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

We study positive solutions of the equationΔu=|x|αu−pin Ω⊂RN(N⩾2), where p>0p>0, α>−2α>−2, and Ω   is a bounded or unbounded domain. We show that there is a critical power p=pc(α)p=pc(α) such that this equation with Ω=RNΩ=RN has no stable positive solution for p>pc(α)p>pc(α) but it admits a family of stable positive solutions when 0pc(α−)p>pc(α−)(α−=min{α,0})(α−=min{α,0}), we further show that this equation with Ω=Br∖{0}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∖BRΩ=RN∖BR. Among other results, we also classify the positive solutions over Br∖{0}Br∖{0} which are not bounded near 0.