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} -Δu_1 = 2ρ_1(h_1e^{u_1}-1) - ρ_2(h_2e^{u_2}-1),\ -Δu_2 = 2ρ_2(h_2e^{u_2}-1) - ρ_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.