A refined result on sign changing solutions for a critical elliptic problem

Yuxin Ge; Monica Musso; A. Pistoia; Daniel Pollack

doi:10.3934/cpaa.2013.12.125

Article Contents

2013, Volume 12, Issue 1: 125-155. Doi: 10.3934/cpaa.2013.12.125

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A refined result on sign changing solutions for a critical elliptic problem

1.
Laboratoire d'Analyse et de Mathématiques Appliquées, CNRS UMR 8050, Département de Mathématiques, Université Paris Est Créteil, 61 avenue du Général de Gaulle, 94010 Créteil Cedex
2.
Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago
3.
Dipartimento di Metodi e Modelli Matematici, Università di Roma "La Sapienza", Via Scarpa, 16 - 00166 Roma
4.
University of Washington

Received: December 31, 2010

Revised: March 31, 2012

Published: December 2012

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Abstract

In this work, we consider sign changing solutions to the critical elliptic problem $\Delta u + |u|^{\frac{4}{N-2}}u = 0$ in $\Omega_\varepsilon$ and $u=0$ on $\partial\Omega_\varepsilon$, where $\Omega_\varepsilon:=\Omega-\left(\bigcup_{i=1}^m (a_i+\varepsilon\Omega_i)\right)$ for small parameter $\varepsilon>0$ is a perforated domain, $\Omega$ and $\Omega_i$ with $0\in \Omega_i$ ($\forall i=1,\cdots,m$) are bounded regular general domains without symmetry in $\mathbb{R}^N$ and $a_i$ are points in $\Omega$ for all $i=1,\cdots,m$. As $\varepsilon$ goes to zero, we construct by gluing method solutions with multiple blow up at each point $a_i$ for all $i=1,\cdots,m$.

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Mathematics Subject Classification: Primary: 35J60, 35J25.

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