Stability of the Caffarelli-Kohn-Nirenberg inequality along Felli-Schneider curve: critical points at infinity

In this paper, we consider the following Caffarelli-Kohn-Nirenberg (CKN for short) inequality eqnarray* (ₑᵈ|x|^-b (p+1) |u|^p+1dx) ^2{p+1} S₀, ₁ ₑᵈ|x|^-2a| u|²dx, eqnarray* where u D^1, 2₀ (Rᵈ), d2, p=d+2 (1+a-b) d-2 (1+a-b) and eqnarrayeq0003 \ &a<b<a+1, d=2, \\ &a b<a+1, d3. . eqnarray Based on the ideas of DSW2024, FP2024, 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 d2, b=b₅ₒ (a) (Felli-Schneider curve) and a<0: let { N and u D^1, 2₀ (Rᵈ) be an nonnegative function such that eqnarrayeqqqnew0001 (-12) (S₀, ₁^-1) ^p+1{p-1}<\|u\|²₃^₁, ₂ₐ ({ Rᵈ) }< (+12) (S₀, ₁^-1) ^p+1{p-1} eqnarray Then we have the following sharp inequality eqnarray* _ ({ R_+) ^, _ R^}\|u-₉=₁^ⱼ W䲛\|\|-div (|x|^-a u) -|x|^-b (p+1) |u|^p-1u\|ₖ^-₁, ₂ₐ ({ Rᵈ) }^1{3} eqnarray* as \|-div (|x|^-a u) -|x|^-b (p+1) |u|^p-1u\|ₖ^-₁, ₂ₐ ({ Rᵈ) }0. The significant finding in our result is that in the degenerate case, the power of the optimal stability is an absolute constant 1/3 (independent of p and) which is quite different from the non-degenerate case DSW2024, WW2022.