Stability for the Sobolev inequality: Existence of a minimizer

Abstract

We prove that the quantitative stability inequality associated to Sobolev’s inequality, due to Bianchi and Egnell, admits a minimizer attaining the best constant $c_{\mathrm{BE}}$ for every $d\ge 3$. 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 $c_{\mathrm{BE}}<2-2^{\frac{d-2}{d}}$, 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 $s\in (0,\frac{d}{2})$ and dimension $d\ge 2$.