Solutions of an anisotropic nonlocal problem involving variable exponent

Abstract

The present paper deals with an anisotropic Kirchhoff problem under homogeneous Dirichlet boundary conditions, set in a bounded smooth domain ? of ?N (N ? 3). The problem studied is a stationary version of the original Kirchhoff equation, involving the anisotropic p?(.)-Laplacian operator, in the framework of the variable exponent Lebesgue and Sobolev spaces. The question of the existence of weak solutions is treated. Applying the Mountain Pass Theorem of Ambrosetti and Rabinowitz, the existence of a nontrivial weak solution is obtained in the anisotropic variable exponent Sobolev space W0 1p?(.) (?), provided that the positive parameter ? that multiplies the nonlinearity f is small enough. © de Gruyter 2013.