07-74 Jean DOLBEAULT, Maria J. ESTEBAN, Gabriella TARANTELLO
Weighted Moser-Trudinger and Hardy-Sobolev inequalities]{A weighted Moser-Trudinger inequality and its relation to the Caffarelli-Kohn- Nirenberg inequalities in two space dimensions. (1283K, Postscript) Mar 29, 07
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Abstract. 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.

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