Kazdan-Warner equation on graph in the negative case

TL;DR

This paper investigates the solvability of the Kazdan-Warner equation on finite graphs in the negative case, establishing conditions under which solutions exist at critical parameter values and extending previous results.

Contribution

It proves the existence of solutions at the critical parameter $c=c_-(h)$, answering an open question and generalizing prior work on the negative case.

Findings

Solutions exist at $c=c_-(h)$ when $c_-(h)>- finite$.

If $c_-(h)=-inite$, then $h\

Solutions are characterized by the sign and nullity of $h$.

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 $c=c_-(h)$.