Symmetry results for semilinear elliptic systems in the whole space
type: Item
instance of: scholarly article
title: Symmetry results for semilinear elliptic systems in the whole space
zbMATH Open document ID: 0952.35033
author: Jérôme Busca
author: Boyan Sirakov
published in: Journal of Differential Equations
publication date: 2001-01-04T00:00:00Z
review text: By means of the ``moving plane'' method, the symmetry of classical solutions for the elliptic system: \[ \begin{cases} \Delta u_{i}+f_{i}(r,u_{1},....,u_{n})=0 & \text{ in }\mathbb{R}^{N},\;i=1,...,n, \\ u_{i}>0 & \text{ in }\mathbb{R}^{N}, \\ u_{i}(x)\rightarrow 0 & \text{ as } r=|x|\rightarrow \infty ,\end{cases} \] is proved, where \(n\geq 1\) and \(N\geq 2\). It generalizes related results from the case of bounded domains. For instance: \textit{W. C. Troy} [J. Differ. Equations 42, 400-413 (1981; Zbl 0486.35032)], \textit{D. G. de Figueiredo} [NoDEA, Nonlinear Differ. Equ. Appl. 1, No.~2, 119-123 (1994; Zbl 0822.35039)], and \textit{A. W. Shaker} [Appl. Anal. 41, No.~1-4, 1-9 (1991; Zbl 0724.35022)]. Furthermore, a few simple examples for the nonexistence of positive solutions are given when one of the main assumptions to \(f_{i}\) is not satisfied.
reviewed by: Mervan Pašić
Mathematics Subject Classification ID: 35J45
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1454725
zbMATH Keywords: semilinear elliptic systems
zbMATH Keywords: radial symmetry
zbMATH Keywords: moving plane method
zbMATH Keywords: nonexistence
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3701
OpenAlex ID: W1969965512
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DOI: 10.1006/JDEQ.1999.3701
description: scientific article; zbMATH DE number 1454725