LARGE SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATION OF MIXED TYPE†

Journal of Applied Mathematics & Informatics.
2014.
Sep,
32(5_6):
721-736

- Received : July 19, 2013
- Accepted : May 21, 2014
- Published : September 30, 2014

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We consider the equation △
_{m}u
=
p
(
x
)
u^{α}
+
q
(
x
)
u^{β}
on
R
^{N}
(
N
≥ 2), where
p, q
are nonnegative continuous functions and 0 <
α
≤
β
. Under several hypotheses on
p
(
x
) and
q
(
x
), we obtain existence and nonexistence of blow-up solutions both for the superlinear and sublinear cases. Existence and nonexistence of entire bounded solutions are established as well.
AMS Mathematics Subject Classification : 35J65, 35J50.
where 1 <
p
<
N
, 0 <
α
≤
β
, and the nonnegative functions
p
and
q
are locally Hölder continuous, having the property that min{
p
(
x
),
q
(
x
)} is c-positive in Ω (i.e.,if min{
p
(
x
),
q
(
x
)} vanishes at any point
x
_{0}
, then there is an open set Ω
_{0}
containing
x
_{0}
for which
and it is positive for all x on the boundary of Ω
_{0}
.
By a positive large solution of (1),we mean a positive function
u
∈
C
^{1}
(Ω) which satisfy (1) at every point of Ω and
u
→ ∞ as
x
→ ∂Ω. In the case Ω =
R
^{N}
, if lim
_{r}
_{→∞}
u
(
r
) = ∞, we call it a positive entire large solution (PELS) of (1).
Such problems arise in Riemannian geometry when studying conformal deformation of a metric with prescribed scalar curvature
[1]
and in the study of large solutions of elliptic systems. Our purpose is to establish conditions on
p
and
q
which ensure the existence and nonexistence of positive solutions of (1).
Equations of the above form are mathematical models occuring in the studies of the p-Laplace equation,generalized in reaction-diffusion theory,non-newtonian fluid theory
[3
,
4]
, non-Newtonian filtration
[5]
and the turbulent flow of a gas in porous medium
[6]
. In the non-Newtonian fluid theory, the quantity p is characteristic of the medium. Media with
p
> 2 are called dilatant fluids and those with
p
< 2 are called pseudoplastics. If
p
= 2, they are Newtonian fluids.
Large solutions of the problem
where Ω is a bounded domain in
R^{N}
have been extensively studied. A problem with
f
(
u
) =
e^{u}
and
N
= 2 was first considered by Bieberbach
[16]
. He showed that if Ω is a bounded domain in
R
^{2}
such that ∂Ω is a
C
^{2}
submanifold of
R
^{2}
, then there exists a unique
u
∈
C
^{2}
(Ω) such that Δ
u
=
e^{u}
in Ω and |
u
(
x
) −
ln
(
d
(
x
))
^{−2}
| is bounded on Ω. Here
d
(
x
) denotes the distance from a point
x
to ∂Ω. The result was extended by Rademacher
[20]
to smooth bounded domain in
R
^{3}
. Keller
[17]
and Osserman
[19]
provided necessary and sufficient conditions of f for the existence of solutions of Eq (2). The existence, but not uniqueness of solutions of the problem (2) with f monotone was studied by Keller
[17]
. For
f
(
u
) = −
u^{a}
with
a
> 1, problem (2) is of interest in the study of the sub-sonic motion of a gas when
a
= 2 and is related to a problem involing super-diffusion, particularly for 1 <
a
≤ 2 (see
[18]
).
Quasilinear elliptic problem with boundary blow-up
have been studied, see
[11
,
13]
and the references therein. Diaz and Letelier
[13]
proved the existence and uniqueness of large solutions to the problem (3) both for
f
(
u
) =
u^{γ}
,
γ
>
m
− 1 (super-linear case) and ∂Ω being of the class
C
^{2}
. Recently, Lu et al.
[9]
proved the existence of large solutions to the problem (3) both for
f
(
u
) =
u^{γ}
,
γ
>
m
− 1, Ω =
R
^{N}
or Ω being a bounded domain (super-linear case), and
γ
≤
m
− 1,Ω =
R
^{N}
(sub-linear case), respectively.
The vast majority of papers studying nonnegative entire large solutions for quasi-linear elliptic equations consider equations of the form
where
f
is nondecreasing. (See, for example,
[7
,
8
,
9
,
10]
and references therein)
Recently, Caisheng Chen et al.
[12]
studied the following problem
where
h
(
x
),
H
(
x
):
R
^{N}
→ (0,∞) are the locally Hölder continuous function, and 0 <
m
≤
p
−1 <
n
. They use the test function method to prove the nonexistence of nontrival solution of the problem (5).
Motivated by the results of the above cited papers, we shall attempted to treat such equation (1). The results of the semilinear equations are extended to the quasilinear ones. We can find the related results for
p
= 2 in
[14]
and
[23]
.
We first give some important lemmas which will be used in our proves.
Lemma 1.1.
If v
(
x
) :
I
⊆
R
→
R is a locally integrable nonnegative function, then
for all a, b
∈
I, a
<
b and
1 ≤
h
< ∞;
when
0 <
h
< 1,
the inverse inequality holds
.
Proof
. This lemma can be easily proved using Jessen’s inequality.
Lemma 1.2
(Weak comparison principle
[10]
).
Let
Ω
be a bounded domain in
R
^{N}
(
N
≥ 2)
with a smooth boundary
∂Ω
and θ
: (0,
_{∞}
) → (0
_{,∞}
)
be a continuous and non-decreasing function. Let u
_{1}
,
u
_{2}
∈
w
^{1,p}
(Ω)
satisfy
for any non-decreasing function
Then the inequality
implies that
Lemma 1.3.
If v
(
x
) :
I
⊆
R
→
R is a locally integrable nonnegative function, then
for all a, b
∈
I, a
<
b and
1 ≤
h
< ∞;
when
0 <
h
< 1,
the inverse inequality holds
.
Proof
. This lemma can be easily proved using Jessen’s inequality.
In the sequel proof, we will use the fact that for any nonnegative
a
and
b
,
Denote
Combinations of the following conditions on the nonnegative continuous functions
p
and
q
will be used:
Remark 1.1.
Condition (
M_{pq}
) can be changed into
Remark 1.2.
If
m
≥ 2, condition (
m_{pq}
) can be changed into
Proof
. In fact, condition (8) implies condition (
m_{pq}
). Let 2 ≤
m
< ∞, it follows that
we have
In the first inequality, we used lemma 1.1.
Theorem 2.1.
Suppose
Ω
is a bounded domain in
R
^{N}
with smooth boundary. If p and q are c-positive, locally Hölder continuous
, 0 <
α
≤
β, β
>
m
−1,
then (1) has a large positive solution in
Ω.
Proof
. First we consider the existence of positive solutions to
From Theorem 2.1 of
[10]
, there exists a positive solution to the boundary value problem for each
k
∈
N
Clearly,
Again from
[10]
, we have that for each
k
∈
N
there exists a unique nonnegative classical solution
w_{k}
to the boundary value problem
Then
From the weak comparison principle, we have
w_{k}
≤
v_{k}
for all
k
∈
N
, and hence
v_{k}
and
w_{k}
are ordered upper and lower solutions of (9), respectively. From
[9]
, we concluded that (9) has a positive solution
u_{k}
. Using the weak comparison principle again, we have
u
_{k−1}
≤
u_{k}
in Ω. So
w
_{1}
≤
u
_{k−1}
≤
u_{k}
≤
v_{k}
. From
[10]
, the sequence
v_{k}
converges on Ω to a large solution v, and v satisfies Δ
_{m}v
=
q
(
x
)
v^{β}
on Ω. It follows that
w
_{1}
≤
u
_{k−1}
≤
u_{k}
≤
v
. Thus,
u_{k}
is bounded. Therefore the sequence
u_{k}
converges on Ω to some function u. Standard bootstrap argument shows that the function u(x) is indeed a solution to (1).
We are left to show that u is a large solution. To see this, we let
x
_{0}
∈ ∂Ω, and let
x_{j}
be a sequence in Ω such that
x_{j}
→
x
_{0}
as
j
→ ∞. Let
k
∈
N
, since
u_{k}
is monotone, choose
N_{k}
∈
N
such that
u_{k}
(
x_{j}
) >
k
− 1 for
j
≥
N_{k}
. Thus,
u_{n}
(
x_{j}
) >
k
−1 for
n
≥
k
and
j
≥
N_{k}
. Therefore, given any
A
> 0,
k
and
N_{k}
can be chosen large enough so that
u
(
x_{j}
) ≥
A
for
j
≥
N_{k}
. Thus, lim
_{j→∞}
u
(
x_{j}
) = ∞, and hence, lim
_{x→x0}
u
(
x
) = ∞. Since
x
_{0}
is arbitrary, it is now apparent that
u
is a large solution of (1).
Theorem 2.2.
Suppose p, q are c-positive and satisfy
(
M_{pq}
),
then (1) has an entire large positive solution.
Proof
. From Theorem 2.1, we have that for each
k
∈
N
, there exists a positive solution to the boundary value problem
Clearly, for any
k
and |
x
| ≥
k, v
_{k+1}
≤
v_{k}
= ∞. The maximum principle gives that
in
R
^{N}
.
To show that
v_{k}
converges to some
v
∈
C
(
R
^{N}
) and that
v
→ ∞ as |
x
| → ∞, we observe that condition (
M_{pq}
) implies that
Thus,
where
is the unique positive solution of
We claim that
on |
x
| ≤
k
. Clearly, when |
x
| =
k
,
and thus for
Let
then
From the weak comparison principle, we get
So
if |
x
| ≤
k
.
Let
and note that
v_{k}
≥
w
for all
k
. Thus, {
v_{k}
} converges to some
v
and v ≥
w
in
C
(
R
^{N}
). Since
w
→ ∞ as |
x
| → ∞,
v
→ ∞ as |
x
| → ∞.
This completes the proof.
Next we consider the following nonexistence results.
Theorem 2.3.
Suppose the equation
has no PELS, then Eq.(1) has no PELS.
Proof
. Suppose (1) has a PELS
u
. Let
v_{k}
be a nonnegative solution of
Then the sequence {
v_{k}
} is decreasing and
u
≤
v_{k}
on |
x
| ≤
k
for all
k
∈
N
. Thus {
v_{k}
} converges to a function
v
, on
R
^{N}
and
u
≤
v
on
R
^{N}
. Since
u
is nonnegative and
u
(
x
) → ∞ as |
x
| → ∞, the function
v
has the same properties. A standard regularity argument can be used to show that the function
v
is a nonnegative entire large solution of (12), which is a contradiction. This completes the proof.
We now establish some important results for the purely sublinear problem (0 <
α
≤
β
≤
m
− 1).
Theorem 3.1.
Suppose m
≥ 2, 0 <
α
≤
β
≤
m
− 1,
h
(
r
) ≡
ϕ
_{1}
(
r
) +
ϕ
_{2}
(
r
) −
ψ
_{1}
(
r
) −
ψ
_{2}
(
r
)
and
Then Eq.(1)has a PELS if condi-tion (8) holds.
Proof
. The proof hinges on an upper and lower solution argument and a key part of the proof is to show that the equation
have PELS for which
v
≤
w
. To show that equation (13) have PELS, we show that there exists a number
b
> 1, such that the integral equations
have positive solutions valid for all
r
≥ 0 with
v
≤
w
. We note that condition (8) means that these entire solutions will be PELS. Let
v
_{1}
= 1, define the sequence
v_{k}
iteratively by
The sequence
v_{k}
is monotonically increasing and
v_{k}
≥ 1 for all
k
≥ 1, so that we have
Next we show that
v_{k}
(
r
) ≤
e^{Mr}
for 0 ≤
r
≤
R, k
≥ 1 by induction, where
When
k
= 1, it is obviously true. Suppose
v_{k}
(
r
) ≤
e^{Mr}
, 0 ≤
r
≤
R
, then
we note that
It follows that
Thus the sequence {
v_{k}
} is increasing and locally bounded and therefore converges on
R
^{N}
. Furthermore, its limit
v
is a solution to (14). A similar argument shows that (15) has a solution
w
. Then the only thing left is to show the existence of
b
> 1 such that
v
≤
w
on
R
^{N}
. In fact, we choose
and show that this work. To do this, we define
and show that
R
_{∞}
= ∞.
Suppose
R
_{∞}
< ∞ and note the definition of h to get
From the Gronwall inequality,
Thus there must exist
R
>
R
_{∞}
such that
v
<
w
on [0,
R
], contradicting the definition of
R
_{∞}
. Therefore we must have
R
_{∞}
= ∞ and hence
v
≤
w
on
R
^{N}
.
Then from theorem 1 of
[11]
, we know there exists an entire solution
u
(
x
) satisfying
w
(
x
) ≤
u
(
x
) ≤
v
(
x
),
x
∈
R
^{N}
. This completes the proof.
Theorem 3.2.
Suppose p and q are nonnegative and locally Hölder continuous in
R
^{N}
and satisfy
(
M_{pq}
).
Then (1) has a nonnegative nontrival entire bounded solution in
R
^{N}
.
Proof
. We define
and consider the equation
Let
z
_{0}
=
c
≥ 0 for
r
≥ 0 and define the sequence
for
k
= 1, 2, · · ·. Radial solutions of (17) will be solutions of (18). It is clear that the sequence {
z_{k}
} is monotonically increasing. We will show that the sequence {
z_{k}
} is uniformly bounded. Indeed, for the case
m
≥ 2, we have
Now we split the domain of [0,∞) into two intervals. Note that
We choose
r_{k}
such that
It is possible that
r_{k}
= 0 or
r_{k}
= ∞. Indeed, if our central value
c
> 1, then
r_{k}
= 0. If 0 ≤
c
< 1, then either
r_{k}
= ∞ with
z_{k}
(
r
) ≤ 1 for all
r
> 0 or
r_{k}
< ∞ with
z_{k}
(
r_{k}
) = 1. Since our goal is to bound {
z_{k}
}, without loss of generality, we assume that
r_{k}
< ∞. With the split domains, we have
Condition (
M_{pq}
) shows there exists
R
> 0, so that
Gronwall’s inequality then gives
The same argument can be used for the case 1 <
m
< 2 and we omit it here.
We have now shown that {
z_{k}
} is uniformly bounded monotonic sequence. Let z be the pointwise limit of the sequence. It is easy to show that the sequence {
z_{k}
} is actually equicontinuous. Hence the convergence is uniform and the limit function u in in
C
^{1}
[0,∞). Let
u
_{1}
=
M
≥
z
≡
u
_{2}
, then
That is,
u
_{1}
and
u
_{2}
are upper and lower solutions of (1). Standard upper/lower solution method then gives a solution of (1) such that
u
_{1}
≥
u
≥
u
_{2}
.
Theorem 3.3.
Suppose
0 <
α
≤
β
≤
m
− 1,
and suppose that p
(
x
) =
p
(|
x
|),
q
(
x
) =
q
(|
x
|) ∈
C
(
R
)
are nonnegative and continuous. Then the equation
has a large positive radial solution if and only if m_{pq} holds for p or q.
Proof
. To prove the necessity, we assume the contrary. That is
We will show that (19) has no large positive radial solution. To do this, suppose (19) does have a positive radial solution
u
(
r
), then it satisfy
for
r
≥
r
_{0}
> 0. Again, using the fact that (20) implies
We can choose
r
_{0}
large enough so that
Hence
So
Therefore
u
cannot be a large solution. This completes the proof of necessity. To proof the sufficiency, we assume that
m_{pq}
is true for
p
or
q
. We show that the equation
has a positive solution v such that
v
(
r
) → ∞ as
r
→ ∞. Again, it suffices to show that for any fixed
c
> 0, the operator defined by
has a fixed point in
C
[0,∞). We observe that
m_{pq}
implies
Hence, any fixed point of (22) is large. We now show that T has a fixed point in
C
[0,∞). To do this, we first establish a fixed point in [0,
R
] for any
R
> 0. We consider successive approximation. Let
u
_{0}
=
c
, and define
u
_{k+1}
=
Tu_{k}
,
k
= 0, 1, 2, · · · Notice that
u_{k}
≥
c,k
= 0, 1, 2, · · ·, and
u′_{k}
≥ 0. It is clear by induction that this sequence is monotonically increasing. We will show it is uniformly bounded on [0,
R
]. Since
u′_{k}
(
r
) ≥ 0, we may split the domains of the relevant functions into two intervals as in the proof of the previous theorem. Choosing
r_{k}
such that
and defining
We begin as we did in Theorem 3.3 and choose
R
>
r
≥
r_{k}
,
Thus, the sequence {
u_{k}
} is uniformly bounded on [0,R]. It is easy to show that {
u_{k}
} is also equicontinuous on this interval. Hence by Arzela-Ascoli theorem, {
u_{k}
} has a uniformly convergent subsequence on [0,
R
]. Assuming then that
u_{kj}
→
u
on [0,
R
], it is clear that
u
∈
C
([0,
R
]) and
Tu
=
u
on [0,
R
]. To prove that
T
has a fixed point in
C
([0,
R
]), we let {
w_{k}
} be defined as follows:
As in the previous proof, it can be shown that {
w_{k}
} is bounded and equicontinuous on [0, 1]. Thus {
w_{k}
} has a subsequence,
which converges uniformly on [0, 1]. Let
Likewise, the subsequence
is bounded and equicontinuous on [0, 2] so that it has a subsequence
which converges uniformly on [0, 2]. Let
Note that
on [0, 1] since
is a subsequence of
Thus
v
_{2}
=
v
_{1}
on [0, 1]. Continuing this line of reasoning, we obtain a sequence {
v_{k}
} with the following properties:
Therefore, it is clear that {
v_{k}
} converges to
v
where
and the convergence is uniform on bounded sets. Hence
v
∈
C
([0,∞)) and satisfy
Next we give the following nonexistence results.
Theorem 3.4.
Suppose p and q are locally Hölder continuous in
R
^{N}
.
If m_{pq} holds for p or q, then (1) has no nonnegative bounded entire solution for
0 <
α
≤
β
≤
m
− 1.
Proof
. Without loss of generality, assume that
m_{pq}
holds for
p
and suppose
u
is a nonnegative entire bounded solution of (1). Consider the equation
Note that
Let
then
Thus, by the upper/lower solution method, there exists a nontrival nonnegative entire bounded solution to (23). But this contradicts Theorem 3.1 of
[9]
. Therefore,
u
must not exist. This completes the proof.
Yuan Zhang received M.Sc. from Nanjing Normal University. Her research interest is quasilinear elliptic equation (system).
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China.
Zuodong Yang is currently a professor at Nanjing Normal University. His research interests are elliptic and parabolic problems.
a. Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China.
b. School of Teacher Education, Nanjing Normal University, Jiangsu Nanjing 210097, China.
e-mail:zdyang_jin@263.net

quasi-linear elliptic equation
;
large solution
;
existence
;
non-existence
;
sublinear
;
superlinear

1. Introduction

We consider the existence and nonexistence of large solutions of the equation
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2. Superlinear/mixed case (0 < α ≤ β, β > m − 1)

First, we consider the following existence results.
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3. Sublinear case (0 < α ≤ β ≤ m − 1)

First, we consider existence results.
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Citing 'LARGE SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATION OF MIXED TYPE†
'

@article{ E1MCA9_2014_v32n5_6_721}
,title={LARGE SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATION OF MIXED TYPE†}
,volume={5_6}
, url={http://dx.doi.org/10.14317/jami.2014.721}, DOI={10.14317/jami.2014.721}
, number= {5_6}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={ZHANG, YUAN
and
YANG, ZUODONG}
, year={2014}
, month={Sep}