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

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.