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

Druet [6] proved that if $(f_\gamma)_\gamma$ is a sequence of Moser-Trudinger type nonlinearities with critical growth, and if $(u_\gamma)_\gamma$ solves $$ \begin{cases} &\Delta u =f_\gamma(x,u)\,,~~ u>0\text{ in }\Omega\,,\\ &u =0\text{ on }\partial\Omega\,, \end{cases} $$ and converges weakly in $H^1_0$ to some $u_\infty$, then the Dirichlet energy is quantified, namely there exists an integer $N\ge 0$ such that the energy of $u_\gamma$ converges to $4\pi N$ plus the Dirichlet energy of $u_\infty$. 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, the loss of compactness (i.e. $N>0$) implies that $u_\infty\equiv 0$. In contrast, we prove here that there exist sequences $(f_\gamma)_\gamma$ of Moser-Trudinger type nonlinearities which admit a noncompact sequence of solutions $(u_\gamma)_\gamma$ having a nontrivial weak limit.