Glueing a peak to a non-zero limiting profile for a critical Moser–Trudinger equation

Druet proved that for any given sequence of Moser–Trudinger type nonlinearities with critical growth, and any sequence of solutions to the corresponding semilinear problem for the laplacian operator which converges weakly in H^1_0 to some u_∞, then the Dirichlet energy is quantified, namely there exists an integer N≥0 such that the energy of the solutions converges to 4πN plus the Dirichlet energy of u_∞. As a crucial step to get the general existence results of [7], it was more recently proved in [8] that, for a specific class of nonlinearities (see (2)), the loss of compactness (i.e. N>0) implies that u_∞≡0. In contrast, we prove here that there exist sequences of Moser–Trudinger type nonlinearities which admit a noncompact sequence of solutions having a nontrivial weak limit.