Vol. 11, No. 4, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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
Abstract

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

  1. The pair of local masses (σ1,σ2) at each blowup point has the expression
    σi = 2(Ni1μ1 + Ni2μ2 + Ni3),

    where Nij , i = 1,2, j = 1,2,3.

  2. At each vortex point pt if (αt1,αt2) are integers and ρi4π, then all the solutions of Toda systems are uniformly bounded.
  3. If the blowup point q is a vortex point pt and αt1,αt2 and 1 are linearly independent over Q, then
    uk(x) + 2log|x p t| 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