论文标题
在否定情况下，喀兹丹 - 瓦纳方程的多个解决方案
Multiple solutions of Kazdan-Warner equation on graphs in the negative case
论文作者
论文摘要
令$ g =（v，e）$为有限的连接图，让$κ：v \ rightarrow \ mathbb {r} $成为一个函数，以至于$ \int_vκDμ<0 $。我们在$ g $上考虑以下Kazdan-Warner方程：\ [ΔU+κ-k_λe^{2U} = 0，\]其中$k_λ= k+λ$ and $ k：v \ rightArrow \ rightarrow \ mathbb {r} $是一种非contant函数的$ \ max_ $ \ max_ = 0 \ mathbb {r} $。通过一种变异方法，我们证明存在$λ^*> 0 $，这样，当$λ\ in（ - \ infty，λ^*] $中，上述方程式没有解决方案，当$λ\ leq 0 $λ$λ<λ< / efly niment^oste;尤其只有一个解决方案时，没有解决方案。 $λ=λ^\ ast $。该结果补充了Grigor'yan-lin-yang \ cite {gly16}的早期工作，并被视为ding-liu \ cite {dl95}和Yang-Zhu \ cite {yz19}的离散类似物。
Let $G=(V,E)$ be a finite connected graph, and let $κ: V\rightarrow \mathbb{R}$ be a function such that $\int_Vκdμ<0$. We consider the following Kazdan-Warner equation on $G$:\[Δu+κ-K_λe^{2u}=0,\] where $K_λ=K+λ$ and $K: V\rightarrow \mathbb{R}$ is a non-constant function satisfying $\max_{x\in V}K(x)=0$ and $λ\in \mathbb{R}$. By a variational method, we prove that there exists a $λ^*>0$ such that when $λ\in(-\infty,λ^*]$ the above equation has solutions, and has no solution when $λ\geq λ^\ast$. In particular, it has only one solution if $λ\leq 0$; at least two distinct solutions if $0<λ<λ^*$; at least one solution if $λ=λ^\ast$. This result complements earlier work of Grigor'yan-Lin-Yang \cite{GLY16}, and is viewed as a discrete analog of that of Ding-Liu \cite{DL95} and Yang-Zhu \cite{YZ19} on manifolds.
