School of Science, Jiangnan University, Wuxi, 214122, People’s Republic of China

School of Mathematics Science, Nanjing Normal University, Nanjing, 210097, People’s Republic of China

Abstract

In this note, we study the existence and multiplicity of solutions for the quasilinear elliptic problem as follows:

where

1 Introduction

In this note, we discuss the existence and multiplicity of solutions of the following boundary value problem:

where

is an increasing homeomorphism from

(a) nonlinear elasticity:

(b) plasticity:

(c) generalized Newtonian fluids:

So, in the discussions, some special techniques are needed, and the problem (1.1) has been studied in an Orlicz-Sobolev space and received considerable attention in recent years; see, for instance, the papers

Motivated by their results, in this note, we discuss the problem (1.1) when

The paper is organized as follows. In Section 2, we present some preliminary knowledge on the Orlicz-Sobolev spaces and give the main result. In Section 3, we make the proof.

2 Preliminaries

Obviously, the problem (1.1) allows a nonhomogeneous function

Let

then

Throughout this paper, we assume the following condition on

Under the condition (

and is equipped with the (Luxemburg) norm,

We will denote by

and define

Now, we introduce the Orlicz-Sobolev conjugate

where we suppose that

Let

(

and

Let

then we can obtain complementary

Similar to the condition (

In order to prove our results, we now state some useful lemmas.

**Lemma 2.1**

**Lemma 2.2**

**Lemma 2.3**

(1)

(2)

(3)

(4)

**Lemma 2.4**

(

(

(_{c}

Using the version of the symmetric mountain pass theorem mentioned above, we can state our result as follows.

**Theorem 2.1**

(

(

3 Main results and proofs

In this section, we assume that

Set

and we know that the critical points of

For

Now, we set

**Lemma 3.1**

**Lemma 3.2**

Consequently, considering

Now, taking

**Lemma 3.3**

Since

From the condition (

Consequently, for

and

where

**Lemma 3.4**

Noting that

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

Project supported by Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK201209), the China Scholarship Council, the Fundamental Research Funds for the Central Universities (No. JUSRP11118) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).