On the solutions of Liouville systems
type: Item
instance of: scholarly article
title: On the solutions of Liouville systems
zbMATH Open document ID: 0902.35039
published in: Journal of Differential Equations
publication date: 1998-12-02T00:00:00Z
review text: In the case when \(\Omega \) is a bounded domain in \({\mathbb R}^{2}\), necessary and sufficient conditions for the existence of solutions of the Liouville system are obtained, using Pokhozaev's identity and a variational argument based on a dual formulation of the problem. In the case when \(\Omega ={\mathbb{R}}^{2}\), some results on the symmetry of entire solutions and on a ``Liouville solution'' as a ``special'' solution, both recently studied by \textit{S. Chanillo} and \textit{M. K.-H. Kiessling} [Geom. Func. Anal. 5, 924-947 (1995; Zbl 0858.35035)], are improved. The authors provide completely new proofs based on the moving plane method, developed by \textit{B. Gidas, W. M. Ni} and \textit{L. Nirenberg} [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)]. The main results are illustrated by suitable examples, the proofs are presented in detail, and a few interesting references on applications of the Liouville systems in various fields of Physics, Chemistry and Ecology are given.
reviewed by: Mervan Pašić
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 1080198
zbMATH Keywords: bounded and unbounded domains
zbMATH Keywords: nonexistence
zbMATH Keywords: symmetry of solutions
zbMATH Keywords: Pohozaev identity
zbMATH Keywords: moving plane method
zbMATH Keywords: existence
author: Michel Chipot
author: Itai Shafrir
author: Gershon Wolansky
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3316
OpenAlex ID: W1982522909
DOI: 10.1006/JDEQ.1997.3316
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description: scientific article; zbMATH DE number 1080198