On $SU(3)$ Toda system with multiple singular sources

Abstract

We consider the singular $SU(3)$ Toda system with multiple singular sources \begin{align*} \left\{\begin{array}{ll}-\Delta w_1=2e^{2w_1}-e^{w_2}+2\pi\sum_{\ell=1}^m\beta_{1,\ell}\delta_{P_{\ell}}\quad\text{in }\mathbb{R}^2\\ \rule{0cm}{.5cm} -\Delta w_2=2e^{2w_2}-e^{w_1}+2\pi\sum_{\ell=1}^m\beta_{2,\ell}\delta_{P_{\ell}}\quad\text{in }\mathbb{R}^2 \\ w_i(x)=-2\log|x|+O(1)\quad\text{as }|x|\to\infty,\, i=1,2, \end{array}\right.\end{align*} with $m\geq 3$ and $\beta_{i,\ell}\in [0,1)$. We prove the existence and non-existence results under suitable assumptions on $\beta_{i,\ell}$. This generalizes Luo-Tian's \cite{Luo-Tian} result for a singular Liouville equation in $\mathbb{R}^2$. We also study existence results for a higher order singular Liouville equation in $\mathbb{R}^n$.