Multiple positive solutions for a semilinear equation with prescribed singularity
type: Item
instance of: scholarly article
title: Multiple positive solutions for a semilinear equation with prescribed singularity
zbMATH Open document ID: 1141.35371
author: J. Chen
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-21T00:00:00Z
review text: The author deals with the study of the existence and multiplicity of positive solutions for the following problem \[ -\Delta u-\mu V(x)u= K(x)|u|^{2^*-2} u+\theta h(x),\quad u\in H^1_0(\Omega),\tag{1} \] where \(\Omega\subset\mathbb{R}^N\) \((N\geq 3)\) is an open bounded domain with smooth boundary \(\partial\Omega\), \(0\leq\mu< \widetilde\mu= {(N-2)^2\over 4}\) and \(\widetilde\mu\) is the best constant in the Hardy inequality. \(2^*= {2N\over N-2}\) is the so-called critical Sobolev exponent. The weight \(V(x)\) has finitely many singular points. The author adresses to the following question: If \(V(x)\) has finitely many singular points, say \(m\) in \(\Omega\), does it produce at least \(m\) really positive solutions to (1)? The result of the author shows that the answer is affirmative. To this end, the author uses a variational method.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 2159780
zbMATH Keywords: Variational methods
zbMATH Keywords: Multiple positive solutions
zbMATH Keywords: Singular potential
zbMATH Keywords: Critical Sobolev exponent
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.10.057
OpenAlex ID: W2085585240
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DOI: 10.1016/J.JMAA.2004.10.057
description: scientific article; zbMATH DE number 2159780