The supercritical Lane-Emden-Fowler equation in exterior domains



Juan Dávila
Universidad de Chile, CMM (UMI CNRS)
email: jdavila@dim.uchile.cl



Abstract: We consider the so called Lane-Emden-Fowler equation,

$\displaystyle \Delta u + u^p=0\, , \ u>0 \quad \hbox{ in } \Omega\, , $

$\displaystyle u= 0\quad \hbox{on } \partial\Omega \, , $

where $ \Omega$ is a domain with smooth boundary in $ \mathbb{R}^N$, $ N\ge 3$ and $ p>1$.

We are interested in the case when $ p$ is super-critical, i.e. $ p> \frac{N+2}{N-2}$, and when the domain is an exterior domain, that is, $ \Omega$ is of the form $ \Omega= \mathbb{R}^N\setminus\overline {\mathcal D}$ where $ {\mathcal D}$ be a bounded open set with smooth boundary. Thus we consider

$\displaystyle \Delta u + u^p=0\, , \ u>0 \quad \hbox{ in } \mathbb{R}^N \setminus \bar {\mathcal D}\, ,$ (1)

$\displaystyle u= 0\ \ \hbox{on } \partial{\mathcal D}\, ,\quad \lim_{\vert x\vert\to +\infty} u(x) = 0 .$ (2)

The main result is that for any $ p> \frac{N+2}{N-2}$ there is a continuum of solutions $ u_\lambda $, $ \lambda >0$ small, to problem (1)-(2) such that

$\displaystyle \hbox{ $ u_\lambda (x) \to 0 \quad \hbox{as } \lambda \to 0$, } $

uniformly in $ \mathbb{R}^N \setminus \overline {\mathcal D}$, and all have the same slow decay at infinity:

$\displaystyle u_\lambda (x) = \beta^{\frac{1}{ p-1}} \vert x\vert^{-\frac 2{p-1}} ( 1 + o(1)) \quad\hbox{as } \vert x\vert \to \infty,$    

where $ \beta>0$.

This is joint work with Manuel del Pino (Universidad de Chile), Monica Musso (Politecnico di Torino & Universidad Católica) and Juncheng Wei (Chinese University of Hong Kong).