Classification of solutions for a class of singular integral system

In this paper, we consider the following integral system: {u(x) = ∫ℝnυq(y)|x − y|n−αdy,u(x) = ∫ℝnυp(y)|x − y|n−μdy,where 0 < α, μ < n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of (0.1) are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].