Symmetry for positive critical points of Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper classifies positive solutions of Caffarelli-Kohn-Nirenberg inequalities involving a weighted p-Laplace operator, establishing symmetry results and extending previous findings for the case p=2 to a broader range of p values.

Contribution

It extends the symmetry classification of extremals for Caffarelli-Kohn-Nirenberg inequalities from p=2 to general p in (1,d), providing a complete solution in certain parameter ranges.

Findings

Complete classification of solutions in a certain parameter range.

Symmetry results for positive solutions.

Extension of previous p=2 results to general p.

Abstract

We consider positive critical points of Caffarelli-Kohn-Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for positive solutions. The governing operator is a weighted $p$-Laplace operator, which we consider for a general $p \in (1,d)$. For $p=2$, the symmetry breaking region for extremals of Caffarelli-Kohn-Nirenberg inequalities was completely characterized in [J. Dolbeault, M. Esteban, M. Loss, Invent. Math. 44 (2016)]. Our results extend this result to a general $p$ and are optimal in some cases.