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/2u =K(x)f(u),x ∈ Rn+,u≡0,x ￠ Rn+,where 0 ＜ o ＜ 2,Rn+:={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) ∈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.