Higher order nonlinear elliptic equations

HIGHER ORDER NONLINEAR ELLIPTIC EQUATIONS UDC 517.956.25

I. V. Skrypnik

Methods of solution of boundary problems for divergent and nondivergent higher order nonlinear elliptic equations are described. Applications of topological methods to the study of general nonlinear boundary problems are given. Results on a priori estimates and properties of generalized solutions are cited.

PREFACE The theory of higher order quasilinear elliptic equations is one of the most actively developing directions of the theory of partial differential equations at the present time. The goal of the present paper is a survey of the results on the solvability of nonlinear ellitic boundary problems. The author aimed at describing the basic methods of proof of theorems on the existence of solutions of boundary problems for various classes of equations, from divergent quasilinear equations to essentially nonlinear equations of arbitrary order. The study of second order quasilinear elliptic equations has an almost age-old history and the basic directions of study, regularity of solutions, solvability of boundary problems, are specifically the nineteenth and twentieth problems of Hilbert. The research of S. N. Bernshtein, Leray, Schauder, Morrey, de Giorgi, O. Ao Ladyzhenskaya, N. N. Ural'tseva, and other authors led not only to the solution of Hilbert's problems, but also to the creation of many methods which play a fundamental role both in the theory of differential equations and in adjacent areas of mathematics. There is a survey of these studies in [37]. In solving problems of regularity of equations and systems of arbitrary order, basic results were obtained by Petrovskii [47] who singled out the class of systems now called Petrovskii elliptic, all of whose sufficiently smooth solutions are analytic. A broad range of research on quasilinear higher order elliptic equations has appeared since the beginning of the sixties. The first results were obtained by Vishik [11-13] who, by a modifiction of Galerkin's method proved the solvability of boundary problems. Subsequent progress is connected with the application of the theory of monotone and more general operators. The methods of the theory of monotone operators applied to nonlinear boundary problems are described in the first chapter of the survey. We note the solvability of problems with coercive or noncoercive operators, of boundary problems for equations with strongly growing coefficients, of higher order equations. The chapter contains a survey of the results of M. I. Vishik, Browder, Leray, Lions, Yu. A. Dubinskii, S. I. Pokhozhaev, Gossez, and other authors. Essential progress in the study of nonlinear boundary problems is connected with the creation of topological methods for mappings of monotone type. The foundations of these methods are laid in the papers of Browder, Petryshyn for powers of A-proper maps, of the author for powers [in different terminology, rotation (curl) of a field] of maps of class (S)+ of topological characteristics for essentially nonlinear boundary problems. The solvability of boundary problems for operators with linear noninvertible principal part is also considered in the second chapter. Results on the estimation of the number of solutions based on the method of spherical fibration and application of the theory of powers of maps are also cited in this same chapter. At the same time, due to the restricted size of the paper we are unable to include some questions which are closely related to the theme under discussion. In particular, this Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 37, pp. 3-87, 1990.

0090-4104/91/5604-2505512.50

9 1991 Plenum Publishing Corporation

2505

concerns results on regularity of solutions, ramification of solutions, and averaging nonlinear problems in perforated domains. The basic publications in this direction are included in the literature cited, which facilitates the reader's search for them. To a considerable degree, results on the regularity of generalized solutions are reflected in [58]. In conclusion we note that in citing the literature we sometimes use double references of the form [a:b]. This denotes the paper cited as number b in the literature cited at the end of paper a cited at the end of the survey. CHAPTER I SOLVABILITY OF BOUNDARY PROBLEMS FOR QUASILINEAR ELLIPTIC EQUATIONS OF DIVERGENT FORM In this chapter we give a survey of the basic results concerning the solvability of quasilinear variational and nonvariational problems for divergent elliptic equations. Here, as a rule, on the boundary of the domain Dirichlet conditions will be given although most of the results are valid for other boundary conditions, in particular, nonlinear Neumann conditions which arise n a t u r a l l y . The first studies of the solvability of boundary problems for quasilinear elliptic equations of order 2m were made by Vishik [11-12] on the basis of a modification of Galerkin's method. In proving the convergence of the Galerkin approximations strong Lp-convergence in subdomains of the m-th order derivatives which follows from the a priori estimates of the (m + l)-st derivatives obtained by Vishik and the compactness of the imbedding of Sobolev spaces was established. In the present survey the results on the solvability of boundary problems obtained by Vishik and following him by Browder, Yu. A. Duhinskii, Leray and Lions, S. I. Pokhozhaev, Petryshyn, and many other authors are described on the basis of the theory of monotone operators. Vishik's research, although based on the compactness of the Galerkin approximations, on the whole seemed fundamental in the study of quasilinear problems thanks to the development for such problems of Galerkin's method and thanks to the proof of the a priori estimates with which we shall be concerned later. The reduction of boundary problems for divergent elliptic equations to nonlinear operator equations with operators satisfying specific monotonicity conditions suggested by Browder and developed by a large number of authors is now the simplest and longest range route for studying these problems. The fact is that for equations with monotone operators in applying Galerkin's method it is not necessary to establish the strong convergence of the approximate solutions since it is proved that the limit of weakly convergent approximations is a solution of the original equation. In application to differential problems this removes the need for laboriously finding a priori estimates which are connected with additional restrictions. In this connection basic attention is given to the study of operator equations with operators of monotonic type, the creation of general methods of study of operator equations, and on this basis finding results for differential problems. i.

Formulation of Boundary Problems and Their Reduction to

Operator Equations i.I. Reduction of the Problem to an Operator Equation. Throughout the paper we use the following notation: ~ is a bounded open set in n-dimensional Euclidean space Rn with boundary 0~, x='~1 ..... xn)~Rn; ~=(al ..... ~n) is a multi-index with nonnegative integral components ~i, lal

= =1 + . . .

+ ~n

O~u(x)= (\Oxd ~ ""kOxnl ( ~ u(x), Dku={D~u:l~l=k}. For simplicity f u n c t i o n s w i l l be a s s u m e d t o be r e a l - v a l u e d , Lp(~) is the Banach space of functions which are summable to the p-th power, i < p < ~ over ~. Functions from Lp(~) are defined up to sets of measure zero. For a natural number m, W~(~) is the Sobolev space consisting of functions belonging to Lp(~) and having all generalized derivatives to order m inclusive, summable over ~ to the power p; the norm in W~(~) is defined by

II~ II,~,p=

2506

{!

~

l,,l<m

11

ID~u (x)lp dx ~.

0m

Wp(fl) is the subspace of the space W~(~) obtained by taking the closure in the norm ![Oilm,p of the set of all infinitely differentiable functions with supports in ~. In the presence of specific smoothness of 8~ (cf. imbeddings of the Sobolev spaces

Wpm(~)cW~(Q),

if

w~(~)cw$(~), wy(f~)cc,~,~(6),

i>

q

1

m--~>O,

p

q< ~,

~f

[39]) and w h e n 0 ~ . ~ < m - - I there are

n

]- = m--k ,

p

(~)

n

n-<m-Ck+~), oOO 1~[=m

(i0)

and (10) holds uniformly with respect to xG~, ~60, where G is an arbitrary bounded subset of

RM'. To conclude the point we comment on conditions AI)-A 4) for the function A~(x, ~). First of all, conditions Al) , A2) guarantee the existence for arbitrary functions u(x), of the integral on the left side of the integral identity (6). These conditions, in particular, guarantee that the map A~ defined for I~[ ->-m - n/p by

~(x)6W~(Q)

i = (u (x)) = A~ (x, u (x) . . . . . acts from W~(~) to Lq~(~) and acts continuously. continuity of a Nemytskii operator [34]. The inequality

(')

r=v

-~

X

S/=(~x'u..... mmu)m%dx, ~V.

(13)

I~l-.~mQ The following proposition give the connection of the problem of finding minimum points of the functional F and the boundary problem corresponding to V for the equation

(-- 1)laID=/=(x, ~ ..., Dmu)=O

(14)

l~l<m Proposition i.I. Let the functional F :X-+RII defined on the Banach space X have Gateau derivative F'(u 0) at the local minimum point F'(uo). Then F'(u 0) = 0. The problem of finding the minimum of integral functionals comes from the proposition asserted by Hilbert (twentieth problem [58:95]) on the solvability of each regular variational problem. Here Hilbert admitted that the need can arise of giving an extended interpretation of the concept of solution. Direct methods of the calculus of variations coming from the works of Lebesgue, Tonelli, Morrey [139], let one prove the existence of a minimum of regular functionals. They are based on the lower semicontinuity of integral functionals with respect to some weak convergence, weak compactness of bounded sets in spaces of functions with generalized derivatives. A broad development of direct methods is obtained in the papers of Vainberg and his students (cf. [9] for a survey of results and literature), Browder [58:159], the Bergers [58:145] and others where various propositions about the existence of a minimum of nonlinear functionals in Banach spaces, the convergence of minimizing sequences are established. One of the results about the solvability of variational problems is given below. To formulate a result about the existence of minimum points we need Definition 1.3. Let X be a Banach space, D be a subset of it. The functional F:D-+R is called weakly lower semicontinuous if for any sequence un~D, which converges weakly to one has u0~D, F (u0) < llm F (u.).

I

/2-+oo

THEOREM 1 . 1 [ 9 ] . L e t X be a r e f l e x i v e Banach s p a c e and F : X - + R 1 be a w e a k l y l o w e r s e m i continuous functional satisfying lira F ( u ) = + oo. ( 15 ) 11ulf-~oo

Then there exists a local minimal point of the functional F. It is simple to get (cf. [9]) the lower semicontinuity of a convex functional F. We indicate the connection of the properties of a functional with the properties of its gradient expressed in terms of monotonicity. C x.

2510

Proposition 1.2. Then :

Let D be an open convex set in X and

F:D-->R*

be a functional of class

a) in order that the functional F be convex it is necessary and sufficient that its gradient F' be a monotone map of D into X*; b) in order that the functional F be weakly lower semicontinuous it suffices that F' be a bounded and pseudomonotone map. We note the further fact of the strong convergence of a minimizing sequence of the functional~F:X-+R 1, i.e., a sequence u~X, such that F(u~)~in[{F(u):uEX}. This point is important in finding minimal points. Definition 1.4. The operator A:X + X* is called demicontinuous if for an arbitrary sequence un6X such that un-~u0, one has Aun-~AUo. LEMMA 1.3 [58:155]. Let X be a reflexive Banach space and F : X - + R I be a Gateau differentiable functional, F(u) + +~ as llull § ~ the gradient F' be demicontinuous and satisfy condition (S)+. Then any weakly convergent minimizing sequence converges strongly. The next theorem follows from the results of the present point and Lemmas 1.2 and 1.3. THEOREM 1.2. Let the functional F : V - + R l be defined by (12) and satisfy (15). Let us assume that f(x,O)~Li(~)and the functions f~(x, $) satisfy conditions AI)-A3), A88 Then the boundary problem for (14) corresponding to the subspace V has at least one generalized solution. If in addition condition A 4) holds for f~(x, $), then this generalized solution can be obtained as the strong limit of a minimizing sequence for the functional F. 1.3. Examples of Nonlinear Problems of Mechanics. We indicate here examples of specific nonlinear problems described by divergent equations satisfying the conditions formulated above. The first such problem is the problem of elasto-plastic bending of a rigidly fastened plate. The flexion of the plate w(x, y) satisfies the equation \ 02

2

02

O~e

02w

(16)

i0o= !0o=~

(17)

02w

1

and the boundary conditions

where

g being a function which is characteristic for the given material. The problem (16), (17) was posed by L. M. Kachanov [56:74] and was considered in [56:70] and [56:87] where under certain conditions the existence of a weak solution is proved. .

.

.

.

0

.

2

Under weak assumptions the exlstence and unlqueness of a generallzed solutlon !n Wp(~) p > 1 is proved in [56] as well as the convergence of Galerkin's method for the problem (16i, (17). It is known that F(H 2) = g(H2)H is an increasing function of H for H ~ 0 .

Let us assume

that: 1) g(t) is a continuous function for t > 0; 2) for t > 0 one has --P-I ao+c6t 2 ~0, a ~ > 0

~ Alt 2

,

and f o r p < 2, A o = 0. 0~

,

0

Problem (16), (17) will be considered in the space W~(~) under the assumption f~[W~(Q)]*. Analogously to point i.i we can reduce the problem to the operator equation Au= I, 0

where the operator

(18)

0

A:W~(f~)-+[W~(f~)]* is

defined by 2511

< au,~ > =

a

g(H2(u))

TZ q 2

Oy, ] ex" + exOy" OxOy + ( ~ v , - P T

One can verify that for the natural choice of the functions point i.i hold for them. Thus the operator A defined by (19) is satisfies condition (S)+. In addition, the operator A satisfies plays a fundamental role in the proof of the existence theorem. dition

lim

II#ll~

Ox= ]-~v~j dxdv"

A a conditions AI)-A 4) of continuous, bounded, and another condition which This is the coercivity con-

= + o o .

(20)

)!ult

As a second example we consider the system of strong flexion of thin membranes. First we note that the construction of point i.i carries over trivially to the case of systems of differential equations. We shall not do this just restricting ourselves to the example of a system describing a problem of nonlinear mechanics. The problem reduces to finding a solution of a boundary problem for the flexion w and the stress function F:

A 2 w _ L ( ~ , F ) = I q(x, 9),

D

(21)

A

2

A2F + L (~, w) =0, (x, y)6~cR 2,

E

wl = ~ IS

(22)

I =FI = oe I On Is [s On'Is = 0 '

where A is the Laplacian,

L(w,F)---- a~

Ox~

O'F + a,~ Oy2 Oy'

O2F --2 a~ O~F Ox2 axe)y" OxOy'

fl is a bounded domain in the plane with boundary S; h, E, and D are positive constants. We consider the Hilbert space

/-/= {u= (~, F): ~,

F

0

6W22(~)}

with scalar product

(u, X)-----l:i~==I~ {D=wD=~+ D=FD=~;} dxdy, where X=(r

4).

By a generalized solution of the problem (21), (22) we mean a vector-function

uEH, which for all x~H satisfies the equation D.AwA~p.-L(w,F)~+TAFAcp. Q

+L(w,w)~

dxdy= T

h

q(x,g)~p(x,y)dxdy. o

We define an operator A:H + H and an element Q 6 H

(Au, x)= S~

such that

(w, F)~ + E2. AFAr+ L(w, w ) , } d x d y , !

(Q, z)---'K- IS q~dxdy. Here i t is assumed that q(x, y)6Lt(~). Analogously to point i.i one can see that A is a bounded continuous operator satisfying condition (S)+.

It is easy to verify that for

Q

From this one gets that (20) holds for the operator considered. The system of strong flexion of thin membranes was considered in [17, 56], [56:39], [56:72], its solvability is guaranteed by the results of the next section. One can consider the complete system of equations of strong flexion of a membrane [56:52], the problem of equilibrium of sloping thin hulls [56:40], and the boundary problems for equations of yon Karman type [56:4] analogously. 2512

2.

Existence of Solutions of Operator Equations and Boundary Problems

2.1. Solvability of Coercive Operator Equations. A large number of papers starting with those of Browder [87], [58:161] are devoted to the study of the solvability of equations with operators satisfying a monotonicity condition or various generalizations of this condition. There are surveys of these results in [14, 58]. Here we cite one of the existence theorems. THEOREM 2.1. Let X be a reflexive, separable Banach space and A:X § X* be a bounded, demicontinuous, pseudomonotone operator satisfying the coercivity condition (201}. Then the equation Au = h has a solution for any h6X*. We note the basic points of the proof. Let {vi} be a complete countable system of the space X. We define Galerkin approximations u n = clv I + ... + CnV n as solutions of the equations

/g~(]~lol)~

In=l;-g,!(l,.noq)

in, l

(29)

m--~ ~ - - ~ , r ~ , r~ a r e determined, by the same c o n d i t i o n s as r~x, r 7 in c o n d i t i o n A~) but

with p replaced by p,

N0

={~=:;~l<m-~},~(x)~Z~i(fl), gx, g~being

positive, continuous, re-

spectively nonincreasing and nondecreasing functions.

, [wm-'(fl)] * by We d e f i n e an o p e r a t o r ~: Wm--~,~ ~ ~ )-~[ b<Bu,~)----

~

I B~(x,u ..... Dra-'u)D~q)dx"

(30)

We also assume that for the operator B the coercivity condition -I II u lira_,,;
= -l-

holds. THEOREM 2.5. Let the f u n c t i o n s Aa(x, $) s a t i s f y the c o n d i t i o n s A1)-A3), A~) the funct i o n s B~(x, q) s a t i s f y AI) and (27) and (28). Let us assume t h a t c o e r c i v i t y c o n d i t i o n s hold for the operators A and B. Then for any functions f~(x) belonging to Lp~(fl) for I~I = m and to L ~ ( ~ ) for l~l-.<m--lthe boundary problem for (26) corresponding to V = V N ~ 7 ~ - I ( ~ ) has at least one solution in V. The assertion of the theorem follows from Theorem 2.1 provided we prove the pseudomonotonicity of the operator A + B. We outline the verification of this property. Let u n be an arbitrary sequence of the space ~ which is weakly convergent to u 0 such that |i-m < AUn-~Bun, un--u0 ) ~ 0 .

Using (27) and (28), from this we have li'-m ( Au n,

Un--Uo>-. ... n, j < m - n / 2 . 2515

In what follows we assume the following conditions hold: i) for xG~, ~6~ My , the functions A=(x,~), I ~ I = m are continuous and have continuous first derivatives with respect to $; here M 3 is the number of different multi-indices of length at most j ; 2) for x ~ , ~]E~m'[ the functions B~(x,I]), I~].. 0 one has(D(20 ~O

,

We give imbedding theorems from Donaldson and Trudinger [98] for the case of sufficiently smooth bounded ~. Let c o be an arbitrary N-function and let us assume that n~2. By varying the value of c o on a bounded subset of R I, one can guarantee that I0(1) < +~, where t S

!

4(t)= c;'(~)~

_1__

"d~.

0

If I0(+=) = +~ we define a new N-function c I by c~l(t) = 10(t). Continuing this process we get a finite sequence of N-functions co, cl, .... Cq, whereq=q(co) 0 and sufficiently large t, ~(0 0, p(t) § +~ as t § +~ and we define t

~(t)= l p(*)d~. 0

We consider the Dirichlet problem for the equation

(-- |)I='D=p(D=u)= ~ ( _ l=l~m

I)'=IO~f=(x).

(40)

]alUm

On the basis of simple inequalities relating the functions r and ~ (cf. [35, Chap~ i, Sec. 2]), one can verify that for the problem (40), (7) conditions Al), A~), ~2), and (39) hold. So the next theorem follows from Theorem 3.3. THEOREM 3.4.

Under the conditions formulated above for the function p:RI-+R I and arbi0

trary functions Lr

f=(x)6E~(~), [~I-. 0 and any elements u, v6DNB ~ one has

~--c(R, Ilu--~ll'), where I1"11' is a norm which is compact compared with It'it. When c(R, t) ~ 0 an operator A satisfying Definition 1.3 is called monotone and we were concerned with them in the first chapter. The definition of semiboundedness of variation of an operator is due to Dubinskii [16].

2523

An example of an operator satisfying condition (S)+ can be a monotone operator A satisfying the following stronger condition than the monotonicity condition:

>>-c(llu-vll), where

c(r)>~O i s a c o n t i n u o u s n o n d e c r e a s i n g f u n c t i o n w h i c h i s e q u a l t o z e r o o n l y f o r r = O.

Numerous examples of operators satisfying the cited definitions are given by boundary problems for nonlinear elliptic operators. Similar examples were considered in the first chapter. We return to an operator A satisfying the condition ~0(SD) and for it we introduce the concept of degree. In addition we shall assume that this operator is demicontinuous. Definition 1.4. The operator A:D + X* is called demicontinuous if for an arbitrary sequence uo~D, which converges strongly to un~D, one has Aun~Auo. Let {vi} , i = l, 2,... be an arbitrary complete system of the space X and let us assume that for each n the elements vl,...,v n are linearly independent. We denote by F n the linear span of the elements vl,...,v n. In what follows D is an arbitrary bounded open set of the space X and we denote by 3D the boundary of the set D. We introduce the classes of maps for which the degree will be defined. Definition 1.5. For F c D we denote by Aa(D, F) [respectively A(D, F)] the set of bounded demicontinuous maps A:D + X* satisfying condition s0(F) [respectively, condition ~(F)]. If F = D, we shall write A0(D), A(D) instead of A0(D, D), A(D, D). We shall define Deg (A, D, 0), the degree of the map A of the set D with respect to the zero of the space X* under the following conditions:

a)

A~Ao(D, OD);

b) f o r an a r b i t r a r y F o r e a c h n = 1, 2 , . . . follows :

element

u~OD Au=/=O.

we d e f i n e

finite-dimensional

approximations

An o f t h e map A as

A n u = s ~& with

some p o s i t i v e We i n t r o d u c e

number 5.

for

u~OD

(5)

H e r e tl'll i s t h e norm i n X*.

t h e map EA0u + Au + Tu f o r

f o r a map EA 0 + A + T s a t i s f y i n g degree

all

~619.

the hypotheses

Let

ll4=supllAotlll.. Then f o r 0 < s < ~/M

of Theorem 1.3,

there

is defined

the

Deg(eAoq-Aq-T,/5, 0). One c a n show t h a t f o r 0 < ~ < 5/M t h e d e g r e e s the following limit exists: lim

defined

(6) by ( 6 ) do n o t d e p e n d on E and h e n c e

Deg(sAo+ A-}-T, D, 0}.

8-+0

One proves that this limit is independent of the choice of the map A 0. introduce the following concept:

Thus one can

For a demicontinuous operator A with semibounded variation and a completely continuous operator T by the degree of the map A + T of the set D with respect to the point 00X* we mean the following number:

DEQ(A-~T,

D, O)=limD~g(sAo+ Aq-T, D, 0), 840

where

AofiA(D, OD),and Deg i s t h e d e g r e e i n t r o d u c e d

by D e f i n i t i o n

1.7. 2525

One can also define the degree of pseudomonotone maps analogously. 1.3. Properties of the Degree of Generalized Monotone M@ps. The degre of a map introduced above in various cases has all the natural properties of the degree of finite-dimensional maps. We show this on the example of the degree defined in point i.i. In the present point X is a separable reflexive Banach space, D is an arbitrary bounded open set in X. Let ~,I]={t0R~:0~t~.~I}, At:D-','X*, tE [0, I] be a parametric family of nonlinear maps. Definition 1.8. sequences

We say that the family A t satisfies condition

u.EOD, t~s

aoC~

if for arbitrary

1] froml u=-~.uo, Atn(u~)--*O

and

1T~ < At.(u.)u.--uo > < 0 the strong convergence of u n to u 0 follows. Definition 1.9. Let A', A":D § X* be maps of class A0(D, 3D) and let A'u ~ 0, A"u ; 0 for uOaD. We call the maps A' and A" homotopic on D if there exists a parametric family of maps At:D.-+X*, t~, I], satisfying condition ao(~ such that:

a) Atu ~ 0 f o r ueOD, t0[0, 1];

Ao=A', A,=A";

b) f o r any s e q u e n c e s t n, u n s u c h t h a t /.G[0, 1],U.fib, t.-+to, un-+Uo, t h e s e q u e n c e AtnUn converges weakly to At0u 0THEOREM 1.4.a. Let A r : D _ + X *, A~:D-+X * be maps of class A0(D, 8D). Let us assume that A'u r 0, A"u ~ 0 for u00D and the maps A' and A" are homotopic on D. Then

Deg(A', s

0)-----Deg(A u,/9, 0).

(7)

The degree of a map defined in point i.i is, under certain conditions, the unique homotopy invariant. The theorem cited below generalizes the classical Hopf theorem for finite-dimensional maps. In the case of maps which are the sums of an identity and completely continuous operators the corresponding theorem was proved by Krasnosel'skii [34]. THEOREM 1.5. Let D be a convex bounded open set in the space X and Ao:D-+X *, A,: ~)-+X" be maps of class A(D, 3D) such that A0u ~ 0, Azu ~ 0 for u~aD and Deg(A0, D, 0) = D e K ( A I, D, 0). Let us assume that the spaces X and X* are uniformly convex. Then the maps A 0 and A~ are homotopic on D. Remark i.i. The assumption of uniform convexity of the spaces X and X* in the formulation of Theorem 1.5 can be replaced by the condition that there exists a demicontinuous operator A0:D + X* satisfying condition ~(SD) and such that > 0 for u ; 0. Numerous applications of the theory of the degree of maps to the study of the solvability of nonlinear operator equations and nonlinear boundary problems are based on applications of Theorem 1.4 and the following assertion. LEMMA i.I. Let A:D + X* be a map of class A0(D) and let Au ~ 0 for u~OD. In order that the equation

Au=O

(8) have a solution in D it suffices that Deg (A, D, 0) ~ 0. We cite two frequently used tests for the difference of the degree of a map from zero; other similar tests are connected with the calculation of the index of a critical point with which we shall be concerned in the next point. THEOREM 1.6. Let A:D + X* be a map of class A0(D, 8D). that for uOaD one has

Let us assume that

OOD\aD and

~O.

(9)

Then Deg(A, D, 0) = 1. THEOREM 1 . 7 . L e t B g = {u~X : [[u]]~R} and A : BR(O)--+X* be a map of class A(B R, 8BR). us assume t h a t Au ~ 0 f o r u6OBR and 2526

Let

Au A(--u) for u~0BR. il Au % =/=HA (--u) II,

(i0)

Then Deg (A, BR, 0) i s an odd number. 1.4. Calculation a separable, reflexive Definition 1.5. Definition i.i0.

of the Index of a Critical Point. L e t D be a b o u n d e d open s e t i n Banach s p a c e X, A:D + X* be a map o f t h e c l a s s A0(D) i n t r o d u c e d i n The point uo~D

is called a critical point of the map A if Au 0 = 0.

The term "critical point," which is natural in the case of a potential operator A, which is the gradient of a functional, is applied in this paper for general nonpotential maps also. Let u 0 be an isolated critical point of the map A, i.e., there exists a ball B~~ {u~X: which contains no other critical points than u 0 of the map A. One can show that for 0 < r < r 0 one has

Ilu-uoH>0

for u=~0

(13)

and the operator L = (A' + F)-zF:X + X is defined and completely continuous; 3) for sufficiently small e > 0 the weak closure of the set

{

~

z~= v=~:Au=---

,,Aui,, T Tu:If, --A'u, i[ A

o 0 and positive constant c one has

RJ Definition 1.13. The functional ,9": X--~R l is called increasin~ if for any C~Rl the set {u : u0X, ~'(u)~.c} i is bounded in X. THEOREM 1.12 [27]. Let ~ ' : X - + R 1 be a continuous functional which at each point u~X has Gateau derivative ~"(u). Let us assume that ~ is a nondegenerate increasing functional and the map ~ " belongs to A(B R) for each R. Then

lira Deg(9 r', B~, 0 ) ~ 1 . We note another generalization of Theorem i.ii obtained in [7] by Bobylev which relates to the case when the functional realizes a local minimum on a finite-dimensional manifold. Let H be a real, separable Hilbert space, D be a bounded open set in H, and ~ : D - + R I be a functional of class C I. THEOREM 1.13. Let us assume that the set of critical points of the map 8r' contained in is a finite-dimensional compact connected smooth manifold M without boundary. If~" belongs to the class A(D), M realizes a local minimum of the functional ~', and~4NdD-----~," then Deg(~",\D,O) =X(M),~ where x(M) is the Euler-Poincar6 characteristic of the manifold M. 1.6. Uniqueness of the Degree of Maps of Monotone Type. In recent papers of Browder [89-91] conditions defining the degree of a map uniquely are discussed. We restrict ourselves to consideration of maps of class (S)+ in the case of uniformly convex spaces X and X*. Let D be a bounded open set of the space X and we consider the family of maps A(D) defined in point i.i. A dual operator J which makes correspond to an element u~X a functional luOX*,satisfying the conditions

2528

= Itull 2, IIYuII.= Ilull. belongs to this family for uniformly convex spaces X and X*. Let us assume that for arbitraryA~A(D), open subsets G c D andhEX*, h~A~OO) there is defined an integer-valued function d(A, G, h) satisfying the following conditions: I) normalization: if d(A, G, h) ~ 0 then where J is a dual operator;

h6A(G);

for each h ~ 7 ( G ) \ ( d ~

d(f,G,h)=l,

2) additivity with respect to the domain: let Gl, G 2 be disjoint open subsets G c and let hCA(G\(GIUG=), then d(A, G, h) = d(A, G~, h) + d(A, G 2, h); 3) invariance with respect to homotopy: if A t , ht,t~,l] are, respectively, parametric families of maps from A(D)_and elements of X* such that the map (A t - ht)(u) = Atu - h t is a homotopy on G c D in the sense of Definition 1.9, then d(A t, G, h t) has constant value for t~, I]. THEOREM 1.14.

If the integer=valued function d(A, G, h) defined for A@A([)), OCl), 1)-3), then d(A, G, h) D e g ( A - h, G, 0), where D e g ( A - h, G, 0) is the degree of the map (A - h)u = Au - h introduced in point 1.2.

h6X*\A(OG),satisfies

Remark 1.3. The assumption of the invariance of the function d(A, H, h) with respect to homotopies in the sense of definition 1.9 can be relaxed, replacing it by invariance with respect only to linear homotopies A t = tA 0 + (i - t)A I. 2.

Application of the Theory of the Degree of a Map to the Proof of

the Solvability of Nonlinear Equations 2.1. Solvability of General Operator Equationsm The application to operator equations of topological methods based on the concept of degree of a map let one simply generalize the results of the first chapter on the solvability of equations with coercive or odd operators and get new existence theorems. The topological approach lets one, in particular, include operator equations in a parametric family of equations of the same form and reduce the study of solvability to establishing the difference from zero of the degree of simpler maps and getting a priori estimates. First we note a simple corollary of Theorem 1.4 and Lemma i.I. THEOREM 2.1. Let X be a reflexive, separable Banach space, D be a bounded domain in X with boundary 8D, A:D • [0, i] + X* be a bounded demicontinuous operator and let us a s s ~ e that: i) the family of operators A t = A(., t) satisfies condition A I satisfies condition ~0(D); 2) A(u, t) ~ 0 for

uE#D, t~,

ao~~

and the operator

i];

3) Deg(A0, D, 0) ~ 0. Then the equation A1u = 0 has at least one solution in D. The next theorem follows directly from Theorems 2.1, 1.6, and 1.7. THEOREM 2.2. Let X be a reflexive, separable Banach space, t6[0, |]~ As: %-+%* be a family of bounded demicontinuous operators satisfying condition ~(X) and such that for an arbitrary bounded set BcXAt~u) depends continuously on t, uniformly with respect to u~B. Let us assume that condition i) and one of the two conditions 2), 3) hold: i) for any h6X* there exists an R = R(h) such that from

Atu-~h,t~[O, I]

it follows that

2) the operator A 0 is coercive; 3) the operator A 0 is odd. Then the equation A1u = h is solvable for arbitrary

h~X*.

We note another corollary of Theorem 2.1 which can be useful in proving the solvability of boundary problems. Definition 2.1. The operator A:X + X* is called asymptotically homogeneous if it can be represented in the form A = A 0 + Al, where A 0 is a positively homogeneous operator of order 2529

k, i . e . ,

A0(tu) = TkAou f o r t > 0 and lira

r[A,ul[,ll[lull~k=o.

Definition 2.2. An asymptotically homogeneous operator A is called regular at infinity if k > 0 and the equation A0u = 0 has only the zero solution. THEOREM 2.3. Let A:X § X* be an asymptotically homogeneous operator which is regular at infinity. Let us assume that A and A 0 are operators of class A(X) and that the index of zero of the map A 0 is nonzero. Then the equation Au = h is solvable for any h~X*. We note one of the consequences relating to the solvability of equations with odd homogeneous operators. COROLLARY 2.1. Let A:X § X* and B:X + X* be an odd completely Then for arbitrary %6R ], for which equation Au + %Bu = h is solvable

be an odd positive operator of order k > 0 of class A(X) continuous positively homogeneous operator of order k > 0. the equation Au + XBu = 0 has only the zero solution, the for any A6X*

The first such result was obtained by Pokhazhaev [48]. Subsequently similar results were established by Browder, Petryshyn, Necas, the author, etc. (cf. [166]). Remark 2.1. In applying the degree of pseudomonotone maps, the condition that maps belong to the class ~(X) can naturally be replaced by the condition of pseudomonotonicity. We show that the concept of degree permits one to prove the invariance of domain for the maps considered. The first such results for operators of the form "identity plus completely continuous" were obtained by Schauder [58:271]. For locally A-proper maps having a special local A-proper homotopy these questions were studied by Petryshyn [58:257]. Below we give an improvement of a theorem of Petryshyn obtained in [58]. Definition 2.3. We say an operator A defined on an open set D of a Banach space X satisfies condition ~) locally and is locally one-to-one if for each point u0~D there exists a ball B % (u0)~---{u:[]U--uoll 0; laJ=m 4) f o r XC~, ~lGRM', ~CRM~ one h a s I~l~m

l~[=m

[Y(K,

(19)

, m,p

then for solutions of the equation Atu = tf one has the a priori estimate

llullm,p < K.

Existence theorems under conditions of the form (18) were obtained by many authors Browder, Fitzpatrick, Petryshyn, etc. Cf. [155] about these results with the corresponding bibliographical references. Remark 2.3. The assertions of points 2.2 and 2.3 can be proved under relaxed hypotheses on the coefficients As(x , ~), namely replacing condition A 4) by condition A~) of Chap. 1 which is connected with the application of the theory of degrees of pseudomonotone maps. 2.4. Convergence of Galerkin Approximations. Here we show that the theory of degrees of generalized monotone maps lets one establish the strong convergence of the Galerkin approximations for nonlinear problems. We start with theorems of convergence for Galerkin approximations of an operator equation. Let X be separable, reflexive real Banach space and let vi, i = i, 2,... be a system of elements in X such that UF,~=X, where F n is the n-dimensional linear span of the elements v~,...,v n and the dash denotes closure. By an approximate solution of the equation Au = 0 we shall mean an element

u,~F~

such

that

----0, i = 1 . . . . . n.

(20)

THEOREM 2 . 9 . L e t D be a b o u n d e d domain o f t h e s p a c e X and A:D + X* be a b o u n d e d d e m i continuous operator satisfying condition a0(D). L e t u s a s s u m e t h a t i n t h e domain D t h e map A has a unique critical point, Au ~ 0 for Deg (A, D, 0) ~ 0. T h e n approximate solutions u n of the equation Au = 0 exist for n larger than some n o and as n + ~ the approximate solutions u n converge strongly to a solution of the equation Au = 0. Remark 2.4. Convergence of Galerkin's method for an equation of the form u + Fu = 0, where F is a completely continuous operator in a Banach space, is obtained in Krasnosel'skii [34]. R. I. Kachurovskii and M. M. Vainberg were concerned with the justification of Galerkin's method for equations with monotone operators. In [166:77] equations with uniformly monotone operators were considered. The results contained in this paper follow directly from Theorem 2.9. We note that many papers of Petryshyn are devoted to questions of convergence of Galerkin approximations (cf., in particular [158]). These same questions are discussed in Landes [1281. As an example we cite the application of Theorem 2.9 to establishing the convergence of the Galerkin approximations for (14). THEOREM 2.10. 1 and for ~, ~ ' ~ M

2532

Let the functions

A=(x, ~), lal~m

satisfy conditions Al), Ai), A~) of Chap.

~ [A=(x,~)--A=(x,

~ ' ) ] ( ~ = - - ~ ) > O.

Let us assume that f=(x)~L~=(~) and (18) holds. Then the boundary problem for (14)corresponding to the subspace V has strongly convergent Galerkin approximations Un(X) defined according

to (20). .

Solvability of Weakly Nonlinear Boundary Problem~

In this section we shall consider the Dirichlet problem for a divergent elliptic equation although on the whole the results also hold for nondivergent equations and genera] linear boundary conditions subject to the condition of Ya. B. Lopatinskii. We shall consider the solvability of the Dirichlet boundary problem J

(--l)'=ID~[a=~(x)D~u]-r - ~ (--1)'='D~b~(x, u. . . . . D~u)i= ~(--1)I=!Daf~(x), r I,ID]<m

J=l 0 one has

a=~(x)~=~>~,j~? '' fo,:

l~l=l131=,n

b z) the functions b=(x, to. . . . . ~m), [ ~ [ ~ p and the estimates

xe~,

~R";

for xEf~, ~ R N(~I satisfy Caratheodory conditions m

j=0 with positive constant ~, (1610,I), f~(x)EL~(~). 0zn~

We shall establish the solvability of the problem (21), (22) for f=(x)6L2(Q) in ~ 2 (Q), by reducing it to the solvability of the operator equation

Lou + Nou --~f (x),

( 24 )

where the operators Lo, No:l~"~"(n)-+~7~n(fl ), the functionsfeW~(Q) the conditions

are defined, respectively,

(Lou, (p)~= X I a='~(x)D~uD=q~dx' f"l,l 31~K~llu]]

2.1.

Under t h e h y p o t h e s e s (29)

with K E > 0. Introducing the homotopy A t = L 0 + EC + tN we get the solvability for any E ~ 0 of the equation Lou+sCu+tNu =f. Passing to the limit as ~ § 0 using condition 2), we arrive at the solvability of (28). In [153] sufficient conditions are given for assumption 2) of the preceding theorem to hold. THEOREM 3.3. Let L:H ~ H be a linear map of class (S)+ of index zero and let the demicontinuous map N:H + H be such that L + N is pseudomonotone. In order that the set Z + be bounded under the hypotheses of Theorem 3.2 it is sufficient that one of the following three conditions hold: I) Nu = o(lluii) as L[u[l + ~ and for an arbitrary sequence uk~H such that [luhl]-+oo, I]Uk]l--lUk-+ vEKerL one h a s l i m ( N ~ , M y ) > ( / , 2dr); (30)

2534

2) one has

llNull 0 for ~Ker(L0), 0

able in wm(~) for arbitrary functions

ll~[Im,2=|, then the problem (21), (22) is solv-

f=(x)GLf(f~);

3) if o = 0 and H(w) > (f, w) m for ~ K e r L 0, [l~OlIm,2-----l, then the problem (21), (22) has a 0

solution in wm(~). Like Theorem 3.4, Theorem 3.5 follows from Theorem 3.2. Here, in each of the cases considered one verifies that condition i) of Theorem 3.3 holds. 3.5. Existence of Multiple Solutions of Weakly Nonlinear Boundary Problems-- Here we give one of the results of Ambrosetti and Mancini [79] which gives the existence of at least two solutions of the equation I~l,l~]],

~
1 for m = n/2, p = 2n/(n + 2m) for m < n/2. THEOREM 3.8 [164]. trary

Let conditions al)-a~), h') hold and let us assume that for an arbiThen the problem (34), (22) has a solution u(x)

functionw(x)~KerLo, w=/=0 (35) holds.

~m(~). F o l l o w i n g [135, 156] we g i v e a r e s u l t for the equation

Z

about the solvability

of t h e D i r i c h l e t

( -1)I=ID~[a~(x)Dpu]=g(x' u . . . . . Dmu)u+f(x).

problem (48)

lal,iPl.~m

We preserve conditions al)-a~) and we denote by Xj the eigenvalues of the operator L 0 numbered in increasing order. Let ~(x), ~(x) be measurable functions such that for some k

mes {xO~ : tz(x) >~,h} > 0 ,

(49)

mes{xOfl : [~(x) 0 . THEOREM 3.9. Let conditions al)-aa) hold, the function theodory condition for x6~j@l~NCJ~, ]=0, ..., m and

g(X,~o . . . . . ~m)

satisfy a Cara-

,~(x)~g(x, ~o. . . . . ~ ) ~ ( x ) with functions ~(x), ~(x) satisfying (49). 2538

Then the problem (48), (22) has a solution ~m(~).

The proof is based (cf. [156]) on a linear homotopy of Eq. (48) to the equation obtained from (48) by replacing g(x, u ..... Dmu) by g(x, 0 ..... 0). We note that in [156] Theorem 3.9 is obtained for larger classes of equations obtained, in particular, by replacing g(x, u ..... Dmu), f(x), respectively, in (48) by g(x, u, .... D2m-~u), f(x, u ..... D=mu). 4.

Topological Characteristics of General Nonlinear El_liptic Problems

4.1. Reduction of General Nonlinear Elliptic Problems to Operator Equations. Let ~ be a bounded domain inRn with infinitely differentiable boundary 8~, n o = [n/2] + i, m, ml, .... m m be nonnegative integers, m > i~ We denote by M(q) the number of different multi-indices = (~i ..... ~n) with nonnegative integral coordinates ~i of length ]~[=~l-~...q-~., at most q. Let s be an integer no less than max(2m, m I + 1/2,...,m m + 1/2) and let us assmne that there are defined functions

/~':~XRM(2"~

~.i:~XRM(m:)-+RL

7---1 . . . . . m,

having continuous derivatives with respect to all arguments, respectively, to orders s 2m + i, s - m j + i, where ~ is an integer satisfying l>~i0q-n0. We shall also represent the functions ~-(x,~), Gj(X,N), ~={~=: l=i~.2m}, N={N~: l}l!~mj} in the form

~qx, ~)=~r(x, ~0, .... ~=~), Oj(x, n)=Oi(x, no..... %@ w h e r e ~k={~= : {=[ =k}, ~tk= {r[= :

t~z[

=k}.

We shall use the previously introduced notation D~u, Dku and let

_~__ OGI(x, ~i) o~(.,~) o~ ' G ,~(x,u~-~.

~r.(x,g)= In the present

section

we s h a l l

reduce the boundary problem

,..~-(X, U. . . . .

D2mu)=[(X),

O~(x, u, . . . . D ~ u)=g~(x), t o an o p e r a t o r

(50)

j=l

(51)

x6~d,

. . . . . m, xEOQ

(52)

equation.

L e t u s assume t h e f o l l o w i n g

conditions

hold:

I) for an arbitrary function v(x)6Ht(~) = W ~t( Q) t h e o p e r a t o r I

I

u(~):m(~)~m(a, oa) =H,-~Ca)x~-~'-r(oa)x... XH

m

(oa),

defined by

(53)

U(v)u=(L(v)u, & (v)u . . . . . B.,(v)u), where

L(v)u(x)=

~.d ~=(x, v . . . . , D~mv)D=u(x), xfiQ, I=]~2m

Bj(v) u ( x ) =

~

O1,~(x, v , . . . , D'~Jv)Dl~u(x), xeO~,

t={<mj

is elliptic

and h a s i n d e x z e r o ;

2) t h e r e e x i s t s a f u n c t i o n H:~XRM(=m+I)-+R 1 o f c l a s s function vfHt(fi) the problem

L(v)u+M(v)u=O, B~(v)u=0, has only the zero solution in Hs

] = 1. . . . .

C t-2=§

x~-fa, m,

such that for an arbitrary

(54)

x6afa

Here

H~ (x, v . . . . . D 2'~-I'o) OVtt, IVl. = ( F ( x ,

u. . . . .

D ~ u ) - - f ( x ) , L(tOq~+M(u){p)~_~m,n +

m

+ X~=~(O~(x,u. . . . . D m;tt)--g~(x), Bs(tz)~)~_m~_~_ o~, w h e r e (., .)~, n,

(',')~, on a r e ,

respectively,

the scalar

products

(55)

in the spaces

One can show that the right side of (55) is a continuous linear functional on the space Hs which makes the definition of the operator A~ proper. The proof of the boundedness of the functional so arising is based on the Nirenberg-Galliardo inequalities. The operator A~ introduced lets one reduce the study of the solvability of the nonlinear boundary problem (51), (52) to the solvability of the nonlinear operator equation A~u = 0. THEOREM 4.1. In order that the function u(x)~HZ(f~)'~ be a solution of the problem (51), (52) it is necessary and sufficient that Amu = 0. Questions of the solvability of a nonlinear problem (51), (52) will be studied on the basis of the degree of the map A m which we can define by virtue of the following theorem. THEOREM 4.2. If conditions i) and 2) hold and l>~10~-n~, the operator A~ defined by (55) is continuous, bounded, and satisfies condition =(Ht(~)). The a priori estimates of linear elliptic problems [75] are key in the verification of condition ~. If now D is an arbitrary bounded domain in the space H~(~) such that Alu ~ 0 for u~OD then Deg(Al, D, 0) defined in Sec. i, is the topological characteristic of the problem (5i), (52) sought. 2540

The operator AI introduced by (55) can be defined for sufficiently high smoothness of the functions ~r(X, ~), Oj(x,~j). One can relax the assumption about the differentiability of these functions by introducing into consideration corresponding operators in " V/pt,+0(~) for p>n. Let S be a bounded domain in R n with boundary 8~ of class Cl,+ 2 and we define a finite number of open sets U l ..... U I covering ~ and diffeomorphisms 9~:UI-+R n of class C t0+=,for which

cpz(U~)= B 1= {g~R" :[ Y I < 1}, i f U~ c ~ , and ~Pt (Ul)A ~)-----B1+~-- {vERn:I 9' [ < 1, 9n > 0}, ~ (U~ N 0 ~ ) = B(--{V~ Rn:]y,[ 0 or m > 1 (57) may not hold. It follows from the examples given in [166:65] that there exist fourth order linear operators L and M with real coefficients for which (57) is invalid for s = 0, there exist second order operators L and M for which (57) does not hold for s = i. It is also shown in this same paper that (57) may not hold for m = i, s = 0 for operators L and M with complex-valued coefficients. In connection with the counterexamples indicated the following problem is solved in [61]: construct for a given family of elliptic linear operators {L} a linear operator M with special scalar product [., .]s in W2s such that an inequality of the form (57) holds with (Lu, Mu)s replaced by [Lu, Mu]z and such that the norm generated by [', "]s is equivalent with II"IIs These results are given in the present point. These results are recounted in detail in [166]. Subsequently in this point ~ is a bounded domain in the Euclidean space R ~ with boundary 3~ of class C ~ although it is obvious that the infinite differentiability of the boundary can be replaced by a specific smoothness condition. The positive number A is called an ellipticity constant of the linear operator

L(X, D)----

if for

X6~, ~=(~t . . . . .

X a=(x)D=, I= lAl~T ~". lal=2~ Here as(x) the coefficients of the operator L(x, D), are real-valued functions, ~ = = ~

... ~=~.

For a nonnegative integer s and 0 < I < 1 we denote by Sfl,X am t ,A , B, f~) a family of regularly elliptic linear operators L(x, D) of order 2m with one ellipticity constant A, and coefficients as(x) of class CI,X(~) satisfying the conditions

IIa,, (x) Ilc~,~(~) --< B,

I c~l -< 2ra,

where ll'llc,.~(Q)denotes the norm in the space Ct'~(~) 9 THEOREM 4.4.

Let A and B be arbitrary positive numbers, 0 < ~ < i.

operator 7%4(x,m)----- X

be(x) m = w i t h

lal c l

(SS)

II ~ll~-c2llul[~

fl

0m

for any function

u(x)~W~m(f~)nW2 (f~).

Definition 4.1. We call a linear operator M(x, D) satisfying the conditions of Theorem 4.4, directing for the family ~0,zla2m ~ , B, fl). We proceed to a coercive estimate for pairs of elliptic linear operators for s > 0. For zA , B,~) we shall indicate a special choice of the scalar product in W~(~) for a family ~r 2 ~, ~,~ which a coercive estimate holds for an arbitrary operator L(x, D)6 2~ ( , B, Q) and directing operator M(x, D) defined by Theorem 4.4. Examples show the necessity of special choice of the scalar product. THEOREM 4.5. For arbitrary positive numbers A and B, nonnegative integer s and ~6(0, |) there are infinitely differentiable real-valued functions C~8(x), l~I, l~I~ Kx II~ IlL+z-- K2 IIu 15

(59)

Ka It~' II~~< [v, vl, < K= [Iv lift,

(6o)

where the operator M(x, D) is defined by Theorem 4.4, the scalar product [v x, v2] ~ for v~(x), ~=(X)6V/~(~) is defined as follows:

[v~, v2],= 4.3. Reduction of Operator Equation. Now differential problem in strict ourselves to the

X

i C~p(x)D=vt(x)D~v2(x)dx"

(61)

the Dirichlet Problem for a General Nonlinear E l l ~ u a t i o n to an we give a simple construction of a nonlinear map corresponding to the the case of a Dirichlet boundary condition. For simplicity we recase of a problem with homogeneous boundary data.

Let ~ be a bounded domain in R ,~ with infinitely differentiable boundary 8Q and let us assume that the function fr : ~ X R ~ 2 m u + R ~ has continuous derivatives with respect to all its arguments to order s + i, s > n o . Here n o , M(2m) mean the same thing as in point 4.1. We shall reduce the boundary problem

Or(x, u . . . . .

Dmu)=f(x),

x~g2, (62)

D~'u(x)=O,

[~[~m--1,

x00~,

to an o p e r a t o r e q u a t i o n under t h e assumption t h a t f(x)~HZ(~) and t h e f u n c t i o n or(x, ~) s a t i s fies the condition. A) t h e r e e x i s t s a p o s i t i v e c o n s t a n t A such t h a t f o r a r b i t r a r y

~ R ~r

2~ ~r~(x, ~ ) n ~ > A l ~ j ~

~ R n one has

(63)

]al=2ra

where Sz'~(x, ~) is d e f i n e d by (50). o

Let D be an arbitrary bounded domain in the space:X=H~m+z(Q)NHm(f~). We assume the norm in X coincides with the norm in H2m+z(f~) and let B R be a ball in X with center at zero containing D. We consider a family of uniformly elliptic operators

2~-----{Lv=i=, n 2543

For the constants A and B, the numbers m and s ing operator

AJ(x, D ) =

Z

and the domain ~ one can define a direct-

ba(x) D=

(64)

["[t~0=[ on--" L~A conditions hold:

Let us assume that the following

I) there exists a %fi(0,I) and a positive constant K such that for t~[0,I] from 0

F (t, X, v . . . . .

it follows that

D ~ m ~ ) = 0, ~ 6 X = H m ( e ) ~

H 2ra+t(e),

( 74 )

HvHc2m,~,(n).~ I we denote by V/~'n't't~(f~n)the closure of the set of infinitely differentiable functions on ~hwith respect to the norm I

2545

where p

~ h)

l~l~2m

I~l~t

Here E' denotes summation over all those multi-indices whose last coordinate is equal to zero. The space cs,r,k(~h) is defined for nonnegative integers s and r and for % belonging to the segment [0, i] as the set of functions with finite norm

where

and If" II~ '~ is the usual norm in the space of functions satisfying a HSlder condition with exponent I. We note an auxiliary estimate for functions belonging to the spaces just defined: 0

I

For an arbitrary function U(X)@~'~m't'O(~h)NW$(~h)

for I > n n u|,

O =

S ~2~(Y){O~ [t$'~(y, u~. . . . . DmuO+ Sh

+ (,1 -- t) L , u , - tf~ (y)] D~Mvi + p t=t,~t THEOREM 4.9.

The operators At,

D ~ [t~'~ (y, ut . . . . . DWuO +(1 -- t) L#~ -6 tf~ (y)] D=Mv~} dy. tE[0,1],satisfy condition a(00 of Sec. i.

The applicability of Theorem 2.1 to the proof of solvability of the operator equation I

A1u = 0 follows from Theorem 4.9. Choosing h so thatA'h 2 (N+|)~ m - p ,

n

l~[<m----~,

l~lq-1~l 0

Deg{~, BR, 0)~I

by

~

Theorem 1.12, BR = { u Q W o ( ):ll~llm,0~/~}. iIf the local minimum points u I and u 2 are nonisolated critical points of the map ~', then the assertion being proved is valid. If the local minimal points ul, u 2 are isolated critical points then by Theorem i.ii, Ind(~",ul)-----Ind(SZ",uz)=l and the assertion being proved is a corollary of Theorem 1.8. 2549

Definition 5.1. The critical point u 0 of the functional $r is called nondegenerate the equation ~r#(u0)h=0 has only the zero solution.

if

THEOREM 5.6. Let ~ be a bounded domain in R 2 with boundary 8~ of class C ~ and let conditions a) and b) hold for the functional ~'. Let us assume that for the increasing func0 tional ~ there exists a nondegenerate critical point u I and a function u2(x)EWpm(~), u2=/=ul for which 8r(u2)o fi

h o l d s f o r ~s

2550

0 2

~=/=0.

In this case the problem (98), (99) has a countable number of eigenvalues An, where 0