Kazdan-Warner equation on graph

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).

Journal

Publication date Created: 2016-07-15

Article

ArXiV ID: 1607.04540

License:

http://creativecommons.org/licenses/by-nc-sa/4.0/

Custom metadata

ACM classes 34B45, 35A15, 58E30

Comments 13 pages

Categories math.AP