A regularity criterion for positive weak solutions of \(- {\Delta}u=u^{\alpha}\)
type: Item
instance of: scholarly article
title: A regularity criterion for positive weak solutions of \(- {\Delta}u=u^{\alpha}\)
zbMATH Open document ID: 0799.35070
DOI: 10.1007/BF02565810
published in: Commentarii Mathematici Helvetici
publication date: 1994-11-14T00:00:00Z
full work available at URL: https://eudml.org/doc/140294
review text: Let \(\Omega\subset \mathbb{R}^ n\) be open, \(\alpha> n/(n-2)\) and \(u\in L^ \alpha (\Omega)\) a positive weak solution to \(-\Delta u= u^ \alpha\). The author shows: If the map \[ x\in \Omega \mapsto \int_ \Omega {{u(y)^{\alpha-1}} \over {| x-y|^{n-2}}} dy\in (- \infty,\infty] \] is continuous then the Hausdorff dimension of the singular set of \(u\) is less than or equal to \(n- 2\alpha/ (\alpha-1)\).
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 218288
zbMATH Keywords: partial regularity
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: singular set
author: Frank Pacard
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 218288