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

Let $(M,g)$ be a compact Riemann surface with area $1$, we shall study the Toda system $$\{\begin{array}{l}-\mathrm{\Delta }{u}_1=2{\rho }_1(h_1{e}^{u_1}-1)-{\rho }_2(h_2{e}^{u_2}-1),\\ -\mathrm{\Delta }{u}_2=2{\rho }_2(h_2{e}^{u_2}-1)-{\rho }_1(h_1{e}^{u_1}-1),\end{array}$$ on $(M,g)$ with ${\rho }_1=4\pi$, ${\rho }_2\in (0,4\pi )$, $h_1$ 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 this Toda system 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 the blowup can only happen at one point where $h_1$ is positive.