Monotonicity and symmetry of solutions to fractional Laplacian equation

Tingzhi Cheng

doi:10.3934/dcds.2017154

Abstract

Let \begin{document} $0 be any real number and let \begin{document} $\Omega$ \end{document} be an open domain in \begin{document} $\mathbb R^{n}$ \end{document} . Consider the following Dirichlet problem of a semi-linear equation involving the fractional Laplacian: 1 \begin{document}$\begin{equation}\left\{\begin{array}{ll}(-\Delta)^{\alpha/2} u(x)=f(x,u,\nabla{u}),~u(x)>0,&\qquad x\in{\Omega}, \\u(x)\equiv0,&\qquad x\notin{\Omega}.\end{array}\right. \tag{1}\label{p1}\end{equation}$\end{document} 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.

Highlights

The monotonicity and symmetry properties of the solutions play important roles in the research of semi-linear elliptic equations
The authors used the above extension method to reduce the nonlocal problem of u into a local one U (x, y)
Compared with [7], here we remove the restriction α ≥ 1 and we extend the results of Berestycki and Nirenberg([6]) to the fractional Laplacian

Summary

Introduction

The monotonicity and symmetry properties of the solutions play important roles in the research of semi-linear elliptic equations. In [1], [19], [8], those properties played an essential role to derive some priori bounds for solutions of some semi-linear elliptic equations. In [17], [22], [23], [26], [27], those properties are used to derive the uniqueness of solutions of some semi-linear elliptic equations. There have been a lot of papers which investigate the monotonicity and symmetry properties of the solutions of different types of the semi-linear equations with local operators. Monotonicity, symmetry, fractional Laplacian, Dirichlet problem, positive solutions, direct method of moving planes for fractional Laplacian.

TINGZHI CHENG

Then we will have

Assume f

It follows that