Abstract
We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem
(
-
Δ
)
α
/
2
u
=
a
1
(
x
)
u
σ
1
+
a
2
(
x
)
u
σ
2
, in
D
lim
x
→
z
∈
∂
D
(
δ
(
x
)
)
1
-
(
α
/
2
)
u
(
x
)
=
0
,
where
0
<
α
<
2
,
σ
1
,
σ
2
∈
(
-
1,1
)
,
D
is a bounded
C
1,1
-domain in
ℝ
n
,
n
≥
2
,
and
δ
(
x
)
denotes the Euclidian distance from
x
to the boundary of
D
.
The nonnegative weight functions
a
1
,
a
2
are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution.