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|^{-β}|\mathrm{div} (|x|^α\nabla u)|^2 \mathrm{d}x \geq \mathcal{S}\left(\int_{\mathbb{R}^N} |x|^β|u|^{p^*_{α,β}} \mathrm{d}x\right)^{\frac{2}{p^*_{α,β}}},\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 α>2-N,\quad α-2<β\leq \frac{N}{N-2}α,\quad p^*_{α,β}=\frac{2(N+β)}{N-4+2α-β}. \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 $2-N<\alpha <0$...