Liouville theorems for stable at infinity solutions of Lane-Emden system

TL;DR

This paper proves Liouville-type theorems for stable solutions of the Lane-Emden system in Euclidean space, establishing nonexistence results under certain conditions related to the Sobolev hyperbola.

Contribution

It introduces new nonexistence results for stable solutions of the Lane-Emden system outside compact sets, extending previous Liouville theorems.

Findings

No smooth positive solutions are stable outside compact sets under the Sobolev hyperbola.

The results apply for all positive exponents p, θ.

The paper generalizes Liouville theorems for the Lane-Emden system.

Abstract

We consider the Lane-Emden system $-\Delta u = v^p$, $-\Delta v= u^\theta$ in $\mathbb{R}^N$, and we prove the nonexistence of smooth positive solutions which are stable outside a compact set, for any $p, \theta > 0$ under the Sobolev hyperbola.