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

TL;DR

This paper provides a new proof, using variational methods and the maximum principle, for a known result about the minimization of a functional related to compact Riemann surfaces, extending previous work by Sun and Zhu.

Contribution

It offers an alternative proof of Sun and Zhu's result on the minimization of the functional under sign-changing conditions of h, using variational techniques and maximum principle.

Findings

New proof of Sun and Zhu's result on functional minimization.

Validation of the sufficiency of Ding-Jost-Li-Wang condition for sign-changing h.

Application of variational method and maximum principle in the proof.

Abstract

Let $(M,g)$ be a compact Riemann surface, $h$ be a positive smooth function on $M$. It is well known the functional $$J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8\pi\int_M udv_g-8\pi\log\int_Mhe^{u}dv_g$$ 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 (arXiv:2012.12840) showed Ding-Jost-Li-Wang condition is also sufficient for $J$ achieves its minimum when $h$ changes sign, which was reproved later by Wang and Yang (J. Funct. Anal. 282: Paper No. 109449, 2022) and Li and Xu (Calc. Var. 61: Paper No. 143, 2022) respectively using 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.