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

TL;DR

This paper proves the existence of solutions for a critical Toda system on a compact Riemann surface with sign-changing functions, extending previous results to the case where parameters reach critical values and functions change signs.

Contribution

It extends the existence results for Toda systems to the critical case with sign-changing functions, using improved inequalities and analytical techniques.

Findings

Existence of solutions when parameters are at critical values with sign-changing functions.

Validation of previous sufficient conditions in the critical case.

Development of an improved Moser-Trudinger inequality for the Toda system.

Abstract

Let $(M, g)$ be a compact Riemann surface with area $1$. We investigate the Toda system \begin{align} \begin{cases} -\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{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.