Abstract
Abstract
In this paper, we study k-vortex solutions of the form
ψ
(
t
,
x
)
=
e
i
(
μ
t
+
k
θ
(
x
1
,
x
2
)
)
u
(
x
)
of the pseudo-relativistic Hartree equation
1
i
ψ
t
(
x
,
t
)
=
(
−
Δ
+
m
2
−
m
)
ψ
(
x
,
t
)
−
(
|
x
|
−
1
∗
|
ψ
|
2
)
ψ
(
x
,
t
)
,
(
x
,
t
)
∈
R
3
×
R
,
under the constraint
∫
R
3
|
u
|
2
d
x
=
N
.
Such solutions are obtained as minimizers of the problem
2
e
k
(
N
)
=
inf
{
E
k
(
u
)
:
u
∈
H
s
∖
{
0
}
,
∫
ℝ
3
|
u
(
x
,
0
)
|
2
d
x
=
N
>
0
}
with the associated functional
E
k
(
u
)
of (1). We show that there is a threshold value
N
c
(
k
)
>
0
such that problem (2) admits a nonnegative minimizer u
N
if
0
<
N
<
N
c
(
k
)
, and there exists no minimizer for
e
k
(
N
)
if
N
⩾
N
c
(
k
)
. Moreover, the stability of the vortex solution is considered, and the limiting behavior of the minimizer u
N
as
N
→
N
c
(
k
)
−
is described.
Funder
Jiangxi Provincial Natural Science Foundation
National Natural Science Foundation of China
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics