Caffarelli-Kohn-Nirenberg-type inequalities related to weighted $p$-Laplace equations

TL;DR

This paper studies sharp constants and optimizers for Caffarelli-Kohn-Nirenberg inequalities linked to weighted p-Laplace equations, providing classifications and extending previous work on gradient remainder terms, especially in the radial case.

Contribution

It introduces a transform related to Sobolev inequality to analyze sharp constants, classifies linearized problem solutions, and extends spectral estimates for gradient remainders in the radial setting.

Findings

Determined sharp constants and optimizers for the inequalities.

Classified solutions to the linearized problem for radial extremals.

Extended spectral estimates for gradient remainders in the radial case.

Abstract

We use a suitable transform related to Sobolev inequality to investigate the sharp constants and optimizers for some Caffarelli-Kohn-Nirenberg-type inequalities which are related to the weighted $p$-Laplace equations. Moreover, we give the classification to the linearized problem related to the radial extremals. As an application, we investigate the gradient type remainder term of related inequality by using spectral estimate combined with a compactness argument which extends the work of Figalli and Zhang (Duke Math. J. 2022) at least for radial case.