Stability for the Sobolev inequality: Existence of a minimizer

We prove that the quantitative stability inequality associated to Sobolev’s inequality, due to Bianchi and Egnell, admits a minimizer attaining the best constant c₁₄ for every d 3. Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality c₁₄< 2 - 2^d-2{d}, which means that a sequence of two asymptotically non-interacting bubbles cannot be minimizing. Our arguments cover in fact the analogous stability inequality for the fractional Sobolev inequality for arbitrary fractional exponent s (0, d2) and dimension d 2.