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

Abstract: Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space, $J: W^{1,2}(M)\rightarrow \mathbb{R}$ 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 $h$.