SYMMETRY OF SINGULAR SOLUTIONS FOR A WEIGHTED CHOQUARD EQUATION INVOLVING THE FRACTIONAL p-LAPLACIAN

Let u is an element of L-sp boolean AND C-loc(1, 1) (R-n \ {0}) be a positive solution, which may blow up at zero, of the equation (-Delta)(p)(s)u = (1/vertical bar x vertical bar(n-beta) * u(q)/vertical bar x vertical bar(alpha)) u(q-1)/vertical bar x vertical bar(alpha) in R-n\{0}, where 0 < s < 1, 0 < beta < n, p > 2, q >= 1 and alpha > 0. We prove that if u satisfies some suitable asymptotic properties, then u must be radially symmetric and monotone decreasing about the origin. In stead of using equivalent fractional systems, we exploit a direct method of moving planes for the weighted Choquard nonlinearity. This method allows us to cover the full range 0 < beta < in our results.