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 .