The Adimurthi–Druet inequality is an improvement of the standard Moser–Trudinger inequality by adding a L2-type perturbation, quantified by α∈[0, λ1), where λ1 is the first Dirichlet eigenvalue of the Laplacian on a smooth bounded domain. It is known that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter αapproaches λ1. Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler–Lagrange equation, which take into account the fact that the problem becomes singular as α→λ1.

Non-existence of extremals for the Adimurthi–Druet inequality / Mancini, Gabriele; Thizy, Pierre-Damien. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 266:2-3(2019), pp. 1051-1072. [10.1016/j.jde.2018.07.065]

Non-existence of extremals for the Adimurthi–Druet inequality

Mancini, Gabriele;
2019

Abstract

The Adimurthi–Druet inequality is an improvement of the standard Moser–Trudinger inequality by adding a L2-type perturbation, quantified by α∈[0, λ1), where λ1 is the first Dirichlet eigenvalue of the Laplacian on a smooth bounded domain. It is known that this inequality admits extremal functions, when the perturbation parameter α is small. By contrast, we prove here that the Adimurthi–Druet inequality does not admit any extremal, when the perturbation parameter αapproaches λ1. Our result is based on sharp expansions of the Dirichlet energy for blowing sequences of solutions of the corresponding Euler–Lagrange equation, which take into account the fact that the problem becomes singular as α→λ1.
File allegati a questo prodotto
File Dimensione Formato  
4 - Non-existence of extremals for the Adimurthi–Druet inequality.pdf

solo gestori archivio

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 369.13 kB
Formato Adobe PDF
369.13 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

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: http://hdl.handle.net/11573/1272756
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 15
  • ???jsp.display-item.citation.isi??? 13
social impact