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|^{-\beta}|\mathrm{div} (|x|^{\alpha}\nabla u)|^2 \mathrm{d}x \geq \mathcal{S}\left(\int_{\mathbb{R}^N}|x|^{\gamma} |u|^{2^{**}_{\alpha,\beta}} \mathrm{d}x\right)^{\frac{2}{2^{**}_{\alpha,\beta}}}, \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 $N\geq 5$, $\alpha>2-N$, $\frac{N-4}{N-2}\alpha-4 \leq \beta\leq\alpha -2$ and \begin{align*} 2^{**}_{\alpha,\beta}:=\frac{2(N+\gamma)}{N+2\alpha-\beta-4} \quad \mbox{with}\quad (N+\beta)(N+\gamma)=(N+2\alpha-\beta-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.