Stability of the Caffarelli–Kohn–Nirenberg inequality along the Felli–Schneider curve: Critical points at infinity

Abstract In this paper, we consider the following Caffarelli–Kohn–Nirenberg (CKN) inequality where , with is the optimal constant and with Based on the ideas of Deng, Sun, and Wei ( Duke Math . J 174 (2025)) and Frank and Peteranderl ( Calc. Var . 63 (2024)), we develop a suitable strategy to derive the following sharp stability of the critical points at infinity of the above CKN inequality in the degenerate case with (the Felli–Schneider curve): let and be a nonnegative function such that then provided that is sufficiently small, where with being the dual space of , with being the unique extremal function of the above CKN inequality which is positive and radial up to dilations and scalar multiplications and . The above stability is sharp in the sense that the power of the right‐hand side cannot be improved any more. The significant finding in our result is that in the degenerate case, the power of the optimal stability is an absolute constant (independent of and ) which is quite different from the nondegenerate case considered in Deng, Sun, and Wei ( Duke Math. J ) and Wei and Wu ( Math. Ann . 384 (2022), 1509–1546). We also believe that our proof‐based strategy might be useful in studying many other degenerate problems.