临界Hartree方程组基态解的存在性  

The Ground State Solution of Critical Hartree System

作　　者：郑雨 沈自飞 ZHENG Yu;SHEN Zifei(College of Mathematics and Computer Science,Zhejiang Normal University,Jinhua,Zhejiang,321004,P.R.Chino)

机构地区：[1]浙江师范大学数学与计算机科学学院,金华浙江321004

出　　处：《数学进展》2020年第1期53-63,共11页Advances in Mathematics（China）

基　　金：supported by NSFC(No.11671364)。

摘　　要：本文考虑临界耦合的Hartree方程组{-△+λu=∫Ω|u(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,-△+νu=∫Ω|ν(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,其中Ω是RN中带有光滑边界的有界区域,N≥3,λ,v是常数,且满足λ,v>-λ1(Ω),λ1(Ω)是(-△,H01(Ω))的第一特征值,β> 0是耦合参数,临界指标2μ*=(2N-μ)/(N-2)来源于Hardy-LittlewoodSobolev不等式,利用变分的方法证明了临界Hartree方程组基态正解的存在性.In this paper,we are interested in the following critical coupled Hartree system{-△+λu=∫Ω|u(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,-△+νu=∫Ω|ν(z)|^2*μ/|x-z|μdz|u|^2*μ-2u+βν,x∈Ω,where Ω■R^N(N≥3) is a smooth bounded domain,λ,v> λ1(Ω) are constants,λ1(Ω) is the first eigenvalue of(-△,H01(Ω)),β> 0 is a coupling parameter,2μ*=(2 N-μ)/(N-2) due to the HardyLittle wood-Sobolev inequality.By using the variational method,the existence of positive ground state solution of this system is proved.

关 键 词：Hartree方程组 Brezis-Nirenberg问题 Hardy-Littlewood-Sobolev不等式 临界指标 

分 类 号：O177.91[理学—数学]