Radial symmetry results for systems of integral equations on \(\Omega \subset {\mathbb{R}}^n\)
type: Item
instance of: scholarly article
title: Radial symmetry results for systems of integral equations on \(\Omega \subset {\mathbb{R}}^n\)
zbMATH Open document ID: 1246.45013
author: Xiaotao Huang
author: Dongsheng Li
author: Lihe Wang
published in: Manuscripta Mathematica
publication date: 2012-02-14T00:00:00Z
review text: The authors investigate positive solutions of the integral equations with weight Riesz potential on the bounded domain \(\Omega\subseteq\mathbb{R}^n\): \[ \begin{gathered} u(x)= \int_\Omega|x|^{-\alpha} |x-y|^{-\mu}|y|^{-\beta} u^a(y) v^b(y)\,dy,\\ v(x)= \int_\Omega |x|^{-\gamma} |x-y|^{-\nu} |y|^{-k} u^c(y) v^d(y)\,dy,\\ u(x)> 0,\quad u(x)\equiv A,\end{gathered}\tag{1} \] where \[ \begin{gathered} a\geq 0,\;b\geq 0,\;c\geq 0,\;d\geq 0,\;A> 0,\;B> 0,\;0\leq\alpha< n,\;0\leq\beta< n,\\ 0\leq\gamma< n,\;0\leq k< n,\;0<\mu< n,\;0<\nu< n.\end{gathered}\tag{2} \] The authors prove that if \(\Omega\) is a \(C^1\) domain, \(0\in\Omega\), the system (1), with conditions (2), have positive solutions \((u,v)\in L^p(\Omega)\times L^q(\Omega)\), such that \(p> 1\), \(q> 1\), \[ \begin{aligned} (a- 1)p^{-1}+ bq^{-1} &= (n-\mu-\alpha-\beta) n^{-1},\\ cp^{-1}+ (d- 1)q^{-1} &= (n-\nu-\gamma- k)n^{-1},\end{aligned} \] then \(\Omega\) must be ball, \(u\) and \(v\) must be radially symmetric and monotonic with the radius of the ball. The authors, also investigate the symmetry for system of integral equation weight Riesz potential on an exterior domain.
reviewed by: Dan-Mircea Borş
Mathematics Subject Classification ID: 45M20
Mathematics Subject Classification ID: 45G15
zbMATH DE Number: 6006487
zbMATH Keywords: systems
zbMATH Keywords: positive solution
zbMATH Keywords: symmetry
zbMATH Keywords: Riesz potential
MaRDI profile type: Publication
full work available at URL: s00229-011-0465-6
OpenAlex ID: W1982541190
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DOI: 10.1007/S00229-011-0465-6
description: scientific article; zbMATH DE number 6006487