R3中非线性Schrodinger方程组的爆破解  

Blow-up solutions for nonlinear Schr0dinger systems in R^(3)

作　　者：陆璐 Lu Lu

机构地区：[1]中南财经政法大学统计与数学学院,武汉430073

出　　处：《中国科学：数学》2022年第3期269-290,共22页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11601523)资助项目。

摘　　要：本文研究如下非线性Schrodinger方程组:{-△u_(1)+V_(1)(x)u_(1)=μ_(1)u_(1)+α_(1)|x|u_(1)^(3)+β|x|u_(2)^(2)u_(1),x∈R^(3),-△u_(2)+V_(2)(x)u_(2)=μ_(2)u_(2)+α_(2)|x|u_(2)^(3)+β|x|u_(1)^(2)u_(2),x∈R^(3)的L^(2)标准化解的存在性与质量集中性,其中a_(i)>0,μ_(i)∈R,i=1,2,β> 0,并且V_(1)和V_(2)是两个非负位势函数.通过比较(a_(1),a_(2))和(a^(*),a^(*)),本文得到L^(2)标准化解的存在性,其中a^(*)=‖Q‖_(2)^(2)且Q是R^(3)中方程-△u+u-|x|u^(3)=0唯一(不计平移)且关于原点对称的正解.特别地,当(a_(1),a_(2))=(a^(*)-β,a^(*)-β时,L^(2)标准化解的存在性与非存在性都可能发生,此时完全由V_(1)(0)+V_(2)(0)的性质决定.当V_(1)(0)=V_(2)(0)=0时,本文同样分析了当(a_(1),a_(2))↗(a^(*)-β,a^(*)-β)时,L^(2)标准化解的爆破行为.This paper is concerned with the existence and mass concentration of L2 normalized solutions for the nonlinear Schr?dinger systems {-△u_(1)+V_(1)(x)u_(1)=μ_(1)u_(1)+α_(1)|x|u_(1)^(3)+β|x|u_(2)^(2)u_(1),x∈R^(3),-△u_(2)+V_(2)(x)u_(2)=μ_(2)u_(2)+α_(2)|x|u_(2)^(3)+β|x|u_(1)^(2)u_(2),x∈R^(3) where a_(i)> 0,μ_(i)∈ R,i=1,2,β> 0 and V_(1)and V_(2)are two nonnegative trapping potentials.We address the existence and nonexistence of L^(2)normalized solutions,which relate generally to the comparison between(a_(1),a_(2))and(a^(*),a^(*)),where a^(*)=‖Q‖_(2)^(2)and Q is the unique positive radial solution of-Δu+u-|x|u^(3)=0 in R^(3).Specially,our analytic results show that both the existence and nonexistence may occur at the threshold where(a_(1),a_(2))=(a^(*)-β,a^(*)-β),which depends heavily on the behavior of V_(1)(0)+V_(2)(0).When V_(1)(0)=V_(2)(0)=0,the limit behavior of L^(2)normalized solutions as(a_(1),a_(2))↗(a^(*)-β,a^(*)-β) is also analyzed.

关 键 词：约束极小 质量集中性 爆破 

分 类 号：O175[理学—数学]