Liouville-type theorems for the fourth order nonlinear elliptic equation

TL;DR

This paper establishes Liouville-type theorems for certain classes of solutions to a fourth order nonlinear elliptic equation in unbounded domains, using advanced analytical techniques to extend existing nonexistence results.

Contribution

It introduces new Liouville-type theorems for stable and finite Morse index solutions of a fourth order nonlinear elliptic equation, employing Pohozaev identities and monotonicity formulas.

Findings

Liouville-type theorems for stable solutions

Liouville-type theorems for finite Morse index solutions

Sharper nonexistence results using blowing down sequences

Abstract

In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equation {equation*} \Delta^2 u=|x|^a |u|^{p-1}u\;\ {in}\;\ \Omega, {equation*}where $a \ge 0$, $p>1$ and $\Omega \subset \mathbb{R}^n$ is an unbounded domain of $\mathbb{R}^n$, $n \ge 5$. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the {\it Pohozaev-type identity}, {\it monotonicity formula} of solutions and a {\it blowing down} sequence, which is used to obtain sharper results.