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

This paper answers the conjecture of Adimurthi and Struwe [4], that the semilinear Trudinger-Moser functional(0.1)J(u)=12∫Ω|∇u|2dx−18π∫Ω(e4πu2−1)dx (as well as functionals with more general critical nonlinearities) satisfies the Palais-Smale condition at all levels except n2, n∈N. In this paper we construct critical sequences at any level c>12 corresponding to a large family of distinct concentration profiles, indexed by closed subsets C of (0,1), that arise in the two-dimensional case instead of the “standard bubble” in higher dimensions. The paper uses the notion of concentration of [2,5] developed in the spirit of Solimini [14] and of [15].