Qihu Zhang,^1,2 Xiaopin Liu,² and Zhimei Qiu²

Abstract

This paper investigates the existence of solutions for weighted p(r)-Laplacian ordinary boundary value problems. Our method is based on Leray-Schauder degree. As an application, we give the existence of weak solutions for p(x)-Laplacian partial differential equations.

1. Introduction

In this paper, we consider the existence of solutions for the following weighted -Laplacian ordinary equation with right-hand terms depending on the first-order derivative:(P) with one of the following boundary value conditions: (1.1)(1.2)(1.3)(1.4)where and ; satisfies and ; is called the weighted -Laplacian; the notation means exists and(1.5)similarly(1.6) where and are continuous and increasing in for any fixed , respectively.

The study of differential equations and variational problems with nonstandard -growth conditions is a new and interesting topic. Many results have been obtained on these kinds of problem, for example, [1–18]. If (a constant), (P) is the well-known -Laplacian problem. Because of the nonhomogeneity of -Laplacian, -Laplacian problems are more complicated than those of -Laplacian, many methods and results for -Laplacian problems are invalid for -Laplacian problems. For example,

(1) if is an open bounded domain, then the Rayleigh quotient(1.7)is zero in general, and only under some special conditions (see [4]), but the fact that is very important in the study of -Laplacian problems. In [19], the author considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system:(S)the first eigenfunction is used to constructing the subsolution of problem (S) successfully. On the -Laplacian problems, maybe -Laplacian does not have the first eigenvalue and the first eigenfunction. Because of the nonhomogeneity of -Laplacian, the first eigenfunction cannot be used to construct the subsolution of -Laplacian problems, even if the first eigenfunction of -Laplacian exists. On the existence of solutions for -Laplacian equations Dirichlet problems via subsuper solution methods, we refer to [13, 14];

(2) if (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 typical -Laplacian problem:(1.8)because of the nonhomogeneity of -Laplacian, when we use critical point theory to deal with the existence of solutions, we usually need the corresponding functional is coercive or satisfy Palais-Smale conditions. If , then the corresponding functional is coercive, if , then the corresponding functional satisfies Palais-Smale conditions (see [3]). But if , one can see that the corresponding functional is neither coercive nor satisfying Palais-Smale conditions, the results on this case are rare.

There are many papers on the existence of solutions for -Laplacian boundary value problems via subsuper solution method (see [20–24]). But results on the sub-super-solution method for -Laplacian equations and systems are rare. In this paper, when is a general function, we establish several sub-super-solution theorems for the existence of solutions for weighted -Laplacian equation with Dirichlet, Robin, and Periodic boundary value conditions. Moreover, the case of is discussed. Our results partially generalize the results of [13, 14, 20, 25].

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(1.9)

We set , is continuous in (), and exist. Denote and The spaces and will be equipped with the norm and , respectively.

We say a function is a solution of (P), if and is absolutely continuous and satisfies (P) almost every on .

Functions are called subsolution and supersolution of (P), if and are absolutely continuous and satisfy(1.10)

Throughout this paper, we assume that are subsolution and supersolution, respectively. Denote(1.11)

We also assume that

() , for all , where () are positive value and continuous on , () are positive value and continuous on .

() There exist positive numbers and such that , for , where is increasing and satisfies , where .

Our main results are as the following theorem.

Theorem 1.1. If is Caratheodory and satisfies () and (), and satisfy , , then (P) with (1.1) possesses a solution.

Theorem 1.2. If is Caratheodory and satisfies () and (), and satisfy and (1.12) then (P) with (1.2) possesses a solution.

Theorem 1.3. If is Caratheodory and satisfies () and (), and satisfy (1.13) then (P) with (1.3) possesses a solution.

Theorem 1.4. If is Caratheodory and satisfies () and (), and satisfy (1.14) then (P) with (1.4) possesses a solution.

As an application, we consider the existence of weak solutions for the following -Laplacian partial differential equation:(1.15)where is a bounded symmetric domain in , is radially symmetric. We will write , and satisfies , is radially symmetric with respect to , namely, , and satisfies the Caratheodory condition.

2. Preliminary

Denote , . Obviously, has the following properties.

Lemma 2.1. is a continuous function and satisfies (i) for any , is strictly increasing; (ii) is a homeomorphism from to for any fixed .

For any fixed , denote as(2.1)

It is clear that is continuous and send bounded sets into bounded sets.

Let us now consider the simple problem(2.2)with boundary value condition (1.1), where . If is a solution of (2.2) with (1.1), by integrating (2.2) from to , we find that(2.3)

Denote(2.4)then(2.5)

The boundary conditions imply that(2.6)

For fixed , we denote(2.7)

We have the following lemma.

Lemma 2.2. The function has the following properties. For any fixed , the equation (2.8) has a unique solution .

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

Proof. (i) Obviously, for any fixed , is continuous and strictly increasing, then, if (2.8) has a solution, it is unique.

Since and , it is easy to see that(2.9)

It means the existence of solutions of .

In this way, we define a function , which satisfies(2.10)

(ii) We claim that(2.11)

If it is false. Without loss of generality, we may assume that there are some such that(2.12)then(2.13)

It is a contradiction. Thus, (2.11) is valid. It mens that sends bounded sets to bounded sets.

Finally, to show the continuity of , let be a convergent sequence in and , as . Obviously, is a bounded sequence, then it contains a convergent subsequence . Let as . Since(2.14)letting , we have(2.15)from (i), we get , it means is continuous.

This completes the proof.

Now, we define is defined by(2.16)

It is clear that is a continuous function which send bounded sets of into bounded sets of , and hence it is a complete continuous mapping.

We continue now with our argument previous to Lemma 2.2. By solving for in (2.3) and integrating, we find(2.17)

Let us define(2.18)

We denote by , the Nemytsky operator associated to defined by(2.19)

It is easy to see the following lemma.

Lemma 2.3. is a solution of (P) with boundary value condition (1.1) if and only if is a solution of the following abstract equation: (2.20)

Lemma 2.4. The operator is continuous and sends equi-integrable sets in into relatively compact sets in .

Proof. It is easy to check that . Since , and(2.21)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(2.22)

We want to show that is a compact set.

Let be a sequence in , then there exist a sequence such that . For , we have that(2.23)

Hence, the sequence is uniformly bounded and equicontinuous, then there exists a subsequence of which is convergent in , and we name the same. Since the operator is bounded and continuous, we can choose a subsequence of (which we still denote ) that is convergent in , then(2.24)is convergent in . Since(2.25)according to the continuous of and the integrability of in , then is convergent in . Then, we can conclude that convergent in .

Lemma 2.5. Let be subsolution and supersolution of (P), respectively, which satisfies for any , then there exists a positive constant such that, for any solution of (P) with (1.1) which satisfies one has .

Proof. We denote(2.26)then there exists a such that(2.27)

From (), there exist positive numbers and such that(2.28)

Assume that our conclusion is not true, combining (2.27), then there exists such that keeps the same sign on and(2.29)or inversely(2.30)

For simplicity, we assume that the former appears. Hence,(2.31)which is impossible. The proof is completed.

Let us consider the auxiliary SBVP of the form(2.32)where(2.33)where(2.34)where is defined in Lemma 2.5, and(2.35)where .

Lemma 2.6. Let the conditions of Lemma 2.5 hold, and let be any solution of SBVP with (1.1) satisfies and , then for any .

Proof. We will only prove that for any . The argument of the case of for any is similar.

Assume that for some , then there exist a and a positive number such that , for any . Hence,(2.36)

There exists a positive number such that for any . From the definition of and we conclude that(2.37)where is small enough. For any , we have(2.38)

From (2.36) and (2.38), we have(2.39)it means that(2.40)

It is a contradiction to the definition of , so for any .

3. Proofs of Main Results

In this section, we will deal with the proofs of main results.

Proof of Theorem 1.1. From Lemmas 2.5 and 2.6, we only need to prove the existence of solutions for SBVP with (1.1). Obviously, is a solution of SBVP with (1.1) if and only if is a solution of(3.1)

We set(3.2)

Obviously, sends into equi-integrable sets in . Similar to the proof of Lemma 2.4, we can conclude that sends equi-integrable sets in into relatively compact sets in , then is compact continuous.

Obviously, for any , we have , and is bounded. By virtue of Schauder fixed point theorem, has at least one fixed point in . Then, is a solution of SBVP with (1.1). This completes the proof.

Proof of Theorem 1.2. Let with be fixed. According to Theorem 1.1, (P) with the following boundary value condition:(3.3)possesses a solution such that(3.4)

Since exists, we have(3.5)

Similarly,(3.6)

Obviously(3.7)then, we can conclude that(3.8)

Since , and is increasing in , we have(3.9)

We may assume that , or we get a solution for (P) with (1.2).

Since is a solution of (P), it is also a subsolution of (P). Similarly, (P) with boundary value condition(3.10)possesses a solution such that(3.11)which satisfies(3.12)then(3.13)

Obviously, and are subsolution and supersolution of (P) with (1.2), respectively. According to Theorem 1.1, (P) with boundary value condition(3.14)possesses a solution such that(3.15)

We may assume that , or we get a solution for (P) with (1.2).

If , then denote and ; if , then denote and . It is easy to see that and both are solutions of (P) and satisfy(3.16)

Repeated the step, we get two sequences and , all are solutions of (P), and satisfy(3.17)(3.18)(3.19)(3.20)

According to Lemma 2.5, and both are bounded in , then is a bounded set and has a convergent subsequence. Note that are solutions of (P) and satisfy(3.21)where(3.22)

Similar to the proof of Lemma 2.4, possesses a convergent subsequence in , and then is bounded. From [2], we can see that and have uniform regularity. We may assume that in and in .

It is easy to see that both are solutions of (P). From the definition of and , we can see that(3.23)

Combining (3.18) and (3.20), we have(3.24)

Similar to (3.7), we have(3.25)

From (3.17) and the continuity of , we can see that(3.26)

From (3.25), (3.26), and the increasing property of with respect to , we have(3.27)

Thus, and both are solutions of (P) with (1.2). This completes the proof.

Proof of Theorem 1.3. According to Theorem 1.2, (P) possesses a solution such that(3.28)

Similar to the proof of (3.7), we have(3.29)

Obviously, . We may assume that(3.30)or we get a solution for (P) with (1.3), then is a subsolution of (P) with (1.3).

According to Theorem 1.2, (P) possesses a solution such that(3.31)

Similarly, . We may assume that(3.32)or we get a solution for (P) with (1.3), then is a supersolution of (P) with (1.3).

According to Theorem 1.2, (P) possesses a solution such that(3.33)

We may assume that , or we get a solution for (P) with (1.3). If , then denote and , if , then denote and . It is easy to see that and both are solutions of (P) and satisfy(3.34)

Repeating the step, similar to the proof of Theorem 1.2, we get two sequences and , all are solutions of (P), and satisfy(3.35)

Similar to the proof of Theorem 1.2, and possess convergent subsequence and in , respectively. We may assume that in , and similar in . It is easy to see that both are solutions of (P) with (1.3). This completes the proof.

Proof of Theorem 1.4. According to Theorem 1.1, (P) possesses solution which satisfies(3.36)

We may assume that , or we get a solution for (P) with (1.4), then , and is a subsolution of (P). According to Theorem 1.1, (P) possesses solutions which satisfies(3.37)

We may assume that , or we get a solution for (P) with (1.4), then , and is a supersolution of (P). According to Theorem 1.1, (P) possesses solutions and satisfies(3.38)

Similar to the proof of Theorem 1.2, we obtain and that are solutions of (P), which satisfy(3.39)(3.40)(3.41)(3.42)

From (3.39) and (3.40), we have(3.43)

From (3.41), (3.42), and (3.43), we can conclude that (P) with (1.4) possesses a solution. This completes the proof.

On the case of , we consider(I)where , , is a positive constant. Denote(3.44)

We have the following corollary.

Corollary 3.1. If is even, satisfies (3.45) then (I) possesses at least a nontrivial solution.

Proof. It is easy to see that is a subsolution of (I). Denote(3.46)where is a positive constant satisfying . Since is even, then . It is easy to see that , and(3.47)where . Then, is a supersolution of (I). From Theorem 1.1, one can see that (I) possesses at least a nontrivial solution.

4. Applications in PDE

Let be an open bounded domain. In this section, we always denote(4.1)

Let us now consider (1.15) with one of the following boundary value conditions:(4.2)(4.3)

If is a radial solution of (1.15), then it can be transformed into(4.4)and the boundary value condition will be transformed into (1.1), (1.2), or (1.3), respectively.

Theorem 4.1. If (4.4) has subsolution and supersolution and respectively, satisfying for any , and is continuous and satisfies ()-(), in each of the following cases: (i) , , , and ; (ii) , , and ; , ; (iii) , , and ; , ; then (1.15) with (4.2) has at least one weak radially symmetric solution .

Proof. Notice that and satisfies We can conclude the existence of solutions for (4.4) with (1.1), (1.2), or (1.3), from Theorems 1.1, 1.2, and 1.3. If , notice that(4.5)then we have . This completes the proof.

Similarly, we have the following theorem.

Theorem 4.2. If (4.4) has subsolution and supersolution and respectively, satisfying for any , and (4.6) and is continuous and satisfies ()-(), in each of the following cases: (i) ; ; (ii) ; or ; and then (1.15) with (4.3) has at least one weak radially symmetric solution .

On the case of , we consider(II)where , , , is a positive constant.

We have the following corollary.

Corollary 4.3. If is radial, and satisfies (4.7) then (II) possesses at least a nontrivial solution.

Proof. It is easy to see that is a subsolution of (II). Denote(4.8)where is a positive constant satisfying . It is easy to see that , and(4.9)where . Then, is a supersolution of (II). From Theorem 4.1, one can see that (II) possesses at least a nontrivial solution.

Acknowledgments

This work is partly supported by the National Science Foundation of China (10701066 and 10671084), China Postdoctoral Science Foundation (20070421107), and the Natural Science Foundation of Henan Education Committee (2007110037).

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