图上的爱因斯坦-标量场李奇诺维茨方程

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2024-06-05 DOI:10.1007/s00526-024-02737-1

Leilei Cui, Yong Liu, Chunhua Wang, Jun Wang, Wen Yang

{"title":"图上的爱因斯坦-标量场李奇诺维茨方程","authors":"Leilei Cui, Yong Liu, Chunhua Wang, Jun Wang, Wen Yang","doi":"10.1007/s00526-024-02737-1","DOIUrl":null,"url":null,"abstract":"In this article, we consider the Einstein-scalar field Lichnerowicz equation $$\\begin{aligned} -\\Delta u+hu=Bu^{p-1}+Au^{-p-1} \\end{aligned}$$on any connected finite graph \$G=(V,E)\$, where A, B, h are given functions on V with \$A\\ge 0\$, \$A\\not \\equiv 0\$ on V, and \$p>2\$ is a constant. By using the classical variational method, topological degree theory and heat-flow method, we provide a systematical study on this equation by providing the existence results for each case: positive, negative and null Yamabe-scalar field conformal invariant, namely \$h>0\$, \$h<0\$ and \$h=0\$ respectively.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"25 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Einstein-scalar field Lichnerowicz equations on graphs\",\"authors\":\"Leilei Cui, Yong Liu, Chunhua Wang, Jun Wang, Wen Yang\",\"doi\":\"10.1007/s00526-024-02737-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this article, we consider the Einstein-scalar field Lichnerowicz equation $$\\\\begin{aligned} -\\\\Delta u+hu=Bu^{p-1}+Au^{-p-1} \\\\end{aligned}$$on any connected finite graph \\\$G=(V,E)\\\$, where A, B, h are given functions on V with \\\$A\\\\ge 0\\\$, \\\$A\\\\not \\\\equiv 0\\\$ on V, and \\\$p>2\\\$ is a constant. By using the classical variational method, topological degree theory and heat-flow method, we provide a systematical study on this equation by providing the existence results for each case: positive, negative and null Yamabe-scalar field conformal invariant, namely \\\$h>0\\\$, \\\$h<0\\\$ and \\\$h=0\\\$ respectively.\",\"PeriodicalId\":9478,\"journal\":{\"name\":\"Calculus of Variations and Partial Differential Equations\",\"volume\":\"25 1\",\"pages\":\"\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-06-05\",\"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/s00526-024-02737-1\",\"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/s00526-024-02737-1","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

在本文中，我们考虑在任意连通的有限图$G=(V,E)$上的爱因斯坦-标量场李奇诺维茨方程 $$begin{aligned} -\Delta u+hu=Bu^{p-1}+Au^{-p-1} \end{aligned}$$，其中 A、B、h 是 V 上的给定函数，V 上有$A\ge 0$、$A\not \equiv 0$ 和$p>；2）是一个常数。通过使用经典的变分法、拓扑度理论和热流法，我们对该方程进行了系统的研究，提供了正Yamabe-scalar场保角不变性、负Yamabe-scalar场保角不变性和空Yamabe-scalar场保角不变性三种情况下的存在结果，即分别为\(h>0$、$h<0$和$h=0$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

The Einstein-scalar field Lichnerowicz equations on graphs

In this article, we consider the Einstein-scalar field Lichnerowicz equation

$$\begin{aligned} -\Delta u+hu=Bu^{p-1}+Au^{-p-1} \end{aligned}$$

on any connected finite graph $G=(V,E)$, where A, B, h are given functions on V with $A\ge 0$, $A\not \equiv 0$ on V, and $p>2$ is a constant. By using the classical variational method, topological degree theory and heat-flow method, we provide a systematical study on this equation by providing the existence results for each case: positive, negative and null Yamabe-scalar field conformal invariant, namely $h>0$, $h<0$ and $h=0$ respectively.

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.