Existence and multiplicity solutions for a singular elliptic p(x)-Laplacian equation

Abstract

This paper deals with the existence and multiplicity of nontrivial weak solutions for the following equationinvolving variable exponents:\begin{align*}\begin{cases}-\vartriangle_{p(x)}u+\dfrac{\vert u\vert^{r-2}u}{|x|^{r}}=\lambda h(x,u),&in ~\Omega,\\u=0,&on~\partial\Omega,\end{cases}\end{align*}where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ with smooth enough boundary which is subject to Dirichlet boundary condition.Using a variational method and Krasnoselskii's genus theory, we would show the existence andmultiplicity of the solutions. Next, we study closedness of set of eigenfunctions, such that $p(x)\equiv p$.