### 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 maximizers of a Hénon type functional in dimension two. Namely, we study the symmetry of the functions that realize the maximum $$\sup_{u\in {{H^{1}(\Omega)}}\atop ||u||\le 1} {{\int_\Omega}} \left(e^{\gamma u^2} - 1\right)|x|^\alpha {{\,dx}},$$ where $\Omega$ is the unit ball of $\mathbb R^2$ and $\alpha,\, \gamma>0$. We identify and study the limit functional $$\sup_{u\in {{H^{1}(\Omega)}}\atop ||u||\le 1} {{\int_{\partial\Omega}}} \left(e^{\gamma u^2} - 1\right)\,ds,$$ which is the main ingredient to describe the behavior of maximizers as $\alpha\to \infty$. We also consider the limit functional as $\alpha\to 0$ and the properties of its maximizers.

#### Article information

Source
Adv. Differential Equations, Volume 13, Number 1-2 (2008), 105-138.

Dates
First available in Project Euclid: 18 December 2012