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

Ali Hyder, Changshou Lin and Juncheng Wei

Vol. 305 (2020), No. 2, 645–666

DOI: 10.2140/pjm.2020.305.645

Abstract

We consider the singular SU$\left(3\right)$ Toda system with multiple singular sources:

$$\begin{array}{lll}\hfill -\Delta w_1& =2e^{2w_1}-e^{2w_2}+2\pi \sum _{\ell =1}^m{\beta }_{1,\ell }{\delta }_{P_{\ell }}\text{ in }{ℝ}^2,& \hfill \text{ (1)}\\ \hfill -\Delta w_2& =2e^{2w_2}-e^{2w_1}+2\pi \sum _{\ell =1}^m{\beta }_{2,\ell }{\delta }_{P_{\ell }}\text{ in }{ℝ}^2,\\ \hfill w_i\left(x\right)& =-2log\left|x\right|+O\left(1\right)\text{ as }\left|x\right|\to \infty ;i=1,2,& \hfill \text{ (2)}\end{array}$$

with $m\ge 3$ and ${\beta }_{i,\ell }\in \left[0,1\right)$. We prove existence and nonexistence results under suitable assumptions on ${\beta }_{i,\ell }$. This generalizes Luo–Tian’s (1992) result for a singular Liouville equation in ${ℝ}^2$. We also study existence results for a higher order singular Liouville equation in ${ℝ}^n$.

Keywords

Toda system, blow-up analysis

Mathematical Subject Classification 2010

Primary: 35B40, 35B44, 35J61

Milestones

Received: 13 March 2019

Revised: 19 May 2019

Accepted: 11 October 2019

Published: 29 April 2020