On compactness in the Trüdinger-Moser inequality

• Adimurthi Tintarev ^[2] ; Cyril Tintarev ^[1]
  1. [1] Uppsala University

     Uppsala University

     Uppsala domkyrkoförs., Suecia

     
  2. [2] TIFR CAM, Bangalore, India
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 13, Nº 2, 2014, págs. 399-416
• Idioma: inglés
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• Resumen

  • We show that the Moser functional J (u) = R (e4⇡u2 − 1) dx on the set B = {u 2 H1 0 () : kruk2  1}, where  ⇢ R2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Define gsw(r ) = s−12 w(r s ) for s > 0. If uk * u in B while liminf J (uk ) > J (u), then, with some sk !0, uk = gsk  (2⇡)−12 min ⇢ 1, log 1 |x| $% , up to translations and up to a remainder vanishing in the Sobolev norm. In other words, the weak continuity fails only on translations of concentratingMoser functions.

    The proof is based on a profile decomposition similar to that of Solimini [16], but with different concentration operators, pertinent to the two-dimensional case.