Regularity and symmetry of solutions of an integral system

In this paper, we are concerned with the regularity and symmetry of positive solutions of the following nonlinear integral system u(x)=∫RnGα(x−y)v(y)q|y|βdy,      v(x)=∫RnGα(x−y)u(y)p|y|βdy for x∈ℝnx∈ℝn, where Gα(x) is the kernel of Bessel potential of order α, 0 ≤ β < α < n, 1 <, p, q   < n−ββ and 1p+1+1q+1>n−α+βn. We show that positive solution pairs (u, v) ∈ Lp+1(ℝn)×Lq+1(ℝn) are Hölder continuous, radially symmetric and strictly decreasing about the origin.