On nondegeneracy of solutions to $ \mathit{SU}(3)$ Toda system

Juncheng Wei; Chunyi Zhao; Feng Zhou

doi:10.1016/j.crma.2010.11.025

Partial Differential Equations

On nondegeneracy of solutions to

SU (3)

Toda system
[Sur la nondégénérescence de solutions de

SU (3)

système de Toda]

Présenté par : Haïm Brezis

Juncheng Wei ¹ ; Chunyi Zhao ² ; Feng Zhou ²

¹ Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
² Department of Mathematics, East China Normal University, Shanghai, 200241, PR China

Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 185-190.

Résumé
Abstract

On montre que pour toute solution de $SU (3)$ système de Toda suivant $Δ u + 2 e^{u} - e^{v} = 0$ , $Δ v - e^{u} + 2 e^{v} = 0$ dans $R^{2}$ , $\int_{R^{2}} e^{u} < \infty$ , $\int_{R^{2}} e^{v} < \infty$ , le noyau de l'opérateur linéarisé associé est exactement de dimension huit, i.e., ce qu'on appelle la nondégénérescence.

We prove that the solution to the following $SU (3)$ Toda system

{\begin{matrix} Δ u + 2 e^{u} - e^{v} = 0, Δ v - e^{u} + 2 e^{v} = 0 in R^{2}, \\ \int_{R^{2}} e^{u} < \infty, \int_{R^{2}} e^{v} < \infty, \end{matrix}

is nondegenerate, i.e., the kernel of the associated linearized operator is exactly eight-dimensional.

Reçu le : 2010-11-23
Publié le : 2011-01-12

DOI : 10.1016/j.crma.2010.11.025

Affiliations des auteurs :

Juncheng Wei ¹ ; Chunyi Zhao ² ; Feng Zhou ²

¹ Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
² Department of Mathematics, East China Normal University, Shanghai, 200241, PR China

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     author = {Juncheng Wei and Chunyi Zhao and Feng Zhou},
     title = {On nondegeneracy of solutions to $ \mathit{SU}(3)$ {Toda} system},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {185--190},
     publisher = {Elsevier},
     volume = {349},
     number = {3-4},
     year = {2011},
     doi = {10.1016/j.crma.2010.11.025},
     language = {en},
}

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Juncheng Wei; Chunyi Zhao; Feng Zhou. On nondegeneracy of solutions to $ \mathit{SU}(3)$ Toda system. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 185-190. doi : 10.1016/j.crma.2010.11.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.025/

Bibliographie
Cité par

[1] C.C. Chen; C.S. Lin Sharp estimates for solutions of multi-bubbles in compact Riemann surface, Comm. Pure Appl. Math., Volume 55 (2002) no. 6, pp. 728-771

[2] G. Dunne Self-dual Chern–Simons Theories, Lecture Notes in Physics, vol. 36, Springer, Berlin, 1995

[3] J. Jost; G.F. Wang Analytic aspects of the Toda system. I. A Moser–Trudinger inequality, Comm. Pure Appl. Math., Volume 54 (2001) no. 11, pp. 1289-1319

[4] J. Jost; G.F. Wang Classification of solutions of a Toda system in $R^{2}$ , Int. Math. Res. Not., Volume 2002 (2002) no. 6, pp. 277-290

[5] J. Jost; C.S. Lin; G.F. Wang Analytic aspects of Toda system. II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math., Volume 59 (2006) no. 4, pp. 526-558

[6] A.N. Leznov; M.V. Saveliev Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems, Progress in Physics, vol. 15, Birkhäuser, 1992

[7] J.Y. Li; Y.X. Li Solutions for Toda systems on Riemann surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 5 (2005) no. 4, pp. 703-728

[8] A. Malchiodi; C.B. Ndiaye Some existence results for the Toda system on closed surfaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Math. Appl., Volume 18 (2007) no. 4, pp. 391-412

[9] H. Ohtsuka; T. Suzuki Blow-up analysis for $SU (3)$ Toda system, J. Differential Equations, Volume 232 (2007) no. 2, pp. 419-440

[10] Y. Yang Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics, Springer, New York, 2001

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