Liouville-type theorems for a nonlinear fractional Choquard equation

In this paper, we are concerned with the fractional Choquard equation on the whole space R-N (-Delta)(s)u = ( 1 |x|(N-2s)*mu p)mu(p-1) with 0 < s < 1,N > 2s and p is an element of R. We first prove that the equation does not possess any positive solution for p <= 1. When p > 1, we establish a Liouville type theorem saying that if N < 6s + 4s(1 + root p(2) -p)p - 1, then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation.