  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.
