Liouville-type theorems for the fourth order nonlinear elliptic equation

Abstract

In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equation {equation*} Δ^2 u=|x|^a |u|^{p-1}u\;\ {in}\;\ Ω, {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.

Cited by in corpus (1)

• On stable solutions of the fractional Henon-Lane-Emden equation