We study the Dirichlet energy of non-negative radially symmetric critical points uμ of the Moser–Trudinger inequality on the unit disc in R2, and prove that it expands as (Formula presented.),where μ= uμ(0) is the maximum of uμ. 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 π 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.

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

Mancini G.;Martinazzi L.
2017

Abstract

We study the Dirichlet energy of non-negative radially symmetric critical points uμ of the Moser–Trudinger inequality on the unit disc in R2, and prove that it expands as (Formula presented.),where μ= uμ(0) is the maximum of uμ. 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 π 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.
File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3358053
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 37
  • ???jsp.display-item.citation.isi??? 34
social impact