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

Abstract

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 $$\{\begin{array}{ll}& \Delta u=f_{\gamma }(x,u),u>0\text{ in }\Omega ,\\ & u=0\text{ on }\partial \Omega ,\end{array}$$ and converges weakly in $H_0^1$ 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.