We establish the existence of positive solution to the critical nonlocal elliptic system
(S)\hskip10mm \left\{ \begin{aligned} & (-\Delta)^{s}_p u+a(x)|u|^{p-2} u+ c(x) |v|^{p-2} v = \tfrac{1}{p^{*}_s}K_u(u,v) \ \ \text{in} \ \ \mathbb{R}^{N},\\ & (-\Delta)^{s}_p v+c(x)| u|^{p-2} u+ b(x)|v|^{p-2} v = \tfrac{1}{p^{*}_s}K_v(u,v) \ \ \text{in} \ \ \mathbb{R}^{N},\\ &\ u, v>0 \ \text{in} \ \mathbb{R}^{N},\ u, v \in D^{s, p}(\mathbb{R}^{N}),\ N> ps,\ s\in (0,1). \end{aligned} \right.
(
S
)
{
(
−
Δ
)
p
s
u
+
a
(
x
)
∣
u
∣
p
−
2
u
+
c
(
x
)
∣
v
∣
p
−
2
v
=
1
p
s
∗
K
u
(
u
,
v
)
in
R
N
,
(
−
Δ
)
p
s
v
+
c
(
x
)
∣
u
∣
p
−
2
u
+
b
(
x
)
∣
v
∣
p
−
2
v
=
1
p
s
∗
K
v
(
u
,
v
)
in
R
N
,
u
,
v
>
0
in
R
N
,
u
,
v
∈
D
s
,
p
(
R
N
)
,
N
>
p
s
,
s
∈
(
0
,
1
)
.
Here
(-\Delta)^{s}_p
(
−
Δ
)
p
s
denotes the fractional
p
p
-Laplacian,
a,b
a
,
b
and
c
c
are suitable functions and
K
K
is a
p^{*}_s
p
s
∗
-homogeneous function,
p^{*}_s= (pN)/(N-ps)
p
s
∗
=
(
p
N
)
/
(
N
−
p
s
)
,
N > ps
N
>
p
s
. One of the main tools is to apply the global compactness result for the associated energy functional similar to that due to M. Struwe [A global compactness result for elliptic boundary value problems involving limiting nonliarities, Math. Zeitschrift 187/4 (1984) 511–517] combined with some information on a limit system of
(S)
(
S
)
with
a=b=c=0
a
=
b
=
c
=
0
, the concentration compactness due to P. L. Lions [The concentration-compactness principle in the calculus of variations. I: The limit case, Rev. Mat. Iberoamericana 1/1 (1985) 145–201] and the Brouwer degree theory.