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)
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.