On the classification of extremals of Caffarelli-Kohn-Nirenberg inequalities

TL;DR

This paper classifies positive solutions of critical elliptic equations linked to Caffarelli-Kohn-Nirenberg inequalities in convex cones, focusing on solutions without finite energy in certain dimensional ranges.

Contribution

It provides a classification of extremals for Caffarelli-Kohn-Nirenberg inequalities without finite energy assumptions in specific dimensions.

Findings

Classified positive solutions in the specified dimension range.

Extended understanding of extremals beyond finite energy solutions.

Applicable to convex cone domains.

Abstract

We consider a family of critical elliptic equations which arise as the Euler-Lagrange equation of Caffarelli-Kohn-Nirenberg inequalities, possibly in convex cones in $\mathbb{R}^d$, with $d\geq 2$. We classify positive solutions without assuming that the solution has finite energy and when the intrinsic dimension $n \in (\frac{3}{2},5]$.