Abstract
In this paper, we study the existence of multiple normalized solutions to the following dipolar Gross–Pitaevskii equation with a mass subcritical perturbation
{
−
1
2
Δ
u
+
μ
u
+
V
(
ε
x
)
u
+
λ
1
|
u
|
2
u
+
λ
2
(
K
∗
|
u
|
2
)
u
+
λ
3
|
u
|
p
−
2
u
=
0
,
in
R
3
,
∫
R
3
|
u
|
2
d
x
=
a
2
,
where
a
,
ε
>
0
,
2
p
10
3
,
μ
∈
R
is the Lagrange multiplier,
λ
3
0
,
(
λ
1
,
λ
2
)
∈
{
(
λ
1
,
λ
2
)
∈
R
2
:
λ
1
4
π
3
λ
2
≤
0
or
λ
1
−
8
π
3
λ
2
≤
0
}
,
V
(
x
)
is an external potential,
∗
denotes convolution,
K
(
x
)
=
1
−
3
cos
2
θ
(
x
)
|
x
|
3
and
θ
(
x
)
is the angle between the dipole axis determined by
(
0
,
0
,
1
)
and the vector
x
. Under suitable assumptions of
V
, we use variational methods to prove that the number of normalized solutions is not less than the number of global minimum points of
V
if
ε
>
0
is sufficiently small.