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

Jean Dolbeault (1) , Maria J. Esteban (1) , Gabriella Tarantello (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DIPMAT - Dipartimento di Matematica [Roma II]

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

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Maria J. Esteban, Gabriella Tarantello. 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, 2008, 7, pp.313-341. ⟨hal-00139062⟩