具有线性和非线性耦合的薛定谔系统的归一化解的存在性

IF 1.7 4区 数学 Q1 Mathematics
Zhaoyang Yun, Zhitao Zhang
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引用次数: 0

摘要

本文研究了非线性玻色-爱因斯坦凝聚态薛定谔系统 $$ \textstyle\begin{cases} -\Delta u_{1}-\lambda _{1} u_{1}=\mu _{1} u_{1}^{3}+\beta u_{1}u_{2}^{2}+ \kappa (x) u_{2}\quad\text{in }\mathbb{R}^{3}、-Delta u_{2}-\lambda _{2} u_{2}=\mu _{2} u_{2}^{3}+\beta u_{1}^{2}u_{2}+ \kappa (x) u_{1}\quad\text{in }\mathbb{R}^{3}、\\ u_{1}^{2}=a_{1}^{2},\qquad \int _\{mathbb{R}^{3}} u_{2}^{2}=a_{2}^{2}, \end{cases} $$ 其中 $a_{1}$ 、$a_{2}$ , $\mu _{1}$ , $\mu _{2}$ , $\kappa =\kappa (x)>0$ , $\beta <0$ , $\lambda _{1}$ , $\lambda _{2}$ 是拉格朗日乘数。我们利用埃克兰变异原理和流形上的最小值法来证明这个系统有一个径向对称的正解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Existence of normalized solutions for Schrödinger systems with linear and nonlinear couplings
In this paper we study the nonlinear Bose–Einstein condensates Schrödinger system $$ \textstyle\begin{cases} -\Delta u_{1}-\lambda _{1} u_{1}=\mu _{1} u_{1}^{3}+\beta u_{1}u_{2}^{2}+ \kappa (x) u_{2}\quad\text{in }\mathbb{R}^{3}, \\ -\Delta u_{2}-\lambda _{2} u_{2}=\mu _{2} u_{2}^{3}+\beta u_{1}^{2}u_{2}+ \kappa (x) u_{1}\quad\text{in }\mathbb{R}^{3}, \\ \int _{\mathbb{R}^{3}} u_{1}^{2}=a_{1}^{2},\qquad \int _{\mathbb{R}^{3}} u_{2}^{2}=a_{2}^{2}, \end{cases} $$ where $a_{1}$ , $a_{2}$ , $\mu _{1}$ , $\mu _{2}$ , $\kappa =\kappa (x)>0$ , $\beta <0$ , and $\lambda _{1}$ , $\lambda _{2}$ are Lagrangian multipliers. We use the Ekeland variational principle and the minimax method on manifold to prove that this system has a solution that is radially symmetric and positive.
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来源期刊
Boundary Value Problems
Boundary Value Problems MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.00
自引率
5.90%
发文量
83
审稿时长
4 months
期刊介绍: The main aim of Boundary Value Problems is to provide a forum to promote, encourage, and bring together various disciplines which use the theory, methods, and applications of boundary value problems. Boundary Value Problems will publish very high quality research articles on boundary value problems for ordinary, functional, difference, elliptic, parabolic, and hyperbolic differential equations. Articles on singular, free, and ill-posed boundary value problems, and other areas of abstract and concrete analysis are welcome. In addition to regular research articles, Boundary Value Problems will publish review articles.
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