Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$

Alberto Farina

doi:10.1016/j.crma.2005.07.006

Partial Differential Equations

Liouville-type results for solutions of

- Δ u = {| u |}^{p - 1} u

on unbounded domains of

R^{N}

[Résultats de type Liouville pour des solutions de

- Δ u = {| u |}^{p - 1} u

dans des domaines non-bornés de

R^{N}

]

Présenté par : Philippe G. Ciarlet

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 341 (2005) no. 7, pp. 415-418.

Résumé
Abstract

Cette Note porte sur l'étude des solutions, éventuellement non-bornées et de signe quelconque, de l'équation $- Δ u = {| u |}^{p - 1} u$ dans des domaines non-bornés de $R^{N}$ avec $N ⩾ 2$ et $p > 1$ . Nous démontrons des résultats de type Liouville ainsi que des théorèmes de classification pour les solutions régulières appartenant à une des classes suivantes : solutions stables, solutions d'indice de Morse fini et solutions stables à l'extérieur d'un compact. Nous étendons aussi, au cas d'un épigraphe coercif régulier, les célèbres résultats de Gidas et Spruck concernant les solutions positives de l'équation considérée.

In this Note we study solutions, possibly unbounded and sign-changing, of the equation $- Δ u = {| u |}^{p - 1} u$ on unbounded domains of $R^{N}$ with $N ⩾ 2$ and $p > 1$ . We prove some Liouville-type results and a classification theorem for $C^{2}$ solutions belonging to one of the following classes: stable solutions, finite Morse index solutions and solutions which are stable outside a compact set. We also extend, to smooth coercive epigraphs, the well-known results of Gidas and Spruck concerning non-negative solutions of the considered equation.

Reçu le : 2005-06-27
Accepté le : 2005-06-29
Publié le : 2005-09-19

DOI : 10.1016/j.crma.2005.07.006

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 = {Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {415--418},
     publisher = {Elsevier},
     volume = {341},
     number = {7},
     year = {2005},
     doi = {10.1016/j.crma.2005.07.006},
     language = {en},
}

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Alberto Farina. Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$. Comptes Rendus. Mathématique, Volume 341 (2005) no. 7, pp. 415-418. doi : 10.1016/j.crma.2005.07.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.006/

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