Kazdan-Warner equation on graph

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).