Existence of Solutions to Mean Field Equations on Graphs

An Huang, Yong Lin, Shing-Tung Yau

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 finit…