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A remark on a result of Ding-Jost-Li-Wang


Authors: Yunyan Yang and Xiaobao Zhu
Journal: Proc. Amer. Math. Soc. 145 (2017), 3953-3959
MSC (2010): Primary 46E35
DOI: https://doi.org/10.1090/proc/13515
Published electronically: March 27, 2017
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Abstract: Let $ (M,g)$ be a compact Riemannian surface without boundary, $ W^{1,2}(M)$ be the usual Sobolev space, and $ J: W^{1,2}(M)\rightarrow \mathbb{R}$ be the functional defined by

$\displaystyle J(u)=\frac {1}{2}\int _M\vert\nabla u\vert^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^udv_g,$

where $ h$ is a positive smooth function on $ M$. In an inspiring work (Asian J. Math., vol. 1, pp. 230-248, 1997), Ding, Jost, Li and Wang obtained a sufficient condition under which $ J$ achieves its minimum. In this note, we prove that if the smooth function $ h$ satisfies $ h\geq 0$ and $ h\not \equiv 0$, then the above result still holds. Our method is to exclude blow-up points on the zero set of $ h$.

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Additional Information

Yunyan Yang
Affiliation: Department of Mathematics, Renmin University of China, Beijing 100872, People’s Republic of China
Email: yunyanyang@ruc.edu.cn

Xiaobao Zhu
Affiliation: Department of Mathematics, Renmin University of China, Beijing 100872, People’s Republic of China
Email: zhuxiaobao@ruc.edu.cn

DOI: https://doi.org/10.1090/proc/13515
Keywords: Kazdan-Warner problem, Trudinger-Moser inequality, blow-up analysis
Received by editor(s): January 17, 2016
Received by editor(s) in revised form: September 30, 2016
Published electronically: March 27, 2017
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society