Symmetry-breaking for positive solutions of semilinear elliptic equations
type: Item
instance of: scholarly article
title: Symmetry-breaking for positive solutions of semilinear elliptic equations
zbMATH Open document ID: 0629.35040
DOI: 10.1007/BF00251359
published in: Archive for Rational Mechanics and Analysis
publication date: 1986-01-01T00:00:00Z
review text: Consider the Dirichlet problem \[ (*)\quad \Delta u(x)+f(u(x))=0,\quad x\in B_ R;\quad u(x)=0,\quad x\in \partial B_ R \] where \(f\in C^ 1\) and \(B_ R\subset {\mathbb{R}}^ n\) denotes the open ball of radius \(R>0\). From the results of Gidas, Ni and Nirenberg all positive solutions u of (*) are radial functions such that \(u_ r(r)<0\), \(0<r<R\) and consequently they solve the problem \[ u_{rr}+((n-1)/r)u_ r+f(u)=0,\quad 0<r<R;\quad u_ r(0)=u(R)=0. \] So it is natural to think of solutions of (*) as symmetric functions. In the paper a few results providing necessary and also sufficient conditions for symmetry breaking and infinitesimal symmetry breaking are given. A bifurcation Theorem of Vanderbauwhede which extends an earlier result of Crandall and Rabinowitz is used.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 4023839
zbMATH Keywords: semilinear
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: positive solutions
zbMATH Keywords: radial functions
zbMATH Keywords: symmetry breaking
zbMATH Keywords: bifurcation
author: Joel Smoller
author: Arthur G. Wasserman
reviewed by: José Valdo A. Gonçalves
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00251359
OpenAlex ID: W2063357204
description: scientific article; zbMATH DE number 4023839