Classification of positive solutions to a system of Hardy-Sobolev type equations

In this paper, we are concerned with the following Hardy-Sobolev type system {(-Δ)^{[Formula presented]}u(x)=[Formula presented](-Δ)^{[Formula presented]}υ(x)=[Formula presented],x=(y,z)∈(ℝ^k\{0})×ℝ^n-kwhere 0<α<n, 0<t₁,t² < min{α,k}, and1<p≤τ₁:=[Formula presented],1<q≤τ₂:=[Formula presented].. We first establish the equivalence of classical and weak solutions between PDE system {u(x)=∫_ℝⁿG_α(x,ξ)[Formula presented]dξυ(x)=∫_ℝⁿG_α(x,ξ)[Formula presented]dξwhereG_α(x,ξ)=[Formula presented] is the Green's function of(-Δ)^{[Formula presented]} in ℝⁿ. Then, by the method of moving planes in the integral forms, in the critical case p = τ₁ and q = τ₂, we prove that each pair of nonnegative solutions(u,v) of (0.1) is radially symmetric and monotone decreasing about the origin in ℝ^k and some point z₀ in ℝ^n-k. In the subcritical case[Formula presented]+[Formula presented]>n-α,1<p≤τ₁ and 1 < q ≤ τ₂, we derive the nonexistence of nontrivial nonnegative solutions for (0.1).