o f (1.1) w i t h s>O on Ct, s<O on C- ; z ~ E N , a(zN,w) = 0

and (b,,wi) + 0 as s i + 0.

Moreover, e i t h e r k 5 kg, u = u0 + s zN i s a l o c a l l y v e r t i c a l s o l u t i o n branch

o f ( l . l ) , o r (0 ,0 ,0) i s an accumula t ion p o i n t o f n o n t r i v i a l s o l u t i o n s o f ( 5 . 7 ) .

Proof : ( f o r s > 0 ) ; i n t h e non v e r t i c a l case, Lemma 5.4 i m p l i e s

t h a t (0 ,0 ,0) , wh ich i s obv ious l y a s o l u t i o n o f ( 5 . 7 ) , i s a l i m i t p o i n t

o f s o l u t i o n s (si,ai,wi) o f (5.7) si > 0, (6i,wi) # (0,O). The re fo re

t h e connected component o f s o l u t i o n s o f ( 5 .7 ) c o n t a i n i n g (0,O ,0) i s

n o t empty, an e v i d e n t f a c t i n t h e v e r t i c a l case. Then we aga in app ly

a s tandard r e s u l t o f Rab inowi tz [ 3 5 ]

Remark 5.4 :

Cond i t i on (9) i s s a t i s f i e d i f z > 0. Such a s i t u a t i o n occurs, f o r :I'

i ns tance , i f , i n t h e case o f r e g u l a r d a t a andco inc i dence se t , ko i s t h e p r i n c i p a l e i genvalue of t h e l i n e a r i z e d O i r i c h l e t p rob lem o f

( 1 .1 ) i n a\E [ 2 6 ]

6.A DISCUSSION CONC.ERNING THE CONDITION ( s )

L e t ( i , ~ ) be a s o l u t i o n o f ( 1 . 1 ) . We s e t

; ( A , u ) = i n f ia(w,w) - i ( F ' ( u ) w, w) / w G Su - Su ; l l w l = 11

I n S e c t i o n 4, we have proved t h a t t h e l o c a l behav iour o f t h e s o l u t i o n s

(A ,u ) near (h0,u0) i s g iven, up t o t h e second o rde r i n i - ho, by t h e c o n i c a l

d i r e c t i o n a t l e a s t a t p o i n t s (A ,u ) s a t i s f y i n g c o n d i t i o n ( S ) . ( S ) i m p l i e s

t h a t (k0,u0) i s r e g u l a r and i s e q u i v a l e n t t o ; ( ~ , u ) > 0. Dow

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On t h e o t h e r hand, i n S e c t i o n 5, we have c o n s i d e r e d a " l i n e a r " s i n -

g u l a r case i n ( A , u ) where o b v i o u s l y ; ( A , u ) 5 0. Our aim i n t h i s s e c t i o n

i s t o ana lyze c o n d i t i o n IS ) i n some ( r a t h e r g e n e r a l ) cases i n o r d e r t o see

when i t i s s a t i s f i e d , and when n o t .

F i r s t , by a s u i t a b l e n o r m a l i z a t i o n , i t i s easy t o see t h a t ; ( A , u ) > 0

i s e q u i v a l e n t t o v ( i , u ) > 0 ( s e e [ 1 3 ] ) where

v ( A , u ) = i n f i a(w,w) - i ( F 1 ( u ) w, w ) / w E SU - SU ; I w ( = 1 )

( t h a t i s , 1 1 1 can be r e p l a c e d by t h e L 2 ( s ) norm) .

Then , we n o t i c e t h a t i n t h e case o f an e q u a t i o n ( o r on t h e e q u a t i o n

branch o f ( ] . I ) , we have Su = H;(n) and c l e a r l y u ( i , u ) i s t h e f i r s t e i g e n -

v a l u e o f t h e l i n e a r i z e d D i r i c h l e t p rob lem o f ( 1 . 3 ) on a :

I Aw - A F 1 ( u ) w = v w i n a

( 6 . 1 ) w - 0 on a s

Thus t h e c l a s s i c a l r e s u l t o f C r a n d a l l and R a b i n o w i t z about t h e exchange

o f s t a b i l i t y [ 1 6 ] g i v e s i n f o r m a t i o n on t h e s i g n o f v ( h , u ) . ( s ) i s i n t h i s

case e x a c t l y a l i n e a r i z e d s t a b i l i t y c o n d i t i o n ( r e l a t i v e l y t o t h e a s s o c i a t e d

e v o l u t i o n p r o b l e m ) .

Consider now t h e case o f an o b s t a c l e p rob lem o f t h e f o r m ( 1 . I ) .

A t a t r a n s i t i o n p o i n t ( x , u ) , Su - Su = H;(s) , therefore v(a.,u) i s

t h e fundamental e i g e n v a l u e o f t h e 1 i n e a r i z e d prob lem ( 6 . 1 ) .

F i n a l l y , c o n s i d e r a p o i n t ( A , u ) o f t h e E . V . I . b ranch such t h a t Su

i s l i n e a r (we r e c a l l t h a t S ? c t i o n 3.3 g i v e s c o n d i t i o n s where t h i s s i t u a t i o n

o c c u r s ) . Then i f E = i u = ~ l c s i s r e g u l a r , v ( a , u ) i s t h e fundamental e i g e n -

v a l u e o f t h e l i n e a r i z e d D i r i c h l e t p rob lem :

( 6 . 2 )

We see

A w - A F ' ( u ) w = v w ~

w = O o n a n U a E

t h a t V ( A , U ) i s s t r o n g l y r

n s \ E

e l a t e d t o t h e fundamental e i g e n v a l u e

v1 o f t h e l i n e a r i z e d D i r i c h l e t p rob lem on t h e complementary o f t h e c o i n c i -

dence s e t ( i f r e g u l a r ) except p o s s i b l y a t t r a n s i t i o n p o i n t s , and ( s ) appears

t h e r e f o r e as a k i n d o f s t a b i l i t y c o n d i t i o n ( i n some sense wh ich needs t o

be made more p r e c i s e ) .

L e t us i l l u s t r a t e t h i s f a c t on two examples.

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CONRAD ET AL.

6.1. A n a l y t i c s tudy on a c l a s s o f one d imens iona l problems

L e t f : R + R be a non decreas ing f u n c t i o n wh ich s a t i s f i e s

f ( 0 ) > 0, f U ( t ) > m > 0 when t 0.

We choose A = - A and Y ( x ) z a> 0 f o r xE n = 3-1,1[ . Then

t he o b s t a c l e problem can be w r i t t e n as :

The assoc ia ted equa t i on i s

It i s a w e l l known f a c t t h a t t h e r e e x i s t s A* <- such t h a t

( 6 .4 ) admi ts s o l u t i o n s o n l y i f A 9 A* 117, 321. We cons ide r a

c l a s s i c a l case when A* i s a t u r n i n g p o i n t o f (6.4) and suppose

t h a t t h e t u r n i n g p o i n t i s un ique.

The genera l f e a t u r e o f t h e b i f u r c a t i o n diagram o f (6 .3) i s

g i v e n on F igu re 3 ( s i n c e u i s non i n c r e a s i n g on [ 0 ,1 ] , t h e c o i n -

c idence s e t o f u i s n e c e s s a r i l y o f t h e f o rm [ - r , r ] o r 0); t h e

coo rd i na tes a re : (A, u ( 0 ) ) f o r t h e equa t i on branch, ( h , r ) f o r

t he E.V.1, branch.

L e t (ha, ua) be t h e t r a n s i t i o n p o i n t .

I n t h e case o f dimension 1, we have an a n a l y t i c exp ress ion o f

t he diagram ( A , r ) [13 ] , namely r = 1 - WA when A > A,, f o r

t h e E.V. I . branch, and no bend ing occu rs on t h i s branch.

L e t u ,u) be t h e fundamental e i genva lue o f t h e 1 i n e a r i z e d

problem o u t s i d e t h e co inc i dence s e t E f o r A > h a ( f o r A = A ,, E = ( 0 1 i n t he case o f d imens ion 1 ) .

Then a s t r a i g h t f o r w a r d c a l c u l a t i o n [ 131 u s i n g s h i f t i n v a r i a n -

ce oT t h e Lap lace o p e r a t o r and r e - n o r m a l i z a t i o n shows t h a t

V ~ ( A , U ) > h/ha p2(",ua) where p2(Aa,ua) i s t h e second e i gen -

va lue o f t h e problem :

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EIGENVALUE PROBLEMS

F i g u r e 3

Since we a re i n t he case o f a unique bending, uZ(ha,ua) > 0

f o r any cons tan t a > 0 117,321 and vl > 0 on t he whole s o l u t i o n

branch i n c l u d i n g (ha,ua) .

On the o t h e r hand, o u t s i d e t he t r a n s i t i o n p o i n t , we have

vl(h,u) = v(A,u), b u t a t t he t r a n s i t i o n p o i n t , v(Aa.ua)< 0 i f

( l a ,ua ) i s n o t on t he min imal equa t i on branch.

For t he equa t i on branch, however, vl(ha,ua) = v(Aa,ua) c 0.

I t seems t h a t t he good s t a b i l i t y c o n d i t i o n has t o be expres-

sed by : v1(A,u) > 0, which co inc ides w i t h ($) o u t s i d e t h e t r a n s i -

t i o n p o i n t b u t i s weaker t han (S) a t t h i s p o i n t ( x ) . We expect

t he c o n d i t i o n vl>O t o be s u f f i c i e n t t o g e t l o c a l ex i s tence of an

E . V . I . branch.

When ua i s n o t t he min imal s o l u t i o n , (Xa,ua) i s a gene ra l i zed

t u r n i n g p o i n t w i t h exchange o f s t a b i l i t y s ince, on t he equa t i on

branch,

( x ) Reca l l t h a t E = I 0 1 has p o s i t i v e capac i t y , and vl i s t he f i r s t

e igenva lue o f (6.5) completed by t he c o n d i t i o n w(0) = 0.

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CONRAD ET AL.

v; = l i m vl(X,u) < 0 and on t he E . V . 1 , branch.

X + ha

v; = l i m vl(h,u) > 0 (see F igu re 4)

h + ha

Note t h a t t he jump of vl(A,u) a t ( Ia ,ua) i s ve ry r e l a t e d t o

t he d i s c o n t i n u i t y of t he co inc idence s e t a t t h i s p o i n t .

As a p a r t i a l conc lus ion o f t h i s subsect ion, f o r our c l a s s o f

problems, on t he E.V. I . branch, exc lud ing t he t r a n s i t i o n p o i n t ,

v = v 1 > 0 and no t u r n i n g p o i n t occurs : t h i s branch i s , i n some

sense, s tab le , f o r any va lue o f a > 0.

6 . 2 . A numer ica l s tudy i n h i ghe r dimension

We choose A = -a, f ( t ) = et ,Y = a > 0 on Q, u n i t b a l l o f R ~ ,

n 1. The obs tac le problem can be so lved by a shoot ing techn ique

(see [ 13 ] ) . We have c a l c u l a t e d t he f i r s t e igenva lue o f t he l i n e a -

r i z e d D i r i c h l e t problem ou ts i de t he co inc idence s e t . Since t he

r e g u l a r i t y assumptions o f Sec t i on 3.3 a re s a t i s f i e d o u t s i d e t he

t r a n s i t i o n p o i n t , (and even a t t h i s p o i n t f o r n > I ) , we have i n

t h i s case "1 = v '

F i gu re 4

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2 4 6 7 8 9 11 > +

10 '. .. -. .-.. .. .... '.. -. -. .. -.

.*. F I R S T E I G E N Y A L U E .-. .. *.. - 5 0 . , >o .-.. -

- * - ---. -. . --. -.. '.- --. -. -. -'I

B I F ' J R C A T I O N l i A G R A 3 F 3 9 THE E . V . I . BRANCH

( r 1 s t h e r a d i u s o f t h e c o i n c i d e n c e s e t )

2 4, 5 6. 7 8 9 10 11 D

Figu re 5

S l F U R C A T l O N D I A G R A Y F 0 9 T H E E . Y . ; . BRANCH

( r i s t h e radius o f t h e coincidence s e t )

I 1 1.5 >

0,. 5 C

Figu re 6

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CONRAD ET AL. 188

The p r e c i s e numer ica l r e s u l t s appear on F i g u r e s 5 and 6, and

can be schematized as f o l l o w s (same coo rd i na tes as i n F i g u r e 3 ) :

Other obs tac l es , s p e c i f i c a l l y Y (x ) = a + b1x12, a - b1x12,

a + b l x / 4, a + b x l , have been t e s t e d and l e a d t o t h e same con-

c l u s i o n s : c o n d i t i o n (S) : v (h ,u ) > 0 i s g e n e r a l l y connected t o a

s t a b i l i t y c o n d i t i o n on t h e i n e q u a t i o n branch and a t a t u r n i n g

( s i n g u l a r ) p o i n t we observe an exchange o f s t a b i l i t y .

ACKNOWLEDGEMENTS

We would l i k e t o thank P ro f . L u i s C a f f a r e l l i f o r s t i m u l a t i n g d i scuss ions . We are ve ry g r a t e f u l t o P r o f . Miche l P i e r r e who brought L . I . Hedberg's work t o our a t t e n t i o n . We are a l s o indeb- t e d t o t h e Referee f o r h i s va l uab le remarks.

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Received J u l y 1984

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