Existence of Kazdan-Warner equation with sign-changing prescribed function
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arXiv
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2020
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| author | Sun, Linlin Zhu, Jingyong |
| author_facet | Sun, Linlin Zhu, Jingyong |
| contents | 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 $J_{8π}$ 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 $J_{8π}$ 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 $u_{\varepsilon}$ of critical points of $J_{8π-\varepsilon}$ with $\int_Σhe^{u_{\varepsilon}}=1, \lim\limits_{\varepsilon\searrow 0}J_{8π-\varepsilon}\left(u_{\varepsilon}\right)<\infty$ 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 $p_\varepsilon$ and $λ_\varepsilon$ 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 $2\ln h^{+}+A$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_12840 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Existence of Kazdan-Warner equation with sign-changing prescribed function Sun, Linlin Zhu, Jingyong Analysis of PDEs 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 $J_{8π}$ 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 $J_{8π}$ 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 $u_{\varepsilon}$ of critical points of $J_{8π-\varepsilon}$ with $\int_Σhe^{u_{\varepsilon}}=1, \lim\limits_{\varepsilon\searrow 0}J_{8π-\varepsilon}\left(u_{\varepsilon}\right)<\infty$ 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 $p_\varepsilon$ and $λ_\varepsilon$ 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 $2\ln h^{+}+A$. |
| title | Existence of Kazdan-Warner equation with sign-changing prescribed function |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2012.12840 |