Degenerate stability of critical points of the Caffarelli–Kohn–Nirenberg inequality along the Felli–Schneider curve

Abstract In this paper, we investigate the validity of a quantitative version of stability for the critical Hardy–Hénon equation which is well known as the Euler–Lagrange equation of the classical Caffarelli–Kohn–Nirenberg inequality. Establishing quantitative stability for this equation amounts to finding a non‐negative function such that the estimate holds for any non‐negative function satisfying Here, and denotes the set of positive solutions of the equation. When falls above the Felli–Schneider curve, Wei and Wu [28] found an optimal . Their proof relies heavily on the fact that is non‐degenerate. When falls on the Felli–Schneider curve, due to the absence of the non‐degeneracy condition, it becomes complicated and technical to find a suitable . In this paper, we focus on this case. When , we obtain an optimal . When and is not too degenerate, we also derive an optimal . To our knowledge, the results in this paper provide the first instance of degenerate stability in the critical point setting. We believe that our methods will be useful in other works on degenerate stability.