Monotonicity and symmetry of solutions to fractional Laplacian equation

Tingzhi Cheng

doi:10.3934/dcds.2017154

Article Contents

2017, Volume 37, Issue 7: 3587-3599. Doi: 10.3934/dcds.2017154

This issue Previous Article Singular perturbations of Blaschke products and connectivity of Fatou components Next Article On a definition of Morse-Smale evolution processes

Monotonicity and symmetry of solutions to fractional Laplacian equation

Tingzhi Cheng^,

School of Mathematical Sciences, Shanghai Jiaotong University, Shanghai 200240, China

^* Corresponding author: Tingzhi Cheng
^* Corresponding author: Tingzhi Cheng

Received: December 2015

Revised: February 2017

Published: August 2017

The author is supported by Research Grant for Graduate Students of Shanghai Jiaotong University 201701

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Abstract

Let $0 < \alpha < 2$ be any real number and let $\Omega$ be an open domain in $\mathbb R^{n}$ . Consider the following Dirichlet problem of a semi-linear equation involving the fractional Laplacian:

$\begin{equation}\left\{\begin{array}{ll}(-\Delta)^{\alpha/2} u(x)=f(x,u,\nabla{u}),~u(x)>0,&\qquad x\in{\Omega}, \\u(x)\equiv0,&\qquad x\notin{\Omega}.\end{array}\right.\label{p1}\end{equation}$

In this paper, instead of using the conventional extension method introduced by Caffarelli and Silvestre, we employ a direct method of moving planes for the fractional Laplacian to obtain the monotonicity and symmetry of the positive solutions of a semi-linear equation involving the fractional Laplacian. By using the integral definition of the fractional Laplacian, we first introduce various maximum principles which play an important role in the process of moving planes. Then we establish the monotonicity and symmetry of positive solutions of the semi-linear equations involving the fractional Laplacian.

Keywords:

Monotonicity,
symmetry,
fractional Laplacian,
Dirichlet problem,
positive solutions,
direct method of moving planes for fractional Laplacian.

Mathematics Subject Classification: Primary: 35S15, 35B06, 35J61.

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