Entire solutions of singular elliptic equations
type: Item
instance of: scholarly article
title: Entire solutions of singular elliptic equations
zbMATH Open document ID: 0679.35003
DOI: 10.1016/0022-247X(89)90126-1
author: Allan L. Edelson
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: This paper is concerned with the existence and asymptotic properties of solutions to the singular elliptic equation \[ -\Delta u=f(x)u^{- \lambda},\quad \lambda >0. \] It is assumed that f is positive and locally \(\alpha\)-Hölder continuous in \(R^ n\). The main results give sufficient conditions for the existence of positive, entire solutions having asymptotic behavior \[ (i)\quad u(x)\sim | x|^{2- n},\quad n\geq 3;\quad (ii)\quad u(x)\sim \log (x),\quad n=2. \] The method is based on the Schauder fixed point theorem and classical integral operator equations in \(R^ n\). It is of particular interest because it gives better information about rate of decay as \(| x| \to \infty\), and it generalizes to higher order equations where methods based on the maximum principle do not apply.
reviewed by: Allan L. Edelson
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 4113100
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: asymptotic equivalence
zbMATH Keywords: singular equation
zbMATH Keywords: Schauder fixed point theorem
zbMATH Keywords: integral operator equations
zbMATH Keywords: higher order equations
MaRDI profile type: Publication
cites work: Q3672366
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full work available at URL: 0022-247x(89)90126-1
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description: scientific article; zbMATH DE number 4113100