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

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

$$\begin{aligned} 4\pi +\frac{4\pi }{\mu ^{4}}+o(\mu ^{-4})\le \int _{B_1}|\nabla u_\mu |^2dx\le 4\pi +\frac{6\pi }{\mu ^{4}}+o(\mu ^{-4}),\quad \text {as }\mu \rightarrow \infty , \end{aligned}$$

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.