论文标题
具有签名规定功能的卡兹丹 - 瓦纳方程的存在
Existence of Kazdan-Warner equation with sign-changing prescribed function
论文作者
论文摘要
在本文中，我们研究了以下喀兹 - 瓦纳方程，并使用签名更改的规定功能$ h $ \ begin {align*} -ΔU=8π\ left（\ frac {he^{u}} {\int_σhe^{u}} - 1 \ right）\ end End {align*}，其面积等于一个。解决方案是\ begin {align*}定义的功能$ j_ {8π} $的关键点 J_ {8π}（u）= \ frac {1} {16π} \int_σ| \ nabla u |^2+\int_σu-\ ln \ ln \ left | \int_σhe^{u} { \ end {align*}我们通过假设\ begin {equation*}来证明$ j_ {8π} $的最小化器的存在 δ\ ln H^++8π-2K> 0 \ end {equation*}在每个最大点的$ 2 \ ln h^++ a $，其中$ k $是高斯曲率，$ h^+$是$ h $的正部分，$ h $，$ a $是绿色功能的常规部分。这概括了Ding，Jost，Li和Wang的存在结果[Asian J. Math。 1（1997），230-248]到签名规定的功能案例。我们也对$ j_ {8π-\ varepsilon} $的序列$ u _ {\ varepsilon} $的爆炸行为感兴趣0} j_ {8π-\ varepsilon} \ left（u _ {\ varepsilon} \ right）<\ infty $，并在爆破过程中获得以下身份 - \ varepsilon = \ frac {16π} {（8π-\ varepsilon）h（p_ \ varepsilon）} \ left [δ\ ln h（p_ \ varepsilon）+8π-2K（p_ \ varepsilon）\ right]λ_ {\ varepsilon} e^{ - λ_ {\ v varepsilon}}}+o \ left（e^{ - λ_{ - λ_{\ varepsilon}} $ p_ \ varepsilon $和$λ_\ varepsilon $分别是$ u_ \ varepsilon $的最大点和最大值。此外，$ p _ {\ varepsilon} $收敛到爆破点，这是函数$ 2 \ ln h^{+}+a $的关键点。
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 $2\ln h^{+}+A$.
