Kazdan-Warner equation on graph

TL;DR

This paper investigates the existence of solutions to the Kazdan-Warner equation on finite graphs using variational methods and upper-lower solutions, extending results from the manifold case to graph structures.

Contribution

It introduces conditions for solvability of the Kazdan-Warner equation on graphs and explores higher order derivative equations, bridging graph theory and geometric analysis.

Findings

Established conditions for solutions on finite graphs

Extended analysis to higher order derivative equations

Connected graph results with classical manifold cases

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).