On compactness in the Trudinger-Moser inequality
  The paper studies continutity of Moser nonlinearity in two dimensions with respect to weak convergence. Unlike the critical nonlinearity in the Sobolev inequality, which lacks weak continuity at any point, Moser functional fails to be weakly continuous only in an exceptional case of a concentrating sequence of functions from the Moser family (up to translations and the remainder vanishing in Sobolev norm). The argument is based on a structural description of the defect of weak converegence, analogous to the profile decomposition established by Solimini for Sobolev inequalities, but involving gauge operators specific for the two-dimensional case. 
Adimurthi, Kyril Tintarev
math.AP