A note on Liouville type equations on graphs

In this note, we study the Liouville equation $\Delta u = -e^u$ on a graph G satisfying certain isoperimetric inequality. Following the idea of W. Ding, we prove that there exists a uniform lower bound for the energy, $\Sigma_G e^u$ of any solution $u$, to the equation. In particular, for the 2-dimensional lattice graph $Z^2$; the lower bound is given by 4.