Abstract and Applied Analysis

Volume 2012 (2012), Article ID 971243, 19 pages

doi:10.1155/2012/971243

Research Article

Multiple Solutions for a Class of Differential Inclusion System Involving the -Laplacian

Bin Ge and Ji-Hong Shen

Department of Mathematics, Harbin Engineering University, Harbin 150001, China

Received 29 December 2011; Revised 28 February 2012; Accepted 2 April 2012

Academic Editor: Kanishka Perera

Copyright © 2012 Bin Ge and Ji-Hong Shen. 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.

Abstract.

We consider a differential inclusion system involving the -Laplacian with Dirichlet boundary condition on a bounded domain and obtain two nontrivial solutions under appropriate hypotheses. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

1. Introduction

In recent years, the study of differential equations and variational problems with -growth conditions has been a new and interesting topic, which arises from nonlinear electrorheological fluids (see [1]) and elastic mechanics (see [2]). The study on variable exponent problems attracts more and more interest in recent years, and many results have been obtained on this kind of problems, for example [3–11].

Elliptic systems with standard growth conditions have been the subject of a sizeable literature. We refer to the excellent survey article of de Figueiredo [12].

In [11], the author obtained the existence and multiplicity of solutions for the following problem:

where is a bounded domain with a smooth boundary , , , , , for every . The function is assumed to be continuous in and of class in . More precisely, the author was able to prove that, under suitable conditions, the system might have at least one solution or have infinite number of solutions.

Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, now a question arises: whether there exist solutions for system () in the case where there is no continuously differentiable hypothesis required on the potential function with respect to . See, for example, is locally Lipschitz with respect to . That is the main problem which we want to solve in the present paper.

To this end, we mainly discuss the existence and multiplicity of solutions for the following nonlinear differential inclusion system involving the -Laplacian:

where is a bounded domain with -boundary , is the parameter, , , , is a function such that is measurable in for all , and is locally Lipschitz with respect to (in general it can be nonsmooth), is the subdifferential with respect to the -variable in the sense of Clarke [13].

We emphasize that the operator is said to be -Laplacian, which becomes -Laplacian when (a constant). The -Laplacian possesses more complicated nonlinearities than the -Laplacian, for example, it is inhomogeneous and, in general, it does not have the first eigenvalue. In other words, the infimum of the eigenvalues of -Laplacian equals 0 (see [14]).

Specially, if for a.a. , and , , then the problem () becomes the following problem:

There have been a large number of papers that study the existence of the solutions to (). For instance, when , Li and Tang [15] ensured the existence of three solutions to this problem. In [16], Kristály studied the multiplicity of solutions of the quasilinear elliptic systems (), where is a strip-like domain and is the parameter. Under some growth conditions on , the author guaranteed the existence of an open interval , such that for each problem () has at least two distinct nontrivial solutions; when and are real numbers larger than 1, , Boccardo and Guedes de Figueiredo [17] obtained the existence of solutions of the system ().

But up to now, to the best of our knowledge, no paper discussing the solutions of problem () with nonsmooth potential via nonsmooth critical point theory can be found in the existing literature. In order to fill in this gap, we study problem () from a more extensive viewpoint. More precisely, we would study the existence of at least two nontrivial solutions for the problem () as the parameter for some constant .

This paper is divided into three sections: in the second section we introduce some necessary knowledge on the nonsmooth analysis, basic properties of the generalized Lebesgue-space and the generalized Lebesgue-Sobolev space . In the third section, we give the assumptions on the nonsmooth potential and prove the multiplicity results for problem ().

2. Preliminary

2.1. Variable Exponent Sobolev Space

In order to discuss problem (), we need some theories on which we call variable exponent Sobolev space. Firstly we review some facts on variable exponent spaces and . For the details see [4, 18–20].

Firstly, we need to give some notations, which we shall use through this paper:

Obviously, .

Denote by the set of all measurable real functions defined on . Two functions in are considered to be one element of , when they are equal almost everywhere.

For , define

and with the norm with the norm .

Denote as the closure of in .

Hereafter, let

Lemma 2.1 (see [19]). (1) The spaces , , and are separable and reflexive Banach spaces. Moreover, is uniform convex.

(2) Poincare inequality in holds; that is, there exists a positive constant such that

(3) If and for any , then the embedding from to is compact and continuous.

By (2) of Lemma 2.1, we know that and are equivalent norms on . We will use to replace in the following discussions.

Lemma 2.2 (see [4]). The conjugate space of is , where . For any and , one has

Lemma 2.3 (see [4]). Set . For , one has

(1)for ;(2); (3)if , then ;(4)if , then ;(5); (6).

In this paper, the space will be endowed with the following equivalent norm:

where

Similar to Lemma 2.3, we have the following.

Lemma 2.4. Set . For , one has

(1)for ;(2); (3)if , then ;(4)if , then ;(5); (6).

Consider the following function:

We know that (see [21]) and -Laplacian operator is the derivative operator of in the weak sense. We denote , then .

Lemma 2.5 (see [19]). Set , is as above, then

(1) is a continuous, bounded and strictly monotone operator;(2) is a mapping of type , if (weak) in and, then in ;(3) is a homeomorphism.

2.2. Generalized Gradient

Let be a Banach space, its topological dual space and we denote as the duality bracket for pair . A function is said to be locally Lipschitz, if for every we can find a neighbourhood of and a constant (depending on ), such that .

The generalized directional derivative of at the point in the direction is

The generalized subdifferential of at the point is defined by the

which is a nonempty, convex and -compact set of . We say that is a critical point of , if . For further details, we refer the reader to [12].

Finally we have the following Weierstrass Theorem and Mountain Pass Theorem.

Theorem 2.6. If is a reflexive Banach space and satisfies

(1) is weak lower semicontinuous, that is,

(2) is coercive, that is, ,

then there exists such that .

Theorem 2.7 (see [22]). Let be locally Lipschitz function and . If there exists a bounded open neighbourhood of , such that , and satisfies the nonsmooth C-condition at level , where , then is a critical value of and .

3. Existence Results

For each , define

By a solution of (2), we mean function to which there corresponds mapping with , for almost every having the property that for every , the function and

Our hypotheses on nonsmooth potential is as follows.

H(F): is a function such that a.e. on and satisfies the following facts:

(1)for all , , is measurable;(2)for almost all , and are locally Lipschitz.

Lemma 3.1. Suppose and the following conditions hold:

(f₁) there exists and , such that

for almost all , all and ;(f₂) there exists and , such that

for almost all , all and ;

then is locally Lipschitz on .

Proof. We need only to prove that is locally Lipschitz on .

By we have

where.

Let.

Note that is -compact. Then we obtain that there exists a positive constant , such that with , for sufficiently small .

Therefore, for any , we have

Fixing , by (f₁) and Lebourg mean value theorem, we have

Hence,

where.

It is immediate that

are bounded.

So,

since is a compact embedding.

Therefore, is locally Lipschitz. Similarly, we can prove that is locally Lipschitz.

Theorem 3.2. Suppose that , with , with and the following conditions - hold:

(f₃) there exists with and , such that

uniformly for almost all and all ;(f₄) there exist and , such that

where .

Then there exists such that, for each , the problem (2) has at least two nontrivial solutions.

Proof. The proof is divided into four steps as follows.

Step 1. We will show that is coercive in the step.

Firstly, for almost all , by H(F)(2), is differentiable almost everywhere on and we have

Moreover, from (f₁), (f₂) and Young inequality, we can get that

for almost all and .

Note that and , then, by Lemma 2.1, we have and (compact embedding). Furthermore, there exists such that and .

So, for any and ,

and, for any and ,

By (3.14), (3.15), (3.31), the Hölder inequality and the Sobolev embedding theorem, we have

Step 2. We will show that the is weakly lower semicontinuous.

Leting weakly in , weakly in by Lemma 2.1(3), we obtain the following results:

; ; in in ; for a.a. for a.a. for a.a. .

By Fatou’s Lemma,

Thus,

Hence, by Theorem 2.6, we deduce that there exists a global minimizer such that

Step 3. We will show that there exists such that for each , .

By the condition (f₄), there exists such that , a.e. . It is clear that

Now we denote

where and is given in the condition (f₄). A simple calculation shows that the function is positive whenever and . Thus is well defined and .

We will show that, for each , the problem () has two nontrivial solutions. In order to do this, for , let us define

By conditions (f₁) and (f₃) we have

Hence, for ,

so that whenever .

Step 4. We will check the C-condition in the following.

Suppose such that and . Let be such that . The interpretation of is that and . We know that

with and . From Chang [23] we know that and , where .

Since is coercive, , are bounded and passed to a subsequence, still denoting and , we may assume that there exist , , such that weakly in and weakly in . Next we will prove that

By , , we have in and in . Moreover, since , we get , , so , .

From (3.26), we have

Moreover, and , since in , in , in and in are bounded, where . Therefore,

From Lemma 2.5, we have , as . Thus satisfies the nonsmooth C-condition.

Step 5. We will show that there exists another nontrivial weak solution of problem ().

From Lebourg Mean Value Theorem, we obtain

for some , , and . Thus by the condition (f₃), there exists such that

for all .

It follows from the conditions (f₁), (f₂) and that for all , and a.e. ,

this together with (3.31) yields that, for all and a.e. ,

for positive constants .

Note that , , then, by Lemma 2.1, we have and . Furthermore, there exist such that

For all , , and , from (3.33) we have

So, for small enough, there exists a such that

and . So by the Nonsmooth Mountain Pass Theorem (cf. Theorem 2.7), we can get   which satisfies

Therefore, is another nontrivial solution of problem ().

Remark 3.3. Let , and consider the following nonsmooth locally Lipschitz function:

where , , and .

Obvious, and are locally Lipschitz. Then,

Hence, for any and , we have

Therefore,

uniformly for almost all , all and .

Thus far the results involved potential functions exhibiting -sublinear. The next theorem concerns problems where the potential function is -superlinear.

Theorem 3.4. Suppose that H(F), (f₁) with , (f₂) with , (f₃), (f₄) and the following condition (f₅) hold.

(f₅)For almost all and all , one has .

Then there exists a such that, for each , the problem () has at least two nontrivial solutions.

Proof. The steps are similar to those of Theorem 3.2. In fact,we only need to modify Step 1 and Step 5 as follows: Step 6 shows, that is coercive under the condition (f₅); Step 7 shows, that there exists second nontrivial solution under the conditions (f₁), (f₂), and (f₃). Then from Steps 6, 2, 3, 4, and 7 above, the problem () has at least two nontrivial solutions.

Step 6. By (f₅), for all , , we have

Step 7. Because of hypothesis (f₁), (f₂) and mean value theorem for locally Lipschitz functions, we have

for a.e. , all , with .

Combining (3.31) and (3.43), it follows that

for a.e. and all .

Thus, for all , , , and , we have

So, for small enough, there exists a such that

and . Arguing as in proof of Step 4 of Theorem 3.2, we conclude that satisfies the nonsmooth C-condition. So by the Nonsmooth Mountain Pass Theorem (cf. Theorem 2.7), we can get   which satisfies

Therefore, is second nontrivial of problem ().

Remark 3.5. Consider the following nonsmooth locally Lipschitz function:

In the following, we will show that satisfies hypotheses H(F) and (f₁)–(f₅).

It is clear that for a.e. ,thenhypotheses H(F) is satisfied. A direct verification shows that conditions (f₄) and (f₅) are satisfied. Note that

So,

which shows that assumptions (f₁), (f₂), and (f₃) are fulfilled.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (nos. 11126286, 11001063, 10971043), the Fundamental Research Funds for the Central Universities (no. 2012), China Postdoctoral Science Foundation Funded Project (no. 20110491032), and the National Natural Science Foundation of Heilongjiang Province (no. A200803).

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