Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities: the singular case
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909336275517440 |
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| author | Deng, Shengbing Tian, Xingliang |
| author_facet | Deng, Shengbing Tian, Xingliang |
| contents | Let us consider the following Caffarelli-Kohn-Nirenberg type inequality \begin{equation}\label{nsckn} \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|^γ |u|^{2^{**}_{α,β}} \mathrm{d}x\right)^{\frac{2}{2^{**}_{α,β}}}, \quad \mbox{for all}\quad u\in C^\infty_0(\mathbb{R}^N\setminus\{0\}), \end{equation} for some $\mathcal{S}=\mathcal{S}(N,α,β)>0$, where $N\geq 5$, $α>2-N$, $\frac{N-4}{N-2}α-4 \leq β\leqα-2$ and \begin{align*} 2^{**}_{α,β}:=\frac{2(N+γ)}{N+2α-β-4} \quad \mbox{with}\quad (N+β)(N+γ)=(N+2α-β-4)^2. \end{align*} A crucial element is that the functional $\int_{\mathbb{R}^N}|x|^{-β}|\mathrm{div} (|x|^α\nabla u)|^2 \mathrm{d}x$ is equivalent to $\int_{\mathbb{R}^N}|x|^{2α-β}|Δu|^2 \mathrm{d}x$. Firstly, we obtain a symmetry result (with partial translation invariant) when $α=0$ and $β=-4$, then existence and non-existence of extremal functions for the best constant $\mathcal{S}$ in \eqref{nsckn} under different conditions are completely given. Moreover, by a result of linearized problem related to radial solution of \eqref{Pwhs0}, we obtain a symmetry breaking conclusion: when $α>0$ and $\frac{N-4}{N-2}α-4<β<β_{\mathrm{FS}}(α)$ where $β_{\mathrm{FS}}(α):= N+2α-4-\sqrt{(N-2+α)^2+4(N-1)}$, the extremal functions for $\mathcal{S}$ are nonradial. Finally, we give a partial symmetry result when $β=\frac{N-4}{N-2}α-4$ and $2-N<α<0$, and we also study the stability of extremal functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18154 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities: the singular case Deng, Shengbing Tian, Xingliang Analysis of PDEs Let us consider the following Caffarelli-Kohn-Nirenberg type inequality \begin{equation}\label{nsckn} \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|^γ |u|^{2^{**}_{α,β}} \mathrm{d}x\right)^{\frac{2}{2^{**}_{α,β}}}, \quad \mbox{for all}\quad u\in C^\infty_0(\mathbb{R}^N\setminus\{0\}), \end{equation} for some $\mathcal{S}=\mathcal{S}(N,α,β)>0$, where $N\geq 5$, $α>2-N$, $\frac{N-4}{N-2}α-4 \leq β\leqα-2$ and \begin{align*} 2^{**}_{α,β}:=\frac{2(N+γ)}{N+2α-β-4} \quad \mbox{with}\quad (N+β)(N+γ)=(N+2α-β-4)^2. \end{align*} A crucial element is that the functional $\int_{\mathbb{R}^N}|x|^{-β}|\mathrm{div} (|x|^α\nabla u)|^2 \mathrm{d}x$ is equivalent to $\int_{\mathbb{R}^N}|x|^{2α-β}|Δu|^2 \mathrm{d}x$. Firstly, we obtain a symmetry result (with partial translation invariant) when $α=0$ and $β=-4$, then existence and non-existence of extremal functions for the best constant $\mathcal{S}$ in \eqref{nsckn} under different conditions are completely given. Moreover, by a result of linearized problem related to radial solution of \eqref{Pwhs0}, we obtain a symmetry breaking conclusion: when $α>0$ and $\frac{N-4}{N-2}α-4<β<β_{\mathrm{FS}}(α)$ where $β_{\mathrm{FS}}(α):= N+2α-4-\sqrt{(N-2+α)^2+4(N-1)}$, the extremal functions for $\mathcal{S}$ are nonradial. Finally, we give a partial symmetry result when $β=\frac{N-4}{N-2}α-4$ and $2-N<α<0$, and we also study the stability of extremal functions. |
| title | Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities: the singular case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.18154 |