Liouville theorems involving the fractional Laplacian on a half space

Let R+n be the upper half Euclidean space and let α be any real number between 0 and 2. Consider the following Dirichlet problem involving the fractional Laplacian:equation(1){(−Δ)α/2u=up,x∈R+n,u≡0,x∉R+n.Instead of using the conventional extension method of Caffarelli and Silvestre [8], we employ a new and direct approach by studying an equivalent integral equationequation(2)u(x)=∫R+nG(x,y)up(y)dy. Applying the method of moving planes in integral forms, we prove the non-existence of positive solutions in the critical and subcritical cases under no restrictions on the growth of the solutions.