A Liouville theorem for a fourth order H\'enon equation

TL;DR

This paper proves a Liouville-type theorem for a fourth order Hénon equation in five dimensions, showing the nonexistence of positive bounded solutions under certain conditions on the exponent p.

Contribution

It establishes a new nonexistence result for positive solutions of a fourth order Hénon equation in dimension five, extending Liouville theorems to this higher-order PDE.

Findings

No positive bounded solutions exist for 1 < p < p_4(α) in dimension N=5.

The result applies to the entire space a0a0a0a0.

Defines the Hardy-Sobolev exponent p_4(b5) for the problem.

Abstract

We examine the following fourth order H\'enon equation \label{pipe} \Delta^2 u = |x|^\alpha u^p \qquad \text{in}\ \IR^N, where $ 0 < \alpha$. Define the Hardy-Sobolev exponent $ p_4(\alpha):= \frac{N+4 + 2 \alpha}{N-4}$. We show that in dimension N=5 there are no positive bounded classical solutions of (\ref{pipe}) provided $ 1 < p < p_4(\alpha)$.