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 $$4\pi +\frac{4\pi }{{\mu }^4}+o({\mu }^{-4})\le {\int }_{B_1}\mathrm{\mid }\mathrm{\nabla }{u}_{\mu }{\mathrm{\mid }}^{2}dx\le 4\pi +\frac{6\pi }{{\mu }^4}+o({\mu }^{-4}),\text{as }\mu \to \mathrm{\infty },$$ 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.