A note on Liouville type equations on graphs

Abstract

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.

6 pages,accepted and to be published in Proceedings of the American Mathematical Society

References in corpus (5)

• Kazdan-Warner equation on graph in the negative case
• Kazdan-Warner equation on infinite graphs
• The existence and nonexistence of global solutions for a semilinear heat equation on graphs
• The Kazdan-Warner equation on canonically compactifiable graphs
• Blow-up problems for nonlinear parabolic equations on locally finite graphs