Liouville theorems for stable at infinity solutions of m-triharmonic equation in $\mathbb{R}^N$

TL;DR

This paper establishes Liouville theorems for stable solutions at infinity of a specific m-triharmonic equation in Euclidean space, identifying conditions under which solutions must be trivial.

Contribution

It extends Liouville theorems to m-triharmonic equations with stability at infinity, introducing new critical exponent bounds for such solutions.

Findings

Liouville theorems hold for solutions with certain stability conditions

Critical exponent for stability is identified as rac{N(m-1)+3m}{N-3m}

Results generalize known theorems for bi-harmonic equations

Abstract

In this paper we prove the Liouville type theorem for stable at infinity solutions of the following equation $$\Delta_{m}^{3}u =|u|^{\theta-1}u\;\;\; \mbox{in}\,\, \mathbb{R}^N,$$ for $1<m-1<\theta<\theta_{s, m}:=\frac{N(m-1)+3m }{N-3m}.$ Here $\theta_{s, m}$ is a the classic critical exponent for $m-$ bi-harmonic equation.