Singular minimization problems
type: Item
instance of: scholarly article
title: Singular minimization problems
zbMATH Open document ID: 0954.35074
author: Zhi-Qiang Wang
author: Michel Willem
published in: Journal of Differential Equations
publication date: 2001-02-01T00:00:00Z
review text: It is considered the existence of ground state solutions of the following two degenerate elliptic equations on \({\mathbb{R}}^N\): \[ -\text{div} (|x|^{-2a} \nabla u)=|x|^{-bp} |u|^{p-2} u, \quad x\in{\mathbb{R}}^N, \] and \[ -\text{div} (|x|^{-2a} \nabla u)+\lambda |x|^{-2(1+a)} u = |x|^{-bp} |u|^{p-2} u,\quad x\in{\mathbb{R}}^N, \] where \(N\geq 3, 0\leq a \leq(N-2)/2, \) \(a\leq b\leq a+1, \) \(p=\frac{2N}{N-2+2(b-a)},\) and \(\lambda\) is a real parameter. On the strength of an inequality established by Caffarelli, Kohn and Nirenberg, the problems under consideration allow variational formulations for the above parameter range, and the corresponding minimization problems with constraints are formulated. It is proved that minimizing sequences for stated minimization problems are relatively compact up to dilation under \(0\leq a<(N-2)/2, a\leq b<a+1\), and some another conditions on \(a,b,\lambda.\)
reviewed by: Dmitry V. Larin
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 1437329
zbMATH Keywords: degenerate elliptic equation
zbMATH Keywords: singular minimization problem
zbMATH Keywords: existence
zbMATH Keywords: ground state solution
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3699
OpenAlex ID: W1974549290
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DOI: 10.1006/JDEQ.1999.3699
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description: scientific article; zbMATH DE number 1437329