## Advances in Differential Equations

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

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

Angela Pistoia and Olivier Rey

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867689

**Mathematical Reviews number (MathSciNet)**

MR2238023

**Zentralblatt MATH identifier**

1166.35333

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

#### Citation

Pistoia, Angela; Rey, Olivier. Multiplicity of solutions to the supercritical Bahri-Coron's problem in pierced domains. Adv. Differential Equations 11 (2006), no. 6, 647--666. https://projecteuclid.org/euclid.ade/1355867689