Liouville-type theorems for the Lane-Emden equation in the half-space and cones

TL;DR

This paper establishes Liouville-type theorems for the Lane-Emden equation in half-spaces and cones, showing uniqueness of solutions under stability and weak solution conditions.

Contribution

It extends Liouville theorems to broader geometric settings including cones and weak solutions, under stability assumptions.

Findings

Uniqueness of stable solutions in half-space

Liouville theorem for weak solutions in cones

Extension to general cone geometries

Abstract

We prove that 0 the only classical solution of the Lane-Emden equation in the half-space which is stable outside a compact set. We also consider weak solutions and the case of general cones.