A note on Liouville type equations on graphs

Huabin Ge; Bobo Hua; Wenfeng Jiang

A note on Liouville type equations on graphs

Analysis of PDEs 2018-03-13 v1

Authors: Huabin Ge , Bobo Hua , Wenfeng Jiang

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.

Keywords

liouville theorem graph energy parabolic equations

Cite

@article{arxiv.1803.04181,
  title  = {A note on Liouville type equations on graphs},
  author = {Huabin Ge and Bobo Hua and Wenfeng Jiang},
  journal= {arXiv preprint arXiv:1803.04181},
  year   = {2018}
}

Comments

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

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