标题： 带有变号预定函数的Toda系统的存在性结果：第二部分 显示英文标题

标题： Existence results for Toda systems with sign-changing prescribed functions: Part II

摘要： 设$(M, g)$是一个面积为$1$的紧致黎曼曲面。 我们研究在$(M, g)$上的 Toda 系统\begin{align} \begin{cases} -\Delta u_1 = 2\rho_1(h_1e^{u_1}-1) - \rho_2(h_2e^{u_2}-1),\\ -\Delta u_2 = 2\rho_2(h_2e^{u_2}-1) - \rho_1(h_1e^{u_1}-1), \end{cases} \end{align}，其中$\rho_1, \rho_2 \in (0,4\pi]$，$h_1$和$h_2$是$M$上的两个光滑函数。当某些$\rho_i$等于$4\pi$时，Toda 系统相对于其的 Moser-Trudinger 不等式变得临界，使得存在性问题显著更具挑战性。 在他们具有开创性的文章（Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558）中，Jost、Lin 和 Wang 在假设$h_1$和$h_2$均为正的情况下，建立了当$\rho_1=4\pi$、$\rho_2 \in (0,4\pi)$或$\rho_1=\rho_2=4\pi$时 Toda 系统解存在的充分条件。 在我们之前的论文中，我们将这些结果扩展，以允许$h_1$和$h_2$在情况$\rho_1=4\pi$，$\rho_2 \in (0,4\pi)$下改变符号。 在本文中，我们进一步扩展了研究，以证明 Jost-Lin-Wang 的充分条件即使在$h_1$和$h_2$可以改变符号以及$\rho_1=\rho_2=4\pi$的情况下仍然有效。 我们的证明依赖于 Toda 系统的 Moser-Trudinger 不等式的改进版本，以及类似于 Brezis-Merle 类型的专门分析和 Pohozaev 恒等式的使用。 显示英文摘要

摘要： Let $(M, g)$ be a compact Riemann surface with area $1$. We investigate the Toda system \begin{align} \begin{cases} -\Delta u_1 = 2\rho_1(h_1e^{u_1}-1) - \rho_2(h_2e^{u_2}-1),\\ -\Delta u_2 = 2\rho_2(h_2e^{u_2}-1) - \rho_1(h_1e^{u_1}-1), \end{cases} \end{align} on $(M, g)$ where $\rho_1, \rho_2 \in (0,4\pi]$, and $h_1$ and $h_2$ are two smooth functions on $M$.When some $\rho_i$ equals $4\pi$, the Toda system becomes critical with respect to the Moser-Trudinger inequality for it, making the existence problem significantly more challenging. In their seminal article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526--558), Jost, Lin, and Wang established sufficient conditions for the existence of solutions the Toda system when $\rho_1=4\pi$, $\rho_2 \in (0,4\pi)$ or $\rho_1=\rho_2=4\pi$, assuming that $h_1$ and $h_2$ are both positive. In our previous paper we extended these results to allow $h_1$ and $h_2$ to change signs in the case $\rho_1=4\pi$, $\rho_2 \in (0,4\pi)$. In this paper we further extend the study to prove that Jost-Lin-Wang's sufficient conditions remain valid even when $h_1$ and $h_2$ can change signs and $\rho_1=\rho_2=4\pi$. Our proof relies on an improved version of the Moser-Trudinger inequality for the Toda system, along with edicated analyses similar to Brezis-Merle type and the use of Pohozaev identities.