Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$$4\pi+\frac{4\pi}{\mu^{4}}+o(\mu^{-4})\le \int_{B_1}|\nabla u_\mu|^2dx\le
4\pi+\frac{6\pi}{\mu^{4}}+o(\mu^{-4}),\quad \text{as }\mu\to\infty,$$ 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.

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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: bayes-an