On a class of Kazdan--Warner equations

Authors: YU FANG, MENGJIE ZHANG

Abstract: Let $(\small{\Si},g)$ be a compact Riemannian surface without boundary and $W^{1,2}(\Si)$ be the usual Sobolev space. For any real number $p>1$ and $\alpha\in\mathbb{R}$, we define a functional $$ J_{\alpha,8\pi}(u)=\frac{1}{2}\le( \int_\Si |\nabla_g u|^2dv_g-\alpha (\int_\Si |u|^pdv_g)^{2/p}\ri)-8\pi\log\int_\Si he^u dv_g $$ on a function space $\mathcal{H}=\le\{u\in W^{1,2}(\Si):\int_{\Si}u dv_{g}=0\ri\}$, where $h$ is a positive smooth function on $\Si$. Denote $$\lambda_{1,p}(\Si)=\inf_{u\in \mathcal{H},\,\int_\Si |u|^p dv_g=1}\int_{\Si}|\nabla_{g}u|^{2}\mathrm{d}v_{g}. $$ If $\alpha<\lambda_{1,p}(\Si)$ and $J_{\alpha,8\pi}$ has no minimizer on $\mathcal{H}$, then we obtain the exact value of $\inf_{\mathcal{H}}J_{\alpha,8\pi}$ by using a method of blow-up analysis. Hence, if $\inf_{\mathcal{H}}J_{\alpha,8\pi}$ is not equal to that value, then $J_{\alpha,8\pi}|_\mathcal{H}$ has a critical point that satisfies a Kazdan-Warner equation. This recovers a recent result of Yang and Zhu (DOI: 10.1007/s11425-017-9086-6).

Keywords: Trudinger-Moser inequality, blow-up analysis, Kazdan-Warner equation

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