Solutions of Nonlocal -Laplacian Equations

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, [2–4]. 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 [5–9] 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 [10–14] 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