Supercritical problems in domains with thin toroidal holes

Abstract

In this paper we study the Lane-Emden-Fowler equation $$(P)_\epsilon\ \{\Delta u+|u|^{q-2}u=0 \ \hbox{in}\ \mathcal D_\epsilon, u=0 \ \hbox{on}\ \partial\mathcal D_\epsilon.$$ Here $\mathcal D_\epsilon = \mathcal D \setminus \{x \in \mathcal D \ : \ \mathrm{dist}(x,\Gamma_\ell)\le \epsilon\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $\Gamma_\ell$ is an $\ell-$dimensional closed manifold such that $\Gamma_\ell \subset \mathcal D$ with $1\le \ell \le N-3$ and $q={2(N-\ell)\over N-\ell-2}$. We prove that, under some symmetry assumptions, the number of sign changing solutions to $(P)_\epsilon$ increases as $\epsilon$ goes to zero.