This paper deals with the lack of compactness in the nonlinear elliptic problem $-\Delta u+u=|u|^{p-2}u$ in $\Omega,\ u>0$ in $\Omega,\ u=0$ on $\partial \Omega$, when $\Omega$ is un unbounded domain in $\mathbb{R}^n$ and $2<p<2n/(n-2)$.

In particular, a domain $\widetilde\Omega$ is provided where non-converging Palais–Smale sequences exist at every energy level. Nevertheless, it is proved that the problem has infinitely many solutions on $\widetilde\Omega$.