Liouville Theorem for Lane-Emden Equation on the Heisenberg Group

TL;DR

This paper proves Liouville type theorems for the Lane-Emden equation on the Heisenberg group, showing uniqueness of stable solutions under certain conditions related to the exponent p.

Contribution

It extends Liouville theorems to the Heisenberg group setting for the Lane-Emden equation, identifying conditions for solution stability and uniqueness.

Findings

0 is the unique stable solution outside a compact set when p is below the Joseph Lundgren exponent

Established Liouville theorems for solutions on the Heisenberg group

Identified critical exponent ranges for solution stability

Abstract

This paper establishes some Liouville type results for solutions to the Lane Emden equation on the entire Heisenberg group, both in the stable and stable outside a compact set scenarios.Specifically, we prove that when p is smaller than the Joseph Lundgren exponent and does not equal the Sobolev exponent, 0 is the unique solution that is stable outside a compact set.