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A note on Liouville type equations on graphs


Authors: Huabin Ge, Bobo Hua and Wenfeng Jiang
Journal: Proc. Amer. Math. Soc. 146 (2018), 4837-4842
MSC (2010): Primary 35R02; Secondary 58J05
DOI: https://doi.org/10.1090/proc/14155
Published electronically: July 23, 2018
MathSciNet review: 3856150
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Abstract: In this note, we study the Liouville equation $ \Delta u=-e^u$ on a graph $ G$ satisfying a certain isoperimetric inequality. Following the idea of W. Ding, we prove that there exists a uniform lower bound for the energy, $ \sum _G e^u,$ of any solution $ u$ to the equation. In particular, for the 2-dimensional lattice graph $ \mathbb{Z}^2,$ the lower bound is given by $ 4.$


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Additional Information

Huabin Ge
Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
Address at time of publication: School of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China
Email: hbge@bjtu.edu.cn

Bobo Hua
Affiliation: School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, People’s Republic of China
Email: bobohua@fudan.edu.cn

Wenfeng Jiang
Affiliation: School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, People’s Republic of China
Email: wen_feng1912@outlook.com

DOI: https://doi.org/10.1090/proc/14155
Received by editor(s): November 10, 2017
Received by editor(s) in revised form: March 7, 2018
Published electronically: July 23, 2018
Additional Notes: The research was supported by the National Natural Science Foundation of China (NSFC) under grants No. 11501027 (the first author) and No. 11401106 (the second author).
Communicated by: Guofang Wei
Article copyright: © Copyright 2018 American Mathematical Society