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

Abstract

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|^{-\beta}|\mathrm{div} (|x|^{\alpha}\nabla u)|^2 \mathrm{d}x \geq \mathcal{S}\left(\int_{\mathbb{R}^N} |x|^{\beta}|u|^{p^*_{\alpha,\beta}} \mathrm{d}x\right)^{\frac{2}{p^*_{\alpha,\beta}}},\quad \mbox{for all}\ u\in C^\infty_0(\mathbb{R}^N), \end{equation*} for some constant $\mathcal{S}=\mathcal{S}(N,\alpha,\beta)>0$, where \begin{align*} N\geq 5,\quad \alpha>2-N,\quad \alpha-2<\beta\leq \frac{N}{N-2}\alpha,\quad p^*_{\alpha,\beta}=\frac{2(N+\beta)}{N-4+2\alpha-\beta}. \end{align*} We obtain a symmetry breaking conclusion: when $\alpha>0$ and $\beta_{\mathrm{FS}}(\alpha)<\beta< \frac{N}{N-2}\alpha$ where $\beta_{\mathrm{FS}}(\alpha):= -N+\sqrt{N^2+\alpha^2+2(N-2)\alpha}$, then the extremal function for the best constant $\mathcal{S}$, if it exists, is nonradial. Furthermore, we give a symmetry result when $\beta=\frac{N}{N-2}\alpha$ and $2-N<\alpha<0$...