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

TL;DR

This paper investigates the existence of multiple positive solutions for a singular p(x)-Laplace equation using the Nehari manifold method, addressing variable exponent problems with singular and nonlinear terms.

Contribution

It introduces a novel application of the Nehari manifold approach to singular p(x)-Laplace equations, establishing multiplicity results under new hypotheses.

Findings

Proves existence of multiple positive solutions for the problem.

Develops new techniques for handling variable exponents and singularities.

Extends the Nehari manifold method to a broader class of nonlinear PDEs.

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_\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 ${{(\rm P_\lambda)}}$.