Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian

For $N\ge 5$ and $0<\mu <N-4$, we first show a non-degenerate result of the extremal functions for the following Rellich-Sobolev type inequality \begin{align*} \int_{$\mathbb{R}$^N}|Δu|^2 $\mathrm{d}$x -C_{μ,1}\int_{$\mathbb{R}$^N}\frac{|\nabla u|^2}{|x|^2} $\mathrm{d}$x +C_{μ,2}\int_{$\mathbb{R}$^N}\frac{u^2}{|x|^4} $\mathrm{d}$x \geq $\mathcal{S}$μ\left(\int{$\mathbb{R}$^N}|u|^{\frac{2N}{N-4}} $\mathrm{d}$x\right)^\frac{N-4}{N},\quad \forall u\in C^\infty_0($\mathbb{R}$^N), \end{align*} where $C_{\mu ,1}$, $C_{\mu ,2}$ and ${\mathcal{S}}_{\mu }$ are constants depending on $N$ and $\mu$, which is a key ingredient in analyzing the blow-up phenomena of solutions to various elliptic equations on bounded or unbounded domains. Then by using spectral analysis combined with a compactness argument, we consider the stability of this inequality. Furthermore, we derive a remainder term inequality in the weak Lebesgue-norm sense in a subdomain with finite Lebesgue measure.