A remark on a result of Ding-Jost-Li-Wang
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- by Yunyan Yang and Xiaobao Zhu PDF
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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 \[ J(u)=\frac {1}{2}\int _M|\nabla u|^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^udv_g,\] 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 $h$.References
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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
- MR Author ID: 923728
- Email: zhuxiaobao@ruc.edu.cn
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3953-3959
- MSC (2010): Primary 46E35
- DOI: https://doi.org/10.1090/proc/13515
- MathSciNet review: 3665046