Symmetry and nonexistence of positive solutions of integral systems with Hardy terms

Let alpha, s and t be real numbers satisfying 0 < alpha < n and 0 <= s, t < alpha, we consider the following weighted system of partial differential equations {(-Delta)(alpha/2)u(x) = vertical bar x vertical bar(-s)v(q), (-Delta)(alpha/2)v(x) = vertical bar x vertical bar(-t)u(p), where p, q > 1. We first establish the equivalence between partial differential system and weighted integral system {u(x) = integral(n)(R) v(q)(y)/vertical bar x - y vertical bar n-alpha vertical bar y vertical bar(s) dy, v(x) = integral(n)(R) u(p)(y)/vertical bar x - y vertical bar n-alpha vertical bar y vertical bar(t) dy. Then, in the critical case of n-s/q+1 + n-t/p+1 = n - alpha, we show that every pair of positive solutions (u(x), v(x)) is radially symmetric about the origin. While in the subcritical case, we prove the nonexistence of positive solutions. (C) 2014 Elsevier Inc. All rights reserved.