  We investigate here the nonlinear elliptic H\'enon type equation:
  $$\D^{2} u= |x|^a|u|^{p-1}u \; \,\,\mbox{in}\,\,\,\, \R^{n}_{+}, \quad \quad
u =\frac{\partial u}{\partial x_n} = 0 \quad \mbox{in}\,\,\,\, \partial
\R^{n}_{+},$$ with $p>1$ and $n\geq 2$. In particular, we prove some Liouville
type theorems for stable at infinity solutions. The main methods used are the
integral estimates, the Pohozaev-type identity and the monotonicity formula.
