Sign changing solutions to a Bahri-Coron's problem in pierced domains

Monica Musso; A. Pistoia

doi:10.3934/dcds.2008.21.295

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2008, Volume 21, Issue 1: 295-306. Doi: 10.3934/dcds.2008.21.295

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Sign changing solutions to a Bahri-Coron's problem in pierced domains

Monica Musso^1, and
A. Pistoia^2,

1.
Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago
2.
Dipartimento di Metodi e Modelli Matematici, Università di Roma "La Sapienza", Via Scarpa, 16 - 00166 Roma

Received: January 2007

Revised: May 2007

Published: January 2008

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Abstract

We consider the problem

$-\Delta u= |u|^{\frac4{N-2}} u \mbox{ in } \Omega \setminus \{B(\xi_1,\varepsilon)\cup B(\xi_2,\varepsilon)\},$
$ u = 0 \mbox{ on } \partial( \Omega \setminus \{B(\xi_1,\varepsilon)\cup B(\xi_2,\varepsilon)\}),$

where $\Omega$ is a smooth bounded domain in $R^N$, $N\ge 3,$ $\xi_1,$ $\xi_2$ are different points in $\Omega$ and ε is a small positive parameter. We show that, for ε small enough, the equation has at least one pair of sign changing solutions, whose positive and negative parts concentrate at $\xi_1$ and $\xi_2$ as ε goes to zero.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35J25.

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