Stability of Hardy Littlewood Sobolev inequality under bubbling

Abstract

Abstract In this note we will generalize the results deduced in Figalli and Glaudo (Arch Ration Mech Anal 237(1):201–258, 2020) and Deng et al. (Sharp quantitative estimates of Struwe’s Decomposition. Preprint http://arxiv.org/abs/2103.15360

, 2021) to fractional Sobolev spaces. In particular we will show that for

$$s\in (0,1)$$

s ∈ ( 0 , 1 )

,

$$n>2s$$

n > 2 s

and

$$\nu \in \mathbb {N}$$

ν ∈ N

there exists constants

$$\delta = \delta (n,s,\nu )>0$$

δ = δ ( n , s , ν ) > 0

and

$$C=C(n,s,\nu )>0$$

C = C ( n , s , ν ) > 0

such that for any function

$$u\in \dot{H}^s(\mathbb {R}^n)$$

u ∈

H ˙

s

(

R

n

)

satisfying,

$$\begin{aligned} \left\| u-\sum _{i=1}^{\nu } \tilde{U}_{i}\right\| _{\dot{H}^s} \le \delta \end{aligned}$$

u -

∑

i = 1

ν

U ~

i

H ˙

s

≤ δ

where

$$\tilde{U}_{1}, \tilde{U}_{2},\ldots \tilde{U}_{\nu }$$

U ~

1

,

U ~

2

, …

U ~

ν

is a

$$\delta $$

δ

-interacting family of Talenti bubbles, there exists a family of Talenti bubbles

$$U_{1}, U_{2},\ldots U_{\nu }$$

U 1

,

U 2

, …

U ν

such that

$$\begin{aligned} \left\| u-\sum _{i=1}^{\nu } U_{i}\right\| _{\dot{H}^s} \le C\left\{ \begin{array}{ll} \Gamma &{} \text{ if } 2s< n < 6s,\\ \Gamma |\log \Gamma |^{\frac{1}{2}} &{} \text{ if } n=6s, \\ \Gamma ^{\frac{p}{2}} &{} \text{ if } n > 6s \end{array}\right. \end{aligned}$$

u -

∑

i = 1

ν

U i

H ˙

s

≤ C

Γ

if

2 s < n < 6 s ,

Γ | log Γ |

1 2

if

n = 6 s ,

Γ

p 2

if

n > 6 s

for

$$\Gamma =\left\| \Delta u+u|u|^{p-1}\right\| _{H^{-s}}$$

Γ =

Δ u +

u | u |

p - 1

H

- s

and

$$p=2^*-1=\frac{n+2s}{n-2s}.$$

p =

2 ∗

- 1 =

n + 2 s

n - 2 s

.