Some Liouville theorems for Hénon type elliptic equations

We investigate here the nonlinear elliptic equations −Δu=α|x|eu and −Δu=α|x||u|p−1u with α>−2, p>1 and N⩾2. In particular, we prove some Liouville type theorems for weak solutions with finite Morse index in the low dimensional Euclidean spaces or half spaces.