  We are interested in the qualitative properties of solutions of the H\'enon
type equations with exponential nonlinearity. First, we classify the stable at
infinity solutions of $\Delta u +|x|^\alpha e^u=0$ in $\mathbb{R}^N$, which
gives a complete answer to the problem considered in [Wang-Ye]. Secondly,
existence and precise asymptotic behaviors of entire radial solutions to
$\Delta^2 u=|x|^{\alpha} e^u$ are obtained. Then we classify the stable and
stable at infinity radial solutions to $\Delta^2 u=|x|^{\alpha} e^u$ in any
dimension.
