The Brezis-Nirenberg result for the fractional Laplacian
Authors: Servadei Raffaella , Valdinoci Enrico 
Published: September 2014

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column left-parenthesis negative normal upper Delta right-parenthesis Superscript s Baseline u minus lamda u equals StartAbsoluteValue u EndAbsoluteValue Superscript 2 Super Superscript asterisk Superscript minus 2 Baseline u 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u equals 0 2nd Column a m p semicolon in double-struck upper R Superscript n Baseline minus normal upper Omega comma EndLayout">

{

(
−
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column script upper L Subscript upper K Baseline u plus lamda u plus StartAbsoluteValue u EndAbsoluteValue Superscript 2 Super Superscript asterisk Superscript minus 2 Baseline u plus f left-parenthesis x comma u right-parenthesis equals 0 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u equals 0 2nd Column a m p semicolon in double-struck upper R Superscript n Baseline minus normal upper Omega comma EndLayout">

{

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.