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

Abstract

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 $\ddot{I}{}_1,\ddot{I}{}_2\in (0,4\ddot{I}€]$, and $h_1$ and $h_2$ are two smooth functions on $M$.When some $\ddot{I}{}_i$ equals $4\ddot{I}€$, 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 $\ddot{I}{}_1=4\ddot{I}€$, $\ddot{I}{}_2\in (0,4\ddot{I}€)$ or $\ddot{I}{}_1=\ddot{I}{}_2=4\ddot{I}€$, 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 $\ddot{I}{}_1=4\ddot{I}€$, $\ddot{I}{}_2\in (0,4\ddot{I}€)$. 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 $\ddot{I}{}_1=\ddot{I}{}_2=4\ddot{I}€$. 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.

19 pages, no figures, all comments are welcome