Abstract
In this paper, we study the asymptotic behavior of the unique positive classical solution to the following semilinear boundary value problem
Δ
u
+
a
(
x
)
u
α
=
0
,
x
∈
Ω
,
u
>
0
in
Ω
,
u
|
∂
Ω
=
0
.
Here
Ω
is a bounded
C
1
,
1
domain,
α
<
1
and the function
a
is in
C
l
o
c
γ
(
Ω
)
,
0
<
γ
<
1
such that there exists
c
>
0
satisfying for each
x
∈
Ω
,
1
c
≤
a
(
x
)
δ
(
x
)
λ
exp
(
−
∫
δ
(
x
)
η
z
(
t
)
t
d
t
)
≤
c
,
where
λ
≤
2
,
η
>
d
=
d
i
a
m
(
Ω
)
,
δ
(
x
)
=
d
i
s
t
(
x
,
∂
Ω
)
and
z
is a continuous function on
[
0
,
η
]
with
z
(
0
)
=
0
.