Symmetry of extremal functions in moser-trudinger inequalities and a Hénon type problem in dimension two

Abstract

In this paper, we analyze the symmetry properties of maxi-mizers of a Henon type functional in dimension two. Namely, we study the symmetry of the functions that realize the maximum where Ω is the unit ball of R2 and α, γ > 0. We identify and study the limit functional which is the main ingredient to describe the behavior of maximizers as α -> ∞ We also consider the limit functional as α -> 0 and the properties of its maximizers.