Existence of Solutions to Mean Field Equations on Graphs

In this paper, we prove two existence results of solutions to mean field equations Δ u + e u = ρ δ 0 and Δ u = λ e u ( e u - 1 ) + 4 π ∑ j = 1 M 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 distinct vertices on the graph.