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.
Categories: math.AP