Concentration profiles for the Trudinger-Moser functional are shaped like toy pyramids

This paper answers the conjecture of Adimurthi and Struwe that the semilinear Trudinger-Moser functional (as well as functionals with more general critical nonlinearities) satisfies the Palais-Smale condition at all levels except n/2 for integer n. In this paper we construct critical sequences at any level greater than 1/2 corresponding to a large family of distinct concentration profiles, indexed by all closed subsets C of (0,1) that arise in the two-dimensional case instead of the "standard bubble" in higher dimensions. The approach is based on the profile decomposition in the style of Solimini.