The Moser-Trudinger inequality and its extremals on a disk via energy estimates

We study the Dirichlet energy of non-negative radially symmetric critical points $u_\mu$ of the Moser-Trudinger inequality on the unit disc in $\mathbb{R}^2$, and prove that it expands as
\[
4\pi + \frac{4\pi}{\mu^4}+o(\mu^{-4}) \le \int_{B_1} |\grad u_\mu|^2 dx \le 4\pi + \frac{6\pi}{\mu^4}+o(\mu^{-4}), as \mu \to \infty,
\]
where $\mu = u_\mu (0)$ is the maximum of $u_\mu$. As a consequence, we obtain a new proof of the Moser-Trudinger inequality, of the Carleson-Chang result about the existence of extremals, and of the Struwe and Lamm-Robert-Struwe multiplicity result in the supercritical regime (only in the case of the unit disk).
Our results are stable under su ciently weak perturbations of the Moser-Trudinger functional. We explicitly identify the critical level of perturbation for which, although the perturbed Moser-Trudinger inequality still holds, the energy of its critical points converges to $4\pi4 from below. We expect, in some of these cases, that the existence of extremals does not hold, nor the existence of critical points in the supercritical regime.