The differential equation $\mathrm{\Delta }u=8\pi -8\pi h\exp u$ on a compact Riemann surface

Let $M$ be a compact Riemann surface, $h(x)$ a positive smooth function on $M$. In this paper, we consider the functional $$J(u)=1\mathrm{/}2\int \text{sb}M\mathrm{\mid }▽u\mathrm{\mid }\text{sp}2+8\pi \int \text{sb}Mu-8\pi \log \int \text{sb}Mh\exp u\text{.}$$ We give a sufficient condition under which $J$ achieves its minimum.