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

Abstract

For $N\geq 5$ 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}|\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 \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.