- Research Article
- Open Access

# Existence and Asymptotic Behavior of Solutions for Weighted -Laplacian System Multipoint Boundary Value Problems in Half Line

- Zhimei Qiu
^{1}, - Qihu Zhang
^{1, 2}Email author and - Yan Wang
^{1}

**2009**:926518

https://doi.org/10.1155/2009/926518

© Zhimei Qiu et al. 2009

**Received:**5 January 2009**Accepted:**20 June 2009**Published:**20 July 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.

## Keywords

- Boundary Condition
- Growth Condition
- Positive Constant
- Asymptotic Behavior
- Bounded Domain

## 1. Introduction

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.

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 .

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 ;

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 .

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]).

Throughout the paper, we denote .

Lemma 2.2.

The function has the following properties.

has a unique solution .

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

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 .

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.

It is clear that is continuous and sends bounded sets of to bounded sets of , and hence it is a compact continuous mapping.

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 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.

Lemma 2.4.

Proof.

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

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.

with boundary condition (1.2) has no solution on .

has no solution on .

(3^{0})The Brouwer degree
.

Then problems (1.1)-(1.2) have a solution on .

Proof.

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

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 (1^{0}) it follows that (3.7) has no solutions for
. For
(3.3) is equivalent to the problem

Hence

^{0}), 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

Since

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 (3^{0}), this last degree is different from zero. This completes the proof.

where is Caratheodory, is continuous and Caratheodory, and for any fixed if then .

Theorem 3.2.

Assume that the following conditions hold

(1^{0})
for all
and all
where
satisfies

(2^{0})
for
uniformly

has no solution on , where

(4^{0})the Brouwer degree
for large enough
, where

Then problem (3.18) with (1.2) has at least one solution.

Proof.

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

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

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

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):

Since we have and it means that is a solution of

^{0}), (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 (4^{0}), 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.

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.

then

(i)

(ii) as

(iii) as

## 4. Satisfies General Growth Condition

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

(A_{1})
are nonnegative and satisfying
or
;

where and are positive constants.

_{1}), then there exists a positive constant that satisfies

We also assume the following

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 (A_{1})–(A_{4}) 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.

(1^{0}) 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).

- (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

- (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

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 .

(2^{0}) For any
, without loss of generality, we may assume that
and
, then we have

It means that has no solution on .

(3^{0}) 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.

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 (A_{1})–(A_{4}), 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.

## Declarations

### 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).

## Authors’ Affiliations

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