On the Stein-Weiss inequalities and higher-order Caffarelli-Kohn-Nirenberg type inequalities: sharp constants, symmetry of extremal functions

TL;DR

This paper classifies radially symmetric solutions to a weighted fourth-order PDE and derives sharp inequalities of Stein-Weiss, Caffarelli-Kohn-Nirenberg, and Rellich-Sobolev types with extremal functions, including Hardy-Rellich inequalities.

Contribution

It provides a comprehensive classification of solutions and establishes new sharp inequalities with extremal functions, extending classical results to weighted and higher-order cases.

Findings

Classified all radially symmetric solutions of a weighted fourth-order PDE.

Derived sharp Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities with extremal functions.

Established sharp weighted Rellich-Sobolev and Hardy-Rellich inequalities.

Abstract

In this paper, we first classify all radially symmetry solutions of the following weighted fourth-order equation \begin{equation*} \Delta(|x|^{-\gamma}\Delta u)=|x|^\gamma u^{\frac{N+4+3\gamma}{N-4-\gamma}},\quad u\geq 0 \quad \mbox{in}\quad \mathbb{R}^N, \end{equation*} where $N\geq 5$, $-2<\gamma<0$. Then we derive the sharp Stein-Weiss inequality and standard second-order Caffarelli-Kohn-Nirenberg inequality with radially symmetry extremal functions. Moreover, by using standard spherical decomposition, we derive a sharp weighted Rellich-Sobolev inequality. Furthermore, we establish the sharp second-order Caffarelli-Kohn-Nirenberg type inequalities with two variables which have radially symmetry extremal functions. Finally, we derive the weak form Hardy-Rellich inequalities with sharp constants.