Existence of Solutions to Mean Field Equations on Graphs

Abstract

In this paper, we prove two existence results of solutions to mean field equations $$\Delta u+e^u=\rho {\delta }_0$$ and $$\Delta u=\lambda e^u(e^u-1)+4\pi \sum _{j=1}^M{\delta }_{p_j}$$ on an arbitrary connected finite graph, where $\rho >0$ and $\lambda >0$ are constants, $M$ is a positive integer, and $p_1,...,p_M$ are arbitrarily chosen vertices on the graph.