# Multiple Solutions for a Class of -Laplacian Systems

- Yongqiang Fu
^{1}and - Xia Zhang
^{1}Email author

**2009**:191649

**DOI: **10.1155/2009/191649

© Y. Fu and X. Zhang. 2009

**Received: **19 November 2008

**Accepted: **11 February 2009

**Published: **18 February 2009

## 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.

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

(H1) Writing

where is positive constant;

when

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

Theorem 1.1.

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].

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).

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

denote the complement of in by

### 3. The Proof of Theorem 1.1

Definition 3.1.

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

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.

Thus we get Then in as Similarly, in

Lemma 3.4.

There exists such that for all with

Proof.

- (i)If We have(321)

- (ii)If Note that For any there exists which is an open subset of such that(322)

where As is a finite dimensional space, we have for

for and where

Let If we get or

Now we get the result.

Lemma 3.5.

There exist and such that for any with

Proof.

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.

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.

## Declarations

### Acknowledgments

This work is supported by Science Research Foundation in Harbin Institute of Technology (HITC200702) and The Natural Science Foundation of Heilongjiang Province (A2007-04).

## Authors’ Affiliations

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## Copyright

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