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Equations involving fractional Laplacian operator: Compactness and application
Authors
Shusen Yan
Jianfu Yang
Xiaohui Yu
Publication date
2 March 2015
Publisher
Doi
View
on
arXiv
Abstract
In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \, \partial\Omega, \end{equation} where
Ω
\Omega
Ω
is a smooth bounded domain in
R
N
\mathbb{R}^N
R
N
,
ε
∈
[
0
,
2
α
∗
−
2
)
\varepsilon\in [0, 2^*_\alpha-2)
ε
∈
[
0
,
2
α
∗
​
−
2
)
,
0
<
α
<
1
,
 
2
α
∗
=
2
N
N
−
2
α
0<\alpha<1,\, 2^*_\alpha = \frac {2N}{N-2\alpha}
0
<
α
<
1
,
2
α
∗
​
=
N
−
2
α
2
N
​
. We show that for any sequence of solutions
u
n
u_n
u
n
​
of \eqref{eq:0.1} corresponding to
ε
n
∈
[
0
,
2
α
∗
−
2
)
\varepsilon_n\in [0, 2^*_\alpha-2)
ε
n
​
∈
[
0
,
2
α
∗
​
−
2
)
, satisfying
∥
u
n
∥
H
≤
C
\|u_n\|_{H}\le C
∥
u
n
​
∥
H
​
≤
C
in the Sobolev space
H
H
H
defined in \eqref{eq:1.1a},
u
n
u_n
u
n
​
converges strongly in
H
H
H
provided that
N
>
6
α
N>6\alpha
N
>
6
α
and
λ
>
0
\lambda>0
λ
>
0
. An application of this compactness result is that problem \eqref{eq:0.1} possesses infinitely many solutions under the same assumptions.Comment: 34 page
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oai:rune.une.edu.au:1959.11/18...
Last time updated on 09/08/2023
Crossref
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info:doi/10.1016%2Fj.jfa.2015....
Last time updated on 06/11/2020