拟线性薛定谔方程组在有界区域上的正规化解  

Normalized Solutions of the Quasilinear Schrödinger System in Bounded Domains

作　　者：张倩 Qian Zhang(Department of Mathematical Sciences,Tsinghua University,Beijing 100084;School of Mathematics and Statistics,Fujian Normal University,Fuzhou 350117)

机构地区：[1]清华大学数学科学系,北京100084 [2]福建师范大学数学与统计学院,福州350117

出　　处：《数学物理学报(A辑)》2025年第1期1-30,共30页Acta Mathematica Scientia

基　　金：福建省高校数学学科联盟科研项目专项资金(2025SXLMQN04)。

摘　　要：该文关注以下非线性耦合方程组{−Δu_(1)+ω_(1)u_(1)−1/2Δ(u_(1)^(2))u_(1)=μ_(1)|u_(1)|^(p−1)u_(1)+β|u_(2)|p+1/2|u_(1)|p−3/2u_(1),−Δu_(2)+ω_(2)u_(2)−1/2Δ(u_(2)^(2))u_(2)=μ_(2)|u_(2)|^(p−1)u_(2)+β|u_(1)|p+1/2|u_(2)|p−3/2u_(2),∫_(Ω)|u_(i)|^(2) dx=ρ_(i),i=1,2,(u_(1),u_(2))∈H_(0)^(1)(Ω;R^(2))以及线性耦合方程组{−Δu_(1)+ω_(1)u_(1)−1/2Δ(u_(1)^(2))u_(1)=μ_(1)|u_(1)|^(p−1)u_(1)+βu_(2),−Δu_(2)+ω_(2)u_(2)−1/2Δ(u_(2)^(2))u_(2)=μ_(2)|u_(2)|^(p−1)u_(2)+βu_(1),∫_(Ω)|u_(i)|^(2) dx=ρ_(i),i=1,2,(u_(1),u_(2))∈H_(0)^(1)(Ω;R^(2))其中Ω⊂R^(N)(N≥1)是一个有界光滑区域,ω_(i),β∈R,μ_(i),ρ_(i)>0,i=1,2.而且,若p>1,N=1,2且若1<p≤3N+2/N-2,N≥3.应用变量替换,一方面,证明了非线性耦合方程组正规化解的存在性和轨道稳定性,以及当β→−∞时正规化解的极限行为.另一方面,应用极小化约束方法来获得线性耦合方程组的正规化解的存在性.与之前的一些结果相比,将现有结果扩展到了拟线性薛定谔方程组,并获得了线性耦合情形下的正规化解.This paper is concerned with the following nonlinear coupled system−Δu_(1)+ω_(1)u_(1)−1/2Δ(u_(1)^(2))u_(1)=μ_(1)|u_(1)|^(p−1)u_(1)+β|u_(2)|p+1/2|u_(1)|p−3/2u_(1)−Δu_(2)+ω_(2)u_(2)−1/2Δ(u_(2)^(2))u_(2)=μ_(2)|u_(2)|^(p−1)u_(2)+β|u_(1)|p+1/2|u_(2)|p−3/2u_(2)∫_(Ω)|u_(i)|^(2) dx=ρ_(i),i=1,2,(u_(1),u_(2))∈H_(0)^(1)(Ω;R^(2))and linear coupled system−Δu_(1)+ω_(1)u_(1)−1/2Δ(u_(1)^(2))u_(1)=μ_(1)|u_(1)|^(p−1)u_(1)+βu_(2)−Δu_(2)+ω_(2)u_(2)−1/2Δ(u_(2)^(2))u_(2)=μ_(2)|u_(2)|^(p−1)u_(2)+βu_(1)∫_(Ω)|u_(i)|^(2) dx=ρ_(i),i=1,2,(u_(1),u_(2))∈H_(0)^(1)(Ω;R^(2))whereΩ⊂R^(N)(N≥1)is a bounded smooth domain,ω_(i),β∈R,μ_(i),ρ_(i)>0,i=1,2.Moreover,p>1 if N=1,2 and 1<p≤3N+2/N-2 if N≥3.Using change of variables,on the one hand,we prove the existence and stability of normalized solutions in nonlinear coupled system and the limiting behavior of normalized solutions asβ→−∞.On the other hand,we apply the minimization constraint technique to obtain the existence of normalized solutions for linear coupled system.Compared with some previous results,we extend the existing results to the quasilinear Schrödinger system and also obtain normalized solutions for the linear coupling case.

关 键 词：线性与非线性耦合 有界区域 变量替换 正规化解 极限行为 

分 类 号：O177.91[理学—数学]