Existence and multiplicity of solutions for a Dirichlet problem involving perturbed p(x)-Laplacian operator

Abstract

In this article we study the existence of solutions for the Dirichlet problem $$\displaylines{ -\text{div}(| \nabla u |^{p(x)-2}\nabla u)+V(x)|u|^{q(x)-2}u =f(x,u)\quad \text{in }\Omega,\cr u=0\quad \text{on }\partial \Omega, }$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, V is a given function in a generalized Lebesgue space $L^{s(x)}(\Omega)$ and f(x,u) is a Caratheodory function which satisfies some growth condition. Using variational arguments based on "Fountain theorem" and "Dual Fountain theorem", we shall prove under appropriate conditions on the above nonhomogeneous quasilinear problem the existence of two sequences of weak solutions for this problem.

Keywords

• p(x)-Laplacian operator
• generalized Lebesgue-Sobolev spaces
• critical point
• Fountain theorem
• dual Fountain theorem