Stability for the Sobolev inequality: existence of a minimizer

We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers $\mathcal M$ and given by $$\frac{\|\nabla f\|_{L^2(\mathbb R^d)}^2 - S_d \|f\|_{L^\frac{2d}{d-2}(\mathbb R^d)}^2}{ \inf_{h \in \mathcal M} \|\nabla (f - h)\|_{L^2(\mathbb R^d)}^2 } \geq c_{BE} > 0 \qquad \text{ for every } f \in \dot{H}^1(\mathbb R^d),$$ which is due to Bianchi and Egnell, admits a minimizer for every $d \geq 3$. 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 $c_{BE} < 2 - 2^\frac{d-2}{d}$, 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 $s \in (0, d/2)$ and dimension $d \geq 2$.