  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_\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, $\Omega$ is a bounded domain in
$\mathbb{R}^{N\geq2}$ 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 ${{(\rm P_\lambda)}}$.

Analysis of PDEs;Functional Analysis