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

In this paper, we consider the existence and asymptotic behavior of solutions for the following weighted -Laplacian system:

(1.1)

(1.2)

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

(1.3)

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;

(1.4)

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

(1.5)

Throughout the paper, we denote

(1.6)

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

(1.7)

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

(1.8)

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.

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,

(2.1)

(ii)There exists a function as , such that

(2.2)

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

(2.3)

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):

(2.4)

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

(2.5)

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

(2.6)

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

(2.7)

The boundary condition (1.2) implies that

(2.8)

For fixed , we denote

(2.9)

Throughout the paper, we denote .

Lemma 2.2.

The function has the following properties.

(i)For any fixed , the equation

(2.10)

has a unique solution .

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

(2.11)

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

(2.13)

then

(2.14)

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

(2.15)

Let us consider the equation

(2.16)

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

(2.17)

and it means the existence of solutions of .

In this way, we define a function , which satisfies

(2.18)

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

(2.20)

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

(2.21)

Let us define

(2.22)

where and satisfies , and we denote as

(2.23)

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

(2.24)

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

(2.25)

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

(2.26)

Hence the sequence is equicontinuous.

From the definition of we have Thus

(2.27)

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

(2.28)

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

(2.29)

Lemma 2.4.

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

(2.30)

Proof.

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

(2.31)

From (2.31), we have

(2.32)

From , we have

(2.33)

So we have

(2.34)

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

(2.35)

Thus and By the definition of the mapping we have

(2.36)

thus

(2.37)

From (2.30), we have

(2.38)

Obviously from (2.38), we have

(2.39)

Since we have and

(2.40)

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.

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

(3..1)

with boundary condition (1.2) has no solution on .

(2⁰)The equation

(3..2)

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):

(3..3)

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

(3..4)

It means that (3.1) with (1.2) and (3.3) with (1.2) have the same solutions for

We denote defined by

(3..5)

where is defined by (2.29). Let

(3..6)

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

(3..7)

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

(3..8)

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

(3..9)

Hence

(3..10)

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

(3..11)

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

(3..12)

Now it is clear that the problem

(3..13)

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

(3..14)

Since

(3..15)

then

(3..16)

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

(3..17)

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)

(3..18)

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

(3..19)

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

(3..20)

At first, we consider the following problem:

(3..21)

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

(3..22)

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

(3..23)

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

(3..24)

then we have

(3..25)

Denote , then

(3..26)

Thus

(3..27)

Since , from (3.27) we have

(3..28)

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

(3..29)

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

(3..30)

and then from (3.27) we have

(3..31)

Since (as ), and , we have

(3..32)

From (3.28)–(3.32), we have

(3..33)

So we get

(3..34)

where satisfies

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

(3..35)

Since , according to the continuity of we have

(3..36)

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

(3..37)

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)

(3..38)

where

We denote defined by

(3..39)

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

(3..40)

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):

(3..41)

then we have

(3..42)

Since we have and it means that is a solution of

(3..43)

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

(3..44)

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

(3..45)

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

(3..46)

then has only one solution , and

(3..47)

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

(3..48)

then

(i)

(ii) as

(iii) as

In this section, under the condition that satisfies

(4.1)

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

(4.2)

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

(4.3)

We also assume the following

(A₃) satisfies

(4.4)

(A₄) satisfies

(4.5)

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

(4.6)

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

(4.7)

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 .

1. (i)

   Suppose that , then . Since , there exists such that . For any , we have

    

(4.8)

This implies that for each . Since , keeps sign. Since keeps sign, also keeps sign.

Assume that is positive, then

(4.9)

It is a contradiction to (1.2).

Assume that is negative, then

(4.10)

This implies that for some . Since , it is easy to see that

(4.11)

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

(4.12)

then

(4.13)

Since , combining (4.11), we have

(4.14)

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

(4.15)

It means that has no solution on .

(3⁰) Let

(4.16)

Denote

(4.17)

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

(4.18)

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

(4.19)

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.