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

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.

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