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International Journal of Partial Differential Equations

Volume 2013 (2013), Article ID 364251, 7 pages

http://dx.doi.org/10.1155/2013/364251
Research Article

Solutions of Nonlocal -Laplacian Equations

1Faculty of Economics and Administrative Sciences, Batman University, 72000 Batman, Turkey

2Faculty of Education, Bayburt University, 69000 Bayburt, Turkey

Received 5 March 2013; Accepted 10 September 2013

Academic Editor: William E. Fitzgibbon

Copyright © 2013 Mustafa Avci and Rabil Ayazoglu (Mashiyev). 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

In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type equation involving -Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.

1. Introduction

We study the existence and multiplicity of solutions of the nonlocal equation where ( ) is a smooth bounded domain, such that for any , and .

The problem is related to the stationary version of a model, the so-called Kirchhoff equation, introduced by [1]. To be more precise, Kirchhoff established a model given by the equation where , , , , and are constants, which extends the classical D’Alambert’s wave equation, by considering the effects of the changes in the length of the strings during the vibrations. A distinguish feature of the Kirchhoff equation (1) is that the equation contains a nonlocal coefficient which depends on the average of the kinetic energy on , and hence the equation is no longer a pointwise identity. For Kirchhoff-type equations involving the -Laplacian operator, see, for example, [24].

The -Laplacian operator is a natural generalization of the -Laplacian operator where is a real constant. The main difference between them is that -Laplacian operator is homogenous, that is, for every , but the -Laplacian operator, when is not constant, is not homogeneous. This causes many problems; some classical theories and methods, such as the Lagrange multiplier theorem and the theory of Sobolev spaces, are not applicable. For -Laplacian operator, we refer the readers to [59] and references there in. Moreover, the nonlinear problems involving the -Laplacian operator are extremely attractive because they can be used to model dynamical phenomenons which arise from the study of electrorheological fluids or elastic mechanics. Problems with variable exponent growth conditions also appear in the modelling of stationary thermorheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes of filtration of an ideal barotropic gas through a porous medium. The detailed application backgrounds of the -Laplacian can be found in [1014] and the references therein.

In the present paper, by considering the joint effects of different ( )-Laplace operator , we study the existence and multiplicity of solutions for a nonlocal problem, that is, problem via Mountain-Pass theorem and Fountain theorem. As far as we know, there is no paper that deals with a nonlocal problem involving ( )-Laplace operator except [15] in which the authors consider problem for the case and . Therefore, our paper deals with more general results than those obtained in [15]. Moreover, if we choose the functions in problem , we get the equation which is the well-known anisotropic -Laplacian problem (see, e.g., [16] and references therein) in the case , that is, As mentioned above, the -Laplacian can be applied to describe the physical phenomenon with pointwise different properties which earliest arose from the nonlinear elasticity theory. In that context, the systems involving the -Laplacian (or -Laplacian) can be good candidates for modeling phenomena which ask for distinct behavior of partial differential derivatives in various directions. For a mathematical model of a real physical phenomenon, one can consider the mean curvature operator It is obvious that problem is a degenerate version of (4) when .

2. Preliminaries

We state some basic properties of the variable exponent Lebesgue-Sobolev spaces and , where is a bounded domain (for details, see, e.g., [1719]).

Set For any , denote and define the variable exponent Lebesgue space by We define a norm, the so-called Luxemburg norm, on by the formula and then becomes a Banach space.

Define the variable exponent Sobolev space by then it can be equipped with the norm The space is defined as the closure of in with respect to the norm . For , we can define an equivalent norm since Poincaré inequality holds; that is, there exists a positive constant such that for all .

Proposition 1 (see [18, 19]). The conjugate space of is , where . For any and , we have

Proposition 2 (see [18, 19]). Denote , for all ; one has(i) ;(ii) .

Proposition 3 (see [18, 19]). If  , then the following statements are equivalent:(i) ;(ii) ;(iii) in measure and .

Proposition 4 (see [18, 19]). If , then spaces , , and are separable and reflexive Banach spaces.

If and for any if and if , then the embedding is compact and continuous.

Definition 5. Let be a Banach space and a functional. We say that a functional satisfies the Palais-Smale condition ( for short), if any sequence in such that is bounded and as admits a convergent subsequence.

3. Main Results and Proofs

Let us consider the functional where with its norm given by , for all . It is obvious that is also a separable and reflexive Banach space.

By using standard arguments, it can be proved that (see [20]), and the ( )-Laplace operator is the derivative operator of in the weak sense. Denote ; then where is the dual pair between and its dual .

Let us denote By the definition, it is not difficult to see that , . For such that for any , we have , and the imbedding is continuous and compact.

We say that is a weak solution of if for any .

We associate to the problem the energy functional, defined as : where ( .) and . We know that from and (see below) is well defined and in a standard way we can prove that and that the critical points of are solutions of .

Moreover, the derivative of is given by for any .

Now, we are ready to set and prove the first main result of the present paper.

Theorem 6. Assume that the following assumptions hold: are continuous functions and satisfy the conditions for all , where and are positive constants and ; is a Carathéodory function and satisfies the growth condition where and are positive constants and such that , for all .

Then problem has a weak solution.

Proof. Let . By the assumptions and , we have where , and . So, is coercive. Since is sequentially weakly lower semicontinuous, has a minimum point in and is a weak solution of .

Theorem 7. Assume that the following assumptions hold: are continuous functions and satisfy the conditions for all , where , , , , and are positive constants such that and ; is a Carathéodory function and satisfies the growth condition , uniformly  for , where and are positive constants and such that , for all ; ; , such that Then problem has at least one nontrivial weak solution.

To obtain the result of Theorem 7, we need to show Lemmas 8 and 9 hold.

Lemma 8. Suppose , , and hold. Then I satisfies condition.

Proof. Let us assume that there exists a sequence in such that Then by the assumptions (26), , and , we get where . Since , we have for large enough. Therefore, is bounded in . Passing to a subsequence, if necessary, there exists such that . Therefore, we have the embeddings By (26), we have . Thus From and Proposition 1, it follows that If we consider the relations given in (28), we get Hence, From , it follows that Furthermore, since in , we have From (33) and (34), we deduce that Next, we apply the following well-known inequality valid for all ( ). From the relations (35) and (36), we infer that and, consequently, in . We are done.

Lemma 9. Suppose , and hold. Then the following statements hold: (i)there exist two positive real numbers and such that , with ;(ii)there exists such that , .

Proof. (i) Let . Then by and Proposition 2, we have where . Since for all , we have the continuous embeddings and , and also there are positive constants and such that Let be small enough such that . By the assumptions and , we have , for all .

Then, for it follows that Therefore, there exists two positive real numbers and such that , for all with .

(ii) From it follows that , for all and . In the other hand, when from we obtain that Hence, for any fixed and we have which implies .

Proof of Theorem 7. From Lemmas 8 and 9 and the fact that , satisfies the Mountain-Pass theorem (see [20, 21]). Therefore, has at least one nontrivial critical point; that is, has a nontrivial weak solution. The proof is complete.

In the following, we will prove the second main result of the present paper.

Theorem 10. Suppose , , , , and hold. Then has a sequence of critical points such that and has infinite many pairs of solutions.

Since is a reflexive and separable Banach space, then there are and such that For convenience, we write , , and .

Lemma 11. If such that for any , denote Then .

Since the proof of Lemma 11 is similar to that of Lemma 4.9 in [7], we omit it.

Proof of Theorem 10. By the assumptions , , and , satisfies condition and from it is also an even functional. In the sequel, we will show that if is large enough, then there exist such that(i) ;(ii) .

Therefore, to obtain the results of Theorem 10 it is enough to apply Fountain theorem (see [21]).(i) For any with big enough, we have Set . Because and , we have (ii) From , we have . Because and , it is obvious that as for .

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