In this paper, we establish a relationship between the elliptic system
\[
{
−
Δ
u
+
λ
u
=
μ
1
|
u
|
2
p
u
+
β
1
|
v
|
q
1
|
u
|
p
1
−
1
u
,
x
∈
Ω
,
−
Δ
v
+
λ
v
=
μ
2
|
v
|
2
p
v
+
β
2
|
u
|
q
2
|
v
|
p
2
−
1
v
,
x
∈
Ω
,
u
=
v
=
0
on
∂
Ω
,
\left \{ \begin {array}{ll...
