Boundary singularities of positive solutions of some nonlinear elliptic equations

Bidaut-Véron, Marie-Françoise; Ponce, Augusto C.; Véron, Laurent

doi:10.1016/j.crma.2006.11.027

Partial Differential Equations

Boundary singularities of positive solutions of some nonlinear elliptic equations
[Singularités au bord de solutions positives d'équations elliptiques non-linéaires]

Présenté par : Haïm Brezis

Bidaut-Véron, Marie-Françoise ¹ ; Ponce, Augusto C.¹ ; Véron, Laurent ¹

¹ Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Faculté des Sciences, 37200 Tours, France

Comptes Rendus. Mathématique, Tome 344 (2007) no. 2, pp. 83-88.

Résumé
Abstract

Nous étudions le comportement quand x tend vers $x_{0}$ de toute solution positive de (E) $- Δ u = u^{q}$ dans Ω qui s'annule sur $\partial Ω ∖ {x_{0}}$ , où $Ω \subset R^{N}$ est un domaine régulier, $q ⩾ (N + 1) / (N - 1)$ et $x_{0} \in \partial Ω$ . Nos résultats sont fondés sur des estimations a priori des solutions de (E), et des résultats d'existence, de non existence et d'unicité de solutions de certaines équations elliptiques non linéaires sur la demi-sphère unité.

We study the behavior near $x_{0}$ of any positive solution of (E) $- Δ u = u^{q}$ in Ω which vanishes on $\partial Ω ∖ {x_{0}}$ , where $Ω \subset R^{N}$ is a smooth domain, $q ⩾ (N + 1) / (N - 1)$ and $x_{0} \in \partial Ω$ . Our results are based upon a priori estimates of solutions of (E) and existence, non-existence and uniqueness results for solutions of some nonlinear elliptic equations on the upper-half unit sphere.

Accepté le : 2006-11-14
Publié le : 2006-12-19

DOI : 10.1016/j.crma.2006.11.027

Affiliations des auteurs :

Bidaut-Véron, Marie-Françoise ¹ ; Ponce, Augusto C. ¹ ; Véron, Laurent ¹

¹ Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Faculté des Sciences, 37200 Tours, France

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     title = {Boundary singularities of positive solutions of some nonlinear elliptic equations},
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Bidaut-Véron, Marie-Françoise; Ponce, Augusto C.; Véron, Laurent. Boundary singularities of positive solutions of some nonlinear elliptic equations. Comptes Rendus. Mathématique, Tome 344 (2007) no. 2, pp. 83-88. doi : 10.1016/j.crma.2006.11.027. http://www.numdam.org/articles/10.1016/j.crma.2006.11.027/

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