Uniqueness of non-negative solutions to an integral equation of the Choquard type

Let u is an element of L-loc(2n(p-1)/alpha/beta) (R-n) be a non-negative solution of the integral equation u(x) = integral(Rn) u(p-1)(y)/vertical bar x-y vertical bar(n-alpha) integral(Rn)u(p)(Z)/vertical bar y-z vertical bar(n-beta) dz dy, x is an element of R-n, where 0 < alpha,beta < n and p >= 2. We prove that u 0 if 2n/2n-alpha-beta < p < and n+beta/n-alpha must assume an explicit form if p = n+beta/n-alpha. As an application, we obtain a similar result for non-negative distributional solutions of the corresponding static Choquard-type equation. The main tool we use is the method of moving planes in integral forms.