We study the following doubly critical Schrödinger system:
\[
{
−
Δ
u
−
λ
1
|
x
|
2
u
=
u
2
∗
−
1
+
ν
α
u
α
−
1
v
β
,
x
∈
R
N
,
−
Δ
v
−
λ
2
|
x
|
2
v
=
v
2
∗
−
1
+
ν
β
u
α
v
β
−
1
,
x
∈
R
N
,
u
,
v
∈
D
1
,
2
(
R
N
)
,
u
,
v
>
0
in
R
N
∖
{
0
}
,
\begin {cases}-\Delta u -\frac {\lambda _1}{|x|^2}u=u^{2^\ast -1}+ \nu \alpha u^{\alpha -1}v^\beta , \quad x\in \mathbb {R}^N,\\ -\Delta v -\frac {\lambda _2}{|x|^2}v=v^{2^\ast -1} + \nu \beta u^{\alpha }v^{\beta -1}, \quad x\in \mathbb {R}^N,\\ u,\, v\in D^{1, 2}(\mathbb {R}^N),\quad u,\, v>0\,\,\hbox {in $\mathbb {R}^N\setminus \{0\}$},\end {cases}
\]
where
N
≥
3
N\ge 3
,
λ
1
,
λ
2
∈
(
0
,
(
N
−
2
)
2
4
)
\lambda _1, \lambda _2\in (0, \frac {(N-2)^2}{4})
,
2
∗
=
2
N
N
−
2
2^\ast =\frac {2N}{N-2}
and
α
>
1
,
β
>
1
\alpha >1, \beta >1
satisfying
α
+
β
=
2
∗
\alpha +\beta =2^\ast
. This problem is related to coupled nonlinear Schrödinger equations with critical exponent for Bose-Einstein condensate. For different ranges of
N
N
,
α
\alpha
,
β
\beta
and
ν
>
0
\nu >0
, we obtain positive ground state solutions via some quite different variational methods, which are all radially symmetric. It turns out that the least energy level depends heavily on the relations among
α
,
β
\alpha , \,\beta
and
2
2
. Besides, for sufficiently small
ν
>
0
\nu >0
, positive solutions are also obtained via a variational perturbation approach. Note that the Palais-Smale condition cannot hold for any positive energy level, which makes the study via variational methods rather complicated.