## Analysis & PDE

• Anal. PDE
• Volume 11, Number 4 (2018), 873-898.

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

#### 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 $( σ 1 , σ 2 )$ at each blowup point has the expression $σ i = 2 ( N i 1 μ 1 + N i 2 μ 2 + N i 3 ) ,$ where $N i j ∈ ℤ$, $i = 1 , 2$, $j = 1 , 2 , 3$.
2. At each vortex point $p t$ if $( α t 1 , α t 2 )$ are integers and $ρ i ∉ 4 π ℕ$, then all the solutions of Toda systems are uniformly bounded.
3. 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 $u k ( x ) + 2 log | x − p t | ≤ C .$

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

#### Article information

Source
Anal. PDE, Volume 11, Number 4 (2018), 873-898.

Dates
Revised: 17 August 2017
Accepted: 5 December 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.apde/1517454157

Digital Object Identifier
doi:10.2140/apde.2018.11.873

Mathematical Reviews number (MathSciNet)
MR3749370

Zentralblatt MATH identifier
1383.35078

Subjects
Primary: 35J47: Second-order elliptic systems
Secondary: 35J60: Nonlinear elliptic equations 35J55

#### Citation

Lin, Chang-Shou; Wei, Jun-cheng; Yang, Wen; Zhang, Lei. On rank-2 Toda systems with arbitrary singularities: local mass and new estimates. Anal. PDE 11 (2018), no. 4, 873--898. doi:10.2140/apde.2018.11.873. https://projecteuclid.org/euclid.apde/1517454157

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