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We study the Dirichlet energy of non-negative radially symmetric criticalpoints $u_\mu$ of the Moser-Trudinger inequality on the unit disc in$\mathbb{R}^2$, and prove that it expands as$$4\pi+\frac{4\pi}{\mu^{4}}+o(\mu^{-4})\le \int_{B_1}|\nabla u_\mu|^2dx\le4\pi+\frac{6\pi}{\mu^{4}}+o(\mu^{-4}),\quad \text{as }\mu\to\infty,$$ where$\mu=u_\mu(0)$ is the maximum of $u_\mu$. As a consequence, we obtain a newproof of the Moser-Trudinger inequality, of the Carleson-Chang result about theexistence of extremals, and of the Struwe and Lamm-Robert-Struwe multiplicityresult in the supercritical regime (only in the case of the unit disk).  Our results are stable under sufficiently weak perturbations of theMoser-Trudinger functional. We explicitly identify the critical level ofperturbation for which, although the perturbed Moser-Trudinger inequality stillholds, the energy of its critical points converges to $4\pi$ from below. Weexpect, in some of these cases, that the existence of extremals does not hold,nor the existence of critical points in the supercritical regime.

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