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2018 The fibering map approach to a $p(x)$-Laplacian equation with singular nonlinearities and nonlinear Neumann boundary conditions
Kamel Saoudi
Rocky Mountain J. Math. 48(3): 927-946 (2018). DOI: 10.1216/RMJ-2018-48-3-927

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.

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Kamel Saoudi. "The fibering map approach to a $p(x)$-Laplacian equation with singular nonlinearities and nonlinear Neumann boundary conditions." Rocky Mountain J. Math. 48 (3) 927 - 946, 2018. https://doi.org/10.1216/RMJ-2018-48-3-927

Information

Published: 2018
First available in Project Euclid: 2 August 2018

zbMATH: 06917355
MathSciNet: MR3835580
Digital Object Identifier: 10.1216/RMJ-2018-48-3-927

Subjects:
Primary: 35J20, 35J60, 35J70, 46E35, 47J10

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 3 • 2018
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