Symmetry and regularity of positive solutions to integral systems with Bessel potential

In this paper, we are concerned with the symmetry and regularity of positive solutions of the following integral system: u(x) = ∫ Rn Gα (x–y)wr (y)vq(y) |y|β dy, v(x) = ∫ Rn Gα (x–y)up(y)wr (y) |y|β dy, w(x) = ∫ Rn Gα (x–y)up(y)vq(y) |y|β dy, where Gα (x) is the αth-order Bessel kernel, n≥ 3, 0≤ β < α < n, 1 < p,q, r < n–β β and 1 p+1 + 1 q+1 + 1 r+1 > 2n–α+β n . We show that every positive solution triple (u, v,w) of the system is radially symmetric and monotonic decreasing about some point by the moving planes method in integral forms. Moreover, by the regularity lifting method, we prove that (u, v,w) belongs to L∞(Rn)× L∞(Rn)× L∞(Rn) and which is then locally Hölder continuous. MSC: 45E10; 45G05