Liouville Theorem for Lane Emden Equation of Baouendi Grushin operators

TL;DR

This paper proves a Liouville theorem for solutions to the Lane Emden equation with Baouendi Grushin operators, showing uniqueness of stable solutions outside compact sets under certain conditions, extending previous results.

Contribution

It extends Liouville theorems to the Lane Emden equation with Baouendi Grushin operators, identifying conditions for solution uniqueness outside compact sets.

Findings

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

The result applies to solutions stable outside compact sets for specific p ranges

The work generalizes previous Liouville theorems to a broader class of operators

Abstract

In this paper, we establish a Liouville theorem for solutions to the Lane Emden equation involving Baouendi Grushin operators. We focus on solutions that are stable outside a compact set. Specifically, we prove that when p is smaller than the Joseph Lundgren exponent and differs from the Sobolev exponent, 0 is the unique solution stable outside a compact set. This work extends the results obtained by Farina (J. Math. Pures Appl., 87 (5) (2007)).