### Multiplicity of solutions to the supercritical Bahri-Coron's problem in pierced domains

#### Abstract

We consider the supercritical Dirichlet problem $$\left(P_{\varepsilon}\right)\qquad -\Delta u=u^{{N+2\over N-2}+{\varepsilon}}\ \hbox{in \Omega,}\ u>0\ \hbox{in \Omega,}\ u=0\ \hbox{on \partial\Omega}$$ where $N\ge3,$ ${\varepsilon}>0$ and $\Omega\subset{\mathbb R}^N$ is a smooth bounded domain with a small hole of radius $d.$ When $\Omega$ has some symmetries, we show that $\left(P_{\varepsilon}\right)$ has an arbitrary number of solutions for ${\varepsilon}$ and $d$ small enough. When $\Omega$ has no symmetries, we prove the existence, for $d$ small enough, of solutions blowing up at two or three points close to the hole as ${\varepsilon}$ goes to zero.

#### Article information

Source
Adv. Differential Equations Volume 11, Number 6 (2006), 647-666.

Dates
First available in Project Euclid: 18 December 2012