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

TL;DR

This paper investigates the existence and multiplicity of solutions to the Kazdan-Warner equation on finite graphs in the negative case, identifying parameter ranges for which solutions exist and are unique or multiple, using variational methods.

Contribution

It establishes new existence and multiplicity results for the Kazdan-Warner equation on graphs in the negative case, extending previous work and providing a discrete analog of manifold results.

Findings

Solutions exist for certain parameter ranges of λ.

Unique solution when λ ≤ 0.

Multiple solutions when 0 < λ < λ*.

Abstract

Let $G=(V,E)$ be a finite connected graph, and let $\kappa: V\rightarrow \mathbb{R}$ be a function such that $\int_V\kappa d\mu<0$. We consider the following Kazdan-Warner equation on $G$:\[\Delta u+\kappa-K_\lambda e^{2u}=0,\] 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 $\lambda\geq \lambda^\ast$. In particular, it has only one solution if $\lambda\leq 0$; at least two distinct solutions if $0<\lambda<\lambda^*$; 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.