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

摘要

In this article, we consider the fractional Laplacian equation(-△)^(α/2)u = K(x)f(u), x ∈ R_+~n,u ≡ 0, x/∈R_+~n,where 0 0}. When K is strictly decreasing with respect to |x′|, the symmetry of positive solutions is proved, where x′=(x_1, x_2, ···, x_(n-1)) ∈R^(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.