A Liouville-type theorem for an integral equation on a half-space \(\mathbb R_+^n\)
type: Item
instance of: scholarly article
title: A Liouville-type theorem for an integral equation on a half-space \(\mathbb R_+^n\)
zbMATH Open document ID: 1241.45004
author: Linfen Cao
author: Zhaohui Dai
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-03-16T00:00:00Z
review text: The following integral equation on the \(n\)-dimensional upper half Euclidean space \(\mathbb{R}_{+}^n\) is studied \[ u(x)=\int\limits_{\mathbb{R}_{+}^n}\left(\frac{1}{|x-y|^{n-\alpha}}-\frac{1}{|x^*-y|^{n-\alpha}}\right)u^p(y)dy, \quad u(x)>0,\quad x\in \mathbb{R}_{+}^n, \] where \(\alpha\) is an even number satisfying \(0<\alpha<n\); \(x^*=(x_1,x_2,\dots,x_{n-1},-x_n)\) is the reflection of the point \(x\) about the \(\partial \mathbb{R}_{+}^n\). The authors use the moving planes method in integral forms introduced by Chen-Li-Ou to establish a Liouville-type theorem for this equation, which is closely related to the higher-order differential equation with Navier boundary conditions: \[ (-\Delta)^{\alpha/2}u=u^p, \text{ in } \mathbb{R}_{+}^n; \] \[ u=(-\Delta)u=\cdots=(-\Delta)^{\alpha/2-1}, \text{ on } \partial \mathbb{R}_{+}^n. \]
reviewed by: Alexander N. Tynda
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 35B53
zbMATH DE Number: 6015291
zbMATH Keywords: symmetry
zbMATH Keywords: monotonicity
zbMATH Keywords: moving planes method
zbMATH Keywords: Liouville-type theorem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.01.015
OpenAlex ID: W2055266228
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DOI: 10.1016/J.JMAA.2012.01.015
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description: scientific article; zbMATH DE number 6015291