Existence of Kazdan-Warner equation with sign-changing prescribed function

Abstract

In this paper, we study the following Kazdan-Warner equation with sign-changing prescribed function $h$ \begin{align*} -\Delta u=8\pi\left(\frac{he^{u}}{\int_{\Sigma}he^{u}}-1\right) \end{align*} on a closed Riemann surface whose area is equal to one. The solutions are the critical points of the functional $J_{8\pi}$ which is defined by \begin{align*} J_{8\pi}(u)=\frac{1}{16\pi}\int_{\Sigma}|\nabla u|^2+\int_{\Sigma}u-\ln\left|\int_{\Sigma}he^{u}\right|,\quad u\in H^1\left(\Sigma\right). \end{align*} We prove the existence of minimizer of $J_{8\pi}$ by assuming \begin{equation*} \Delta \ln h^++8\pi-2K>0 \end{equation*}at each maximum point of $2\ln h^++A$, where $K$ is the Gaussian curvature, $h^+$ is the positive part of $h$ and $A$ is the regular part of the Green function. This generalizes the existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), 230-248] to the sign-changing prescribed function case. We are also interested in the blow-up behavior of a sequence $u_{\varepsilon}$ of critical points of $J_{8\pi-\varepsilon}$ with $\int_{\Sigma}he^{u_{\varepsilon}}=1, \lim\limits_{\varepsilon\searrow 0}J_{8\pi-\varepsilon}\left(u_{\varepsilon}\right)<\infty$ and obtain the following identity during the blow-up process \begin{equation*} -\varepsilon=\frac{16\pi}{(8\pi-\varepsilon)h(p_\varepsilon)}\left[\Delta \ln h(p_\varepsilon)+8\pi-2K(p_\varepsilon)\right]\lambda_{\varepsilon}e^{-\lambda_{\varepsilon}}+O\left(e^{-\lambda_{\varepsilon}}\right), \end{equation*}where $p_\varepsilon$ and $\lambda_\varepsilon$ are the maximum point and maximum value of $u_\varepsilon$, respectively. Moreover, $p_{\varepsilon}$ converges to the blow-up point which is a critical point of the function $2\ln h^{+}+A$.