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

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 $m\ge 3$ and ${\beta }_{i,\mathrm{\ell }}\in [0,1)$. We prove the existence and non-existence results under suitable assumptions on ${\beta }_{i,\mathrm{\ell }}$. 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 ${\mathbb{R}}^n$.