Symmetry-breaking at positive solutions
type: Item
instance of: scholarly article
title: Symmetry-breaking at positive solutions
zbMATH Open document ID: 0629.35014
DOI: 10.1007/BF00945416
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1987-01-01T00:00:00Z
review text: Boundary value problems of the type \[ (*)\quad \Delta w(x)+f(w(x),\lambda)=0,\quad x\in \Omega;\quad w(x)=0,\quad x\in \partial \Omega, \] are considered, where \(f: {\mathbb{R}}\times {\mathbb{R}}\to {\mathbb{R}}\) is a smooth function, \(\Omega =\{x\in {\mathbb{R}}^ n\); \(\| x\| <1\}\), \(n\geq 2\). If \(\lambda_ 0\), \(w_ 0\) satisfy (*) and \(w_ 0(x)>0\) for all \(x\in \Omega\) then \(w_ 0\) is spherically symmetric. It is proved that under certain assumptions there is a symmetry-breaking bifurcation at \(\lambda_ 0\), \(w_ 0\), i.e. any neighborhood of \(\lambda_ 0\), \(w_ 0\) contains solutions which are not spherically symmetric. Moreover, the bifurcating symmetry-breaking solutions are axisymmetric.
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 4023813
zbMATH Keywords: positive solutions
zbMATH Keywords: spherically symmetric
zbMATH Keywords: symmetry-breaking bifurcation
zbMATH Keywords: axisymmetric
author: A. Vanderbauwhede
reviewed by: Milan Kučera
MaRDI profile type: Publication
cites work: Symmetry breaking for a class of semilinear elliptic problems
cites work: Symmetry and related properties via the maximum principle
cites work: Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations
cites work: Symmetry-breaking for positive solutions of semilinear elliptic equations
cites work: Q3326480
full work available at URL: https://doi.org/10.1007/bf00945416
OpenAlex ID: W1963690981
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description: scientific article; zbMATH DE number 4023813