Article Contents

2018, Volume 38, Issue 9: 4603-4615. Doi: 10.3934/dcds.2018201

This issue Previous Article Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms Next Article Weak-strong uniqueness for the general Ericksen—Leslie system in three dimensions

Moving planes for nonlinear fractional Laplacian equation with negative powers

Miaomiao Cai^1, and
Li Ma^2, ,

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2.
School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, China

* Corresponding author: Li Ma
* Corresponding author: Li Ma

Received: November 2017

Revised: April 2018

Published: September 2018

The research of L.Ma is partially supported by the National Natural Science Foundation of China (No. 11771124, No.11271111)

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Abstract

In this paper, we study symmetry properties of positive solutions to the fractional Laplace equation with negative powers on the whole space. We can use the direct method of moving planes introduced by Jarohs-Weth-Chen-Li-Li to prove one particular result below. If $u∈ C^{1, 1}_{loc}(\mathbb{R}^{n})\cap L_{α}$ satisfies

$(-Δ)^{α/2}u(x)+u^{-β}(x) = 0, \ \ \ x∈ \mathbb{R}^n, $

with the growth/decay property

$u(x) = a|x|^{m}+o(1), \ \ as \ \ |x| \to ∞, $

where $\frac{α}{β+1}<m<1$ , $a>0$ is a constant, then the positive solution $u(x)$ must be radially symmetric about some point in $\mathbb{R}^{n}$ . Similar result is also true for Hénon type nonlinear fractional Laplace equation with negative powers.

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Mathematics Subject Classification: Primary: 35R11, 35J60, 53C21; Secondary: 58J05.

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