Abstract
We investigate the ground states of 3-component Bose–Einstein condensates with harmonic-like trapping potentials in
ℝ
2
{\mathbb{R}^{2}}
, where the intra-component interactions
μ
i
{\mu_{i}}
and the inter-component interactions
β
i
j
=
β
j
i
{\beta_{ij}=\beta_{ji}}
(
i
,
j
=
1
,
2
,
3
{i,j=1,2,3}
,
i
≠
j
{i\neq j}
) are all attractive.
We display the regions of
μ
i
{\mu_{i}}
and
β
i
j
{\beta_{ij}}
for the existence and nonexistence of the ground states, and give an elaborate analysis for the asymptotic behavior of the ground states as
β
i
j
↗
β
i
j
*
:=
a
∗
+
1
2
(
a
∗
-
μ
i
)
(
a
∗
-
μ
j
)
{\beta_{ij}\nearrow\beta_{ij}^{*}:=a^{\ast}+\frac{1}{2}\sqrt{{(a^{\ast}-\mu_{i%
})(a^{\ast}-\mu_{j})}}}
, where
0
<
μ
i
<
a
∗
:=
∥
w
∥
2
2
{0<\mu_{i}<a^{\ast}:=\|w\|_{2}^{2}}
are fixed and w is the unique positive solution of
Δ
w
-
w
+
w
3
=
0
{\Delta w-w+w^{3}=0}
in
H
1
(
ℝ
2
)
{H^{1}(\mathbb{R}^{2})}
. The energy estimation as well as the mass concentration phenomena are studied, and when two of the intra-component interactions are equal, the nondegeneracy and the uniqueness of the ground states are proved.