布雷齐斯-尼伦堡问题在$$N\ge 7$$维上的节点群解

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-04-26 DOI:10.1007/s00526-024-02727-3

Monica Musso, Serena Rocci, Giusi Vaira

{"title":"布雷齐斯-尼伦堡问题在$$N\\ge 7$$维上的节点群解","authors":"Monica Musso, Serena Rocci, Giusi Vaira","doi":"10.1007/s00526-024-02727-3","DOIUrl":null,"url":null,"abstract":"We show that the classical Brezis–Nirenberg problem $$\\begin{aligned} \\Delta u + |u|^{4 \\over N-2} u + \\varepsilon u = 0,\\quad {\\text{ in }} \\quad \\Omega , \\quad u= 0, \\quad {\\text{ on }} \\quad \\partial \\Omega \\end{aligned}$$admits nodal solutions clustering around a point on the boundary of \$\\Omega \$ as \$\\varepsilon \\rightarrow 0\$, for smooth bounded domains \$\\Omega \\subset {\\mathbb {R}^N}\$ in dimensions \$N\\ge 7\$.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"51 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Nodal cluster solutions for the Brezis–Nirenberg problem in dimensions $$N\\\\ge 7$$\",\"authors\":\"Monica Musso, Serena Rocci, Giusi Vaira\",\"doi\":\"10.1007/s00526-024-02727-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that the classical Brezis–Nirenberg problem $$\\\\begin{aligned} \\\\Delta u + |u|^{4 \\\\over N-2} u + \\\\varepsilon u = 0,\\\\quad {\\\\text{ in }} \\\\quad \\\\Omega , \\\\quad u= 0, \\\\quad {\\\\text{ on }} \\\\quad \\\\partial \\\\Omega \\\\end{aligned}$$admits nodal solutions clustering around a point on the boundary of \\\$\\\\Omega \\\$ as \\\$\\\\varepsilon \\\\rightarrow 0\\\$, for smooth bounded domains \\\$\\\\Omega \\\\subset {\\\\mathbb {R}^N}\\\$ in dimensions \\\$N\\\\ge 7\\\$.\",\"PeriodicalId\":9478,\"journal\":{\"name\":\"Calculus of Variations and Partial Differential Equations\",\"volume\":\"51 1\",\"pages\":\"\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-04-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Calculus of Variations and Partial Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00526-024-02727-3\",\"RegionNum\":2,\"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":"Calculus of Variations and Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00526-024-02727-3","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

我们证明，经典的布雷齐斯-尼伦堡问题 $$\begin{aligned}\Delta u + |u|^{4 over N-2} u + \varepsilon u = 0,\quad {text\{ in }}\quad \Omega , \quad u= 0, \quad {\text{ on }}\对于维数为 $N\ge 7$ 的光滑有界域 $\Omega \subset {\mathbb {R}^N}$，允许节点解聚集在 $\Omega $ 边界上的一点为 $\varepsilon \rightarrow 0$.

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

Nodal cluster solutions for the Brezis–Nirenberg problem in dimensions $$N\ge 7$$

We show that the classical Brezis–Nirenberg problem

$$\begin{aligned} \Delta u + |u|^{4 \over N-2} u + \varepsilon u = 0,\quad {\text{ in }} \quad \Omega , \quad u= 0, \quad {\text{ on }} \quad \partial \Omega \end{aligned}$$

admits nodal solutions clustering around a point on the boundary of $\Omega $ as $\varepsilon \rightarrow 0$, for smooth bounded domains $\Omega \subset {\mathbb {R}^N}$ in dimensions $N\ge 7$.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.