Finite Morse index solutions of an elliptic equation with supercritical exponent

We study the behavior of finite Morse index solutions of the equation where p>1, α>−2, and Ω is a bounded or unbounded domain. We show that there is a critical power larger than the usual critical exponent such that this equation with has no nontrivial stable solution for but it admits a family of stable positive solutions when . For a positive solution u with finite Morse index, we classify the singularity of u at the origin when Ω is a punctured ball BR(0) {0}, and we classify its behavior at infinity when . We show that the behavior depends crucially on whether p is below or above the critical power . We also demonstrate how a duality method can be used to obtain sharper results.