Hénon型椭圆系统多个非径向对称解的存在性

Existence of Multiple Non-Radial Positive Solutions of a Hénon Type Elliptic System

该文研究如下椭圆系统{−Δu+μ1u=p/p+q|x|^(α)u^(p−1)v^(q),x∈Ω,−Δv+μ2v=q/p+q|x|^(α)u^(p)v^(q−1),x∈Ω,u,v>0,x∈Ω,u=v=0,x∈∂Ω,此处Ω■R^(N)(N≥4)是一个圆环,μ1,μ2>0,p,q>0且p+q<2N−2/N−3.该文利用变分法和伸缩技巧证明上述系统有多个非径向对称解.

In this paper,we study the following elliptic system{−Δu+μ1u=p/p+q|x|^(α)u^(p−1)v^(q),x∈Ω,−Δv+μ2v=q/p+q|x|^(α)u^(p)v^(q−1),x∈Ω,u,v>0,x∈Ω,u=v=0,x∈∂Ω,whereΩ■R^(N)is an annulus N≥4,μ1,μ2>0,p,q>1 and p+q<2N−2/N−3.By variational method and rescaling method,we prove that the system has many non-radial solutions.