On rank-2 Toda systems with arbitrary singularities: local mass and new estimates

Lin, Chang-Shou; Wei, Jun-cheng; Yang, Wen; Zhang, Lei

doi:10.2140/apde.2018.11.873

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On rank-2 Toda systems with arbitrary singularities: local mass and new estimates

Chang-Shou Lin, Jun-cheng Wei, Wen Yang and Lei Zhang

Vol. 11 (2018), No. 4, 873–898

DOI: 10.2140/apde.2018.11.873

Abstract

For all rank-2 Toda systems with an arbitrary singular source, we use a unified approach to prove:

The pair of local masses $(σ_{1}, σ_{2})$ at each blowup point has the expression $σ_{i} = 2 (N_{i 1} μ_{1} + N_{i 2} μ_{2} + N_{i 3}),$
where $N_{i j} \in ℤ$ , $i = 1, 2$ , $j = 1, 2, 3$ .
At each vortex point $p_{t}$ if $(α_{t}^{1}, α_{t}^{2})$ are integers and $ρ_{i} \notin 4 π ℕ$ , then all the solutions of Toda systems are uniformly bounded.
If the blowup point $q$ is a vortex point $p_{t}$ and $α_{t}^{1}, α_{t}^{2}$ and $1$ are linearly independent over $Q$ , then $u^{k} (x) + 2 log | x - p_{t} | \leq C .$

The Harnack-type inequalities of 3 are important for studying the bubbling behavior near each blowup point.

Keywords

SU$(n{+}1)$-Toda system, asymptotic analysis, a priori estimate, classification theorem, topological degree, blowup solutions, Riemann–Hurwitz theorem

Mathematical Subject Classification 2010

Primary: 35J47

Secondary: 35J60, 35J55

Milestones

Received: 3 November 2016

Revised: 17 August 2017

Accepted: 5 December 2017

Published: 12 January 2018

Authors

Chang-Shou Lin
Department of Mathematics Taida Institute of Mathematical Sciences National Taiwan University Taipei Taiwan

Jun-cheng Wei
Department of Mathematics University of British Columbia Vancouver BC Canada

Wen Yang
Wuhan Institute of Physics and Mathematics Chinese Academy of Sciences Wuhan China

Lei Zhang
Department of Mathematics University of Florida Gainesville, FL United States

Vol. 11, No. 4, 2018