The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation
\[
{
(
−
Δ
)
s
u
−
λ
u
=
|
u
|
2
∗
−
2
u
a
m
p
;
in
Ω
,
u
=
0
a
m
p
;
in
R
n
∖
Ω
,
\left \{ \begin {array}{ll} (-\Delta )^s u-\lambda u=|u|^{2^*-2}u & {\mbox { in }} \Omega ,\\ u=0 & {\mbox { in }} \mathbb {R}^n\setminus \Omega \,, \end {array} \right .
\]
where
(
−
Δ
)
s
(-\Delta )^s
is the fractional Laplace operator,
s
∈
(
0
,
1
)
s\in (0,1)
,
Ω
\Omega
is an open bounded set of
R
n
\mathbb {R}^n
,
n
>
2
s
n>2s
, with Lipschitz boundary,
λ
>
0
\lambda >0
is a real parameter and
2
∗
=
2
n
/
(
n
−
2
s
)
2^*=2n/(n-2s)
is a fractional critical Sobolev exponent.
In this paper we first study the problem in a general framework; indeed we consider the equation
\[
{
L
K
u
+
λ
u
+
|
u
|
2
∗
−
2
u
+
f
(
x
,
u
)
=
0
a
m
p
;
in
Ω
,
u
=
0
a
m
p
;
in
R
n
∖
Ω
,
\left \{ \begin {array}{ll} \mathcal L_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 & \mbox {in } \Omega ,\\ u=0 & \mbox {in } \mathbb {R}^n\setminus \Omega \,, \end {array}\right .
\]
where
L
K
\mathcal L_K
is a general non-local integrodifferential operator of order
s
s
and
f
f
is a lower order perturbation of the critical power
|
u
|
2
∗
−
2
u
|u|^{2^*-2}u
. In this setting we prove an existence result through variational techniques.
Then, as a concrete example, we derive a Brezis-Nirenberg type result for our model equation; that is, we show that if
λ
1
,
s
\lambda _{1,s}
is the first eigenvalue of the non-local operator
(
−
Δ
)
s
(-\Delta )^s
with homogeneous Dirichlet boundary datum, then for any
λ
∈
(
0
,
λ
1
,
s
)
\lambda \in (0, \lambda _{1,s})
there exists a non-trivial solution of the above model equation, provided
n
⩾
4
s
n\geqslant 4s
. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators.