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

Zhu, Xiaobao

doi:10.1090/proc/16506

Another remark on a result of Ding-Jost-Li-Wang
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by Xiaobao Zhu;

Proc. Amer. Math. Soc. 152 (2024), 639-651

DOI: https://doi.org/10.1090/proc/16506

Published electronically: November 7, 2023

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Abstract:

Let $(M,g)$ be a compact Riemann surface, $h$ be a positive smooth function on $M$. It is well known that the functional \begin{equation*} J(u)=\frac {1}{2}\int _M|\nabla u|^2dv_g+8\pi \int _M udv_g-8\pi \log \int _Mhe^{u}dv_g \end{equation*} achieves its minimum under Ding-Jost-Li-Wang condition. This result was generalized to nonnegative $h$ by Yang and the author. Later, Sun and Zhu [Existence of Kazdan-Warner equation with sign-changing prescribed function, arXiv:2012.12840, 2020] showed the Ding-Jost-Li-Wang condition is also sufficient when $h$ changes sign, which was reproved later by Wang and Yang [J. Funct. Anal. 282 (2022), Paper No. 109449] and Li and Xu [Calc. Var. Partial Differential Equations 61 (2022), Paper No. 143] respectively using a flow approach. The aim of this note is to give a new proof of Sun and Zhu’s result. Our proof is based on the variational method and the maximum principle.

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