Qihu Zhang,^1,2,3 Zheimei Qiu,² and Xiaopin Liu²

Abstract

This paper investigates the existence of solutions for weighted p(t)-Laplacian system multipoint boundary value problems. When the nonlinearity term f(t,⋅,⋅) satisfies sub-p−−1 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 of solutions for the following weighted -Laplacian system:(1.1)with the following multipoint boundary value condition:(1.2)where and , is called the weighted -Laplacian; satisfies , for all , and ; , ; , , and , ; , ; 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 problems, for example, [1–14]. We refer to [2, 15, 16] 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, thus is more complicated than . We have the following examples. (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 , if then the corresponding functional is coercive; if then the corresponding functional satisfies Palais-Smale condition (see [4, 7, 12]). If we can see that the corresponding functional is neither coercive nor satisfying Palais-Smale conditions, 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 [17–20]). On the existence of solutions for -Laplacian systems boundary value problems, we refer to [5, 7, 10, 11]. But results on the existence 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 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 the following:(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 For , we set , , and exist. For any we denote and Spaces and will be equipped with the norm and , respectively. Then and are Banach spaces.

We say a function is a solution of (1.1) if with absolutely continuous on which satisfies (1.1) a.e. 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 that satisfies general growth condition, if we do not 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 of solutions of (1.1)-(1.2), when satisfies sub- growth condition. Finally, in Section 4, we will discuss the existence 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 the following: (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 to bounded sets. Let us now consider the following problem with boundary value condition (1.2):(2.4)where . 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 . By solving for in (2.5) and integrating, we find(2.6)

From , we have(2.7)

From , we obtain(2.8)

From (2.7) and (2.8), we have (2.9)

For fixed , we denote(2.10)

Throughout the paper, we denote

Lemma 2.2. The function has the following properties: (i) for any fixed , the equation (2.11) has a unique solution (ii) the function , defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, (2.12) where the notation means (2.13)

Proof. (i) It is easy to see that(2.14)

From Lemma 2.1, it is immediate that(2.15)and hence, if (2.11) has a solution, then it is unique.

Let(2.16)

If since and it is easy to see that there exists an such that the th component of satisfies(2.17)

Thus keeps sign on and(2.18)then(2.19)

Thus, when is large enough, the th component of is nonzero, then we have(2.20)

Let us consider the equation (2.21)

It is easy to see that all the solutions of (2.21) belong to So, we have(2.22)it means the existence of solutions of .

In this way, we define a function , which satisfies(2.23)

(ii) By the proof of (i), we also obtain that sends bounded sets to bounded sets, and(2.24)

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 , it means that is continuous. This completes the proof.

Now, we define as(2.25)

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

If is a solution of (2.4) with (1.2), then(2.26)

The boundary condition (1.2) implies that(2.27)

We denote that(2.28)

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(2.29)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(2.30)

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(2.31)

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

Since(2.32)according to the continuity of and the integrability of in , we can see that is convergent in . Thus is convergent in . This completes the proof.

Let us define as(2.33)

It is easy to see that is compact continuous.

We denote the Nemytski operator associated to defined by(2.34)

Lemma 2.4. is a solution of (1.1)-(1.2) if and only if is a solution of the following abstract equation: (2.35)

Proof. If is a solution of (1.1)-(1.2), by integrating (1.1) from to , we find that(2.36)

From (2.36), we have(2.37) then we have(2.38)

So we have(2.39)

Conversely, if is a solution of (2.35), it is easy to see that(2.40)

By the condition of the mapping ,(2.41)then we have(2.42)thus(2.43) from (2.40) and (2.43), we obtain (1.2).

From (2.35), we have(2.44)

Hence is a solution 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 a such that (2.45)

Proof. For any if there exists such that then (2.45) is valid. If it is false, then is strictly monotone.

(i) If is strictly decreasing in , then(2.46)

Thus(2.47) it means that(2.48)then there exists a such that(2.49)

(ii) If is strictly increasing in , then(2.50)

Thus(2.51) it means that(2.52)then there exists a such that(2.53)

Combining (2.49) and (2.53), then we obtain (2.45).

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.

Theorem 3.1. If satisfies sub- growth condition, then for any fixed parameter , problem (1.1)-(1.2) has at least one solution.

Proof. Denote where is defined in (2.34). We know that (1.1)-(1.2) has the same solution of(3.1)when

It is easy to see that the operator is compact continuous. According to Lemmas 2.2 and 2.3, then we can see that is compact continuous from to for any .

We claim that all the solutions of (3.1) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (3.1) such that as , and for any

Let such that(3.2)

For any fixed there exists an such that(3.3)

Thus, becomes a sequence with respect to .

Since are solutions of (3.1), according to Lemma 2.5, for any there exists such that , then(3.4)

For any , we have(3.5)

Without loss of generality, we assume that .

() If , then(3.6)where is defined in (2.45).

Combining (3.2), (3.3), (3.5), and (3.6), we have(3.7)

Then we have(3.8)

Denoting , we have(3.9)

Thus(3.10)

() If , since we have(3.11)

According to (3.2), (3.3), (3.5), and (3.11), we have(3.12)

Since is a positive constant, (3.12) means that(3.13)

Thus(3.14)

Summarizing this argument, we have(3.15)

Since , then we have (3.16)

Thus(3.17)

Combining (2.38) and (3.17), we have(3.18)

For any , since(3.19)we have(3.20)

Thus(3.21)

Combining (3.15) and (3.21), then we obtain that is bounded.

Thus, there exists a large enough such that all the solutions of (3.1) belong to then the Leray-Schauder degree is well defined for , and(3.22)

Let(3.23)where is defined in (2.25), thus is the unique solution of

It is easy to see that is a solution of if and only if is a solution of the following:(3.24)

Obviously, system possesses only one solution . Since thus the Leray-Schauder degree(3.25)therefore, we obtain that (1.1)-(1.2) has at least one solution. This completes the proof.

4. Satisfies General Growth Condition

In the following, we will deal with the existence of solutions for -Laplacian ordinary system, when satisfies general growth condition.

Denote(4.1)

Assumption 4.1. Let positive constant satisfy , , and , where is defined in (3.23), is defined in (2.25).

It is easy to see that is an open bounded domain in .

Theorem 4.2. Assume that Assumption 4.1 is satisfied. If positive parameter is small enough, then the problem (1.1)-(1.2) has at least one solution on .

Proof. Denote . According to Lemma 2.4, is a solution of(4.2)with (1.2) if and only if is a solution of the following abstract equation:(4.3)

From Lemmas 2.2 and 2.3, then we can see that is compact continuous from to for any . According to Leray-Schauder degree theory, we only need to prove that() has no solution on for any ,().

Then we can conclude that the system (1.1)-(1.2) has a solution on .

() If it is false, then there exists a and is a solution of (4.2) with (1.2). Thus satisfies(4.4)

Since , then there exists an such that .

(i) Suppose that , then . On the other hand, for any , , we have(4.5)

This implies that for each .

Notice that , then , holding . Since is continuous, when is small enough, from Assumption 4.1, we have(4.6)

It is a contradiction to for each .

(ii) Suppose that , then . This implies that for some . Since , it is easy to see that(4.7)

Combining (4.4) and (4.7), we have(4.8)

Since and is Caratheodory, it is easy to see that , thus . According to Lemma 2.2, is continuous, we have(4.9)

Thus, when is small enough, from Assumption 4.1, we can conclude that(4.10)

It is a contradiction. Summarizing this argument, for each , the problem (4.2) with (1.2) has no solution on when positive parameter is small enough.

() According to Assumption 4.1, , (where is defined in (3.23)), thus is the unique solution of , then the Leray-Schauder degree(4.11)

This completes the proof.

Similar to the proof of Theorem 4.2, we have Theorem 4.3.

Theorem 4.3. Assume , where satisfy , . If and are small enough, then the problem (1.1)-(1.2) possesses at least one solution.

On the typical case, we have Corollary 4.4.

Corollary 4.4. Assume that , where satisfy , . On the conditions of Theorem 4.2, the problem (1.1)-(1.2) possesses at least one solution.

Acknowledgments

This work is supported by the National Science Foundation of China (10701066 and 10671084), 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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