Nonexistence and symmetry of solutions to some fractional laplacian equations in the upper half space

In this article, we consider the fractional Laplacian equation

$\{\begin{array}{ll}{\left(?\Delta \right)}^{\alpha /2}u=K\left(x\right)f\left(u\right),\hfill & x\in {?}_{+}^n,\hfill \\ u\equiv 0,\hfill & x?{?}_{+}^n,\hfill \end{array}$

where $0<\alpha <2,{?}_{+}^n:=\left\{x=\left(x_1,x_2,?,x_n\right)|x_n>0\right\}.$ When K is strictly decreasing with respect to |x′|, the symmetry of positive solutions is proved, where x′=(x₁,x₂,???,x_n?1) ∈ ${?}^{n?1}$. When K is strictly increasing with respect to x_n or only depend on x_n, the nonexistence of positive solutions is obtained.