Rigidity of solutions to singular/degenerate semilinear critical equations

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 $d\ge 2$. We prove several rigidity results for positive solutions, in particular we classify solutions with possibily infinite energy when the intrinsic dimension $n$ satisfies $2<n<4$.