Existence of Solutions to Mean Field Equations on Graphs

Abstract

In this paper, we prove two existence results of solutions to mean field equations Δu+eu=ρδ0and Δu=λeu(eu-1)+4π∑j=1Mδpjon an arbitrary connected finite graph, where ρ> 0 and λ> 0 are constants, M is a positive integer, and p1, … , pM are arbitrarily chosen distinct vertices on the graph.