Abstract and Applied Analysis

Volume 2013 (2013), Article ID 575328, 6 pages

http://dx.doi.org/10.1155/2013/575328

Research Article

Three Solutions for Inequalities Dirichlet Problem Driven by -Laplacian-Like

Zhou Qing-Mei¹ and Ge Bin²

¹Library, Northeast Forestry University, Harbin 150040, China

²Department of Mathematics, Harbin Engineering University, Harbin 150001, China

Received 19 April 2013; Revised 20 May 2013; Accepted 4 June 2013

Academic Editor: Rodrigo Lopez Pouso

Copyright © 2013 Zhou Qing-Mei and Ge Bin. 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.

A class of nonlinear elliptic problems driven by -Laplacian-like with a nonsmooth locally Lipschitz potential was considered. Applying the version of a nonsmooth three-critical-point theorem, existence of three solutions of the problem is proved.

1. Introduction

Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions of the problems with discontinuous nonlinearities has been widely investigated in recent years. In 1981, Chang [1] extended the variational methods to a class of nondifferentiable functionals and directly applied the variational methods for nondifferentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities. Soon thereafter, Kourogenis and Papageorgiou [2] extend the nonsmooth critical point theory of Chang [1], by replacing the compactness and the boundary conditions. In [3], by using the Ekeland variational principle and a deformation theorem, Kandilakis et al. obtained the local linking theorem for locally Lipschitz functions. In the celebrated work [4], Ricceri elaborated a Ricceri-type variational principle for Gateaux differentiable functionals. Later, Marano and Motreanu [5] extended Ricceri's result to a large class of nondifferentiable functionals and gave an application to a Neumann-type problem involving the -Laplacian with discontinuous nonlinearities.

In this paper, we consider a nonlinear elliptic problem driven by -Laplacian-like with a nonsmooth locally Lipschitz potential (hemivariational inequality):

where is a bounded domain with -boundary . , ,  , and is a locally Lipschitz with respect to the second variable. By , we denote the generalized subdifferential of the locally Lipschitz function . Our goal is to establish the same results under different assumptions.

The study of differential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, and so forth (see [6, 7]). The study on variable exponent problems attracts more and more interest in recent years. Many results have been obtained on this kind of problems, for example, [8–14]. Neumann-type problems involving the -Laplacian have been studied, for instance, in [15–18].

Recently, Rodrigues [19] has considered the existence of nontrivial solution for the Dirichlet problem involving the -Laplacian-like of the type

where is a bounded domain with smooth boundary , with , for all , and satisfies the Caratheodory condition. We emphasize that, in our approach, no continuity hypothesis will be required for the function with respect to the second argument. So, need not have a solution. To avoid this situation, we consider such function which is locally essentially bounded and fill the discontinuity gap of , replacing by the interval , where

On the other hand, it is well known that if , then become locally Lipschitz and (see [1, 20]).

The aim of the present paper is to establish a three-solution theorem for the nonlinear elliptic problem driven by -Laplacian-like with nonsmooth potential (see Theorem 6) by using a consequence (see Theorem 4) of the three-critical-point theorem established firstly by Marano and Motreanu in [20], which is a non-smooth version of Ricceri's three-critical-point theorem (see [21]). The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function. In Section 3, we give the main result of this paper and use the non-smooth three-critical-point theorem to prove it.

2. Preliminary

In order to discuss problem , we need some theories on and the generalized gradient of the locally Lipschitz function. Firstly we state some basic properties of space which will be used later (for details, see [10–12]). Denote by the set of all measurable real functions defined on . Two functions in are considered as the same element of when they are equal almost everywhere.

Put .

If , then write

with the norm , and

with the norm . Denote by the closure of in .

We remember that the variable exponent Lebesgue spaces are separable and reflexive Banach spaces. Denote by the conjugate Lebesgue space of with ; then the Hölder-type inequality

holds. Furthermore, if we define the mapping by

then the following relations hold:

Proposition 1 (see [12]). In Poincare's inequality holds; that is, there exists a positive constant such that

So is an equivalent norm in .

We will use the equivalent norm in the following discussion and write for simplicity.

Proposition 2 (see [10]). If and for any , then the embedding from to is compact and continuous.

Consider the following function:

We know that (see [1]).

If one denotes , then

for all .

Proposition 3 (see [19]). Set ; is as shown, then

(1)    is a convex, bounded previously; and strictly monotone operator; (2)     is a mapping of type ; that is,   in   and   implies   in ; (3)    is a homeomorphism.

Let be a real Banach space, and let be its topological dual. A function is called locally Lipschitz if each point possesses a neighborhood such that for all , for a positive constant depending on . The generalized directional derivative of at the point in the direction is

The generalized gradient of at is defined by

which is a nonempty, convex, and -compact subset of , where is the duality pairing between and . One says that is a critical point of if .

For further details, we refer the reader to the work of Chang [1].

Finally, for proving our results in the next section, we introduce the following theorem.

Theorem 4 (see [22, 23]). Let be a separable and reflexive real Banach space, and let be two locally Lipschitz functions. Assume that there exists such that and for every and that there exist and such that

(1); (2), and further, one assumes that function is sequentially lower semicontinuous and satisfies the (PS)-condition;(3) for every , where

Then, there exist an open interval and a positive real number such that, for every , the function admits at least three critical points whose norms are less than .

3. Existence Results

In this part, we will prove that there exist three solutions for problem under certain conditions.

Definition 5. We say that satisfies -condition if any sequence , such that and , as , has a strongly convergent subsequence, where .

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

We know that is compactly embedded into (by ). So there is a constant such that , for all .

Set ,   and , for all .

We know that the critical points of are just the weak solutions of .

We consider a non-smooth potential function such that a.e. on satisfying the following conditions:

: 

is measurable for all ; is locally Lipschitz for a.e. ; there exist , such that

where and ; there exists with , such that uniformly a.e. ;, for all .

Theorem 6. Let hold. Then, there are an open interval and a number such that, for every belonging to , problem possesses at least three solutions in whose norms are less than .

Proof. We observe that is Lipschitz on and, taking into account that , is also locally Lipschitz on (see Proposition 2.2 of [15]). Moreover it results in (see [24]). The interpretation of is as follows: to every there corresponds a mapping for almost all having the property that for every the function and (see [24]). The proof is divided into the following five steps.

Step  1. We show that is coercive.

By , for almost all , is differentiable almost everywhere on and we have

From , there exist positive constants such that

for a.e. and .

Note that ; then by Proposition 2, we have (compact embedding). Furthermore, there exists such that .

So, for and , we have .

Hence,

as .

Step  2. We show that is weakly lower semicontinuous.

Let weakly in , and by Proposition 2, we obtain the following results:

By Fatou's lemma, we have

Thus,

Step  3. We show that (PS)-condition holds.

Suppose such that and as . If is such that , then we know that

where the nonlinear operator is defined as

for all . From the work of Chang [1], we know that if , then , where .

Since is coercive, is bounded in and there exists such that a subsequence of , which is still denoted as , satisfies weakly in . Next we will prove that in .

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

Since , we obtain

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

But we know that is a mapping of type (by Proposition 3). Thus we obtain

Step  4. There exists a such that .

By , for each , there is such that .

For , denote by a neighborhood of which is the product of compact intervals. From and , for any , there are , and , such that for all .

Since is bounded, is compact. Then we can find such that and and, also, we can find , and positive numbers such that

Now, set , and , and

Then, we can find a closed set such that

where denote the Lebesgue measure of set . We consider a function such that and for all . For instance, we can set , where and

Then, from (27)–(29), we have

Step  5. We show that , satisfy conditions (1) and (2) of Theorem 4.

Let ; then we can easily find .

From (7) and Proposition 1, we have the following:

if , then

if , then

From , there exist and such that

In view of , if we put

then we have

Fix such that . And when , by Sobolev Embedding Theorem (), we have (for suitable positive constants )

Since , we have

And so, taking into account (32) and (33),

From Step 4, there exists such that . Thanks to (32) and (33), we have

and so

By (32), (33), and (39), there exists such that, for each ,

By choosing , conditions (1) and (2) requested in Theorem 4 are verified and so the proof is complete.

Acknowledgments

This paper is supported by the Fundamental Research Funds for the Central Universities (no. DL12BC10; 2013), the New Century Higher Education Teaching Reform Project of Heilongjiang Province in 2012 (no. JG2012010012), the National Science Foundation of China (nos. 11126286 and 11201095), China Postdoctoral Science Foundation Funded Project (no. 20110491032), and China Postdoctoral Science (Special) Foundation (no. 2012T50303).

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