Boundary Value Problems

, 2009:865408 | Cite as

Infinitely Many Solutions of Strongly Indefinite Semilinear Elliptic Systems

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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

In this paper, we shall study the existence of multiple solutions of the semilinear elliptic systems
where Open image in new window and Open image in new window are positive numbers which are in the range we shall specify later. Let us consider that the exponents Open image in new window , Open image in new window are below the critical hyperbola
so one of Open image in new window and Open image in new window could be larger than Open image in new window ; for that matter, the quadratic part of the energy functional

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 .

In [3], Yang shows that under some assumptions on the functions Open image in new window and Open image in new window there exist infinitely many solutions of the semilinear elliptic systems

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].

We shall work with space Open image in new window , which are obtained as the domains of fractional powers of the operator
Namely, Open image in new window for Open image in new window , and the corresponding operator is denoted by Open image in new window . The spaces Open image in new window , the usual fractional Sobolev space Open image in new window , are Hilbert spaces with inner product
and associates norm

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 .

Using the Cauchy-Schwarz inequality, then it is easy to see that Open image in new window is continuous and symmetric. Hence Open image in new window induces a self-adjoint bounded linear operator Open image in new window such that
Here and in what follows Open image in new window denotes the inner product in Open image in new window induced by Open image in new window and Open image in new window on the product space Open image in new window in the usual way. It is easy to see that

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

which give a natural splitting Open image in new window . The spaces Open image in new window and Open image in new window are orthogonal with respect to the bilinear form Open image in new window , that is,
We can also define the quadratic form Open image in new window associated to Open image in new window and Open image in new window as
for all Open image in new window . It follows then that
Similarly

for Open image in new window .

If Open image in new window where Open image in new window is a number satisfying the condition
and Open image in new window , it follows by (2.13) that Open image in new window and by H Open image in new window lder inequalities that

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.

Observe that, by H Open image in new window lder's inequality and (2.14), we have

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

Then we will claim that Open image in new window is compact. Since Open image in new window , for any Open image in new window , there exists Open image in new window , such that Open image in new window . Now, suppose Open image in new window weakly in Open image in new window . We estimate

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

We consider below the problem of finding multiple solutions of the semilinear elliptic systems

and we assume that

and we let

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 ;

(iii) Open image in new window satisfies the Palais-Smale condition. Furthermore, let
be an orthogonal splitting into Open image in new window -invariant subspaces Open image in new window , Open image in new window such that Open image in new window . Then,
  1. (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 ;

It is known from Section 2 that the operator Open image in new window induced by the bilinear form Open image in new window is an invertible bounded self-adjoint linear operator satisfying Open image in new window . We shall need some finite dimensional subspace of Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , be a complete orthogonal system in Open image in new window . Let Open image in new window denote the finite dimensional subspaces of Open image in new window generated by Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window . Since Open image in new window and Open image in new window are isomorphisms, we know that Open image in new window , Open image in new window , Open image in new window , Open image in new window , is a complete orthogonal system in Open image in new window . Let Open image in new window denote the finite dimensional subspaces of Open image in new window generated by Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window . For each Open image in new window , we introduce the following subspaces of Open image in new window and 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.

Condition (ii) is an immediate consequence of the definition of Open image in new window . For condition (iv), by (2.11) and Theorem 2.1, for Open image in new window ,

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.

Next, let us prove condition (v). Let Open image in new window be fixed, let Open image in new window , and write Open image in new window and Open image in new window . We have
Using the fact that the norms in Open image in new window are equivalent we obtain
with constant Open image in new window independent of Open image in new window . So from (3.13) and (2.11) we obtain

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.

We first prove the boundedness of (PS)-sequences of Open image in new window . Let Open image in new window be a (PS)-sequence of Open image in new window such that
Taking Open image in new window in (3.18), it follows from (3.17), (3.18), that
for all Open image in new window . Using H Open image in new window lder's inequality and by (3.20), we obtain
for all Open image in new window , which implies that
Similarly, we prove that
Adding (3.22) and (3.23) we conclude that
Using this estimate in (3.19), we get

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).

Finally, we show that Open image in new window contains a strongly convergent subsequence. It follows from Open image in new window and Open image in new window which are bounded and Theorem 2.1 that Open image in new window contains a subsequence, denoted again by Open image in new window , such that
It follows from (3.18) that
Therefore,

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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Copyright information

© Kuan-Ju Chen. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Applied ScienceNaval AcademyZuoying, KaohsiungTaiwan

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