## Differential and Integral Equations

- Differential Integral Equations
- Volume 21, Number 5-6 (2008), 477-495.

### On the existence and radial symmetry of maximizers for functionals with critical exponential growth in $\Bbb R^2$

#### Abstract

We investigate the problem of existence and symmetry of maximizers for \[ S(\alpha,4\pi)=\sup_{\|u\|=1 }{\int_B\left(e^{4\pi u^2}-1\right)|x|^{\alpha}dx,} \] where $B$ is the unit disk in $\mathbb{R}^2$ and $\alpha >0$, proposed by Secchi and Serra in [11]. Through the notion of spherical symmetrization with respect to a measure, we prove that supremum is attained for $\alpha$ small. Furthermore, we prove that $S(\alpha,4\pi)$ is attained by a radial function.

#### Article information

**Source**

Differential Integral Equations, Volume 21, Number 5-6 (2008), 477-495.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356038629

**Mathematical Reviews number (MathSciNet)**

MR2483265

**Zentralblatt MATH identifier**

1224.35127

**Subjects**

Primary: 49J10: Free problems in two or more independent variables

Secondary: 35J60: Nonlinear elliptic equations

#### Citation

Tarsi, Cristina. On the existence and radial symmetry of maximizers for functionals with critical exponential growth in $\Bbb R^2$. Differential Integral Equations 21 (2008), no. 5-6, 477--495. https://projecteuclid.org/euclid.die/1356038629