Finite Morse index solutions of an elliptic equation with supercritical exponent

We study the behavior of finite Morse index solutions of the equation−Δu=|x|α|u|p−1uin Ω⊂RN, where p>1, α>−2, and Ω is a bounded or unbounded domain. We show that there is a critical power p=p¯(α) larger than the usual critical exponent N+2N−2 such that this equation with Ω=RN has no nontrivial stable solution for 1