Finite Morse index solutions of an elliptic equation with supercritical exponent

Abstract

We study the behavior of finite Morse index solutions of the equationwhere 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+2 N−2 such that this equation with Ω = R N has no nontrivial stable solution for 1 < p < p(α) but it admits a family of stable positive solutions when p p(α). For a positive solution u with finite Morse index, we classify the singularity of u at the origin when Ω is a punctured ball B R (0)\{0}, and we classify its behavior at infinity when Ω = R N \ B R (0). We show that the behavior depends crucially on whether p is below or above the critical power p(α). We also demonstrate how a duality method can be used to obtain sharper results. ✩ We thank Alberto Farina for pointing out a mistake in an earlier version of the paper. Z.M. Guo would like to thank the University of Sydney for a nice stay.

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