Multiple solutions to the Bahri-Coron problem in the complement of a thin tubular neighbourhood of a manifold

Abstract

We show that the critical problem \[ -\Delta u=|u|^{{{4}}/({{N-2}})}u\quad \text{in }\Omega,\qquad\ u=0\quad \text{on }\partial\Omega, \] has at least \[ \max\{\text{cat}(\Theta,\Theta\setminus B_{r}M),\text{cupl}(\Theta ,\Theta\setminus B_{r}M)+1\}\geq2 \] pairs of nontrivial solutions in every domain $\Omega$ obtained by deleting from a given bounded smooth domain $\Theta\subset\mathbb{R}^{N}$ a thin enough tubular neighborhood $B_{r}M$ of a closed smooth submanifold $M$ of $\Theta$ of dimension $\leq N-2$, where ``cat'' is the Lusternik-Schnirelmann category and ``cupl'' is the cup-length of the pair.