Classification and nondegeneracy of SU(n+1) Toda system with singular sources

• 成果类型: Article
• 署名作者: Lin, Chang-Shou; Wei, Juncheng; Ye, Dong
• 署名单位: National Taiwan University; Chinese University of Hong Kong; Universite de Lorraine
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0378-3
• 发表日期: 2012
• 页码: 169-207
• 关键词: mean-field equations; analytic aspects; EXISTENCE; BEHAVIOR; blow

摘要

We consider the following Toda system Delta u(i) + Sigma(n)(j=1)a(ij)e(uj) = 4 pi gamma(i)delta(0) in R-2, integral(R2)e(ui)dx < infinity, for all 1 <= i <= n, where gamma (i) >-1, delta (0) is Dirac measure at 0, and the coefficients a (ij) form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result: Sigma(n)(j=1)a(ij) integral(R2)e(uj)dx = 4 pi(2 + gamma(i) + gamma(n+1-i)), for all 1 <= i <= n This generalizes the classification result by Jost and Wang for gamma (i) =0, . (ii) We prove that if gamma (i) +gamma (i+1)+a <-+gamma (j) a parts per thousand acurrency sign for all 1a parts per thousand currency signia parts per thousand currency signja parts per thousand currency signn, then any solution u (i) is radially symmetric w.r.t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system.