Kazdan-Warner equation on graph

Alexander Grigor'yan; Yong Lin; Yunyan Yang

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arXiv · 1607.04540

Kazdan-Warner equation on graph

Alexander Grigor'yan·Yong Lin·Yunyan Yang

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Abstract

Let $G=(V,E)$ be a finite graph and $\Delta$ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan-Warner equation $\Delta u=c-he^u$ has a solution on $V$, where $c$ is a constant, and $h:V\rightarrow\mathbb{R}$ is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan-Warner (Ann. Math., 1974).

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Alexander Grigor'yan, Yong Lin, Yunyan Yang. 2016-07-15. Kazdan-Warner equation on graph. https://arxiv.org/abs/1607.04540

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