Symmetry and monotonicity of solutions for equations involving the fractional Laplacian of higher order
type: Item
instance of: scholarly article
title: Symmetry and monotonicity of solutions for equations involving the fractional Laplacian of higher order
zbMATH Open document ID: 1391.35389
DOI: 10.1216/RMJ-2018-48-2-485
author: Xuewei Cui
author: Weijie Song
published in: Rocky Mountain Journal of Mathematics
publication date: 2018-06-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1528077628
review text: The paper under review deals with symmetry and monotonicity of solutions to the problem \[ \begin{cases} (-\Delta)^{\alpha+1}u(x)=f(u(x))+g(x) & x\in B_1,\\ (-\Delta)u(x)>0 & x\in B_1,\\ u(x)=0 & x\in \mathbb{R}^n\setminus B_1, \end{cases} \] where \(B_1\) is the open unit ball in \(\mathbb{R}^n,\) \((-\Delta)^{\alpha+1},\) \(\alpha\in(0,1),\) is the higher order fractional Laplacian \[ (-\Delta)^{\alpha+1}u(x):=C(n,\alpha)\text{P.V.} \int_{\mathbb{R}^n} \dfrac{u(x)+(1/2n)\Delta u(x)|x-y|^2-u(y)}{|x-y|^{n+2\alpha+2}}\;dy, \] \(f\) is increasing and locally Lipschitz continuous, while \(g\) is radially symmetric and decreasing in \(|x|.\) The approach is based on reduction of the equation into a system via composition of lower fractional Laplacians, and then getting symmetry and monotonicity of solutions to the system by means of the moving planes method.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35S15
zbMATH DE Number: 6883477
zbMATH Keywords: fractional Laplacian of higher order
zbMATH Keywords: ABP estimate
zbMATH Keywords: radial symmetry
zbMATH Keywords: monotonicity
zbMATH Keywords: method of moving planes
MaRDI profile type: Publication
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Wikidata QID: Q129732620
description: scientific article; zbMATH DE number 6883477