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.