{"title":"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","authors":"Zhaoyang Yun, Zhitao Zhang","doi":"10.1186/s13661-024-01830-w","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence of normalized solutions for Schrödinger systems with linear and nonlinear couplings\",\"authors\":\"Zhaoyang Yun, Zhitao Zhang\",\"doi\":\"10.1186/s13661-024-01830-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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.\",\"PeriodicalId\":49228,\"journal\":{\"name\":\"Boundary Value Problems\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2024-02-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Boundary Value Problems\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1186/s13661-024-01830-w\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Boundary Value Problems","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1186/s13661-024-01830-w","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
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.
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.