Abstract
In this paper, we are interested in the fractional Yamabe-type equation
A
s
u
=
u
n
+
2
s
n
-
2
s
,
u
>
0
in
Ω
and
u
=
0
on
∂
Ω
.
Here
Ω
is a regular bounded domain of
R
n
,
n
≥
2
and
A
s
,
s
∈
(
0
,
1
)
represents the fractional Laplacian operator in
Ω
with zero Dirichlet boundary condition. Based on the theory of critical points at infinity of Bahri and the localization technique of Caffarelli and Silvestre, we compute the difference of topology induced by the critical points at infinity between the level sets of the variational functional associated to the problem. Our result can be seen as a nonlocal analog of the theorem of Bahri
et al
. (
Cal. Var. Partial. Differ. Equ.
3
(1995) 67–94) on the classical Yamabe-type equation.