Finite Morse index solutions of an elliptic equation with supercritical exponent

We study the behavior of finite Morse index solutions of the equation -Delta u = vertical bar x vertical bar(alpha)vertical bar u vertical bar(p-1)u in Omega subset of R-N. where p > 1, alpha > -2, and Omega is a bounded or unbounded domain. We show that there is a critical power p = (p) over bar(alpha) larger than the usual critical exponent N+2/N-2 such that this equation with Omega = R-N has no nontrivial stable solution for 1 < p < (p) over bar(alpha) but it admits a family of stable positive solutions when p >= (p) over bar(alpha). For a positive solution u with finite Morse index, we classify the singularity of u at the origin when Omega is a punctured ball B-R(0)\{0}, and we classify its behavior at infinity when Omega = R-N\B-R(0). We show that the behavior depends crucially on whether p is below or above the critical power (p) over bar(alpha). We also demonstrate how a duality method can be used to obtain sharper results. (C) 2011 Elsevier Inc. All rights reserved.