Let (Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$(\Sigma ,g)$$\end{document} be a compact Riemannian surface, pj∈Σ\documentclass[12pt]{minimal}
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\begin{document}$$p_j\in \Sigma $$\end{document}, βj>-1\documentclass[12pt]{minimal}
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\begin{document}$$\beta _{j}>-1$$\end{document}, for j=1,⋯,m\documentclass[12pt]{minimal}
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\begin{document}$$j=1,\cdots ,m$$\end{document}. Denote β=min{0,β1,⋯,βm}\documentclass[12pt]{minimal}
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\begin{document}$$\beta =\min \{0,\beta _1,\cdots ,\beta _{m}\}$$\end{document}. Let H∈C0(Σ)\documentclass[12pt]{minimal}
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\begin{document}$$H\in C^0(\Sigma )$$\end{document} be a positive function and h(x)=H(x)dg(x,pj)2βj\documentclass[12pt]{minimal}
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\begin{document}$$h(x)=H(x)\left( d_g(x,p_j)\right) ^{2\beta _j}$$\end{document}, where dg(x,pj)\documentclass[12pt]{minimal}
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\begin{document}$$d_g(x,p_j)$$\end{document} denotes the geodesic distance between x and pj\documentclass[12pt]{minimal}
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\begin{document}$$p_j$$\end{document} for each j=1,⋯,m\documentclass[12pt]{minimal}
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\begin{document}$$j=1,\cdots ,m$$\end{document}. In this paper, using a method of blow-up analysis, we prove that the functional J(u)=12∫Σ|∇gu|2dvg+8π(1+β)1Volg(Σ)∫Σudvg-8π(1+β)log∫Σheudvg\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} J(u)=\frac{1}{2}\int _{\Sigma }|\nabla _g u|^2dv_g + 8\pi (1+\beta )\frac{1}{Vol_g(\Sigma )} \int _{\Sigma }udv_g-8\pi (1+\beta ) \log \int _{\Sigma }he^{u}dv_g \end{aligned}$$\end{document}is bounded from below on the Sobolev space W1,2(Σ,g)\documentclass[12pt]{minimal}
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\begin{document}$$W^{1,2}(\Sigma ,g)$$\end{document}.
