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 $$\frac{\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L^2({\mathbb{R}}^d)}^2-S_d\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{\frac{2d}{d-2}}({\mathbb{R}}^d)}^2}{\underset{h\in \mathcal{M}}{\inf }\mathrm{\parallel }\mathrm{\nabla }(f-h){\mathrm{\parallel }}_{L^2({\mathbb{R}}^d)}^2}\ge c_{BE}>0\text{ for every }f\in {\dot{H}}^1({\mathbb{R}}^d),$$ which is due to Bianchi and Egnell, admits a minimizer 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 $c_{BE}<2-2^{\frac{d-2}{d}}$, 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 $s\in (0,d\mathrm{/}2)$ and dimension $d\ge 2$.