On finite Morse index solutions of two equations with negative exponent

We consider the following equations involving negative exponent: Δ u&=|x|\alpha u-p},&\quad u&>0\text{~in~}\varOmega\subset\ mathbb{R}n, Δ u&=u{-p}-1,&\quad u&>0\text{~in~}\varOmega\ subset\mathbb{R}n, where p > 0. Under optimal conditions on the parameters α >-2 and p > 0, we prove the non-existence of finite Morse index solution on exterior domains or near the origin. We also prove an optimal regularity result for solutions with finite Morse index and isolated rupture at 0.