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2006 Multiplicity of solutions to the supercritical Bahri-Coron's problem in pierced domains
Angela Pistoia, Olivier Rey
Adv. Differential Equations 11(6): 647-666 (2006).

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.

Citation

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Angela Pistoia. Olivier Rey. "Multiplicity of solutions to the supercritical Bahri-Coron's problem in pierced domains." Adv. Differential Equations 11 (6) 647 - 666, 2006.

Information

Published: 2006
First available in Project Euclid: 18 December 2012

zbMATH: 1166.35333
MathSciNet: MR2238023

Subjects:
Primary: 35J60
Secondary: 35J20, 35J25, 47J30, 58E05

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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Vol.11 • No. 6 • 2006
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