Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions
type: Item
instance of: scholarly article
title: Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions
zbMATH Open document ID: 0608.35004
DOI: 10.1007/BF01205935
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: Extending their results on symmetry-breaking for positive solutions for semilinear elliptic equations under Dirichlet conditions [Arch. Ration. Mech. Anal. 95, 217-225 (1986)], the authors study now the problem \(\Delta u+f(u)=0\) on balls \(B_ R\subset {\mathbb{R}}^ n\) with general boundary conditions \(\alpha u+\beta (\partial u/\partial u)=0\) on \(\partial B_ R\). The symmetry is said to break if one has local bifurcation (parameter R) from radially symmetric solutions. Especially for homogeneous Neumann conditions, they give a theorem that under certain technical assumptions on f there is an infinite sequence of radial solutions, at which the symmetry breaks and the dimension of the asymmetric manifold is at least n. In fact the kernels of the corresponding linearized operators have increasing dimensions, showing that the situation is more complicated than in the Dirichlet case. For the general case, bounds for the dimensions of the asymmetric manifolds and theorems on the structure of the kernels of the linearized operators are given.
reviewed by: Michael Wiegner
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 3983695
zbMATH Keywords: symmetry-breaking
zbMATH Keywords: positive solutions
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: Dirichlet conditions
zbMATH Keywords: local bifurcation
zbMATH Keywords: radially symmetric solutions
zbMATH Keywords: Neumann conditions
author: Joel Smoller
author: Arthur G. Wasserman
MaRDI profile type: Publication
cites work: Q5518037
cites work: Le spectre d'une variété riemannienne. (The spectrum of a Riemannian manifold)
cites work: On Non-Radially Symmetric Bifurcation
cites work: Symmetry and related properties via the maximum principle
cites work: Topics in harmonic analysis on homogeneous spaces
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: An existence theorem for positive solutions of semilinear elliptic equations
cites work: Q4746129
cites work: Q3326480
full work available at URL: https://doi.org/10.1007/bf01205935
OpenAlex ID: W2037558777
description: scientific article; zbMATH DE number 3983695