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

**2009**:191649

https://doi.org/10.1155/2009/191649

© Y. Fu and X. Zhang. 2009

**Received: **19 November 2008

**Accepted: **11 February 2009

**Published: **18 February 2009

## Abstract

## 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:

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

Theorem 1.1.

### 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

### 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.

where As is a finite dimensional space, we have for

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

## References

- Kováčik O, Rákosník J:
**On spaces and .***Czechoslovak Mathematical Journal*1991,**41(116)**(4):592–618.MATHGoogle Scholar - Růžička M:
*Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics*.*Volume 1748*. Springer, Berlin, Germany; 2000:xvi+176.MATHGoogle Scholar - Chabrowski J, Fu Y:
**Existence of solutions for -Laplacian problems on a bounded domain.***Journal of Mathematical Analysis and Applications*2005,**306**(2):604–618. 10.1016/j.jmaa.2004.10.028MathSciNetView ArticleMATHGoogle Scholar - Chabrowski J, Fu Y:
**Corrigendum to: "Existence of solutions for -Laplacian problems on a bounded domain".***Journal of Mathematical Analysis and Applications*2006,**323**(2):1483. 10.1016/j.jmaa.2005.11.036MathSciNetView ArticleGoogle Scholar - Fan X, Han X:
**Existence and multiplicity of solutions for -Laplacian equations in .***Nonlinear Analysis: Theory, Methods & Applications*2004,**59**(1–2):173–188.MathSciNetMATHGoogle Scholar - Mihăilescu M, Rădulescu V:
**A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids.***Proceedings of the Royal Society of London. Series A*2006,**462**(2073):2625–2641. 10.1098/rspa.2005.1633MathSciNetView ArticleMATHGoogle Scholar - De Figueiredo DG, Ding YH:
**Strongly indefinite functionals and multiple solutions of elliptic systems.***Transactions of the American Mathematical Society*2003,**355**(7):2973–2989. 10.1090/S0002-9947-03-03257-4MathSciNetView ArticleMATHGoogle Scholar - Felmer PL:
**Periodic solutions of "superquadratic" Hamiltonian systems.***Journal of Differential Equations*1993,**102**(1):188–207. 10.1006/jdeq.1993.1027MathSciNetView ArticleMATHGoogle Scholar - Edmunds DE, Lang J, Nekvinda A:
**On norms.***Proceedings of the Royal Society of London. Series A*1999,**455**(1981):219–225. 10.1098/rspa.1999.0309MathSciNetView ArticleMATHGoogle Scholar - Edmunds DE, Rákosník J:
**Sobolev embeddings with variable exponent.***Studia Mathematica*2000,**143**(3):267–293.MathSciNetMATHGoogle Scholar - Fan X, Zhao Y, Zhao D:
**Compact imbedding theorems with symmetry of Strauss-Lions type for the space .***Journal of Mathematical Analysis and Applications*2001,**255**(1):333–348. 10.1006/jmaa.2000.7266MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.