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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 LeraySchauder 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 wellknown 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 PalaisSmale condition, (see [4, 7]). If there are more difficulties to testify that the corresponding functional is coercive or satisfying PalaisSmale conditions, and the results on this case are rare.
There are many results on the existence of solutions for Laplacian equation with multipoint 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 multipoint 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 multipoint 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 equiintegrable 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 equiintegrable 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 AscoliArzela 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 LeraySchauder'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.
(1^{0})For each the problem
with boundary condition (1.2) has no solution on .
(2^{0})The equation
has no solution on .
(3^{0})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 equiintegrable 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
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 (2^{0}), implies that Thus we have proved that (3.7) has no solution on then we get that for each , the LeraySchauder 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 LeraySchauder 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.
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
(1^{0}) for all and all where satisfies
(2^{0}) for uniformly
(3^{0})for large enough , the equation
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.
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 (3^{0}), (3.38) has no solutions on then we get that for each , the LeraySchauder 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 LeraySchauder 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.
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 LeraySchauder'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 ;
(A_{2}); ; keeps sign on , and satisfies
where and are positive constants.
For any , without loss of generality, we may denote . Denote . According to (A_{1}), then there exists a positive constant that satisfies
We also assume the following
(A_{3}) satisfies
(A_{4}) 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 (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).
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 .
(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.
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 (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.
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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 (200875565), 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