Rigidity of solutions to singular/degenerate semilinear critical equations

TL;DR

This paper establishes rigidity results and classifies positive solutions, including those with infinite energy, for singular/degenerate semilinear critical equations related to Caffarelli-Kohn-Nirenberg inequalities in certain dimensions.

Contribution

It provides new classification and rigidity results for solutions of singular/degenerate critical equations, especially in the case of possibly infinite energy solutions.

Findings

Rigidity results for positive solutions

Classification of solutions with infinite energy

Applicable to equations arising from Caffarelli-Kohn-Nirenberg inequalities

Abstract

This paper deals with singular/degenerate semilinear critical equations which arise as the Euler-Lagrange equation of Caffarelli-Kohn-Nirenberg inequalities in $\mathbb{R}^d$, with $d\geq 2$. We prove several rigidity results for positive solutions, in particular we classify solutions with possibily infinite energy when the intrinsic dimension $n$ satisfies $2<n<4$.