Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities
type: Item
instance of: scholarly article
title: Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities
zbMATH Open document ID: 1014.35032
author: Filomena Pacella
published in: Journal of Functional Analysis
publication date: 2002-09-08T00:00:00Z
review text: This paper is devoted to the symmetry properties of classical \(C^2(\Omega) \cap C(\overline \Omega)\) solutions of elliptic problems fo the type \[ \begin{cases} -\Delta u=f(x,u) \quad & \text{in }\Omega,\\ u=g(x)\quad & \text{on } \partial\Omega,\end{cases} \tag{1} \] where \(\Omega\) is a bounded, somehow symmetric domain in \(\mathbb{R}^N\), \(N\geq 2\), \(f\) and \(g\) are given functions having some symmetry in \(x\). The author shows that all solutions of (1) of index one are axially symmetric when \(\Omega\) is an annulus or a ball, \(g\equiv 0\) and \(f\) is strictly convex in \(u\). To this end, the author uses spectral properties of the linearized operator.
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1797600
zbMATH Keywords: axially symmetric
zbMATH Keywords: spectral properties of the linearized operator
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: jfan.2001.3901
OpenAlex ID: W2018018963
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DOI: 10.1006/JFAN.2001.3901
description: scientific article; zbMATH DE number 1797600