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On Toda system with Cartan matrix $ G_2$


Authors: Weiwei Ao, Chang-Shou Lin and Juncheng Wei
Journal: Proc. Amer. Math. Soc. 143 (2015), 3525-3536
MSC (2010): Primary 35C05; Secondary 35C11, 58A17
DOI: https://doi.org/10.1090/proc/12558
Published electronically: April 9, 2015
MathSciNet review: 3348794
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Abstract: We consider the Toda system

$\displaystyle \Delta u_i + \sum _{j = 1}^2 a_{ij}e^{u_j} = 4\pi \gamma _{i}\delta _{0} \;\;$$\displaystyle \text {in }\mathbb{R}^2, \quad \int _{\mathbb{R}^2}e^{u_i} dx < \infty ,\;\; \mbox { for } i=1,2,$    

where $ \gamma _{i} > -1$, $ \delta _0$ is the Dirac measure at 0, and the coefficients $ a_{ij}$ are of the Cartan matrix of rank 2: $ A_2, B_2(=C_2),G_2$. Previously, the authors have gotten the classification and non-degeneracy results of solutions for Cartan matrix $ A_2$ and $ B_2$. In this paper, we consider the $ G_2$ case, and we completely classify the solutions and obtain the quantization result as well as the non-degeneracy of solutions for the $ G_2$ Toda system.

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Additional Information

Weiwei Ao
Affiliation: Center for Advanced Study in Theoretical Science, National Taiwan University, Taipei, Taiwan
Email: weiweiao@gmail.com

Chang-Shou Lin
Affiliation: Taida Institute of Mathematics, Center for Advanced Study in Theoretical Science, National Taiwan University, Taipei, Taiwan
Email: cslin@math.ntu.edu.tw

Juncheng Wei
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada — and — Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
Email: jcwei@math.ubc.ca

DOI: https://doi.org/10.1090/proc/12558
Keywords: Toda system, Cartan matrix $G_2$, classification, nondegeneracy
Received by editor(s): March 10, 2014
Published electronically: April 9, 2015
Additional Notes: The third author was partially supported by NSERC of Canada
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society