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.