Existence of nontrivial solutions for semilinear problems with strictly differentiable nonlinearity.

Publication: Abstract and Applied Analysis

Publication Date: 01-JAN-06

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COPYRIGHT 2006 Hindawi Publishing Corp.

The existence of a nontrivial solution for semilinear elliptic problems with strictly differentiable nonlinearity is proved. A result of homological linking under nonstandard geometrical assumption is also shown. Techniques of Morse theory are employed.

1. Introduction

Since the paper of Amann and Zehnder [1], the existence of nontrivial solutions u for semilinear elliptic problems of the form

-[DELTA]u = g(u) in [OMEGA], u = on [partial derivative][OMEGA], (1.1)

with g(0) = 0, has been the object of several studies, in which topological and variational methods are successfully applied. We refer the reader to [2, 3, 8, 10]. In particular, since the combination of linking theorems and Morse theory has turned out to be very fruitful, it is customary to impose conditions on g that guarantee that the associated functional f : [H.sup.1.sub.0] ([OMEGA]) [right arrow]R, given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2)

is of class [C.sup.2].

In a recent paper [12], Perera and Schechter have proved a result of Amann-Zehnder type under assumptions that imply f to be only of class [C.sup.1]. More precisely, about the regularity of g, they assume that g is continuous, there exist in R the limits

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

and that

g(s) / s is Lipschitz continuous in a neighbourhood of 0. (1.4)

One could observe that hypothesis (1.4) allows f not to be of class [C.sup.2], but it does not include every g satisfying the usual assumption that g is of class [C.sup.1] and g' is bounded. In particular, condition (1.4) is not stable if we add to g a term of the form

[|s|.sup.3/2] / 1 + [s.sup.2]. (1.5)

The first purpose of this paper is to extend the result of [12] in such a way that also the classical smooth case is included. Our result is the following.

Theorem 1.1. Let [OMEGA] be a bounded open subset of [R.sup.n] and g : R [right arrow] R be a continuous function satisfying g(0) = and

(a) there exists C [greater than or equal to] such that

|g(s)| [less than or equal to] C'(1 + |s|); (1.6)

(b) there exists [alpha] [member of] R such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.7)

If we denote by ([[lambda].sub.m]) the sequence of the eigenvalues of -[nabla] with homogeneous Dirichlet boundary condition, let us assume that [alpha] [not equal to] [[lambda].sub.m] for any m [member of] N. Moreover, let us suppose that g is strictly differentiable at (see Definition 3.1 below) and that there exists m [member of] N with either g'(0) [[lambda].sub.m] > [alpha].

Then (1.1) admits a nontrivial solution.

Theorem 1.1 is in fact a particular case of a more general result, which will be presented in Section 2.

Remark 1.2. If, as in [12], we have g(s) = s[gamma](s), with [gamma] Lipschitz continuous in a neighbourhood of 0, then it is easy to see that g is strictly differentiable at 0.

A second purpose of the paper is to improve the saddle theorem proved in [11, Theorem 1.4], also mentioned in [12], in which the functional is of class [C.sup.2], but nonstandard geometrical assumptions are considered. We will prove the following.

Theorem 1.3. Let H be a Hilbert space such that H = [H.sub.-] [direct sum] [H.sub.+] with dim[H.sub.-] < [infinity] and [H.sub.+] closed in H. Let f : H [right arrow] R be a functional of class [C.sup.2] and assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.8)

f satisfies [(PS).sub.c] for every c [member of] [[c.sub.0], [c.sub.1]], f''(u) is a Fredholm operator at every critical point u in [f.sup.-1]([[c.sup.0], [c.sup.1]]).

Then there exists a critical point u of f with [c.sub.0] [less than or equal to] f(u) [less than or equal to] [c.sub.1] and m(f ,u) [less than or equal to] dim[H.sub.-] [less than or equal to] [m.sup.*](f ,u).

In [11] it is only shown that there exist critical points [u.bar], [bar.u] with [c.sub.0] [less than or equal to] f ([bar.u]) [less than or equal to] f ([u.bar]) [less than or equal to] [c.sub.1] and m(f,[u.bar]) [less than or equal to] dim[H.sub.-] [less than or equal to] [m.sub.*](f,[bar.u]), but one cannot say if there exists a critical point u = [u.bar] = [bar.u], as in the case with standard geometrical assumptions (see [8]), or not. Our improvement is related to the fact that, according to Proposition 4.3 below, also under the nonstandard geometrical assumptions of Theorem 1.3, it is possible to recognize a homological linking structure.

The paper is organized as follows: in Section 2 we state the result of existence of nontrivial solutions; Sections 3 and...

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