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Existence and non-existence of solutions of the mean field equations on flat tori


Authors: Zhijie Chen, Ting-Jung Kuo and Chang-Shou Lin
Journal: Proc. Amer. Math. Soc. 145 (2017), 3989-3996
MSC (2010): Primary 34M55, 35J15
DOI: https://doi.org/10.1090/proc/13543
Published electronically: April 7, 2017
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Abstract: We prove the existence or non-existence of solutions of the mean field equation (1.1) with $ n=1$ or $ n=2$ for a rectangular torus $ E$. It is interesting to see that the results depend on the location of the half period $ \frac {\omega _{k}}{2}$.


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Zhijie Chen
Affiliation: Department of Mathematical Sciences, Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, People’s Republic of China
Email: zjchen@math.tsinghua.edu.cn

Ting-Jung Kuo
Affiliation: Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei 10617, Taiwan
Address at time of publication: Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan
Email: tjkuo1215@ntnu.edu.tw

Chang-Shou Lin
Affiliation: Taida Institute for Mathematical Sciences (TIMS), Center for Advanced Study in Theoretical Sciences (CASTS), National Taiwan University, Taipei 10617, Taiwan
Email: cslin@math.ntu.edu.tw

DOI: https://doi.org/10.1090/proc/13543
Received by editor(s): October 7, 2016
Received by editor(s) in revised form: October 18, 2016
Published electronically: April 7, 2017
Communicated by: Lei Ni
Article copyright: © Copyright 2017 American Mathematical Society