Classification of solutions for an integral system with negative exponents

In this paper, we study the following integral system with negative exponents (Formula presented.) where f (u, v) = λ1u^−p1 + μ1v−^{q1 +} γ1μ−αv−β, g(u, v) = λ2u−^p2 +μ2v−^{q2 +}γ2u−βv−α. We obtain asymptotic behaviour and regularity for positive solutions, and in the critical case, we classify all of them by applying the method of moving spheres. We also consider weighted integral systems with negative exponents and derive asymptotic behaviours for positive solutions.