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Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient
We establish that for n ⩾ 3 and p > 1 , the elliptic equation Δ u + K ( x ) u p = 0 in R n possesses a continuum of positive entire solutions with logarithmic decay at ∞, provided that a locally Hölder continuous function K ⩾ 0 in R n ∖ { 0 } , satisfies K ( x ) = O ( | x | σ ) at x = 0 for some...
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| Published in: | Journal of Differential Equations 2007-06, Vol.237 (1), p.159-197 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | We establish that for
n
⩾
3
and
p
>
1
, the elliptic equation
Δ
u
+
K
(
x
)
u
p
=
0
in
R
n
possesses a continuum of positive entire solutions with logarithmic decay at ∞, provided that a locally Hölder continuous function
K
⩾
0
in
R
n
∖
{
0
}
, satisfies
K
(
x
)
=
O
(
|
x
|
σ
)
at
x
=
0
for some
σ
>
−
2
, and
|
x
|
2
K
(
x
)
=
c
+
O
(
[
log
|
x
|
]
−
θ
)
near ∞ for some constants
c
>
0
and
θ
>
1
. The continuum contains at least countably many solutions among which any two do not intersect. This is an affirmative answer to an open question raised in [S. Bae, T.K. Chang, On a class of semilinear elliptic equations in
R
n
, J. Differential Equations 185 (2002) 225–250]. The crucial observation is that in the radial case of
K
(
r
)
=
K
(
|
x
|
)
, two fundamental weights,
(
log
r
)
p
p
−
1
and
r
n
−
2
(
log
r
)
−
p
p
−
1
, appear in analyzing the asymptotic behavior of solutions. |
|---|---|
| ISSN: | 0022-0396 1090-2732 |
| DOI: | 10.1016/j.jde.2007.03.003 |