{
  "id": "1608.07169",
  "version": "v1",
  "published": "2016-08-25T14:25:49.000Z",
  "updated": "2016-08-25T14:25:49.000Z",
  "title": "The Moser-Trudinger inequality and its extremals on a disk via energy estimates",
  "authors": [
    "Gabriele Mancini",
    "Luca Martinazzi"
  ],
  "comment": "29 pages",
  "categories": [
    "math.AP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-25T14:25:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B33",
      "35B44",
      "35B38",
      "35J61"
    ],
    "keywords": [
      "energy estimates",
      "lamm-robert-struwe multiplicity result",
      "supercritical regime",
      "perturbed moser-trudinger inequality",
      "non-negative radially symmetric critical points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}