#### Vol. 11, No. 4, 2018

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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
##### 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 $\left({\sigma }_{1},{\sigma }_{2}\right)$ at each blowup point has the expression
${\sigma }_{i}=2\left({N}_{i1}{\mu }_{1}+{N}_{i2}{\mu }_{2}+{N}_{i3}\right),$

where ${N}_{ij}\in ℤ$, $i=1,2$, $j=1,2,3$.

2. At each vortex point ${p}_{t}$ if $\left({\alpha }_{t}^{1},{\alpha }_{t}^{2}\right)$ are integers and ${\rho }_{i}\notin 4\pi ℕ$, then all the solutions of Toda systems are uniformly bounded.
3. If the blowup point $q$ is a vortex point ${p}_{t}$ and ${\alpha }_{t}^{1},{\alpha }_{t}^{2}$ and $1$ are linearly independent over $Q$, then
${u}^{k}\left(x\right)+2log|x-{p}_{t}|\le C.$

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