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

Let $(M,g)$ be a compact Riemannian surface without boundary, $W^{1,2}(M)$ be the usual Sobolev space, $J:W^{1,2}(M)\to \mathbb{R}$ be the functional defined by $$J(u)=\frac{1}{2}{\int }_M\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{2}d{v}_g+8\pi {\int }_{M}ud{v}_g-8\pi \log {\int }_{M}h{e}^{u}d{v}_g,$$ 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 $h\ge 0$ and $h\equiv ̸0$, then the above result still holds. Our method is to exclude blow-up points on the zero set of $h$.