## Advances in Differential Equations

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

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

Denis Bonheure, Enrico Serra, and Massimo Tarallo

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867361

**Mathematical Reviews number (MathSciNet)**

MR2482538

**Zentralblatt MATH identifier**

1173.35044

**Subjects**

Primary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals

Secondary: 35A15: Variational methods 35J20: Variational methods for second-order elliptic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

#### Citation

Bonheure, Denis; Serra, Enrico; Tarallo, Massimo. Symmetry of extremal functions in Moser-Trudinger inequalities and a Hénon type problem in dimension two. Adv. Differential Equations 13 (2008), no. 1-2, 105--138. https://projecteuclid.org/euclid.ade/1355867361