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arXiv · 1803.04181

A note on Liouville type equations on graphs

Huabin Ge·Bobo Hua·Wenfeng Jiang
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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.

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Huabin Ge, Bobo Hua, Wenfeng Jiang. 2018-03-12. A note on Liouville type equations on graphs. https://arxiv.org/abs/1803.04181

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