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

Jean Dolbeault, Maria J. Esteban, Gabriella Tarantello (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We first discuss a class of inequalities of Onofri 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 Caffarelli-Kohn-Nirenberg inequality, in two space dimensions. In fact, for suitable sets of parameters (asymptotically...

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