Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909336667684864 |
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| author | Deng, Shengbing Tian, Xingliang |
| author_facet | Deng, Shengbing Tian, Xingliang |
| contents | In this paper we give the first result about the precise symmetry and symmetry breaking regions of extremal functions for weighted second-order inequalities. Firstly, based on the work of C.-S. Lin [Comm. Partial Differential Equations, 1986], a new second-order Caffarelli-Kohn-Nirenberg type inequality will be established, i.e.,
\begin{equation*}
\int_{\mathbb{R}^N}|x|^{-β}|\mathrm{div} (|x|^α\nabla u)|^2 \mathrm{d}x
\geq \mathcal{S}\left(\int_{\mathbb{R}^N}
|x|^β|u|^{p^*_{α,β}} \mathrm{d}x\right)^{\frac{2}{p^*_{α,β}}},\quad \mbox{for all}\ u\in C^\infty_0(\mathbb{R}^N),
\end{equation*}
for some constant $\mathcal{S}=\mathcal{S}(N,α,β)>0$, where
\begin{align*}
N\geq 5,\quad α>2-N,\quad α-2<β\leq \frac{N}{N-2}α,\quad p^*_{α,β}=\frac{2(N+β)}{N-4+2α-β}.
\end{align*}
We obtain a symmetry breaking conclusion: when $α>0$ and $β_{\mathrm{FS}}(α)<β< \frac{N}{N-2}α$ where $β_{\mathrm{FS}}(α):=
-N+\sqrt{N^2+α^2+2(N-2)α}$, then the extremal function for the best constant $\mathcal{S}$, if it exists, is nonradial. Furthermore, we give a symmetry result when $β=\frac{N}{N-2}α$ and $2-N<α<0$... |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07568 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities Deng, Shengbing Tian, Xingliang Analysis of PDEs 35P30, 35J30 In this paper we give the first result about the precise symmetry and symmetry breaking regions of extremal functions for weighted second-order inequalities. Firstly, based on the work of C.-S. Lin [Comm. Partial Differential Equations, 1986], a new second-order Caffarelli-Kohn-Nirenberg type inequality will be established, i.e., \begin{equation*} \int_{\mathbb{R}^N}|x|^{-β}|\mathrm{div} (|x|^α\nabla u)|^2 \mathrm{d}x \geq \mathcal{S}\left(\int_{\mathbb{R}^N} |x|^β|u|^{p^*_{α,β}} \mathrm{d}x\right)^{\frac{2}{p^*_{α,β}}},\quad \mbox{for all}\ u\in C^\infty_0(\mathbb{R}^N), \end{equation*} for some constant $\mathcal{S}=\mathcal{S}(N,α,β)>0$, where \begin{align*} N\geq 5,\quad α>2-N,\quad α-2<β\leq \frac{N}{N-2}α,\quad p^*_{α,β}=\frac{2(N+β)}{N-4+2α-β}. \end{align*} We obtain a symmetry breaking conclusion: when $α>0$ and $β_{\mathrm{FS}}(α)<β< \frac{N}{N-2}α$ where $β_{\mathrm{FS}}(α):= -N+\sqrt{N^2+α^2+2(N-2)α}$, then the extremal function for the best constant $\mathcal{S}$, if it exists, is nonradial. Furthermore, we give a symmetry result when $β=\frac{N}{N-2}α$ and $2-N<α<0$... |
| title | Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities |
| topic | Analysis of PDEs 35P30, 35J30 |
| url | https://arxiv.org/abs/2308.07568 |