A system of integral equations on half space
type: Item
instance of: scholarly article
title: A system of integral equations on half space
zbMATH Open document ID: 1220.45005
author: Ran Zhuo
author: Dongyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-06-06T00:00:00Z
review text: In the half-space \({\mathbb R}_+^n=\left\{x\in{\mathbb R}^n:x_n>0\right\}\) the following system of nonlinear integral equations is studied: \[ u(x)=\int_{{\mathbb R}_+^n}\left({1\over\| x-y\|^{n-\alpha}}- {1\over\| x^*-y\|^{n-\alpha}}\right)v(y)^qdy, \] \[ v(x)=\int_{{\mathbb R}_+^n}\left({1\over\| x-y\|^{n-\alpha}}- {1\over\| x^*-y\|^{n-\alpha}}\right)u(y)^pdy. \] Here, \(x^*\) denotes the reflection of \(x\) about the plane \(x_n=0\), \(p,q>1\), and \[ {1\over q+1}+{1\over p+1}={n-\alpha\over n}. \] It is shown that any pair of positive solutions \(u\in L^{p+1}({\mathbb R}_+^n)\), \(v\in L^{q+1}({\mathbb R}_+^n)\) is bounded and continuous. Moreover, using the method of moving planes, it is shown that \((u,v)\) is rotationally symmetric about some line parallel to the \(x_n\)-axis, and that any smooth solution satisfies on \({\mathbb R}_+^n\) the system \[ (-\Delta)^{\alpha/2}u(x)=v(x)^q \] \[ (-\Delta)^{\alpha/2}v(x)=u(x)^p \] and on \(\partial{\mathbb R}_+^n\) the boundary conditions \[ (-\Delta)^ku(x)=(-\Delta)^kv(x)=0 \] for \(k=0,\ldots,{\alpha\over2}-1\).
Mathematics Subject Classification ID: 45G15
Mathematics Subject Classification ID: 35B07
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 45G05
zbMATH DE Number: 5904478
zbMATH Keywords: system of nonlinear integral equations
zbMATH Keywords: fractional Laplacian
zbMATH Keywords: method of moving planes
zbMATH Keywords: regularity of solutions
zbMATH Keywords: rotational symmetry
zbMATH Keywords: upper half space
zbMATH Keywords: bounded solution
zbMATH Keywords: continuous solution
zbMATH Keywords: positive solutions
zbMATH Keywords: smooth solution
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.02.060
OpenAlex ID: W1981867246
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DOI: 10.1016/J.JMAA.2011.02.060
description: scientific article; zbMATH DE number 5904478