Regularity and symmetry of solutions of an integral system

Abstract

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∈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.