Sharp existence, symmetry and asymptotics results for the singular $SU(3)$ Toda system with critical parameters

Abstract

We study the singular $SU(3)$ Toda system on a torus\[\begin{cases}\Delta u + 2e^u - e^v = 4 \pi n \delta_0 \\\Delta v + 2e^v - e^u = 4 \pi n_2 \delta_0 \\\end{cases}\quad \textrm{on} \quad E_\tau := \mathbb{C}/(\mathbb{Z}+\mathbb{Z} \tau) \: ,\]where $\delta_0$ is the Dirac measure at $0$ and $n, n_2 \in \mathbb{N}$. For the noncritical case $3 \nmid (n_2 - n)$ where solutions always exists by [Battaglia et al. Adv. Math. 2015], we proved recently in [Chen-Lin, J. Differ. Geom. 2023] that the number of solutions is at most $(n + 1)(n_2 + 1)(n + n_2 + 2)/6$. <p>This paper is the first one to study the critical case $n_2 = n+3l$ with $l \in \mathbb{N}$, for which the a priori estimates do not hold and the (non-)existence of solutions has been a longstanding problem. We show that

(i) When $n, l$ are both odd, the Toda system has no even solutions.

(ii) When $n$ is odd and $l$ is even, the Toda system always has even solutions, and each even solution belongs to a unique $2$-parametric family of even solutions $(u_{\lambda,\mu})$, where $\lambda,\mu \gt 0$ can be arbitrary. Moreover, if $\tau \in i \mathbb{R}_{\gt 0}$, i.e. $E_\tau$ is a rectangular torus, then the number of $2$-parametric families of even solutions is exactly $\frac{n+1}{2}$ and all these even solutions are axisymmetric. The asymptotics of $(u_{\lambda,\mu}, v_{\lambda,\mu})$ as $\lambda, \mu \to 0, + \infty$ are also studied.

The case that $n$ is even is quite different and the existence of even solutions depends on the geometry of the torus $E_\tau$. As applications, we obtain sharp existence and non-existence results for the $SU(3)$ Toda system with four singular sources in the plane.