The fibering map approach to a $p(x)$-Laplacian equation with singular nonlinearities and nonlinear Neumann boundary conditions

Abstract

The purpose of this paper is to study the singular Neumann problem involving the p$(x)$-Laplace operator: \begin{equation} (P_\lambda)\qquad\begin{cases} - \Delta_{p(x)} u +|u|^{p(x)-2}u = \frac{\lambda a(x)}{u^{\delta(x)}} &\mbox{in }\Omega, \\ u\gt0 &\mbox{in } \Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial\nu} = b(x) u^{q(x)-2}u &\mbox{on } \partial\Omega, \end{cases} \end{equation} where $\Omega\subset\mathbb{R}^N$, $N\geq 2$, is a bounded domain with $C^2$ boundary, $\lambda$ is a positive parameter, $a, b\in C(\overline{\Omega})$ are non-negative weight functions with compact support in $\Omega­$ and $\delta(x),$ $p(x),$ $q(x) \in C(\overline{\Omega})$ are assumed to satisfy the assumptions (A0)--(A1) in Section 1. We employ the Nehari manifold approach and some variational techniques in order to show the multiplicity of positive solutions for the $p(x)$-Laplacian singular problems.