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}-Δw_1=2e^{2w_1}-e^{w_2}+2π\sum_{\ell=1}^mβ_{1,\ell}δ_{P_{\ell}}\quad\text{in }\mathbb{R}^2\\ \rule{0cm}{.5cm} -Δw_2=2e^{2w_2}-e^{w_1}+2π\sum_{\ell=1}^mβ_{2,\ell}δ_{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 $m\ge 3$ and ${\hat{I}}_{i,\ell }^2\in [0,1)$. We prove the existence and non-existence results under suitable assumptions on ${\hat{I}}_{i,\ell }^2$. 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 ${\mathbb{R}}^n$.

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