Nondegeneracy of positive solutions to nonlinear Hardy-Sobolev equations

In this note, we prove that the kernel of the linearized equation around a positive energy solution in ${\mathbb{R}}^n$, $n\ge 3$, to $-\Delta W-\gamma|x|^{-2}V=|x|^{-s}W^{2^\star(s)-1}$isone-dimensionalwhen$s+\gamma>0$\mathrm{.}Here,$s\in [0,2)$,$0\leq\gamma<(n-2)^2/4$and$2^\star(s)=2(n-s)/(n-2)$.