Positive solutions of an elliptic equation with negative exponent: stability and critical power

Abstract

We study positive solutions of the equation Δu = |x|α u−p in Ω ⊂ RN (N ≥ 2), where p > 0, α > −2, and Ω is a bounded or unbounded domain. We show that there is a critical power p = pc(α) such that this equation with Ω = ℝN has no stable positive solution for p > pc(α) but it admits a family of stable positive solutions when 0 < p ≤ pc(α). If p > pc(α⁻) (α⁻ = min{α, 0}), we further show that this equation with Ω = Br {0} has no positive solution with finite Morse index that has an isolated rupture at 0, and analogously it has no positive solution with finite Morse index when Ω = ℝN BR . Among other results, we also classify the positive solutions over Br {0} which are not bounded near 0.