Resumen de On compactness in the Trüdinger-Moser inequality

Adimurthi Tintarev, Kyril Tintarev

• 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.