Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866910754454634496 |
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| author | Deng, Shengbing Tian, Xingliang |
| author_facet | Deng, Shengbing Tian, Xingliang |
| contents | For $N\geq 5$ and $0<μ<N-4$, 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 $C_{μ,1}$, $C_{μ,2}$ and $\mathcal{S}_μ$ are constants depending on $N$ and $μ$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02232 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian Deng, Shengbing Tian, Xingliang Analysis of PDEs For $N\geq 5$ and $0<μ<N-4$, 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 $C_{μ,1}$, $C_{μ,2}$ and $\mathcal{S}_μ$ are constants depending on $N$ and $μ$, 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. |
| title | Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2306.02232 |