Nonexistence of positive solutions for a semi-linear equation involving the fractional laplacian in ℝN*

In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space ℝn: equation(0.1)(−Δ)α/2u(x)=f(xn)up(x), x∈ℝn(−Δ)α/2u(x)=f(xn)up(x), x∈ℝn in the subcritical case with 1 < p   < n+αn−α. Instead of carrying out direct investigations on pseudo-differential equation (0.1), we first seek its equivalent form in an integral equation as below: equation(0.2)u(x)=∫ℝnG∞(x,y)f(yn)up(y)dy, where G∞(x,y) is the Green's function associated with the fractional Laplacian in ℝn. Employing the method of moving planes in integral forms, we are able to derive the nonexistence of positive solutions for (0.2) in the subcritical case. Thanks to the equivalence, same conclusion is true for (0.1).