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*} -Δu=8π\left(\frac{he^{u}}{\int_Σhe^{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 $J_{8\ddot{I}€}$ which is defined by \begin{align*} J_{8π}(u)=\frac{1}{16π}\int_Σ|\nabla u|^2+\int_Σu-\ln\left|\int_Σhe^{u}\right|,\quad u\in H^1\left(Σ\right). \end{align*} We prove the existence of minimizer of $J_{8\ddot{I}€}$ by assuming \begin{equation*} Δ\ln h^++8π-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 $u_{\epsilon }$ of critical points of $J_{8\ddot{I}€-\epsilon }$ with ${\int }_{\hat{I}}\pounds h{e}^{u_{\epsilon }}=1,\underset{\epsilon \searrow 0}{\lim }{J}_{8\ddot{I}€-\epsilon }\left(u_{\epsilon }\right)<\infty$ and obtain the following identity during the blow-up process \begin{equation*} -\varepsilon=\frac{16π}{(8π-\varepsilon)h(p_\varepsilon)}\left[Δ\ln h(p_\varepsilon)+8π-2K(p_\varepsilon)\right]λ_{\varepsilon}e^{-λ_{\varepsilon}}+O\left(e^{-λ_{\varepsilon}}\right), \end{equation*}where $p_{\epsilon }$ and $\hat{I}{»}_{\epsilon }$ are the maximum point and maximum value of $u_{\epsilon }$, respectively. Moreover, $p_{\epsilon }$ converges to the blow-up point which is a critical point of the function $2\ln h^{+}+A$.