Stable solutions of $ -\mathrm{\Delta }u={\mathrm{e}}^{u}$ on $ {\mathbb{R}}^{N}$

Alberto Farina

doi:10.1016/j.crma.2007.05.021

Partial Differential Equations

Stable solutions of

- Δ u = e^{u}

on

R^{N}

[Solutions stables de

- Δ u = e^{u}

dans

R^{N}

]

Présenté par : Haïm Brezis

Alberto Farina ¹

¹ LAMFA, CNRS UMR 6140, université de Picardie Jules-Verne, faculté de mathématiques et d'informatique, 33, rue Saint-Leu, 80039 Amiens, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 2, pp. 63-66.

Résumé
Abstract

Cette Note porte sur l'étude des solutions de l'équation $- Δ u = e^{u}$ dans $R^{N}$ , $N ⩾ 2$ . Nous démontrons la non-existence de solutions stables en dimension $N ⩽ 9$ . En dimension $N = 2$ , nous prouvons aussi un théorème de classification pour les solutions stables à l'extérieur d'un compact.

In this Note we study $C^{2}$ solutions of the equation $- Δ u = e^{u}$ on the entire Euclidean space $R^{N}$ , with $N ⩾ 2$ . We prove the non-existence of stable solutions for $N ⩽ 9$ . In the two-dimensional case we also demonstrate a classification theorem for solutions which are stable outside a compact set.

Reçu le : 2007-05-07
Publié le : 2007-07-03

DOI : 10.1016/j.crma.2007.05.021

Affiliations des auteurs :

Alberto Farina ¹

¹ LAMFA, CNRS UMR 6140, université de Picardie Jules-Verne, faculté de mathématiques et d'informatique, 33, rue Saint-Leu, 80039 Amiens, France

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     author = {Alberto Farina},
     title = {Stable solutions of $ -\mathrm{\Delta }u={\mathrm{e}}^{u}$ on $ {\mathbb{R}}^{N}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {63--66},
     publisher = {Elsevier},
     volume = {345},
     number = {2},
     year = {2007},
     doi = {10.1016/j.crma.2007.05.021},
     language = {en},
}

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Alberto Farina. Stable solutions of $ -\mathrm{\Delta }u={\mathrm{e}}^{u}$ on $ {\mathbb{R}}^{N}$. Comptes Rendus. Mathématique, Volume 345 (2007) no. 2, pp. 63-66. doi : 10.1016/j.crma.2007.05.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.05.021/

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