Let \(\left( M,g\right) \) be a compact Riemann surface with area 1, we shall study the Toda system 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1 = 2\rho _1\left( h_1e^{u_1}-1\right) - \rho _2\left( h_2e^{u_2}-1\right) ,\\ -\Delta u_2 = 2\rho _2\left( h_2e^{u_2}-1\right) - \rho _1\left( h_1e^{u_1}-1\right) , \end{array}\right. } \end{aligned}\) on \(\left( M,g\right) \) with \(\rho _1=4\pi \) , \(\rho _2\in \left( 0,4\pi \right) \) , \(h_1\) and \(h_2\) are two smooth functions on M. In Jost-Lin-Wang’s celebrated article (Comm. Pure Appl. Math., 59 (2006), no. 4, 526–558), they obtained a sufficient condition for the existence of Eq. (0.1) when \(h_1\) and \(h_2\) are both positive. In this paper, we shall improve this result to the case \(h_1\) and \(h_2\) can change signs. We shall pursue a variational method and use the standard blowup analysis. Among other things, the main contribution in our proof is to show that the blowup can only happen at one point where \(h_1\) is positive.
