NONEXISTENCE AND SYMMETRY OF SOLUTIONS TO SOME FRACTIONAL LAPLACIAN EQUATIONS IN THE UPPER HALF SPACE

In this article, we consider the fractional Laplacian equation {(-Delta)(alpha/2) u = K (x) f (u), x is an element of R+ (n) , u 0 x is not an element of R-+(n) , where 0 < alpha< 2, R-+(n) := {x = (x1, x2, ... , xn) | x(n) > 0}. When K is strictly decreasing with respect to |x'|, the symmetry of positive solutions is proved, where x' (x1, x2, ... , x(n-1)) is an element of Rn-1. When K is strictly increasing with respect to x(n) or only depend on x(n), the nonexistence of positive solutions is obtained.