Kazdan–Warner equation on graph

Abstract

Let G = (V, E) be a finite graph and ∆ 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 ∆u = c − he u has a solution on V, where c is a constant, and h : V → 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).

Existence of Solutions to Mean Field Equations on Graphs

“…By using the variational principle (see the similar approach in [9] and [5]) , we prove the following…”

“…Since V is a finite graph, W 1,2 (V ) is V R , the finite dimensional vector space of all real functions on V . We have the following Sobolev embedding (Lemma 5 in [5]):…”

Existence and convergence of solutions for nonlinear biharmonic equations on graphs

“…In this paper, we use methods in functional analysis to study existence of solutions for fourth order nonlinear elliptic equations on graphs. Our ideas are inspired by works of Grigor'yan, Lin and Yang [10,11,12] where they considered several second order nonlinear elliptic equations on graphs. For example, when domains are finite graphs, they proved existence of solutions for the Kazdan-Warner equation [10] and the Yamabe type equation [11].…”

Positive solutions of $p$-th Yamabe type equations on infinite graphs

“…Recently, the investigations of discrete weighted Laplacians and various equations on graphs have attracted much attention (cf. [1,2,3,4,5,6,7,8]). Grigor'yan, Lin and Yang [3] first studied a Yamabe type equation on a finite graph G as follows − ∆u + hu = |u| α−2 u, α > 2 (1.1) where ∆ is a usual discrete graph Laplacian, and h is a positive function defined on the vertices of G. They show that the above equation (1.1) always has a positive solution.…”