October  2007, 17(4): 751-770. doi: 10.3934/dcds.2007.17.751

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, France, France

2. 

Department of Mathematics, East China Normal University, 200062 Shanghai, China

Received  May 2006 Revised  September 2006 Published  January 2007

We construct solutions of the semilinear elliptic problem

$\Delta u+ |u|^{p-1}u+$ε1/2 f = 0 in Ω
u=ε1/2 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 $g=0$.

Citation: Yuxin Ge, Ruihua Jing, Feng Zhou. Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 751-770. doi: 10.3934/dcds.2007.17.751
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