Symmetry of solutions to semilinear equations
involving the fractional laplacian
Lizhi Zhang  School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China (email) Abstract:
Let $0<\alpha<2$ be any real number. Let $\Omega\subset\mathbb{R}^n$ be a bounded or an unbounded domain which is convex and symmetric in $x_1$ direction. We investigate the following Dirichlet problem involving the fractional Laplacian:
\begin{equation}
\left\{\begin{array}{ll}
(\Delta)^{\alpha/2} u(x)=f(x,u), & \qquad x\in\Omega, (1)\\
u(x)\equiv0, & \qquad x\notin\Omega.
\end{array}\right.
\end{equation}
Keywords: Monotonicity, symmetry, nonexistence of positive solutions, the fractional
Laplacian, Dirichlet problem, semilinear elliptic equation, a direct method of moving planes,
maximum principle for antisymmetric functions, narrow region principle, decay at infinity.
Received: February 2015; Revised: July 2015; Available Online: September 2015. 
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