Rocky Mountain Journal of Mathematics

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

Kamel Saoudi

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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.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 3 (2018), 927-946.

Dates
First available in Project Euclid: 2 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1533230833

Digital Object Identifier
doi:10.1216/RMJ-2018-48-3-927

Mathematical Reviews number (MathSciNet)
MR3835580

Zentralblatt MATH identifier
06917355

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 35J70: Degenerate elliptic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05]

Keywords
Nehari manifold fibering map singular equation Neumann boundary condition weak solution $p(x)$-Laplace operator generalized Lebesgue Sobolev spaces

Citation

Saoudi, Kamel. The fibering map approach to a $p(x)$-Laplacian equation with singular nonlinearities and nonlinear Neumann boundary conditions. Rocky Mountain J. Math. 48 (2018), no. 3, 927--946. doi:10.1216/RMJ-2018-48-3-927. https://projecteuclid.org/euclid.rmjm/1533230833


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