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On the general Toda system with multiple singular points

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Abstract

In this paper, we consider the following elliptic Toda system associated to any general simple Lie algebra with multiple singular sources

$$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta w_i=\sum _{j=1}^na_{i,j}e^{2w_j}+2\pi \sum _{\ell =1}^m\beta _{i,\ell }\delta _{p_\ell } \quad &{}\hbox {in}\quad \mathbb {R}^2,\\ w_i(x)=-\,2\log |x|+O(1)~\hbox {as}~|x|\rightarrow \infty ,\quad &{} i=1,\ldots ,n, \end{array}\right. } \end{aligned}$$

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 and Tian’s (Proc Am Math Soc 116(4):1119–1129, 1992) and Hyder et al. (Pac J Math 305(2):645–666, 2020) results to the general Toda system.

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Notes

  1. Even though the equation satisfied by \(\psi ^k_i\) for \(2\le i\le n-1\) looks slightly different from the one of \(\psi _1^k\), the proof is the same.

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Funding

Ali Hyder is supported by the SNSF Grant No. P2BSP2-172064. Jun-Cheng Wei is partially supported by NSERC. Wen Yang is partially supported by NSFC No.11801550 and NSFC No.11871470

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Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Ali Hyder

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Jun-cheng Wei

  3. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan, 430071, People’s Republic of China

    Wen Yang

  4. Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan, 430071, People’s Republic of China

    Wen Yang

Authors
  1. Ali Hyder
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  2. Jun-cheng Wei
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  3. Wen Yang
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Correspondence to Wen Yang.

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Communicated by M. Del Pino.

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Hyder, A., Wei, Jc. & Yang, W. On the general Toda system with multiple singular points. Calc. Var. 59, 136 (2020). https://doi.org/10.1007/s00526-020-01783-9

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