Curves of sign-changing solutions for semilinear equations
type: Item
instance of: scholarly article
title: Curves of sign-changing solutions for semilinear equations
zbMATH Open document ID: 1173.35475
DOI: 10.1016/S0362-546X(01)00863-X
author: Philip Korman
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-11-28T00:00:00Z
review text: Positive and sign-changing solution curves for the Dirichlet problem: \[ \Delta u+\lambda f(u)=0\quad\text{ for}\,\, | x| <R\,,\quad u=0 \quad\text{ for}\,\, | x| =R, \leqno{(1)} \] \(x\in \mathbb{R}^N, \lambda>0\), are studied. Under the condition \(f(u)/u\to\infty\) as \(u\to\infty\), an asymptotic shape of positive solutions to (1) is found for large \(\lambda\). This result is then used to prove non-existence of positive solutions for large \(\lambda\) if \(f\) has no stable roots, extending significantly results of \textit{K.J. Brown}, \textit{A. Castro} and \textit{R. Shivaji} [Differ. Integral Equ. 2, No. 4, 541--545 (1989, Zbl 0736.35039)] (positive solutions to (1) are always radially symmetric). Similar asymptotic results are also found for radially symmetric sign-changing solution. Using the non-existence results of radial solutions the author study symmetry-breaking off a branch of positive solutions to (1). The main result in this direction is that the radial symmetry is lost forward at some \(\lambda=\overline\lambda\) and that a family of symmetry breaking-solutions bifurcates at \(\lambda=\overline\lambda\). Some refined results for ODE are provided as well.
reviewed by: Petr Girg
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1836616
zbMATH Keywords: solution curves
zbMATH Keywords: Pitchfork bifurcation
zbMATH Keywords: symmetry breaking
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1836616