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

Abstract

In this paper, we are concerned with the following Hardy-Sobolev type system (0.1){(-Δ)α2u(x)=υq(x)|y|t2(-Δ)α2υ(x)=up(x)|y|t1,x=(y,z)∈(ℝk\\{0})×ℝn-k,where 0<α<n, 0<t1,t2 < min{α,k}, and 1<p≤τ1:=n+α-2t1n-α,1<q≤τ2:=n+α-2t2n-α.. We first establish the equivalence of classical and weak solutions between PDE system (0.2){u(x)=∫ℝnGα(x,ξ)up(ξ)|η|t2dξυ(x)=∫ℝnGα(x,ξ)up(ξ)|η|t2dξ,where Gα(x,ξ)=cn,α|x-ξ|n-α is the Green's function of (-Δ)α2 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 ℝk and some point z0 in ℝn-k. In the subcritical case n-t1p+1+n-t2q+1>n-α,1<p≤τ1 and 1 < q ≤ τ2, we derive the nonexistence of nontrivial nonnegative solutions for (0.1).