The Nehari manifold approach for singular equations involving the p(x)-Laplace operator

Abstract

We study the following singular problem involving the p$(x)$-Laplace operator ${\Delta }_{p(x)}u=div(|\nabla u{|}^{p(x)-2}\nabla u)$, where $p(x)$ is a nonconstant continuous function, \begin{equation} \nonumber {(\rm P_λ)} \left\{\begin{aligned} - Δ_{p(x)} u & = a(x)|u|^{q(x)-2}u(x)+ \frac{λb(x)}{u^{δ(x)}} \quad\mbox{in}\,Ω,\\ u &>0 \quad\mbox{in}\,Ω, \\ u & =0 \quad\mbox{on}\,\partialΩ.\end{aligned} \right. \end{equation} Here, $\Omega$ is a bounded domain in ${\mathbb{R}}^{N\ge 2}$ with $C^2$-boundary, $\lambda$ is a positive parameter, $a(x),b(x)\in C(\overline{\Omega })$ are positive weight functions with compact support in $\Omega$, and $\delta (x),$ $p(x),$ $q(x)\in C(\overline{\Omega })$ satisfy certain hypotheses ($A_0$) and ($A_1$). We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem $({\mathrm{P}}_{\lambda })$.