- Fu Yongqiang
^{1}and - Yu Mei
^{1}Email author

**2010**:584843

https://doi.org/10.1155/2010/584843

© F. Yongqiang and Y. Mei. 2010

**Received: **19 August 2009

**Accepted: **7 March 2010

**Published: **14 March 2010

## Abstract

## 1. Introduction

After Kováčik and Rákosník first discussed the spaces and spaces in [1], a lot of research has been done concerning these kinds of variable exponent spaces, for example, see [2–5] for the properties of such spaces and [6–9] for the applications of variable exponent spaces on partial differential equations. Especially in spaces, there are a lot of studies on -Laplacian problems; see [8, 9]. In the recent years, the theory of problems with -growth conditions has important applications in nonlinear elastic mechanics and electrorheological fluids (see [10–14]).

In this paper, we study the existence of the weak solutions for the following -Laplacian problem:

where is a bounded domain, , is Lipschitz continuous on , and

Let be the set of all Lebesgue measurable functions . For all , we denote , , and denote by the fact that .

We impose the following condition on :

(F) , , and for , there exist such that , whenever .

A typical example of (1.1) is the following problem involving subcritical Sobolev-Hardy exponents of the form

where , , , . In fact, take , , and , then it is easy to verify that (F) is satisfied.

Our object is to obtain the existence of solutions in the following four cases:

When , the solution of the -Laplacian equations without singularity has been studied by many researchers. The study of problem (1.1) with variable exponents is a new topic.

The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge of variable exponent Lebesgue and Sobolev spaces. In Section 3, we prove our main results.

## 2. Preliminaries

In this section we first recall some facts on variable exponent Lebesgue space and variable exponent Sobolev space , where is an open set; see [1–4, 8, 15] for the details.

The variable exponent Lebesgue space is the class of functions such that . is a Banach space endowed with the norm (2.1).

For a given we define the conjugate function as

Theorem 2.1.

holds for every and with the constant depending on and only.

Theorem 2.2.

Theorem 2.3.

Suppose that satisfies (2.4). Let , , then necessary and sufficient condition for is that for almost all one has , and in this case, the imbedding is continuous.

Theorem 2.4.

Suppose that satisfies (2.4). Let . If , then

We assume that is a given positive integer.

Given a multi-index , we set and , where is the generalized derivative operator.

The generalized Sobolev space is the class of functions on such that for every multi-index with , endowed with the norm

By we denote the subspace of which is the closure of with respect to the norm (2.5).

In this paper we use the following equivalent norm on :

Theorem 2.5.

The spaces and are separable reflexive Banach spaces, if satisfies (2.4).

Theorem 2.6.

Suppose that satisfies (2.4). Let . If , then

We denote the dual space of by , then we have the following.

Theorem 2.7.

Theorem 2.8.

Let be a domain in with cone property. If is Lipschitz continuous and , is measurable and satisfies a.e. , then there is a continuous embedding .

Theorem 2.9.

where is a constant depending on .

Next let us consider the weighted variable exponent Lebesgue space. Let , and for . Define

with the norm

Theorem 2.10.

Suppose that satisfies (2.4). Let . If , , then

Theorem 2.11.

Assume that the boundary of possesses the cone property and . Suppose that , and , for . If and then there is a compact embedding .

Theorem 2.12.

where , , , is a constant, then the Nemytsky operator from to defined by is a continuous and bounded operator.

## 3. Existence and Multiplicity of Solutions

The critical points of , that is,

for all , are weak solutions of problem (1.1). So we need only to consider the existence of nontrivial critical points of .

Denote by , and the generic positive constants. Denote by the Lebesgue measure of .

To study the existence of solutions for problem (1.1) in the first case, we additionally impose the following conditions.

(B-1) There exists a function , such that and for , .

(C-1) there exist such that for , , where and .

Theorem 3.1.

Under assumptions (F) and (A-1)–(C-1), problem (1.1) admits a nontrivial solution.

Proof.

By (B-1), we get

By (F), we get , so there exist such that on . Note , so we have

By Young's inequality, for , we get

Take sufficiently small so that .

Note that , by Young's inequality again and for , we get

Take sufficiently small so that .

From the above remark, we have

As , and , we have . Since and , by Theorem 2.6 we have

when and is sufficiently large. Then it is easy to see that is bounded in . Next we show that possesses a convergent subsequence (still denoted by ).

Note that

By (F), we have

Because is bounded, we get , as . From the above remark, we conclude , as .

Thus , as . Then we get . As in the proof of Theorem in [6, 7], we divide into the following two parts:

From (F) and (B-1) we have , for all , . So we get , for all , . for all , take , then . Since and , there exists such that , and , for . Let , such that for , for , and in . Then we have

where . So if is sufficiently large, we obtain .

From (F) and (C-1), we have , then . So we get

The mountain pass theorem guarantees that has a nontrivial critical point .

Since is a separable and reflexive Banach space, there exist and such that

Theorem 3.2.

Under assumptions (F), (A-1)–(D-1), problem (1.1) admits a sequence of solutions such that .

Proof.

Let . We first show that is weakly strongly continuous. Let weakly in . By the compact embedding , we have and a.e. on . By the inequality and the Vitali Theorem, we get .

Note that

Note that and . Since the dimension of is finite, any two norms on are equivalent, then . If , it is immediate that . If , then . As in the proof of Theorem 3.1 we can find hypercubes which mutually have no common points such that and , where . Then we need only to consider the case: for every . We have

and as . Hence we obtain that as . Thus for each , there exists such that for . From Theorem 3.1 satisfies condition. In view of (D-1), by Fountain Theorem (see [16]), we conclude the result.

In the second case, we additionally impose the following condition:

Theorem 3.3.

Under assumptions (F), (A-2), (B-1), and (C-1) there exist such that when , problem (1.1) admits a nontrivial solution.

Proof.

From (F) and (C-1), we have and . By Theorems 2.8 and 2.11, there exist positive constants such that , , . When is sufficiently small, we have , , and . As in the proof of Theorem 3.1 we can find hypercubes which mutually have no common points such that , and , where . Then

Let . When , . As in the proof of Theorem in [17], denote , we have and . Let . Applying Ekeland's variational principle to the functional , we find such that , , , and . Thus we get a sequence such that and . It is clear that is bounded in . As in the proof of Theorem 3.1, we get a subsequence of , still denoted by , such that in . So and .

Theorem 3.4.

Under assumptions (F), (A-2), and (B-1)–(D-1), problem (1.1) has a sequence of solutions such that .

Proof.

By Young's inequality, for , we get

As in the proof of Theorem 3.1 we can find hypercubes which mutually have no common points such that and , where . Since the dimension of is finite, any two norms on are equivalent. Then we need only to consider the cases and for every . We have

Hence as . As in the proof of Theorem 3.2, we complete the proof.

In the third case, we additionally impose the following condition:

(B-3) there exist such that for .

Theorem 3.5.

Under assumptions (F), (A-3), and (B-3), problem (1.1) admits a nontrivial solution.

Proof.

where , and , . As in the proof of Theorem 3.2, we obtain that is coercive, that is, as . Thus has a critical point such that and further is a weak solution of (1.1).

Next we show that is nontrivial. Let be the same as that in Theorem 3.3. By (B-3), . Then

In the fourth case, we additionally impose the following condition:

Theorem 3.6.

Under assumptions (F), (A-4), and (D-1), problem (1.1) admits a sequence of solutions such that .

Proof.

First we show that any sequence is bounded. Let and , such that and in . Denote and . We have and .

We can get

As in the proof of Theorem 3.1 we can find hypercubes which mutually have no common points such that and , where . Since the dimension of is finite, any two norms on are equivalent. Then we need only to consider the cases and for every . We have

Hence we obtain as . As in the proof of Theorem 3.2, we complete the proof.

## Declarations

### Acknowledgments

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

## Authors’ Affiliations

## References

- Kováčik O, Rákosník J:
**On spaces****and**.*Czechoslovak Mathematical Journal*1991,**41**(4):592-618.MathSciNetMATHGoogle Scholar - Edmunds D, 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 D, Rákosník J:
**Sobolev embeddings with variable exponent.***Studia Mathematica*2000,**143**(3):267-293.MathSciNetMATHGoogle Scholar - Fan X, Shen J, Zhao D:
**Sobolev embedding theorems for spaces****.***Journal of Mathematical Analysis and Applications*2001,**262**(2):749-760. 10.1006/jmaa.2001.7618MathSciNetView ArticleMATHGoogle Scholar - Fan X, Zhang Q, Zhao D:
**Eigenvalues of****-Laplacian Dirichlet problem.***Journal of Mathematical Analysis and Applications*2005,**302**(2):306-317. 10.1016/j.jmaa.2003.11.020MathSciNetView ArticleMATHGoogle 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".Google Scholar
- Fan X:
**Solutions for****-Laplacian dirichlet problems with singular coefficients.***Journal of Mathematical Analysis and Applications*2005,**312**(2):464-477. 10.1016/j.jmaa.2005.03.057MathSciNetView ArticleMATHGoogle Scholar - Galewski M:
**New variational method for the****-Laplacian equation.***Bulletin of the Australian Mathematical Society*2005,**72**(1):53-65. 10.1017/S0004972700034870MathSciNetView ArticleMATHGoogle Scholar - Acerbi E, Mingione G:
**Regularity results for stationary electro-rheological fluids.***Archive for Rational Mechanics and Analysis*2002,**164**(3):213-259. 10.1007/s00205-002-0208-7MathSciNetView ArticleMATHGoogle Scholar - Acerbi E, Mingione G, Seregin GA:
**Regularity results for parabolic systems related to a class of non-newtonian fluids.***Annales de l'Institut Henri Poincaré. Analyse Non Linéaire*2004,**21**(1):25-60.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 - Růžička M:
*Electrorheological Fluids: Modeling and Mathematical Theory*. Springer, Berlin, Germany; 2000.MATHGoogle Scholar - Zhikov V:
**Averaging of functionals in the calculus of variations and elasticity.***Mathematics of the USSR-Izvestiya*1987,**29:**33-66. 10.1070/IM1987v029n01ABEH000958View ArticleMATHGoogle Scholar - Diening L:
**Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces****and**.*Mathematische Nachrichten*2004,**268:**31-43. 10.1002/mana.200310157MathSciNetView ArticleMATHGoogle Scholar - Willem M:
*Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24*. Birkhäuser, Boston, Mass, USA; 1996:x+162.Google Scholar - Mihăilescu M, Rădulescu V:
**On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent.***Proceedings of the American Mathematical Society*2007,**135**(9):2929-2937. 10.1090/S0002-9939-07-08815-6MathSciNetView 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.