{"title":"局部有限图上一类非线性方程全局解的存在性","authors":"Yanxun Chang, Xiaoxiao Zhang","doi":"10.4134/JKMS.J200221","DOIUrl":null,"url":null,"abstract":". Let G = ( V,E ) be a connected locally finite and weighted graph, ∆ p be the p -th graph Laplacian. Consider the p -th nonlinear equation − ∆ p u + h | u | p − 2 u = f ( x,u ) on G , where p > 2, h,f satisfy certain assumptions. Grigor’yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V . In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m -order differential operator L m,p , we also prove the existence of the nontrivial solution to the analogous nonlinear equation.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"EXISTENCE OF GLOBAL SOLUTIONS TO SOME NONLINEAR EQUATIONS ON LOCALLY FINITE GRAPHS\",\"authors\":\"Yanxun Chang, Xiaoxiao Zhang\",\"doi\":\"10.4134/JKMS.J200221\",\"DOIUrl\":null,\"url\":null,\"abstract\":\". Let G = ( V,E ) be a connected locally finite and weighted graph, ∆ p be the p -th graph Laplacian. Consider the p -th nonlinear equation − ∆ p u + h | u | p − 2 u = f ( x,u ) on G , where p > 2, h,f satisfy certain assumptions. Grigor’yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V . In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m -order differential operator L m,p , we also prove the existence of the nontrivial solution to the analogous nonlinear equation.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2021-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4134/JKMS.J200221\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4134/JKMS.J200221","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 3
摘要
设G=(V,E)是一个连通的局部有限和加权图,∆p是第p个图拉普拉斯算子。考虑G上的第p个非线性方程−∆p u+h | u | p−2 u=f(x,u),其中p>2,h,f满足某些假设。Grigor’yan-Lin-Yang[24]证明了上述非线性方程在有界域中解的存在性Ω ⊂ 五、在本文中,我们通过修改[24]中的一些条件,证明了上述非线性方程在有限集V上存在严格正解。对于m阶微分算子LM,p,我们还证明了类似非线性方程非平凡解的存在性。
EXISTENCE OF GLOBAL SOLUTIONS TO SOME NONLINEAR EQUATIONS ON LOCALLY FINITE GRAPHS
. Let G = ( V,E ) be a connected locally finite and weighted graph, ∆ p be the p -th graph Laplacian. Consider the p -th nonlinear equation − ∆ p u + h | u | p − 2 u = f ( x,u ) on G , where p > 2, h,f satisfy certain assumptions. Grigor’yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V . In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m -order differential operator L m,p , we also prove the existence of the nontrivial solution to the analogous nonlinear equation.