# Infinitely Many Solutions of Strongly Indefinite Semilinear Elliptic Systems

## Abstract

We proved a multiplicity result for strongly indefinite semilinear elliptic systems Open image in new window in Open image in new window , Open image in new window in Open image in new window where Open image in new window and Open image in new window are positive numbers which are in the range we shall specify later.

## Keywords

Bilinear Form Invariant Subspace Bounded Linear Operator Multiple Solution Convergent Subsequence## 1. Introduction

has to be redefined, and we then need fractional Sobolev spaces.

Hence the energy functional Open image in new window is strongly indefinite, and we shall use the generalized critical point theorem of Benci [1] in a version due to Heinz [2] to find critical points of Open image in new window . And there is a lack of compactness due to the fact that we are working in Open image in new window .

We shall propose herein a result similar to [3] for problem (1.1).

## 2. Abstract Framework and Fractional Sobolev Spaces

We recall some abstract results developed in [4] or [5].

It is known that Open image in new window is an isomorphism, and so we denote by Open image in new window the inverse of Open image in new window .

We can then prove that Open image in new window has two eigenvalues Open image in new window and Open image in new window , whose corresponding eigenspaces are

for Open image in new window .

In the sequel Open image in new window denotes the norm in Open image in new window , and we denote by Open image in new window the weighted function spaces with the norm defined on Open image in new window by Open image in new window . According to the properties of interpolation space, we have the following embedding theorem.

Theorem 2.1.

Then the inclusion of Open image in new window into Open image in new window is compact if Open image in new window .

Proof.

where Open image in new window ; hence Open image in new window is well defined.

so that by H Open image in new window lder's inequality, we observe that, for any Open image in new window , we can choose a Open image in new window so that the integral over ( Open image in new window ) is smaller than Open image in new window for all Open image in new window , while for this fixed Open image in new window , by strong convergence of Open image in new window to Open image in new window in Open image in new window on any bounded region, the integral over ( Open image in new window ) is smaller than Open image in new window for Open image in new window large enough. We thus have proved that Open image in new window strongly in Open image in new window ; that is, the inclusion of Open image in new window into Open image in new window is compact if Open image in new window .

## 3. Main Theorem

and we assume that

so that, under assumption (H), Theorem 2.1 holds, respectively, with Open image in new window and Open image in new window , and Open image in new window and Open image in new window ; that is, the inclusion of Open image in new window into Open image in new window and the inclusion of Open image in new window into Open image in new window are compact.

denote the energy of Open image in new window . It is well known that under assumption (H) the energy functional Open image in new window is well defined and continuously differentiable on Open image in new window , and for all Open image in new window we have

and it is also well known that the critical points of Open image in new window are weak solutions of problem (3.1). The main theorem is the following.

Theorem 3.1.

Under assumption (H), problem (3.1) possesses infinitely many solutions Open image in new window .

Since the functional Open image in new window are strongly indefinite, a modified multiplicity critical points theorem Heinz [2] which is the generalized critical point theorem of Benci [1] will be used. For completeness, we state the result from here.

Theorem 3.2.

(see [2]) Let Open image in new window be a real Hilbert space, and let Open image in new window be a functional with the following properties:

where Open image in new window is an invertible bounded self-adjoint linear operator in Open image in new window and where Open image in new window is such that Open image in new window and the gradient Open image in new window is a compact operator;

(ii) Open image in new window is even, that is Open image in new window ;

- (a)
suppose that there is an Open image in new window -dimensional linear subspace Open image in new window of Open image in new window ( Open image in new window ) such that for the spaces Open image in new window , Open image in new window one has

(iv) Open image in new window such that Open image in new window , Open image in new window ;

- (b)
a similar result holds when Open image in new window , and one takes Open image in new window , Open image in new window .

Lemma 3.3.

The functional Open image in new window defined in (3.6) satisfies conditions (ii), (iv), and (v) of Theorem 3.2.

Proof.

and since Open image in new window , Open image in new window , we conclude that Open image in new window for Open image in new window with Open image in new window small.

The same arguments can be applied if Open image in new window . So the result follows from (3.16).

A sequence Open image in new window is said to be the Palais-Smale sequence for Open image in new window (PS)-sequence for short) if Open image in new window uniformly in Open image in new window and Open image in new window in Open image in new window . We say that Open image in new window satisfies the Palais-Smale condition (PS)-condition for short) if every (PS)-sequence of Open image in new window is relatively compact in Open image in new window .

Lemma 3.4.

Under assumption (H), the functional Open image in new window satisfies the (PS)-condition.

Proof.

Since Open image in new window and Open image in new window , we conclude that both Open image in new window and Open image in new window are bounded, and consequently Open image in new window and Open image in new window are also bounded in terms of (3.24).

and by Theorem 2.1, we conclude that Open image in new window strongly in Open image in new window and Open image in new window strongly in Open image in new window .

Proof of Theorem 3.1.

Applying Lemmas 3.3 and 3.4 and Theorem 3.2, we can obtain the conclusion of Theorem 3.1.

## References

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