The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions

Dolbeault, Jean; Esteban, Maria J.; Tarantello, Gabriella

Dolbeault, Jean; Esteban, Maria J.; Tarantello, Gabriella (2008), The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 7, 2, p. 313–341. http://dx.doi.org/10.2422/2036-2145.2008.2.05

Type

Article accepté pour publication ou publié

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http://hal.archives-ouvertes.fr/hal-00139062/en/

Date

2008

Nom de la revue

Annali della Scuola Normale Superiore di Pisa. Classe di Scienze

Volume

Numéro

Éditeur

Le Edizioni della Normale

Pages

313–341

Résumé (EN)

We first prove a weighted inequality of Moser-Trudinger type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than -1. Without symmetry assumption, it holds if and only if the parameter is in the interval (-1,0]. The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Hardy-Sobolev inequality, as established by Caffarelli-Kohn-Nirenberg, in two space dimensions. In fact, for suitable sets of parameters (asymptotically sharp) we prove symmetry or symmetry breaking by means of a blow-up method. In this way, the weighted Moser-Trudinger inequality appears as a limit case of the Hardy-Sobolev inequality.

Mots-clés

Weighted Moser-Trudinger inequality; Hardy-Sobolev inequality; Onofri's inequality; Caffarelli-Kohn-Nirenberg inequality; extremal functions; Kelvin transformation; Emden-Fowler transformation; stereographic projection; radial symmetry; symmetry breaking; blow-up analysis