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

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 $d\ge 2$. We classify positive solutions without assuming that the solution has finite energy and when the intrinsic dimension $n\in (\frac{3}{2},5]$.