Qualitative properties of H\'enon type equations with exponential nonlinearity

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.