In this paper, we prove existence of radial positive solutions for the following boundary value problem
where λ > 0, Ω denotes a ball in
R
^{N}
;
f
has more than one zero and
f
(0) < 0(the nonpositone case).
AMS Mathematics Subject Classification : 65H05, 65F10.
1. Introduction
Let us consider the the existence of radial positive solutions of the problem
where
λ
> 0, Ω denotes a ball in
R
^{N}
;
f
(0) < 0,
f
has more than one zero and is not strictly increasing entirely on [0,∞). △
_{p}u
= div(∇
u

^{p}
^{−2}
∇
u
) (1 <
p
≤
N
) is the
p
Laplacian operator of
u
.
The problem (1.1) arises in the theory of quasiregular and quasiconformal mappings or in the study of nonNewtonian fluids. In the latter case, the quantity
p
is a characteristic of the medium.Media with
p
> 2 are called dilatant fluids and these with
p
< 2 are called pseudoplastics(see
[18
,
19]
). If
p
= 2, they are Newtonian fluid. When
p
≠ 2, the problem becomes more complicated certain nice properties inherent to the case
p
= 2 seem to be lost or at least difficult to verify. The main differences between
p
= 2 and
p
≠ 2 can be founded in
[9
,
11]
.
In recent years, the asymptotic behavior, existence and uniqueness of the positive solutions for the quasilinear eigenvalue problems:
where
λ
> 0;
p
> 1;Ω ∈
R
^{N}
,
N
≥ 2 have been considered by a number of authors, see
[5

15
,
20

24
,
26

28]
and the references therein. In
[11]
, Guo and Webb proved existence and uniqueness results of (1.2) for
λ
large when
f
≥ 0, (
f
(
x
)=
x
^{p}
^{−1}
)′ < 0 for
x
> 0 and
f
satisfies some psublinearity conditions at 0 and ∞, generalizing a result in
[11]
where Ω is a ball. When
p
= 2, uniqueness results for semilinear equations were obtained in
[29
,
30]
where the assumption (
f
(
x
)=
x
)′ < 0 is required only for large
x
. Similar results for systems were discussed in
[31]
. Related results for the superlinear case when
f
≥ 0 can be found in
[26
,
32]
. When
p
= 2,
f
(0) < 0,
f
(
s
) has only one zero and Ω being a unit ball or an annulus in
R
^{N}
the related results have been obtained by Castro and Shivaji
[2]
, Arcoya and Zertiti
[1]
. The case when
f
(0) < 0 and
p
= 2 was treated in
[33]
, in which uniqueness of positive solution to single equation of (1.1) for
λ
large was established for sublinear
f
. See also
[34]
where this result was extended to the case when Ω is any bounded domain with convex outer boundary.
In this paper, we study this problem for
p
≠ 2,
f
(0) < 0 and Ω being a unit ball in
R
^{N}
. It extends and complements previous results in the literature
[1]
.
The paper is organized as follows. In section 2, we recall some facts that will be needed in the paper and give the main results. In section 3, we give the proofs of the main results in this paper.
2. Main results
We consider radial solution of (1.1), then, the existence of radial positive solutions of (1.1) is equivalent to the existence of positive solutions of the problem
where Ω is the unit ball of
R
^{N}
and
λ
> 0. Here
f
: [0,+∞) →
R
satisfies the following assumptions:
(H1)
f
∈
C
^{1}
([0,+∞),
R
) such that
f
^{′}
≥ 0 on [
β
,+∞), where
β
is the greatest zero of
f
;
(H2)
f
(0) < 0;
(H3)
where
p
−1 <
q
<
p
^{∗}
−1,
p
^{∗}
= ∞ if
p
≥
N
;
(H4) For some
k
∈ (0, 1),
where
Remark 2.1.
We note that in hypothesis (H1), there is no restriction on the function
f
(
u
) for 0 <
u
<
β
.
Remark 2.2.
If
f
satisfies (H1), any nonnegative solution
u
of (2.1) is positive in Ω, radial symmetric and radially decreasing, that is
By a modification of the method given in
[1]
, we obtain the following results.
Theorem 2.1.
Let assumptions (H1)(H4) be satisfied. Then there exists a positive real number λ
_{0}
such that if λ
∈ [0,
λ
_{0}
],
problem (1.1) has at least one radial positive solution which is decreasing on [0,1].
The proof of the theorem is based on the following preliminaries and four Lemmas.
Lemma 2.2.
Let u
(
r
)
be a solution of (2.1) in
(
r
_{1}
,
r
_{2}
) ⊂ (0,∞)
and let a be an arbitrary constant, then for each r
∈ (
r
_{1}
,
r
_{2}
)
we have
Remark 2.3.
The identity of Pohozaev type was obtained by Ni and Serrin
[6]
.
By a modification of the method given in
[1]
, we first introduce the notations and the following preliminaries. Let
F
be defined as
and
θ
denotes the greatest zero of
F
.
From (H4), we have
such that
Given
d
∈
R, λ
∈
R
, we define
By Lemma 2.2, we show the following Pohozaev identity on (
r
_{0}
,
r
_{1}
)
Moreover, for
d
≥
γ
, there exists
t
_{0}
such that
Next from (H1), we obtain that
f
is nondecreasing on [
kd, d
] ⊂ (
β
,+∞), and from (2.1) we have
Integrating on [0,
t
_{0}
], which implies
where
Hence, taking
r
_{0}
= 0,
r
_{1}
=
t
_{0}
in (2.4), and using (2.5)(2.6), we find
where
Lemma 2.3.
There exists λ
_{1}
> 0
such that if λ
∈ (0,
λ
_{1}
),
then u
(
r, γ, λ
) ≥
β, for
∀
r
∈ [0, 1].
Proof
. Let
r
^{∗}
= sup{0 ≤
r
≤ 1 :
u
(
r, γ, λ
) ≥
β
}. For
u
is decreasing on [0,
r
^{∗}
], then
β
≤
u
(
r, γ, λ
). ≤
u
(0,
r, λ
) =
γ
, ∀
r
∈ [0,
r
^{∗}
]
Moreover, since
f
≥ 0 on [
β
,+∞) and
we obtain
Then for
we have
Next, by using the mean value theorem and (2.8), there exists
such that
Assume that
r
^{∗}
< 1, we have
which contradicts the definition of
r
^{∗}
. Then, the lemma is proved for
r
^{∗}
= 1. □
Lemma 2.4.
There exists λ
_{2}
> 0
such that for λ
∈ (0,
λ
_{2}
)
Proof
. From Lemma 2.2, we have the following Pohozaev identity on (
r
,
t
_{0}
)
Extending
f
by
f
(
x
) =
f
(0) < 0, for
x
∈ (−∞, 0], then there exists
B
< 0 such that
For sufficiently large
γ
, from (H4), we deduce
By (2.7) and (2.9), we get
Then, there exists
λ_{2}
such that
Hence, for all
λ
∈ (0,
λ
_{2}
) and
r
∈ [0, 1],
H
(
t
) > 0, ∀
d
≥
γ
. This also implies that
u
(
r, d, λ
)
^{2}
+
u′
(
r, d, λ
)
^{2}
> 0, for all
t
∈ [0, 1] and all
d
≥
γ
. □.
Lemma 2.5.
For r
∈ [0, 1],
there exists d
≥
γ such that u
(
r, d, λ
) < 0.
Proof
. By contradiction, let
d
≥
λ
, we assume that
u
(
r, d, λ
) ≥ 0 for ∀
r
∈ [0, 1].
Let
is decreasing on (0,
r
)}. Define
ω
be the solution of the following equation:
where
δ
is chosen such that the first zero of
ω
is
and
ω
satisfies
r
∈ (0; 1).
From (H3), there exists
d
_{0}
≥
γ
such that
Since
Let
v
=
dω
,
Then, we obtain
Suppose
u
(
r, d, λ
) ≥
d
_{0}
for all
from (2.12),
On the other hand, from the quality of
we know that
then
From (2.13)(2.15), we obtain
On the other hand, since
which is contradiction with (2.16).
Hence, there exists
such that
And since
d
_{0}
≥
γ
>
β
, there exists
such that
Now, we consider
t
_{0}
defined in (2.5), also
On [0,
t
_{0}
], from (H1),
F
is nondecreasing on [
β
,+∞) and
u
(
r, d, λ
) ≥
kd
≥
β
∀
r
∈ (0,
t
_{0}
]. We have
On the other hand, since
then
Hence, by (2.10) we get
From (H4), (2.18), (2.19)
Therefore, there exists
d
_{1}
≥
d
_{0}
such that for
d
≥
d
_{1}
, we get
By (2.17), (2.18)
Which implies
The mean value theorem gives us
such that
hence
and since
there exists
T
∈ (0, 1) such that
u
(
T, d, λ
) < 0, which contradicts with the assuming, the lemma is proved. □
3. Proof of the Main Results
The proof of Theorem 2.1.
Let
λ
_{0}
= min{
λ
_{1}
,
λ
_{2}
}, for ∀
λ
∈ (0,
λ
_{0}
). Define
From Lemma 2.3, we obtain that the set {
d
≥
γ
:
u
(
r, d, λ
) ≥ 0, ∀
r
∈ (0, 1]} is nonempty. From Lemma 2.5 implies that
Then we claim that
is the solution of problem (1.1). Moreover, the solution
satisfies the following properties:

(i)for allr∈ [0, 1);

(ii)

(iii)

(iv) u is decreasing in [0,1].
For (i). By contradiction, if there exists 0 ≤
R
_{1}
< 1 such that
from Lemma 2.4,
then we can suppose
Hence from
and
we find there exists
R
_{2}
∈ (
R
_{1}
, 1) such that
which contradicts with the definition of
So
for all
r
∈ [0, 1).
For (ii). By contradiction,we assume
then from (i) there exists
η
such that
for ∀
r
∈ (0, 1), moreover, there exists
δ
> 0 such that
for ∀
t
∈ (0, 1], which is a contradiction with the definition of
(ii) is proved.
For (iii). From (2.1),
Taking into account Lemma 2.3 and (H1), we have for ∀
λ
∈ (0,
λ
_{0}
),
u
(
r, γ, λ
) ≥
β
and
f
(
s
) > 0, for ∀
s
∈ (
β
,+∞), which implies for ∀
λ
∈ (0,
λ
_{1}
),
So
(iv) is also proved.
BIO
Weihui Wang received master degree from Nanjing Normal University. Her research interests include elliptic equations.
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China.
Zuodong Yang received Ph.D. from Beijing University of Areonautics and Astronautics. His research interests are elliptic and parabolic equations.
School of Teacher Education, Nanjing Normal University, Jiangsu Nanjing 210097, China; Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China.
email: zdyangjin@263.net
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