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

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