An existence result for the Kazdan–Warner equation with a sign-changing prescribed function

In this paper, we study the following Kazdan–Warner equation with a sign-changing prescribed function h \(\begin{aligned} -\Delta u=8\pi \left( \dfrac{he^{u}}{\int _{\Sigma }he^{u}}-1\right) \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u=8\pi \left({\displaystyle \frac{h{e}^u}{{\int }_{\mathrm{\Sigma }}h{e}^u}},-,1\right)\end{array}$$ on a closed Riemann surface \(\Sigma \) $\mathrm{\Sigma }$ whose area equals one. The solutions are the critical points of the functional \(J_{8\pi }\) $J_{8\pi }$ which is defined by We prove the existence of the minimizer of \(J_{8\pi }\) $J_{8\pi }$ by assuming \(\begin{aligned} \Delta \ln h^++8\pi -2\kappa >0 \end{aligned}\) $$\begin{array}{c}\hfill \mathrm{\Delta }ln{h}^{+}+8\pi -2\kappa >0\end{array}$$ at each maximum point of \(2\ln h^++A\) $2ln{h}^{+}+A$ , where \(\kappa \) $\kappa$ is the Gaussian curvature, \(h^+\) $h^{+}$ is the positive part of h and A is the regular part of the Green function. This generalizes the existence result of Ding et al. (Asian J Math 1:230–248, 1997) to the sign-changing prescribed function case. We are also interested in the blow-up behavior of a sequence \(u_{\varepsilon }\) $u_{\epsilon }$ of critical points of \(J_{8\pi -\varepsilon }\) $J_{8\pi -\epsilon }$ with \(\int _{\Sigma }he^{u_{\varepsilon }}=1, \lim \limits _{\varepsilon \searrow 0}J_{8\pi -\varepsilon }\left( u_{\varepsilon }\right) <\infty \) ${\int }_{\mathrm{\Sigma }}h{e}^{u_{\epsilon }}=1,\underset{\epsilon \searrow 0}{lim}{J}_{8\pi -\epsilon }\left(u_{\epsilon }\right)<\infty$ and obtain the following identity during the blow-up process \(\begin{aligned} -\varepsilon =\frac{16\pi }{(8\pi -\varepsilon )h(p_\varepsilon )}\left[ \Delta \ln h(p_\varepsilon )+8\pi -2\kappa (p_\varepsilon )\right] \lambda _{\varepsilon }e^{-\lambda _{\varepsilon }}+O\left( e^{-\lambda _{\varepsilon }}\right) , \end{aligned}\) $$\begin{array}{c}\hfill -\epsilon =\frac{16\pi }{(8\pi -\epsilon )h(p_{\epsilon })}\left[\mathrm{\Delta },ln,h,(p_{\epsilon }),+,8,\pi ,-,2,\kappa ,(p_{\epsilon })\right]{\lambda }_{\epsilon }{e}^{-{\lambda }_{\epsilon }}+O\left(e^{-{\lambda }_{\epsilon }}\right),\end{array}$$ where \(u_\varepsilon \) $u_{\epsilon }$ takes its maximum value \(\lambda _\varepsilon \) ${\lambda }_{\epsilon }$ at \(p_\varepsilon \) $p_{\epsilon }$ . Moreover, \(p_{\varepsilon }\) $p_{\epsilon }$ converges to the blow-up point which is a critical point of the function \(2\ln h^{+}+A\) $2ln{h}^{+}+A$ .