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 $\mathbb{R}^n$, $n\geq 3$, to $-ΔW-γ|x|^{-2}V=|x|^{-s}W^{2^\star(s)-1}$ is one-dimensional when $s+γ>0$. Here, $s\in [0,2)$, $0\leqγ<(n-2)^2/4$ and $2^\star(s)=2(n-s)/(n-2)$.

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