Kazdan-Warner equation on infinite graphs

TL;DR

This paper investigates the existence of solutions to the Kazdan-Warner equation on infinite graphs using heat flow methods, extending techniques from smooth manifolds to discrete graph structures.

Contribution

It introduces a heat flow approach to solve the Kazdan-Warner equation on infinite graphs, differing from traditional variational methods used in finite cases.

Findings

Existence of solutions under certain conditions on h and graph structure

Extension of continuous manifold techniques to discrete infinite graphs

New method for solving nonlinear equations on infinite graphs

Abstract

We concern in this paper the graph Kazdan-Warner equation \begin{equation*} \Delta f=g-he^f \end{equation*} on an infinite graph, the prototype of which comes from the smooth Kazdan-Warner equation on an open manifold. Different from the variational methods often used in the finite graph case, we use a heat flow method to study the graph Kazdan-Warner equation. We prove the existence of a solution to the graph Kazdan-Warner equation under the assumption that $h\leq0$ and some other integrability conditions or constrictions about the underlying infinite graphs.