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":"<p>In this article, we consider the Einstein-scalar field Lichnerowicz equation </p><span>$$\\begin{aligned} -\\Delta u+hu=Bu^{p-1}+Au^{-p-1} \\end{aligned}$$</span><p>on any connected finite graph <span>\\(G=(V,E)\\)</span>, where <i>A</i>, <i>B</i>, <i>h</i> are given functions on <i>V</i> with <span>\\(A\\ge 0\\)</span>, <span>\\(A\\not \\equiv 0\\)</span> on <i>V</i>, and <span>\\(p>2\\)</span> 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 <span>\\(h>0\\)</span>, <span>\\(h<0\\)</span> and <span>\\(h=0\\)</span> respectively.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"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\":\"<p>In this article, we consider the Einstein-scalar field Lichnerowicz equation </p><span>$$\\\\begin{aligned} -\\\\Delta u+hu=Bu^{p-1}+Au^{-p-1} \\\\end{aligned}$$</span><p>on any connected finite graph <span>\\\\(G=(V,E)\\\\)</span>, where <i>A</i>, <i>B</i>, <i>h</i> are given functions on <i>V</i> with <span>\\\\(A\\\\ge 0\\\\)</span>, <span>\\\\(A\\\\not \\\\equiv 0\\\\)</span> on <i>V</i>, and <span>\\\\(p>2\\\\)</span> 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 <span>\\\\(h>0\\\\)</span>, <span>\\\\(h<0\\\\)</span> and <span>\\\\(h=0\\\\)</span> respectively.</p>\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2024-06-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00526-024-02737-1\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00526-024-02737-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
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.
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