Let (M, g) be a compact Riemann surface with area 1. We investigate the Toda system 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1 = 2\rho _1(h_1e^{u_1}-1) - \rho _2(h_2e^{u_2}-1),\\ -\Delta u_2 = 2\rho _2(h_2e^{u_2}-1) - \rho _1(h_1e^{u_1}-1), \end{array}\right. } \end{aligned}\) on (M, g) where \(\rho _1, \rho _2 \in (0,4\pi ]\) , and \(h_1\) and \(h_2\) are two \(C^2\) functions on M. When some \(\rho _i\) equals \(4\pi \) , Eq. (0.1) becomes critical with respect to the Moser-Trudinger inequality for the Toda system, 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 to Eq. (0.1) 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 dedicated analyses similar to Brezis-Merle type and the use of Pohozaev identities.
