Liouville theorems for stable Lane-Emden systems and biharmonic problems

TL;DR

This paper establishes Liouville theorems for positive stable solutions of certain elliptic systems and biharmonic equations, showing nonexistence results in low dimensions and extending to higher dimensions.

Contribution

It provides new Liouville theorems for stable solutions of Lane-Emden systems and biharmonic problems, including boundary value problems, in various dimensions.

Findings

No positive stable solutions for the system when N ≤ 10 and 2 ≤ p ≤ θ.

No positive bounded solutions for the half-space problem when N ≤ 11 and 2 ≤ p ≤ θ.

Results extend to higher dimensions, broadening the scope of Liouville theorems.

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.