Existence of Solutions to Mean Field Equations on Graphs

Abstract

In this paper, we prove two existence results of solutions to mean field equations ∆u + e u = ρδ 0 and ∆u = λe u (e u − 1) + 4π M j=1 δ p j on an arbitrary connected finite graph, where ρ > 0 and λ > 0 are constants, M is a positive integer, and p 1 , ..., p M are arbitrarily chosen vertices on the graph.

Calculus of variations on locally finite graphs

“…Let C c (V) be a set of all functions with finite support, and W 1,2 0 (V) be a completion of C c (V) under the norm as in (10). Both of W 1,2 (V) and W 1,2 0 (V) are Hilbert spaces with the same inner product u, v = V (∇u∇v + uv)dµ.…”

“…As a consequence, it makes sense to consider Chern-Simons-Higgs model in locally finite graph. Such a model in finite graph was recently studied by Huang-Lin-Yau [10].…”

“…The Kazdan-Warner equation was extended by Keller-Schwarz [11] to canonically compactifiable graphs, and by Ge-Jiang [5] to certain infinite graph. For other related works, we refer the reader to [10,12,13,9,14,16] and the references therein.…”

Application of Rothe's method to a nonlinear wave equation on graphs

“…Lin and Wu [9] considered a semilinear heat equation, and obtained the existence and nonexistence results of global solution. For more relevant results, please refer to [6,7] and their references.…”

A heat flow for the mean field equation on a finite graph

“…Note that if the mean field equation ( 7) has a solution, so does the Kazdan-Warner equation [14]. For such kind of equations, see for examples [12,22,19,27,33,39]. According to [14], its solvability needs some assumptions.…”