Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion

Abstract

We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u\in {\dot{H}}^s({\mathbb{R}}^N)$ whose energy satisfies $$\frac{1}{2}{S}_{N,s}^{\frac{N}{2s}}\le \Vert u{\Vert }_{{\dot{H}}^s({\mathbb{R}}^N)}\le \frac{3}{2}{S}_{N,s}^{\frac{N}{2s}},$$ where $S_{N,s}$ is the optimal Sobolev constant, the bound $$\Vert u-U[z,\lambda ]{\Vert }_{{\dot{H}}^s({\mathbb{R}}^N)}\lesssim \Vert (-\Delta {)}^{s}u-u^{2_s^{\ast }-1}{\Vert }_{{\dot{H}}^{-s}({\mathbb{R}}^N)},$$ holds for a suitable fractional Talenti bubble $U[z,\lambda ]$. {For functions $u$ which are close to Talenti bubbles, we give the sharp asymptotic value of the implied constant in this inequality.} As an application {of this}, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.

Corrected typos, added some remarks and explanations. To appear in J. Funct. Anal

References in corpus (5)

• Asymptotics of the fast diffusion equation via entropy estimates
• On the sharp constant in the Bianchi-Egnell stability inequality
• Rates of convergence to non-degenerate asymptotic profiles for fast diffusion via energy methods
• Asymptotics near extinction for nonlinear fast diffusion on a bounded domain
• Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension

Cited by in corpus (2)

• An exceptional property of the one-dimensional Bianchi-Egnell inequality
• Quantitative Stability in Fractional Hardy-Sobolev Inequalities: The Role of Euler-Lagrange Equations