Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities: the singular case

Abstract

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 $\mathcal{S}=\mathcal{S}(N,\alpha ,\beta )>0$, where $N\ge 5$, $\alpha >2-N$, $\frac{N-4}{N-2}\alpha -4\le \beta \le \alpha -2$ 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 ${\int }_{{\mathbb{R}}^N}|x{|}^{-\beta }|\mathrm{div}(|x{|}^{\alpha }\nabla u){|}^2\mathrm{d}x$ is equivalent to ${\int }_{{\mathbb{R}}^N}|x{|}^{2\alpha -\beta }|\Delta u{|}^2\mathrm{d}x$. Firstly, we obtain a symmetry result (with partial translation invariant) when $\alpha =0$ and $\beta =-4$, 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 $\alpha >0$ and $\frac{N-4}{N-2}\alpha -4<\beta <{\beta }_{\mathrm{FS}}(\alpha )$ where ${\beta }_{\mathrm{FS}}(\alpha ):=N+2\alpha -4-\sqrt{(N-2+\alpha {)}^2+4(N-1)}$, the extremal functions for $\mathcal{S}$ are nonradial. Finally, we give a partial symmetry result when $\beta =\frac{N-4}{N-2}\alpha -4$ and $2-N<\alpha <0$, and we also study the stability of extremal functions.

40 pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2308.07568