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 $$\Delta u=c-h{e}^u$$ 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 $h\le 0$ and $h\not\equiv 0$, 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 $h\le 0$ and $h\not\equiv 0$. Moreover, if $c_{-}(h)>-\infty$, then there exists at least one solution to the Kazdan-Warner equation with $c=c_{-}(h)$.

7 pages

References in corpus (1)

• p-th Kazdan-Warner equation on graph

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