Existence and symmetry of positive solutions of an integral equation system

In this paper, we investigate positive solutions of the following integral equation system in Rn: {u(x) = ∫Rn |x-y|α-n(y)pdy, υ(x) = ∫Rn |x - y|β-n u(y)qdy, wherep, q > 1, 0 < α, β < n. With the method ofmoving spheres, we show the existence and the exact form of its solution in the case p ≤ (n + α)/(n - β), q ≤ (n + β)/(n - α); and with the method of moving planes, we prove the symmetry and monotonicity of its solution in the case 1 / p + 1 / q + 1 -n-α / 2n + β - α + n- β / 2n + α - β.