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

Let us consider the following Caffarelli-Kohn-Nirenberg type inequality \begin{equation}\label{nsckn} \int_{$\mathbb{R}$^N}|x|^{-β}|$\mathrm{d}\mathrm{i}\mathrm{v}$ (|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 ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }x{\mathrm{\mid }}^{-\beta }\mathrm{\mid }\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }\mathrm{\nabla }u){\mathrm{\mid }}^2\mathrm{d}x$ is equivalent to ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha -\beta }\mathrm{\mid }\mathrm{\Delta }u{\mathrm{\mid }}^2\mathrm{d}x$. 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 $\frac{N-4}{N-2}\alpha -4<\beta <{\beta }_{\mathrm{F}\mathrm{S}}(\alpha )$ where ${\beta }_{\mathrm{F}\mathrm{S}}(\alpha ):=N+2\alpha -4-\sqrt{(N-2+\alpha {)}^2+4(N-1)}$, 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.