Multiple solutions of Kazdan-Warner equation on graphs in the negative case

Abstract

Let $G=(V,E)$ be a finite connected graph, and let $\kappa :V\to \mathbb{R}$ be a function such that ${\int }_V\kappa d\mu <0$. We consider the following Kazdan-Warner equation on $G$:\[Δu+κ-K_λe^{2u}=0,\] where $K_{\lambda }=K+\lambda$ and $K:V\to \mathbb{R}$ is a non-constant function satisfying ${\max }_{x\in V}K(x)=0$ and $\lambda \in \mathbb{R}$. By a variational method, we prove that there exists a ${\lambda }^{\ast }>0$ such that when $\lambda \in (-\infty ,{\lambda }^{\ast }]$ the above equation has solutions, and has no solution when $\lambda \ge {\lambda }^{\ast }$. In particular, it has only one solution if $\lambda \le 0$; at least two distinct solutions if $0<\lambda <{\lambda }^{\ast }$; at least one solution if $\lambda ={\lambda }^{\ast }$. 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.

15 pages