Conference Papers in Mathematics

Volume 2013 (2013), Article ID 231898, 7 pages

http://dx.doi.org/10.1155/2013/231898

Conference Paper

Existence and Multiplicity of Solutions for a Robin Problem Involving the -Laplace Operator

Najib Tsouli, Omar Chakrone, Omar Darhouche, and Mostafa Rahmani

Department of Mathematics, University Mohamed I, P.O. Box 717, Oujda 60000, Morocco

Received 6 June 2013; Accepted 10 July 2013

Academic Editors: G. S. F. Frederico, N. Martins, D. F. M. Torres, and A. J. Zaslavski

This Conference Paper is based on a presentation given by Najib Tsouli at “The Cape Verde International Days on Mathematics 2013” held from 22 April 2013 to 25 April 2013 in Praia, Cape Verde.

Copyright © 2013 Najib Tsouli et al. 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 study the following nonlinear Robin boundary-value problem in , on where is a bounded domain with smooth boundary , is the outer unit normal derivative on , is a real number, is a continuous function on with , with , and is a continuous function. Using the variational method, under appropriate assumptions on , we obtain results on existence and multiplicity of solutions.

1. Introduction

The purpose of this paper is to study the existence and multiplicity of solutions for the following Robin problem involving the -Laplacian:

where is a bounded smooth domain, is the outer unit normal derivative on is a real number, is a continuous function on with , and with . The main interest in studying such problems arises from the presence of the -Laplace operator , which is a natural extension of the classical -Laplace operator obtained in the case when is a positive constant. However, such generalizations are not trivial since the -Laplace operator possesses a more complicated structure than -Laplace operator; for example, it is inhomogeneous.

In the recent years increasing attention has been paid to the study of differential and partial differential equations involving variable exponent conditions. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, and calculus of variations; for information on modelling physical phenomena by equations involving -growth condition we refer to [1–10]. In the past decades a vast amount of literature that deals with the existence for type problems with different boundary conditions (Dirichlet, Neumann, Robin, nonlinear, etc.) has appeared. See, for instance, [11–16] and references therein.

In [14], by applying the subsupersolution method and the variational method, under appropriate assumptions on , the author proves that there exists such that problem (1) has at least two positive solutions if , has at least one positive solution if , and has no positive solution if . Recently in [11], the authors obtain the existence of at least two nontrivial solutions for problem (1) using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functions.

We make the following assumptions on the function :

(H0) satisfies the Carathéodory condition and there exists a constant such that

where and for all .(H1) There exist ,   such that, for all and ,

(H2) One has as and uniformly for .(H3) One has  ,  ,  .(H4) One has  , for all .(H5) There exists such that and

The main results of this paper are as follows.

Theorem 1. If (H0), (H1), and (H2) hold and , then, for any , (1) has at least a nontrivial weak solution.

Theorem 2. If (H0), (H1), and (H3) hold and , then, for any , (1) has infinite many pairs of weak solutions.

Theorem 3. If (H0), (H4), and (H5) hold and , then there exist an open interval and a positive real number such that, for each , (1) has at least three solutions whose norms are less than .

To prove our results, we will use a variational method and the theory of variable exponent Sobolev spaces. For the proof of Theorem 1, we will use the Mountain Pass Theorem (see [17, 18]). For the proof of Theorem 2, we will use the Fountain Theorem (see [18, 19]). For the proof of Theorem 3, we will use Ricceri three critical points Theorem (see [20, 21]).

This paper is organized as follows. First, we will introduce some basic preliminary results and lemmas in Section 2. In Section 3, we will give the proofs of our main results.

2. Preliminaries

For completeness, we first recall some facts on the variable exponent spaces and . Suppose that is a bounded open domain of with smooth boundary and , where

Denote and . Define the variable exponent Lebesgue space by

with the norm

Define the variable exponent Sobolev space by

with the norm

We refer the reader to [14, 22, 23] for the basic properties of the variable exponent Lebesgue and Sobolev spaces.

Lemma 4 (see [23]). Both and are separable, reflexive, and uniformly convex Banach spaces.

Lemma 5 (see [23]). Hölder inequality holds, namely,

where .

Now, we introduce a norm, which will be used later. For , define

Then, by Theorem 2.1 in [14], is also a norm on which is equivalent to .

Hereafter, let

Lemma 6 (see [12, 16, 23]). (1) If and for any , then the imbedding from to is compact and continuous.

(2) If and for any , then the trace imbedding from to is compact and continuous.

An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the mapping defined by the following.

Lemma 7 (see [14]). Denoting

with , then

(1),(2),(3) if and only if (as ),(4) if and only if (as ).

Lemma 8 (see [20, 21, 24]). Let be a separable and reflexive real Banach space; is a continuous Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on ; is a continuous Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that

(i) for all ,(ii)there exist and such that ,(iii)

Then there exist an open interval and a positive constant such that for any the equation has at least three solutions in whose norms are less than .

Theorem 9. Let and be a continuous function with primitive . If the following condition hold:

where is a constant and such that, for all ,  , then and ; moreover, the operator is compact.

Proof. It is easily adapted from Theorem 2.1 in [13].

Let and

where .

Obviously and

Moreover, we have the following.

Proposition 10 (see [11]).    is a continuous, bounded, and strictly monotone operator.

   is a mapping of type ; that is, if in and , then in .

   is a homeomorphism.

Definition 11. One says that is a weak solution of problem (1) if for all

where is the measure on the boundary .

3. Proof of Main Results

To prove Theorem 1, we have to check that the functional satisfies the following compactness condition (PS).

Definition 12. One says that the -functional satisfies the Palais-Smale condition ((PS) condition for short) if any sequence for which is bounded and as has a convergent subsequence.

Lemma 13. If (H0), (H1) hold, then for any the functional satisfies the Palais-Smale condition (PS).

Proof. Suppose that is a (PS) sequence; that is,

We claim that is bounded in . Using hypothesis (H1), since is bounded, we have for large enough

where ,  , and are three positive constants. Hence is bounded in since . Without loss of generality, we assume that , then since is completely continuous. The hypothesis , that is, and the fact that imply that . From Proposition 10, is a homeomorphism, then , and so satisfies the (PS) condition. The proof is complete.

Lemma 14. There exists such that for all such that .

Proof. Conditions (H0) and (H2) assure that

For small enough, we have

Note that , for all ; then, by Lemma 6, we have and with a continuous and compact embedding. Furthermore, there exists , such that

Since is small enough, we deduce

Replacing in (22), it results that

Choosing small enough such that , then we obtain

Since , the function is strictly positive in a neighborhood of zero. It follows that there exist and such that

The proof is complete.

Proof of Theorem 1. To apply the Mountain Pass Theorem [17, 18], we need to prove that as , for a certain . From condition (H1), we obtain

Letting and , we have

The fact implies for any ,   as .

It follows that there exists such that and . According to the Mountain Pass Theorem, admits a critical value which is characterized by

where

This completes the proof.

Since is a separable and reflexive Banach space [22, 25], there exist and such that

For denote

Lemma 15. For , , and , let

Then .

Proof. It is clear that , so converges to . Let such that

Then, there exists a subsequence, noted also by , such that in and

Thus and so in . According to Lemma 6, there is a compact embedding of into , which assure that in . Hence we get as .

Proof of Theorem 2. We will use the Fountain theorem [18, 19]. Obviously, is an even functional and satisfies the (PS) condition according to (H1) and (H2). We will prove that if is large enough, then there exist such that

(A1) as ,(A2) as .

(A1) For such that , by condition (H0), we have

If then . And if then . So we obtain that

If we take , that is, , we obtain

Since and , we have as . Consequently,

So holds.

(A2) Condition (H1) implies that there exist positive constants ,   such that

Let be such that . Then

Note that the space has finite dimension; then all norms are equivalents and we obtain

Finally,

since . The assertion (A2) is then satisfied and the proof of Theorem 2 is complete.

Proof of Theorem 3. We will use Ricceri three critical points Theorem (see [20, 21]). Notice that is a continuous convex functional, so it is weakly lower semicontinuous and its inverse derivative is continuous. From Theorem 9 the precondition of Lemma 8 is satisfied. Now we only need to verify that conditions (i), (ii), and (iii) in Lemma 8 are fulfilled.

Firstly for such that , we have

Using the Hölder inequality and the Sobolev embedding theorem, we have for some positive constants and

where . Combining all together we obtain

On the other hand,

Then for any ,

From we obtain

then (i) of Lemma 8 is verified.

Secondly, letting , then obviously we have .

We claim that there exist and such that and

From (H5), there exits and such that

In view of (H0), if we put

where is a positive constant, then we have

Fix such that . If , then, by the Sobolev embedding theorem is continuous), we have

where and are two positive constants.

Since , then we have

By (H4), we can choose a constant such that .

Fix such that .

If , then we have

From (56) and (57), we deduce that, when , then and

Thus

Since , we get

Then

with which implies that

So we can find , and satisfying (ii) and (iii) of Lemma 8.

If , we have

From (56) and (63), we deduce that, when , then and

Thus

Since , we get

Then

with .

Therefore

Then we obtain

This means that condition (iii) in Lemma 8 is verified. Now since all the assumptions of Lemma 8 are verified, there exist an open interval and a positive constant such that for any the equation has at least three solutions in whose norms are less than .

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