Multiplicity of solutions for a nonlinear degenerate problem in anisotropic variable exponent spaces
type: Item
instance of: scholarly article
title: Multiplicity of solutions for a nonlinear degenerate problem in anisotropic variable exponent spaces
zbMATH Open document ID: 1270.49005
author: Denisa Stancu-Dumitru
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2013-03-11T00:00:00Z
full work available at URL: vol36 1 11.html
review text: The author of this paper studies the following anisotropic problem \[ -\sum_{i=1}^N\partial_{x_i}(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u)=\lambda (u^{\alpha(x)-1}-u^{\beta(x)-1})\,\,\,\, \text{in}\,\,\, \Omega \] \[ u=0 \,\,\,\, \text{on}\,\,\, \partial\Omega, \] \[ u\geq 0 \,\,\,\, \text{in}\,\,\, \Omega\,, \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\), \(N\geq 3\), with smooth boundary \(\partial \Omega\), \(p_i\) are continuous functions on \(\overline \Omega\) such that \(p_i(x)\geq 2\) for any \(x\in \overline \Omega\) and \(i=1, \dots, N\), \(\lambda\) is a real parameter, while \(\alpha, \beta: \overline \Omega \to \mathbb R\) are continuous functions such that \[ 1<\beta^-\leq \beta(x)\leq \alpha(x)\leq \alpha^+<P^-_-\leq P^+_+<P_{-, \infty}\,, \qquad x\in \Omega \] and there exists \(x_0\in \Omega\) such that \[ \beta(x_0) < \alpha(x_0)\,. \] Here \(P_+^+=\max\{p_1^+, \dots p_N^+\}\), \(P_-^+=\max\{p_1^-, \dots p_N^-\}\), \(P_-^-=\min\{p_1^-, \dots p_N^-\}\) and \(P_{-, \infty}=\max\{P_-^+, P_-^*\}\), where \[ P_-^*=\frac{N}{\sum_{i=1}^N 1/p_i^- -1} \] with \(\sum_{i=1}^N 1/p_i^- >1\). In the main result of the paper, the author proves that the problem admits at least two distinct nontrivial nonnegative weak solutions, provided the parameter \(\lambda\) is sufficiently large. The proof of this result is based on variational methods. Precisely, the first solution is found using the direct methods in the calculus of variations, that is as a minimum of the energy functional naturally associated with the problem while the second solution comes from the mountain pass theorem.
reviewed by: Raffaella Servadei
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35D30
zbMATH DE Number: 6143143
zbMATH Keywords: anisotropic problem
zbMATH Keywords: variational methods
zbMATH Keywords: direct methods in the calculus of variations
zbMATH Keywords: Mountain Pass Theorem
zbMATH Keywords: weak solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6143143