Axial symmetry and regularity of solutions to an integral equation in a half-space
type: Item
instance of: scholarly article
title: Axial symmetry and regularity of solutions to an integral equation in a half-space
zbMATH Open document ID: 1239.45005
author: Guozhen Lu
author: Jiuyi Zhu
published in: Pacific Journal of Mathematics
publication date: 2012-03-13T00:00:00Z
full work available at URL: p09.xhtml
review text: This paper is concerned with qualitative properties of solutions corresponding to the integral equation \[ u(x)=\int_{\mathbb R^n_+}G(x,y)f(u(y))dy. \] Here \(\mathbb R^n_+=\{x\in \mathbb R^n:x_n>0\}\), \(G(\cdot,\cdot)\) is the Green function of \((-\Delta)^m\), \(n>2m\) subject to a Dirichlet boundary condition and \(f:[0,\infty)\rightarrow [0,\infty)\) is increasing. The first result of the paper establishes that if \(u\in L^r(\mathbb R^n_+)\), \(r>n/(n-2m)\) and one of the following conditions hold: (i) \(|\partial f(u)/\partial u|\leq c(|u|^{\beta_1}+|u|^{\beta_2})\) where \(C>0\), \(\beta_1\geq 0\geq \beta_2\), \(u^{\beta_1}, u^{\beta_2}\in L^{n/(2m)}(\mathbb R^n_+)\) (ii) \(f'\) is nondecreasing and \(f'(u)\in L^{n/(2m)}(\mathbb R^n_+)\) then \(u\) is the trivial solution. The authors also obtain a corresponding result for systems of integral equations. Next, the following integral equation is considered \[ u(x)=\int_{\mathbb R^n_+}G(x,y)|u|^{p-1}u dy. \] The authors prove that any solution \(u\) satisfying \(u\in L^{(p-1)n/(2m)}(\mathbb R^n_+)\), \(p>n/(n-2m)\) is bounded in \(\mathbb R^n_+\) and bounded in any \(L^s(\mathbb R^n_+)\) for all \(s>n/(n-2m)\). Furthermore, if \(1<p<(n+2m)/(n-2m)\), then any nonnegative solution \(u\) such that \(u \in L^{2n/(n-2m)}(\mathbb R^n_+)\) depends only on the \(x_n\)-variable. The proofs rely on the method of moving planes in integral form combined with lifting method and properties of the Green function for the polyharmonic operator.
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 45G15
Mathematics Subject Classification ID: 35J08
zbMATH DE Number: 6014094
zbMATH Keywords: integral equations in half-space
zbMATH Keywords: Green function for the polyharmonic operator
zbMATH Keywords: moving plane method
zbMATH Keywords: lifting method
zbMATH Keywords: systems of integral equations
MaRDI profile type: Publication
OpenAlex ID: W2150265583
DOI: 10.2140/PJM.2011.253.455
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description: scientific article; zbMATH DE number 6014094