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*} (∫_{{ R}^d}|x|⁻ᵇ⁽ᵖ⁺¹⁾|u|ᵖ⁺¹dx)2/p+1≤ Sa,b∫_{{ R}^d}|x|⁻²ᵃ|∇ u|^2dx, {eqnarray*} where u∈ D1,2ₐ( Rᵈ), d≥2, p=d+2(1+a-b)/d-2(1+a-b) and {eqnarray}{eq0003} \{ &a<b<a+1, d=2,\\ &a≤ b<a+1, d≥3. . {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 d≥2, b=bFS(a) (Felli-Schneider curve) and $a<0$: let ν ∈ N and u∈ D1,2ₐ( Rᵈ) be an nonnegative function such that {eqnarray}{eqqqnew0001} (ν-12)(Sa,b⁻¹)p+1/p-1<\|u\|^2_{D1,2_a({ R}^d)}<(ν+12)(Sa,b⁻¹)p+1/p-1 {eqnarray} Then we have the following sharp inequality {eqnarray*} inf_{{α}ν∈({ R}_+)ν, {λ}ν∈ { R}^ν}\|u-∑ⱼ₌₁να_j Wλ_j\|\|-div(|x|⁻ᵃ∇ u)-|x|⁻ᵇ⁽ᵖ⁺¹⁾|u|ᵖ⁻¹u\|_{W-1,2_a({ R}^d)}1/3 {eqnarray*} as \|-div(|x|⁻ᵃ∇ u)-|x|⁻ᵇ⁽ᵖ⁺¹⁾|u|ᵖ⁻¹u\|_W-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 ν) which is quite different from the non-degenerate case {DSW2024,WW2022}.