MULTIPLE SOLUTIONS OF A $p(x)$-LAPLACIAN EQUATION INVOLVING CRITICAL NONLINEARITIES

Abstract

In this paper, we consider the existence of multiple solutions for the following $p(x)$-Laplacian equations with critical Sobolev growth conditions \[ \begin{cases} -div(|\nabla u|^{p(x)-2} \nabla u) + |u|^{p(x)-2} u = f(x,u) \; \textrm{in } \Omega, \\ u = 0 \; \textrm{on } \partial \Omega.\end{cases} \]

We show the existence of infinitely many pairs of solutions by applying the Fountain Theorem and the Dual Fountain Theorem respectively. We also present a variant of the concentration-compactness principle, which is of independent interest.