Kazdan-Warner equation on graph in the negative case

Abstract: Let $G=(V,E)$ be a connected finite graph. In this short paper, we reinvestigate the Kazdan-Warner equation $$\Delta u=c-he^u$$ with $c<0$ on $G$, where $h$ defined on $V$ is a known function. Grigor'yan, Lin and Yang \cite{GLY} showed that if the Kazdan-Warner equation has a solution, then $\overline{h}$, the average value of $h$, is negative. Conversely, if $\overline{h}<0$, then there exists a number $c_-(h)<0$, such that the Kazdan-Warner equation is solvable for every $0>c>c_-(h)$ and it is not solvable for $c<c_-(h)$. Moreover, if $h\leq0$ and $h\not\equiv0$, then $c_-(h)=-\infty$. Inspired by Chen and Li's work \cite{CL}, we ask naturally: \begin{center} Is the Kazdan-Warner equation solvable for $c=c_-(h)$? \end{center} In this paper, we answer the question affirmatively. We show that if $c_-(h)=-\infty$, then $h\leq0$ and $h\not\equiv0$. Moreover, if $c_-(h)>-\infty$, then there exists at least one solution to the Kazdan-Warner equation with $c=c_-(h)$.