DCDS
Monotonicity and symmetry of solutions to fractional Laplacian equation
Tingzhi Cheng
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 3587-3599 doi: 10.3934/dcds.2017154
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:
$\begin{equation}\left\{\begin{array}{ll}(-\Delta)^{\alpha/2} u(x)=f(x,u,\nabla{u}),~u(x)>0,&\qquad x\in{\Omega}, \\u(x)\equiv0,&\qquad x\notin{\Omega}.\end{array}\right. \tag{1}\label{p1}\end{equation}$
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

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