Affiliation:
1. School of Mathematics , Liaoning Normal University , Dalian 116029 , P. R. China
Abstract
Abstract
In this paper, we investigate the ground states for the following fractional Choquard equation with magnetic fields and critical exponents:
(
-
Δ
)
A
s
u
+
V
(
x
)
u
=
λ
f
(
x
,
u
)
+
[
|
x
|
-
α
∗
|
u
|
2
α
,
s
*
]
|
u
|
2
α
,
s
*
-
2
u
in
ℝ
N
,
(-\Delta)_{A}^{s}u+V(x)u=\lambda f(x,u)+[\lvert x\rvert^{-\alpha}\ast\lvert u%
\rvert^{2^{*}_{\alpha,s}}]\lvert u\rvert^{2^{*}_{\alpha,s}-2}u\quad\text{in }%
\mathbb{R}^{N},
where
λ
>
0
{\lambda>0}
,
α
∈
(
0
,
2
s
)
{\alpha\in(0,2s)}
,
N
>
2
s
{N>2s}
,
u
:
ℝ
N
→
ℂ
{u:\mathbb{R}^{N}\rightarrow\mathbb{C}}
is a complex-valued function,
2
α
,
s
*
=
(
2
N
-
α
)
/
(
N
-
2
s
)
{2^{*}_{\alpha,s}=(2N-\alpha)/(N-2s)}
is the fractional Hardy–Littlewood–Sobolev critical exponent,
V
∈
(
ℝ
N
,
ℝ
)
{V\in(\mathbb{R}^{N},\mathbb{R})}
is an electric potential, V and f are asymptotically periodic in x,
A
∈
(
ℝ
N
,
ℝ
N
)
{A\in(\mathbb{R}^{N},\mathbb{R}^{N})}
is a magnetic potential, and
(
-
Δ
)
A
s
{(-\Delta)^{s}_{A}}
is a fractional magnetic Laplacian operator with
s
∈
(
0
,
1
)
{s\in(0,1)}
. We prove that the equation has a ground state solution for large λ by using the Nehari method and the concentration-compactness principle.
Funder
National Natural Science Foundation of China
Natural Science Foundation of Liaoning Province
Department of Education of Liaoning Province
Cited by
1 articles.
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