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 . Without symmetry assumption, it holds if and only if the parameter is in the interval . 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 sharp) we prove symmetry or symmetry breaking by means of a blow-up method and a careful analysis of the convergence to a solution of a Liouville equation. In this way, the Onofri inequality appears as a limit case of the Caffarelli-Kohn-Nirenberg inequality.
@article{ASNSP_2008_5_7_2_313_0,
author = {Dolbeault, Jean and Esteban, Maria Jesus and Tarantello, Gabriella},
title = {The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 7},
number = {2},
year = {2008},
pages = {313-341},
zbl = {1179.26055},
mrnumber = {2437030},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2008_5_7_2_313_0}
}
Dolbeault, Jean; Esteban, Maria J.; Tarantello, Gabriella. 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, Serie 5, Volume 7 (2008) no. 2, pp. 313-341. http://www.numdam.org/item/ASNSP_2008_5_7_2_313_0/
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