  For $N\geq 5$ and $0<\mu<N-4$, we first show a non-degenerate result of the
extremal for the following Rellich-Sobolev type inequality
  \begin{align*}
  & \int_{\mathbb{R}^N}|\Delta u|^2 \mathrm{d}x
  -C_{\mu,1}\int_{\mathbb{R}^N}\frac{|\nabla u|^2}{|x|^2} \mathrm{d}x
  +C_{\mu,2}\int_{\mathbb{R}^N}\frac{u^2}{|x|^4} \mathrm{d}x
  \geq \mathcal{S}_\mu\left(\int_{\mathbb{R}^N}|u|^{\frac{2N}{N-4}}
\mathrm{d}x\right)^\frac{N-4}{N},\quad u\in C^\infty_0(\mathbb{R}^N),
  \end{align*} where $C_{\mu,1}$, $C_{\mu,1}$ and $\mathcal{S}_\mu$ are
constants depending on $\mu$, furthermore equality only holds for some radial
functions, which is a key ingredient in analyzing the blow-up phenomena of
solutions to various elliptic equations on bounded or unbounded domain.
Moreover, by using spectral estimate combined with a compactness argument, we
give the remainder term of above inequality.
