Symmetry of Solutions to Semilinear Equations Involving the Fractional   Laplacian on $\mathbb{R}^n$ and $\mathbb{R}^n_+$
  Let $0<\alpha<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{\alpha/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and $\mathbb{R}^n_+$.   Applying a direct method of moving planes for the fractional Laplacian, we prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ under mild conditions on $f$. 
Lizhi Zhang and Yongzhong Wang
math.AP