Existence of solutions for the \(p(x)\)-Laplacian problem with the critical Sobolev-Hardy exponent
type: Item
instance of: scholarly article
title: Existence of solutions for the \(p(x)\)-Laplacian problem with the critical Sobolev-Hardy exponent
zbMATH Open document ID: 1250.35107
DOI: 10.1155/2012/894925
author: Yu Mei
author: Fu Yongqiang
author: Li Wang
published in: Abstract and Applied Analysis
publication date: 2012-09-07T00:00:00Z
review text: Summary: This paper deals with the \(p(x)\)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness in \(W^{1,p(x)}_0 (\Omega)\) space is established, then by applying it we obtain the existence of solutions for the following \(p(x)\)-Laplacian problem: \(-\text{div}(|\nabla u|^{p(x)-2} \nabla u) + |u|^{p(x)-2}u = (h(x)|u|^{p^\ast_s(x)-2} u/|x|^{s(x)}) + f(x, u), x \in \Omega, u = 0, x \in \partial\Omega\), where \(\Omega \subset \mathbb R^N\) is a bounded domain, \(0 \in \Omega, 1 < p^- \leq p(x) \leq p^+ < N\), and \(f(x, u)\) satisfies \(p(x)\)-growth conditions.
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 6078829
zbMATH Keywords: \(p(x)\)-Laplacian equation
zbMATH Keywords: critical obolev-Hardy exponent
Wikidata QID: Q58697038
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/894925
OpenAlex ID: W2142455966
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description: scientific article; zbMATH DE number 6078829