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

TL;DR

This paper establishes the first precise symmetry and symmetry breaking regions for extremal functions of weighted second-order inequalities, extending the Caffarelli-Kohn-Nirenberg framework and identifying conditions for nonradial extremals.

Contribution

It introduces a new second-order Caffarelli-Kohn-Nirenberg type inequality and determines when extremal functions are symmetric or exhibit symmetry breaking.

Findings

Symmetry breaking occurs for certain parameter ranges when extremals are nonradial.

A new inequality generalizing previous results is established.

Conditions for symmetry and symmetry breaking are explicitly characterized.

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 $2-N<\alpha<0$...