On the general Toda system with multiple singular points

TL;DR

This paper investigates the existence and non-existence of solutions to a generalized elliptic Toda system with multiple singular points, extending previous results to broader Lie algebra cases.

Contribution

It generalizes prior work on Toda systems by establishing new existence and non-existence criteria for systems with multiple singular sources.

Findings

Established conditions for solution existence.

Proved non-existence under certain parameter regimes.

Extended previous results to general Lie algebra settings.

Abstract

In this paper, we consider the following elliptic Toda system associated to a general simple Lie algebra with multiple singular sources \begin{equation*} \begin{cases} -\Delta w_i=\sum_{j=1}^na_{i,j}e^{2w_j}+2\pi\sum_{\ell=1}^m\beta_{i,\ell}\delta_{p_\ell} \quad&\mbox{in}\quad\mathbb{R}^2,\\ \\ w_i(x)=-2\log|x|+O(1)~\mbox{as}~|x|\to\infty,\quad &i=1,\cdots,n, \end{cases} \end{equation*} where $\beta_{i,\ell}\in[0,1)$. Under some suitable assumption on $\beta_{i,\ell}$ we establish the existence and non-existence results. This paper generalizes Luo-Tian's [19] and Hyder-Lin-Wei's [10] results to the general Toda system.