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Existence and Asymptotic Behavior of Solutions for Weighted
-Laplacian System Multipoint Boundary Value Problems in Half Line
Journal of Inequalities and Applications volume 2009, Article number: 926518 (2009)
Abstract
This paper investigates the existence and asymptotic behavior of solutions for weighted -Laplacian system multipoint boundary value problems in half line. When the nonlinearity term
satisfies sub-(
) growth condition or general growth condition, we give the existence of solutions via Leray-Schauder degree.
1. Introduction
In this paper, we consider the existence and asymptotic behavior of solutions for the following weighted -Laplacian system:


where exists and
,
is called the weighted
-Laplacian;
satisfies
and
; the equivalent
means that
and
both exist and equal;
is a positive parameter.
The study of differential equations and variational problems with variable exponent growth conditions is a new and interesting topic. Many results have been obtained on these kinds of problems, for example, [1–15]. We refer to [2, 16, 17], the applied background on these problems. If and
(a constant),
is the well-known
-Laplacian. If
is a general function,
represents a nonhomogeneity and possesses more nonlinearity, and thus
is more complicated than
. For example, We have the following.
(1)If is a bounded domain, the Rayleigh quotient

is zero in general, and only under some special conditions (see [6]), but the fact that
is very important in the study of
-Laplacian problems;
(2)If and
(a constant) and
, then
is concave; this property is used extensively in the study of one dimensional
-Laplacian problems, but it is invalid for
. It is another difference on
and
.
(3)On the existence of solutions of the following typical problem;

because of the nonhomogeneity of , and if
then the corresponding functional is coercive, if
then the corresponding functional can satisfy Palais-Smale condition, (see [4, 7]). If
there are more difficulties to testify that the corresponding functional is coercive or satisfying Palais-Smale conditions, and the results on this case are rare.
There are many results on the existence of solutions for -Laplacian equation with multi-point boundary value conditions (see [18–21]). On the existence of solutions for
-Laplacian systems boundary value problems, we refer to [5, 7, 10–15]. But results on the existence and asymptotic behavior of solutions for weighted
-Laplacian systems with multi-point boundary value conditions are rare. In this paper, when
is a general function, we investigate the existence and asymptotic behavior of solutions for weighted
-Laplacian systems with multi-point boundary value conditions. Moreover, the case of
has been discussed.
Let and
,
; the function
is assumed to be Caratheodory, by this we mean that
(i)for almost every , the function
is continuous;
(ii)for each , the function
is measurable on
;
(iii)for each there is a
such that, for almost every
and every
with
,
, one has

Throughout the paper, we denote

The inner product in will be denoted by
will denote the absolute value and the Euclidean norm on
. Let
denote the space of absolutely continuous functions on the interval
. For
we set
,
. For any
, we denote
,
and
. Spaces
and
will be equipped with the norm
and
, respectively. Then
and
are Banach spaces. Denote
the norm
We say a function is a solution of (1.1) if
with
absolutely continuous on (
,
), which satisfies (1.1) almost every on
.
In this paper, we always use to denote positive constants, if it cannot lead to confusion. Denote

We say satisfies sub-(
) growth condition, if
satisfies

where , and
. We say
satisfies general growth condition, if we don't know whether
satisfies sub-(
) growth condition or not.
We will discuss the existence of solutions of (1.1)-(1.2) in the following two cases
(i) satisfies sub-(
) growth condition;
(ii) satisfies general growth condition.
This paper is divided into four sections. In the second section, we will do some preparation. In the third section, we will discuss the existence and asymptotic behavior of solutions of (1.1)-(1.2), when satisfies sub-(
) growth condition. Finally, in the fourth section, we will discuss the existence and asymptotic behavior of solutions of (1.1)-(1.2), when
satisfies general growth condition.
2. Preliminary
For any , denote
. Obviously,
has the following properties.
Lemma 2.1 (see [4]).

is a continuous function and satisfies
(i)For any ,
is strictly monotone, that is,

(ii)There exists a function as
, such that

It is well known that is a homeomorphism from
to
for any fixed
. For any
, denote by
the inverse operator of
, then

It is clear that is continuous and sends bounded sets into bounded sets. Let us now consider the following problem with boundary value condition (1.2):

where and satisfies
. If
is a solution of (2.4) with (1.2), by integrating (2.4) from
to
, we find that

Denote . It is easy to see that
is dependent on
. Define operator
as

By solving for in (2.5) and integrating, we find that

The boundary condition (1.2) implies that

For fixed , we denote

Throughout the paper, we denote .
Lemma 2.2.
The function has the following properties.
(i)For any fixed , the equation

has a unique solution .
(ii)The function , defined in
, is continuous and sends bounded sets to bounded sets. Moreover

Proof.
-
(i)
From Lemma 2.1, it is immediate that
(2.12)
and hence, if (2.10) has a solution, then it is unique.
Let . If
, since
and
, it is easy to see that there exists an
such that the
th component
of
satisfies
. Thus
keeps sign on
and

then

Thus the th component
of
is nonzero and keeps sign, and then we have

Let us consider the equation

It is easy to see that all the solutions of (2.16) belong to So, we have

and it means the existence of solutions of .
In this way, we define a function , which satisfies

-
(ii)
By the proof of (i), we also obtain
sends bounded sets to bounded sets, and
(2.19)
It only remains to prove the continuity of . Let
be a convergent sequence in
and
as
. Since
is a bounded sequence, then it contains a convergent subsequence
. Let
as
. Since
, letting
, we have
. From (i), we get
, and it means that
is continuous. This completes the proof.
Now, we define the operator as

It is clear that is continuous and sends bounded sets of
to bounded sets of
, and hence it is a compact continuous mapping.
If is a solution of (2.4) with (1.2), then

Let us define

where and satisfies
,
and we denote
as

Lemma 2.3.
The operator is continuous and sends equi-integrable sets in
to relatively compact sets in
.
Proof.
It is easy to check that . Since
and

it is easy to check that is a continuous operator from
to
.
Let now be an equi-integrable set in
, then there exists
, such that

We want to show that is a compact set.
Let be a sequence in
, then there exists a sequence
such that
. For any
we have that

Hence the sequence is equicontinuous.
From the definition of we have
Thus

Thus is uniformly bounded.
By Ascoli-Arzela theorem, there exists a subsequence of (which we rename the same) being convergent in
. According to the bounded continuous of the operator
, we can choose a subsequence of
(which we still denote
is convergent in
, then
is convergent in
.
Since

from the continuity of and the integrability of
in
, we can see that
is convergent in
. Thus that
is convergent in
.
This completes the proof.
We denote by the Nemytski operator associated to
defined by

Lemma 2.4.

is a solution of (1.1)-(1.2) if and only if is a solution of the following abstract equation:

Proof.
If is a solution of (1.1)-(1.2), by integrating (1.1) from
to
, we find that

From (2.31), we have

From , we have

So we have

Conversely, if is a solution of (2.30), then

Thus and
By the definition of the mapping
we have

thus

From (2.30), we have

Obviously from (2.38), we have

Since we have
and

Hence is a solutions of (1.1)-(1.2). This completes the proof.
Lemma 2.5.
If is a solution of (1.1)-(1.2), then for any
, there exists an
such that
.
Proof.
If it is false, then is strictly monotone in
.
(i)If is strictly decreasing in
, then
; it is a contradiction to
(ii)If is strictly increasing in
, then
; it is a contradiction to
This completes the proof.
3.
Satisfies Sub-(
) Growth Condition
In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)-(1.2), when satisfies sub-(
) growth condition. Moreover, the asymptotic behavior has been discussed.
Theorem 3.1.
Assume that is an open bounded set in
such that the following conditions hold.
(10)For each the problem

with boundary condition (1.2) has no solution on .
(20)The equation

has no solution on .
(30)The Brouwer degree .
Then problems (1.1)-(1.2) have a solution on .
Proof.
Let us consider the following equation with boundary value condition (1.2):

For any observe that if
is a solution to (3.1) with (1.2) or
is a solution to (3.3) with (1.2), we have necessarily

It means that (3.1) with (1.2) and (3.3) with (1.2) have the same solutions for
We denote defined by

where is defined by (2.29). Let

and the fixed point of is a solution for (3.3) with (1.2). Also problem (3.3) with (1.2) can be written in the equivalent form

Since is Caratheodory, it is easy to see that
is continuous and sends bounded sets into equi-integrable sets. It is easy to see that
is compact continuous. According to Lemmas 2.2 and 2.3, we can conclude that
is continuous and compact from
to
for any
. We assume that for
, (3.7) does not have a solution on
; otherwise we complete the proof. Now from hypothesis (10) it follows that (3.7) has no solutions for
. For
(3.3) is equivalent to the problem

and if is a solution to this problem, we must have

Hence

where is a constant. From Lemma 2.5, there exist
such that
,
Hence
, it holds
, a constant. Thus by (3.9)

which together with hypothesis (20), implies that Thus we have proved that (3.7) has no solution
on
then we get that for each
, the Leray-Schauder degree
is well defined for
, and from the properties of that degree, we have

Now it is clear that the problem

is equivalent to problem (1.1)-(1.2), and (3.12) tells us that problem (3.13) will have a solution if we can show that

Since

then

From Lemma 2.2, we have . By the properties of the Leray-Schauder degree, we have

where the function is defined in (3.2) and
denotes the Brouwer degree. By hypothesis (30), this last degree is different from zero. This completes the proof.
Our next theorem is a consequence of Theorem 3.1. As an application of Theorem 3.1, let us consider the following equation with (1.2)

where is Caratheodory,
is continuous and Caratheodory, and for any fixed
if
then
.
Theorem 3.2.
Assume that the following conditions hold
(10) for all
and all
where
satisfies
(20) for
uniformly
(30)for large enough , the equation

has no solution on , where
(40)the Brouwer degree for large enough
, where
Then problem (3.18) with (1.2) has at least one solution.
Proof.
Denote

At first, we consider the following problem:

According to the proof of Theorem 3.1, we know that (3.21) with (1.2) has the same solution of

where
We claim that all the solutions of (3.21) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions
for (3.21) with (1.2) such that
as
, and
for any
.
Since are solutions of (3.21) with (1.2), so
. According to Lemma 2.5, there exist
such that
, then

where means the function which is uniformly convergent to 0 (as
). According to the property of
and (3.23), then there exists a positive constant
such that

then we have

Denote , then

Thus

Since , from (3.27) we have

Denote , then
and
, then
possesses a convergent subsequence (which denoted by
), and then there exists a vector
such that

Without loss of generality, we assume that . Since
, there exist
such that

and then from (3.27) we have

Since (as
),
and
, we have

From (3.28)–(3.32), we have

So we get

where satisfies
Since from(1.2) and (3.34), we have

Since , according to the continuity of
we have

and it is a contradiction to (3.35). This implies that there exists a big enough such that all the solutions of (3.21) with (1.2) belong to
, and then we have

If we prove that , then we obtain the existence of solutions (3.18) with (1.2).
Now we consider the following equation with: (1.2)

where
We denote defined by

Similar to the proof of Theorem 3.1, we know that (3.38) with (1.2) has the same solution of

Similar to the discussions of the above, for any all the solutions of (3.38) with (1.2) are uniformly bounded.
If is a solution of the following equation with (1.2):

then we have

Since we have
and it means that
is a solution of

according to hypothesis (30), (3.38) has no solutions on
then we get that for each
, the Leray-Schauder degree
is well defined, and from the properties of that degree, we have

Now it is clear that So
If we prove that
, then we obtain the existence of solutions (3.18) with (1.2). By the properties of the Leray-Schauder degree, we have

By hypothesis (40), this last degree is different from zero. We obtain that (3.18) with (1.2) has at least one solution. This completes the proof.
Corollary 3.3.
If is Caratheodory, which satisfies the conditions of Theorem 3.2,
where
are positive functions, and satisfies
then (3.18) with (1.2) has at least one solution.
Proof.
Since

then has only one solution
, and

and according to Theorem 3.2, we get that (3.18) with (1.2) has at least a solution. This completes the proof.
Now let us consider the boundary asymptotic behavior of solutions of system (1.1)-(1.2).
Theorem 3.4.
If is a solution of (1.1)-(1.2) which is given in Theorem 3.2, then
(i)
(ii) as
(iii) as
Proof.
Since exists and
,
and
both exist and equal, we can conclude that
. Since
we have
Thus
(i)
(ii)as
(iii)as
This completes the proof.
Corollary 3.5.
Assume that exists,
, and

then
(i)
(ii) as
(iii) as
4.
Satisfies General Growth Condition
In this section, under the condition that satisfies

where are nonnegative,
,
and
almost every in
we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1) with boundary value problems. Moreover the asymptotic behavior has been discussed.
Throughout the paper, assume that
(A1) are nonnegative and satisfying
or
;
(A2);
;
keeps sign on
, and satisfies

where and
are positive constants.
For any , without loss of generality, we may denote
. Denote
. According to (A1), then there exists a positive constant
that satisfies

We also assume the following
(A3) satisfies

(A4) satisfies

Note.
Let , and (A
)-(A
) are satisfied. If
and
are positive small enough, then it is easy to see that (A
)-(A
) are satisfied.
Denote

It is easy to see that is an open bounded domain in
.
Theorem 4.1.
If satisfies (4.1), and (A1)–(A4) are satisfied, then the system (1.1)-(1.2) has a solution on
.
Proof.
We only need to prove that the conditions of Theorem 3.1 are satisfied.
(10) We only need to prove that for each the problem

with boundary condition (1.2) has no solution on .
If it is false, then there exists a and
is a solution of (4.7) with (1.2).
Since , there exists an
such that
.
-
(i)
Suppose that
, then
. Since
, there exists
such that
. For any
, we have

This implies that for each
. Since
,
keeps sign. Since
keeps sign,
also keeps sign.
Assume that is positive, then

It is a contradiction to (1.2).
Assume that is negative, then

It is a contradiction to (1.2).
-
(ii)
Suppose that
, then
.
This implies that for some
. Since
, it is easy to see that

According to the boundary value condition, there exists a such that

then

Since , combining (4.11), we have

It is a contradiction.
Summarizing this argument, for each the problem (4.7) with (1.2) has no solution on
.
(20) For any , without loss of generality, we may assume that
and
, then we have

It means that has no solution on
.
(30) Let

Denote

According to (A), it is easy to see that, for any
,
does not have solution on
, then the Brouwer degree

This completes the proof.
Theorem 4.2.
If is a solution of (1.1)-(1.2) which is given in Theorem 4.1, then
(i)
(ii) as
(iii) as
Proof.
Since exists and
and
both exist and equal, we have
Thus
(i)
(ii) as
(iii) as
We completes the proof.
Corollary 4.3.
Assume that exists,
, and

then
(i)
(ii) as
(iii) as
.
Similar to the proof of Theorem 4.1, we have the following.
Theorem 4.4.
Assume that , where
satisfy
. On the conditions of (A1)–(A4), if
, then problem (1.1)-(1.2) possesses at least one solution.
On the typical case, we have the following.
Corollary 4.5.
Assume that , where
satisfy
. On the conditions of Theorem 4.1, then problem (1.1)-(1.2) possesses at least one solution.
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Acknowlegments
This work is partly supported by the National Science Foundation of China (10701066 and 10671084) and China Postdoctoral Science Foundation (20070421107), the Natural Science Foundation of Henan Education Committee (2008-755-65), and the Natural Science Foundation of Jiangsu Education Committee (08KJD110007).
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Qiu, Z., Zhang, Q. & Wang, Y. Existence and Asymptotic Behavior of Solutions for Weighted -Laplacian System Multipoint Boundary Value Problems in Half Line.
J Inequal Appl 2009, 926518 (2009). https://doi.org/10.1155/2009/926518
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DOI: https://doi.org/10.1155/2009/926518