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

Abstract

Let $G=(V,E)$ be a finite connected graph, and let $\kappa : V\rightarrow {\mathbb {R}}$ be a function such that $\int _V\kappa \mathrm{d}\mu <0$. We consider the following Kazdan–Warner equation on G:

$$\begin{aligned} \varDelta u+\kappa -K_\lambda e^{2u}=0, \end{aligned}$$

where $K_\lambda =K+\lambda$ and $K: V\rightarrow {\mathbb {R}}$ 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 $\lambda ^*>0$ such that when $\lambda \in (-\infty ,\lambda ^*]$ the above equation has solutions, and has no solution when $\lambda \ge \lambda ^*$. In particular, it has only one solution if $\lambda \le 0$; at least two distinct solutions if $0<\lambda <\lambda ^*$; at least one solution if $\lambda =\lambda ^*$. This result complements earlier work of Grigor’yan et al. (Calc Var Partial Diff Equ, 55(4):13 2016), and is viewed as a discrete analog of that of Ding and Liu (Trans Am Math Soc, 347:1059–1066 1995) and Yang and Zhu (Ann Acad Sci Fenn Math, 44:167–181 2019) on manifolds.