Symmetry of solutions for a fractional p-Laplacian equation of Choquard type

Let 0 < s < 1, 0 < ( )alpha < n, p > 2, q > 1, r > 0 and u be a positive solution of the equation (-Delta)(p)(s)u = (1/vertical bar x vertical bar(n-alpha) * u(q)) u(r )in R-n. We prove that if u satisfies some decay assumption at infinity, then u must be radially symmetric and monotone decreasing about some point in R-n. Instead of using equivalent fractional systems, we exploit a generalized direct method of moving planes for fractional p-Laplacian equations with nonlocal nonlinearities. This new approach enables us to cover the full range 0 < alpha < n in our results.