Quantitative stability in fractional Hardy–Sobolev inequalities: the role of Euler–Lagrange equations

This paper investigates sharp stability estimates for the fractional Hardy–Sobolev inequality: {align*}\ \ \ \ \ \ \ \ \ μs,t(R^N) (∫R^N {|u|2^*_s(t)}{|x|^t} \,dx )2/2^*_s(t) ≤ ∫R^N |(-Δ)s/2 u |^2 \,dx, for all u ∈ Ḣ^s(R^N){align*} where s ∈ (0,1) , 0 < t < 2s , N > 2s is an integer, and 2^*ₛ(t) = 2(N-t)/N-2s . Here, μs,t(RN) represents the best constant in the inequality. The primary focus is on the quantitative stability results of the above inequality and the corresponding Euler–Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler–Lagrange equation, offering a thorough characterization of the Palais–Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in RN , originally obtained by Bianchi and Egnell [J. Funct. Anal. 1991] as well as the corresponding critical exponent problem in RN , explored by Ciraolo, Figalli, and Maggi [Int. Math. Res. Not. 2017] in the framework of fractional calculus.