Author:
Abebe Abraham,Chhetri Maya,Sankar Lakshmi,Shivaji R
Abstract
Abstract
We study the existence of a positive radial solution to the nonlinear eigenvalue problem
−
Δ
u
=
λ
K
1
(
|
x
|
)
f
(
v
)
in
Ω
e
,
−
Δ
v
=
λ
K
2
(
|
x
|
)
g
(
u
)
in
Ω
e
,
u
(
x
)
=
v
(
x
)
=
0
if
|
x
|
=
r
0
(>0),
u
(
x
)
→
0
,
v
(
x
)
→
0
as
|
x
|
→
∞
, where
λ
>
0
is a parameter,
Δ
u
=
div
(
∇
u
)
is the Laplace operator,
Ω
e
=
{
x
∈
R
n
∣
|
x
|
>
r
0
,
n
>
2
}
, and
K
i
∈
C
1
(
[
r
0
,
∞
)
,
(
0
,
∞
)
)
;
i
=
1
,
2
are such that
K
i
(
|
x
|
)
→
0
as
|
x
|
→
∞
. Here
f
,
g
:
[
0
,
∞
)
→
R
are
C
1
functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for λ small via degree theory and rescaling arguments. We also discuss a non-existence result for
λ
≫
1
for the single equations case.
MSC: 34B16, 34B18.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis
Cited by
14 articles.
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