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arXiv · 1611.09184

Kazdan-Warner equation on graph in the negative case

Huabin Ge
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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 $c -\infty$, then there exists at least one solution to the Kazdan-Warner equation with $c=c_-(h)$.

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BibTeXRIS

Huabin Ge. 2016-11-28. Kazdan-Warner equation on graph in the negative case. https://doi.org/10.1016/j.jmaa.2017.04.052

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