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

Mancini, Gabriele and Martinazzi, Luca. (2016) The Moser-Trudinger inequality and its extremals on a disk via energy estimates. Preprints Fachbereich Mathematik, 2016 (17).

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Official URL: https://edoc.unibas.ch/69975/

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).