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

We study the following singular problem involving the p$(x)$-Laplace operator ${\mathrm{\Delta }}_{p(x)}u=div(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(x)-2}\mathrm{\nabla }u)$, where $p(x)$ is a nonconstant continuous function, \begin{equation} \nonumber {{(\rm P_\lambda)}} \left{\begin{aligned} - \Delta_{p(x)} u & = a(x)|u|^{q(x)-2}u(x)+ \frac{\lambda b(x)}{u^{\delta(x)}} \quad\mbox{in},\Omega,\ u &>0 \quad\mbox{in},\Omega, \ u & =0 \quad\mbox{on},\partial\Omega.\end{aligned} \right. \end{equation} Here, $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^{N\ge 2}$ with $C^2$-boundary, $\lambda$ is a positive parameter, $a(x),b(x)\in C(\bar{\mathrm{\Omega }})$ are positive weight functions with compact support in $\mathrm{\Omega }$, and $\delta (x),$ $p(x),$ $q(x) \in C(\overline{\Omega})$satisfycertainhypotheses($A_{0}$)and($A_{1}$). We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem $({\mathrm{P}}_{\lambda })$.