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*\ \ \ \ \ \ \ \ \ ₒ, ₓ (RN) (ₑ₍ |u|^2^*ₛ (t) |x|ᵗ \, dx) ^2{2^*ₛ (t) } ₑ₍ | (-) ^s{2} u |² \, dx, for all u Ḣˢ (RN) align* where s (0, 1), 0 t 2s, N 2s is an integer, and 2^*ₛ (t) = 2 (N-t) N-2s. Here, ₒ, ₓ (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.