A weighted Moser-Trudinger inequality and its relation to the Caffarelli-Kohn-Nirenberg inequalities in two space dimensions

TL;DR

This paper establishes a weighted Moser-Trudinger inequality in two dimensions, explores symmetry breaking phenomena for extremal functions of Hardy-Sobolev inequalities, and connects these results through blow-up analysis.

Contribution

It introduces a new weighted Moser-Trudinger inequality depending on a parameter, analyzes symmetry and symmetry breaking of extremals, and links it as a limit case of Hardy-Sobolev inequalities.

Findings

Weighted inequality holds for radial functions if parameter > -1

Without symmetry, the inequality holds iff parameter in (-1,0]

Identifies conditions for symmetry breaking and symmetry in extremal functions

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.