| dc.description.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
. | en_US |
