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

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)$.