Singular solutions for some semilinear elliptic equations
type: Item
instance of: scholarly article
title: Singular solutions for some semilinear elliptic equations
zbMATH Open document ID: 0635.35035
DOI: 10.1007/BF00284508
author: Luc Oswald
author: Haïm Brezis
published in: Archive for Rational Mechanics and Analysis
publication date: 1987-01-01T00:00:00Z
review text: This paper studies solutions \(u\in C\) \(2(B_ R\setminus 0)\) of the equation \(-\Delta u+u\) \(p=0\), \(u\geq 0\) on \(B_ R\setminus 0\), the dimension of the underlying space being N. When \(1<p<N/(N-2)\) there are solutions with a singularity at 0. The presented paper shows that if non- negative smooth Dirichlet boundary data are prescribed there is a unique solution which is \(C^ 2\) on the closure of the domain. Moreover a unique solution can be specified satisfying the boundary conditions by giving an asymptotic form for the singularity at 0. The uniqueness statement as a consequence of the imposition of boundary data extends work of Veron. In addition the present paper removes the restriction to a ball and obtains the analogous results for any smooth domain.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 4034217
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: singular solutions
zbMATH Keywords: singularity
zbMATH Keywords: Dirichlet boundary data
zbMATH Keywords: uniqueness
zbMATH Keywords: smooth domain
reviewed by: John F. Toland
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00284508
OpenAlex ID: W2068257120
description: scientific article; zbMATH DE number 4034217