Profile of bubbling solutions to a Liouville system

Lin, Chang-Shou; Zhang, Lei

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 117-143.

En plusieurs champs de Physique, Chimie et Écologie, quelques modèles sont décrits par les systèmes de Liouville. Dans cet article nous prouvons d'abord un résultat de caractère unique pour un système de Liouville dans ${ℝ}^2$. Alors nous établissons une estimation uniforme pour les solutions d'explosion d'un système de Liouville localement défini prés d'un point d'explosion isolé. Le résultat d'unicité, aussi bien que les estimations uniformes locales sont les ingrédients cruciaux pour obtenir a priori l'estimation, les formules comptant le degré, et l'existence pour les systèmes de Liouville définis sur des surfaces de Reimann.

In several fields of Physics, Chemistry and Ecology, some models are described by Liouville systems. In this article we first prove a uniqueness result for a Liouville system in ${ℝ}^2$. Then we establish a uniform estimate for bubbling solutions of a locally defined Liouville system near an isolated blowup point. The uniqueness result, as well as the local uniform estimates are crucial ingredients for obtaining a priori estimate, degree counting formulas and existence results for Liouville systems defined on Riemann surfaces.

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