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 $f$.