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

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 $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , with $$d\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . We classify positive solutions without assuming that the solution has finite energy and when the intrinsic dimension $$n \in ({\frac{5}{2}},5]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mn>5</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> .