Symmetry of extremal functions in Moser-Trudinger inequalities and a Hénon type problem in dimension two

In this paper, we analyze the symmetry properties of maximizers of a H enon 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 ${\bf R}^2$ and $\alpha, \gamma>0$. We identify and study the limit functional: $$ \sup_{u\in H^1(\Omega)\atop ||u||\le 1} \int_{\partial\Omega} \left(e^{\gamma u^2} - 1\right)\,d\sigma, $$ 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.