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

Abstract

In this article, we consider the fractional Laplacian equation {(−Δ)α/2u=K(x)f(u),x∈ℝ+n,u≡0,x∉ℝ+n,where 0<α<2,ℝ+n:={x=(x1,x2,⋯,xn)|xn>0}. When K is strictly decreasing with respect to |x′|, the symmetry of positive solutions is proved, where x′=(x1,x2,ċċċ,xn−1) ∈ ℝn−1. When K is strictly increasing with respect to xn or only depend on xn, the nonexistence of positive solutions is obtained.