Bubble tower solutions of  slightly supercritical elliptic equations and application in symmetric domains

Yuxin Ge; Ruihua Jing; Feng Zhou

doi:10.3934/dcds.2007.17.751

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2007, Volume 17, Issue 4: 751-770. Doi: 10.3934/dcds.2007.17.751

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Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains

1.
Laboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, Département de Mathématiques, Université Paris XII-Val de Marne, 61 avenue du Général de Gaulle, 94010 Créteil Cedex
2.
Department of Mathematics, East China Normal University, 200062 Shanghai

Received: April 30, 2006

Revised: August 31, 2006

Published: September 2007

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Abstract

We construct solutions of the semilinear elliptic problem

$\Delta u+ |u|^{p-1}u+$ε^1/2 f = 0 in Ω
u=ε^1/2 g on $\partial$Ω

in a bounded smooth domain $\Omega \subset \R^N$ $(N\geq 3)$, when the exponent $p$ is supercritical and close enough to $\frac{N+2}{N-2}$. As $p\rightarrow \frac{N+2}{N-2}$, the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. As applications, we will give some existence results, in particular, when $\O$ are symmetric domains perforated with the small hole and when $f=0$ and $g=0$.

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Mathematics Subject Classification: 35J60, 35J25.

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