Liouville theorems for stable Lane-Emden systems and biharmonic problems

Abstract

We examine the elliptic system given by {equation} \label{system_abstract} -\Delta u = v^p, \qquad -\Delta v = u^\theta, \qquad \{in} \IR^N, {equation} for $ 1 < p \le \theta$ and the fourth order scalar equation {equation} \label{fourth_abstract} \Delta^2 u = u^\theta, \qquad \{in $ \IR^N$,} {equation} where $ 1 < \theta$. We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (\ref{system_abstract}) (resp. (\ref{fourth_abstract})) provided $ N \le 10$ and $ 2 \le p \le \theta$ (resp. $ N \le 10$ and $1 < \theta$). Results for higher dimensions are also obtained. These results regarding stable solutions on the full space imply various Liouville theorems for positive (possibly unstable) bounded solutions of {equation} \label{eq_half_abstract} -\Delta u = v^p, \qquad -\Delta v = u^\theta, \qquad \{in} \IR^{N-1}, {equation} with $ u=v=0$ on $ \partial \IR^N_+$. In particular there is no positive bounded solution of (\ref{eq_half_abstract}) for any $ 2 \le p \le \theta$ if $ N \le 11$. Higher dimensional results are also obtained.