A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities

In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularity. Precisely, let (Σ,D) be such a surface with divisor D=Σi=1mβipi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D=\Sigma_{i=1}^m\beta_{i}p_{i}$$\end{document}, where βi > −1 and pi ∈ Σ for i = 1, …, m, and g be a metric representing D. Denote b0 = 4π(1 + min1⩽i⩽mβi). Suppose ψ : Σ → ℝ is a continuous function with ∫Σψdvg ≠ 0 and define λ1∗∗(∑,g)=infu∈H1(∑,g),∫∑ψudvg=0,∫∑u2dvg=1∫∑|∇gu|2dvg.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1^{**} (\sum ,g) = \mathop {\inf }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi udv_g = 0,\smallint _\sum u^2 dv_g = 1} \int_\sum {\left| {\nabla _g u} \right|^2 dv_g .}$$\end{document} Then for anyα∈[0,λ1**(Σ,g))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha\in[0,\lambda_1^{**}(\Sigma, g))$$\end{document}, we have supu∈H1(∑,g),∫∑ψu=0,∫∑|∇gu|2dvg−α∫∑u2dvg⩽1∫∑eb0u2dvg<+∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {\sup }\limits_{u \in H^1 (\sum ,g),\smallint _\sum \psi u = 0,\smallint _\sum \left| {\nabla _g u} \right|^2 dv_g - \alpha \smallint _\sum u^2 dv_g \leqslant 1} \int_\sum {e^{b_0 u^2 } dv_g < + \infty .}$$\end{document} When b > b0, the integrals ∫∑ebu2dvg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_\sum {e^{bu^2 } dv_g }$$\end{document} are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.