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

Let α, s and t be real numbers satisfying 0<α1. We first establish the equivalence between partial differential system and weighted integral system{u(x)=∫Rnvq(y)|x−y|n−α|y|sdy,v(x)=∫Rnup(y)|x−y|n−α|y|tdy. Then, in the critical case of n−sq+1+n−tp+1=n−α, 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.