Multiple Solutions for a Class of -Laplacian Systems

Abstract

We study the multiplicity of solutions for a class of Hamiltonian systems with the -Laplacian. Under suitable assumptions, we obtain a sequence of solutions associated with a sequence of positive energies going toward infinity.

1. Introduction and Main Results

Since the space and were thoroughly studied by Kováčik and Rákosník [1], variable exponent Sobolev spaces have been used in the last decades to model various phenomena. In [2], Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids.

In recent years, the differential equations and variational problems with -growth conditions have been studied extensively; see for example [3–6]. In [7], De Figueiredo and Ding discussed the multiple solutions for a kind of elliptic systems on a smooth bounded domain. Motivated by their work, we will consider the following sort of -Laplacian systems with "concave and convex nonlinearity":

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where is a bounded domain, is continuous on and satisfies , and is a function. In this paper, we are mainly interested in the class of Hamiltonians such that

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where Here we denote

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and denote by the fact that Throughout this paper, satisfies the following conditions:

(H1) Writing

(H2) there exist such that

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where is positive constant;

(H3) there exist with , and such that

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when

As [8,Lemma 1.1], from assumption (H3), there exist such that

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for any We can also get that there exists such that

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for any In this paper, we will prove the following result.

Theorem 1.1.

Assume that hypotheses (H1)–(H3) are fulfilled. If is even in , then problem (1.1) has a sequence of solutions such that

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as

2. Preliminaries

First we recall some basic properties of variable exponent spaces and variable exponent Sobolev spaces where is a domain. For a deeper treatment on these spaces, we refer to [1, 9–11].

Let be the set of all Lebesgue measurable functions and

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The variable exponent space is the class of all functions such that Under the assumption that is a Banach space equipped with the norm (2.1).

The variable exponent Sobolev space is the class of all functions such that and it can be equipped with the norm

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For if we define

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then and are equivalent norms on

By we denote the subspace of which is the closure of with respect to the norm (2.2) and denote the dual space of by We know that if is a bounded domain, and are equivalent norms on

Under the condition is a separable and reflexive Banach space, then there exist and such that

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In the following, we will denote that where

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For any define the norm For any set and

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denote the complement of in by

3. The Proof of Theorem 1.1

Definition 3.1.

We say that is a weak solution of problem (1.1), that is,

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In this section, we denote that for any , and is positive constant, for any

Lemma 3.2.

Any sequence , that is, and as is bounded.

Proof.

Let be sufficiently small such that

Let be such that as We get

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As by the Young inequality, we can get that for any

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Let be sufficiently small such that

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then

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Note that by the Young inequality, for any we get

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Let be sufficiently small such that and then we get

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Note that

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and for being large enough, we have

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It is easy to know that if and

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thus we get

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then are bounded. Similarly, if or we can also get that are bounded. It is immediate to get that is bounded in

Lemma 3.3.

Any sequence contains a convergent subsequence.

Proof.

Let be a sequence. By Lemma 3.2, we obtain that is bounded in As is reflexive, passing to a subsequence, still denoted by we may assume that there exists such that weakly in Then we can get weakly in . Note that

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It is easy to get that

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and in in as Then

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as By condition (H2), we obtain

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It is immediate to get that are bounded and then we get

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as Similar to [3, 4], we divide into two parts:

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On we have

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then On we have

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Thus we get Then in as Similarly, in

Lemma 3.4.

There exists such that for all with

Proof.

For any we have

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In the following, we will consider

• (i) If We have

  

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• (ii) If Note that For any there exists which is an open subset of such that

  

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then is an open covering of As is compact, we can pick a finite subcovering for Thus there exists a sequence of open set such that and

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for Denote that then we have

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where As is a finite dimensional space, we have for

We denote by the maximum of polynomial on for Then there exists such that

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for and where

Let If we get or

(i)If It is easy to verify that there exists at least such that thus

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(ii)If We obtain

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Now we get the result.

Lemma 3.5.

There exist and such that for any with

Proof.

For By condition (H2), there exists such that

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Let we get

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Denote that

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thus

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Let

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By [5, Lemma 3.3], we get that as then

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when is sufficiently large and It is easy to get that as

Lemma 3.6.

I is bounded from above on any bounded set of

Proof.

For We get

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By conditions (H2) and (H3), we know that if and if Then

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and it is easy to get the result.

Proof.

By Lemmas 3.2–3.6 above, and [7, Proposition 2.1 and Remark 2.1], we know that the functional has a sequence of critical values as Now we complete the proof.