{"title":"Existence results for Kazdan-Warner type equations on graphs","authors":"Pengxiu Yu","doi":"arxiv-2409.10181","DOIUrl":null,"url":null,"abstract":"In this paper, motivated by the work of Huang-Lin-Yau (Commun. Math. Phys.\n2020), Sun-Wang (Adv. Math. 2022) and Li-Sun-Yang (Calc. Var. Partial\nDifferential Equations 2024), we investigate the existence of Kazdan-Warner\ntype equations on a finite connected graph, based on the theory of Brouwer\ndegree. Specifically, we consider the equation \\begin{equation*} -\\Delta\nu=h(x)f(u)-c, \\end{equation*} where $h$ is a real-valued function defined on\nthe vertex set $V$, $c\\in\\mathbb{R}$ and \\begin{equation*} f(u)=\n\\left(1-\\displaystyle\\frac{1}{1+u^{2n}}\\right)e^u \\end{equation*} with $n\\in\n\\mathbb{N}^*$. Different from the previous studies, the main difficulty in this\npaper is to show that the corresponding equation has only three constant\nsolutions, based on delicate analysis and the connectivity of graphs, which\nhave not been extensively explored in previous literature.","PeriodicalId":501165,"journal":{"name":"arXiv - MATH - Analysis of PDEs","volume":"207 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Analysis of PDEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10181","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, motivated by the work of Huang-Lin-Yau (Commun. Math. Phys.
2020), Sun-Wang (Adv. Math. 2022) and Li-Sun-Yang (Calc. Var. Partial
Differential Equations 2024), we investigate the existence of Kazdan-Warner
type equations on a finite connected graph, based on the theory of Brouwer
degree. Specifically, we consider the equation \begin{equation*} -\Delta
u=h(x)f(u)-c, \end{equation*} where $h$ is a real-valued function defined on
the vertex set $V$, $c\in\mathbb{R}$ and \begin{equation*} f(u)=
\left(1-\displaystyle\frac{1}{1+u^{2n}}\right)e^u \end{equation*} with $n\in
\mathbb{N}^*$. Different from the previous studies, the main difficulty in this
paper is to show that the corresponding equation has only three constant
solutions, based on delicate analysis and the connectivity of graphs, which
have not been extensively explored in previous literature.