*preprint*

**Inserted:** 17 jul 2018

**Year:** 2016

**Abstract:**

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
$$4\pi+\frac{4\pi}{\mu^{{4}}+o}(\mu^{{}-4})\le \int_{{B}_{1}\nabla} u_{\mu}^{2dx\le}
4\pi+\frac{6\pi}{\mu^{{4}}+o}(\mu^{{}-4}),\quad \text{as }\mu\to\infty,$$ where
$\mu=u_\mu(0)$ is the maximum of $u_\mu$. As a consequence, we obtain a 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 sufficiently 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 $4\pi$ 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.