On the existence of maximizers for functionals with critical exponential growth in R^2

In this paper we establish the existence of extremal functions for weighted functionals with critical exponential growth in R^2, which arise from Henon-type equations. The proof is based on the notion of spherical symmetrization with respect to a measure, which allows us to reduce the problem to a one dimensional functional as in the proof due to Carleson and Chang for the unweighted case.