# Existence of solutions and nonnegative solutions for a class of p ( t ) -Laplacian differential systems with multipoint and integral boundary value conditions

Guizhen Zhi1, Yunrui Guo2, Yan Wang1 and Qihu Zhang1*

Author Affiliations

1 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan, 450002, China

2 Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan, 453003, China

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Boundary Value Problems 2013, 2013:106  doi:10.1186/1687-2770-2013-106

 Received: 12 November 2011 Accepted: 12 April 2013 Published: 26 April 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper explores the existence of solutions for a class of -Laplacian differential systems with multipoint and integral boundary value conditions via Leray-Schauder’s degree. Moreover, the existence of nonnegative solutions is discussed.

MSC: 34B10.

##### Keywords:
-Laplacian; Leray-Schauder degree; fixed point

### 1 Introduction

In this paper, we consider the existence of solutions for the following system:

where , (); is called -Laplacian; , ; , () and , ; , they are both nonnegative, , ; ; and are nonnegative constants; and are positive parameters.

The study of differential equations and variational problems with variable exponent growth conditions has attracted more and more attention in recent years. Many results have been obtained on these problems, for example, [1-16]. We refer to [3,12,16] for the applied background of these problems. If (a constant), becomes the well-known p-Laplacian. If is a general function, represents a non-homogeneity and possesses more nonlinearity, thus is more complicated than (see [7]).

In recent years, because of the wide mathematical and physical background (see [17-19]), the existence of positive solutions for the p-Laplacian equation group has received extensive attention. Especially, when , the existence of positive solutions for the equation group boundary value problems has been obtained (see [20-25]). On the integral boundary value problems, we refer to [26-30]. But as for the -Laplacian equation group, there are few papers dealing with the existence of solutions, especially the existence of solutions for the systems with multipoint and integral boundary value problems. Therefore, when is a general function, this paper mainly investigates the existence of solutions for a class of -Laplacian differential systems with multipoint and integral boundary value conditions. Moreover, we discuss the existence of nonnegative solutions.

Let and , the function , () is assumed to be Carathéodory, by which we mean:

(i) For almost every , the function is continuous;

(ii) For each , the function is measurable on J;

(iii) For each , there are 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 . For , we set , ; . For any , we denote , and . For any , we denote . Spaces C, and W will be equipped with the norm , and , respectively. Then , and are Banach spaces. Denote with the norm .

We say a function is a solution of (P) if satisfies the differential equation in (P) a.e. on J and the boundary value conditions.

In this paper, we always use to denote positive constants if this does not lead to confusion. Denote

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

where , and . We say satisfies a general growth condition if does not satisfy a sub- growth condition.

We will discuss the existence of solutions for (P) in the following two cases:

(i) satisfies a sub- growth condition for ;

(ii) satisfies a general growth condition for .

This paper is organized as follows. In Section 2, we do some preparation. In Section 3, we discuss the existence of solutions of (P). Finally, in Section 4, we discuss the existence of nonnegative solutions for (P).

### 2 Preliminary

For any , denote (). Obviously, has the following properties.

Lemma 2.1 (see [5])

is a continuous function and satisfies the following:

(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:

(1)

with the boundary value condition

(2)

where . If u is a solution of (1) with (2), by integrating (1) from 0 to t, we find that

(3)

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

From (3), we have

(4)

By integrating (4) from 0 to t, we find that

From (2), we have

and

For fixed , we define as

(5)

It is easy to obtain the following lemma.

Lemma 2.2is continuous and sends bounded sets ofto bounded sets of. Moreover,

(6)

It is clear that is a compact continuous mapping.

Let us now consider another problem

(7)

with the boundary value condition

(8)

where . Similar to the discussion of the solutions of (1) with (2), we have

and

where , for any .

From , we have

(9)

From , we have

(10)

From (9) and (10), we have

For fixed , we denote

Lemma 2.3The functionhas the following properties:

(i) For any fixed, the equation

(11)

has a unique solution.

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

Proof (i) It is easy to see that

From Lemma 2.1, it is immediate that

and hence, if (11) has a solution, then it is unique.

Let . Suppose . Since , it is easy to see that there exists an such that the ith component of satisfies

Thus keeps sign on J and

Obviously, , then

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

Let us consider the equation

(12)

It is easy to see that all the solutions of (12) belong to . So, we have

which implies the existence of solutions of .

In this way, we define a function , which satisfies

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

It only remains to prove the continuity of . Let be a convergent sequence in C 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. The proof is completed. □

Now, we define the operator as

(13)

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

If u is a solution of (1) with (2), we have

and

If u is a solution of (7) with (8), we have

and

We denote

Lemma 2.4The operators () are continuous and send equi-integrable sets into relatively compact sets in.

Proof We only prove that the operator is continuous and sends equi-integrable sets in to relatively compact sets in , the rest is similar.

It is easy to check that for all . Since

it is easy to check that is a continuous operator from to .

Let now U 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

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

From the definition of and the continuity of , we can see that is convergent in C. Thus, is convergent in . This completes the proof. □

Let us define as

It is easy to see that and are both compact continuous.

We denote () the Nemytskii operator associated to defined by

Lemma 2.5is a solution of (P) if and only ifis a solution of the following abstract equation:

Proof If is a solution to (P), according to the proof before Lemma 2.5, it is easy to obtain that is a solution to (S).

Conversely, if is a solution to (S), then

which implies

It follows from (S) that

then

By the condition of the mapping , we have

and then

From (S), we have

and

then

Thus

From (S), we have

By the condition of the mapping , we have

which implies that

Since , then we have

Moreover, from (S), it is easy to obtain

and

Hence is a solution of (P).

This completes the proof. □

### 3 Existence of solutions

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for (P), when satisfies a sub- growth condition or a general growth condition ().

We denote (S) as

where

Theorem 3.1Ifsatisfies a sub-growth condition, then the problem (P) has at least one solution for any fixed parameter ().

Proof Denote

where

According to Lemma 2.5, we know that (P) has the same solution of

(14)

when .

It is easy to see that the operators and are compact continuous. According to Lemma 2.2, Lemma 2.3 and Lemma 2.4, we can see that and are compact continuous from to , thus is compact continuous from to W.

We claim that all the solutions of (14) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (14) such that as n.

From Lemma 2.2, we have

which together with the sub- growth condition of implies that

(15)

From (14), we have

then

Denote . From the above inequality we have

(16)

It follows from (14) and (15) that

For any , we have

which implies that

Thus

(17)

It follows from (16) and (17) that .

Similarly, we have , where .

Thus, is bounded.

Thus, we can choose a large enough such that all the solutions of (14) belong to . Therefore, the Leray-Schauder degree is well defined for each , and

Denote

(18)

where and are defined in (5) and (13), then 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 system:

(19)

Obviously, (19) possesses a unique solution . Note that , we have

Therefore (P) has at least one solution. This completes the proof. □

In the following, we investigate the existence of solutions for (P) when satisfies a general growth condition.

Denote

Assume the following.

(A1) Let a positive constant ε be such that , , and , , where is defined in (18), and are defined in (5) and (13), respectively.

It is easy to see that is an open bounded domain in W. We have the following theorem.

Theorem 3.2Assume that (A1) is satisfied. If positive parametersandare small enough, then the problem (P) has at least one solution on.

Proof Similarly, we denote . By Lemma 2.5, is a solution of

with (2) and (8) if and only if is a solution of the following abstract equation:

(20)

From the proof of Theorem 3.1, we can see that is compact continuous from to W. According to Leray-Schauder’s degree theory, we only need to prove that

(1) has no solution on for any ,

(2) ,

then we can conclude that the system (P) has a solution on .

(1) If there exists a and is a solution of (20), then and λ satisfy

and

Since , there exists an i such that or .

(i) If .

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

This implies that for each .

Note that , then , holding . Since is continuous, when is small enough, from (A1), we have

It is a contradiction to for each .

() Suppose that , then . This implies that for some , and we can find

(21)

Since and is Carathéodory, it is easy to see that

thus

From Lemma 2.2, is continuous, then we have

When is small enough, from (A1) and (21), we can conclude that

It is a contradiction. Thus .

(ii) If . Similar to the proof of (i), we get a contradiction. Thus .

Summarizing this argument, for each , has no solution on when positive parameters and are small enough.

(2) Since (where is defined in (18)) is the unique solution of , and (A1) holds , we can see that the Leray-Schauder degree

This completes the proof. □

As applications of Theorem 3.2, we have the following.

Corollary 3.3Assume that, where; satisfy, . Ifandare small enough, then the problem (P) possesses at least one solution.

Proof It is easy to have

From and the definition of , we have

Since , then there exists a small enough ε such that

From Lemma 2.2 and the small enough , we have

then is valid.

Similarly, we have .

Obviously, it follows from , and the small enough that , , and .

Thus, the conditions of (A1) are satisfied, then the problem (P) possesses at least one solution. □

Corollary 3.4Assume that, where; satisfy. If, andare small enough, then the problem (P) possesses at least one solution.

Proof From Lemma 2.2, we have

Since is dependent on the small enough and , then it follows from the continuity of that is small enough, which implies that

Similarly, we have .

From , and the small enough and , it is easy to have that , , and .

Thus, the conditions of (A1) are satisfied, then the problem (P) possesses at least one solution. □

We denote

Assume the following.

(A2) Let positive constants and be such that , , and , , where is defined in (18), and are defined in (5) and (13), respectively.

It is easy to see that is an open bounded domain in W. We have the following.

Corollary 3.5Assume that

whereϵ, are positive constants; satisfy, and, . Then the problem (P) possesses at least one solution.

Proof Similar to the proof of Theorem 3.2, we only need to prove that (A2) is satisfied, then we can conclude that the problem (P) possesses at least one solution.

From and the definition of , it is easy to have that

where we suppose . Since , then there exists a big enough such that

From Lemma 2.2, we have

then is valid.

From and the definition of , we have

Since , then there exists a such that (where ), which implies that

From Lemma 2.3, we have

then is valid.

Obviously, it follows from and that , , and .

Thus, the conditions of (A2) are satisfied, then the problem (P) possesses at least one solution. □

Corollary 3.6Assume that

whereϵ, are positive constants; satisfy, , and, . Ifandare small enough, then the problem (P) possesses at least one solution.

Proof Similar to the proof of Corollary 3.5, we conclude that (A2) is satisfied. Then the problem (P) possesses at least one solution. □

### 4 Existence of nonnegative solutions

In the following, we deal with the existence of nonnegative solutions of (P). For any , the notation () means () for any . For any , the notation means , the notation means .

Theorem 4.1We assume that

(10) , ;

(20) , ;

(30) ;

(40) .

Then every solution of (P) is nonnegative.

Proof (i) We shall show that is nonnegative.

If is a solution of (P), from Lemma 2.5, we have

which together with (5), (10) and (40) implies that

Thus for any . Holding is decreasing, namely for any with .

According to the boundary value condition (2) and condition (30), we have

then

Thus is nonnegative.

(ii) We shall show that is nonnegative.

If is a solution of (P), From Lemma 2.5, we have

We claim that . If it is false, then there exists some such that , which together with condition (20) implies that

(22)

Similar to the proof of Lemma 2.3, the boundary value condition (8) implies

(23)

From (22) and , we get a contradiction to (23).

Thus .

We claim that

(24)

If it is false, then there exists some such that

which together with condition (20) implies

(25)

From (25) and , we get a contradiction to (23). Thus (24) is valid.

Denote

Obviously, , , and is decreasing, i.e., for any with . For any , there exist such that

We can conclude that is increasing on , and is decreasing on . Thus

For any fixed , if

which together with (8) implies that

(26)

From , we have

(27)

It follows from (26) and (27) that

If

(28)

from (8) and (28), we have

(29)

Since , we have

(30)

Combining (29) and (30), we have

Thus is nonnegative.

Combining (i) and (ii), we find that every solution of (P) is nonnegative. □

Corollary 4.2We assume that

(10) , with;

(20) , with;

(30) ;

(40) .

Then we have

(a) Under the conditions of Theorem 3.1, (P) has at least one nonnegative solution;

(b) Under the conditions of Theorem 3.2, (P) has at least one nonnegative solution.

Proof (a) Define

where

Denote

where , then satisfies the Carathéodory condition, and .

We assume the following.

(A2) for uniformly, where and .

Obviously, satisfies a sub- growth condition.

Let us consider the existence of solutions of the following system:

(31)

According to Theorem 3.1, (31) has at least a solution . From Theorem 4.1, we can see that is nonnegative. Thus, is a nonnegative solution of (P).

(b) It is similar to the proof of (a).

This completes the proof. □

### 5 Examples

Example 5.1 Consider the following problem:

where , , .

Obviously, and are Caratheodory, , , , then the conditions of Theorem 3.1 are satisfied, then () has a solution.

Example 5.2 Consider the following problem

where , , , .

Obviously, and are Caratheodory, , , , the conditions of Corollary 4.2 are satisfied, then () has a nonnegative solution.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors typed, read and approved the final manuscript.

### Acknowledgements

Partly supported by the National Science Foundation of China (10701066 & 10971087).

### References

1. Acerbi, E, Mingione, G: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal.. 156, 121–140 (2001). Publisher Full Text

2. Antontsev, S, Shmarev, S: Anisotropic parabolic equations with variable nonlinearity. Publ. Mat.. 53, 355–399 (2009)

3. Chen, Y, Levine, S, Rao, M: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math.. 66(4), 1383–1406 (2006). Publisher Full Text

4. Diening, L: Maximal function on generalized Lebesgue spaces . Math. Inequal. Appl.. 7(2), 245–253 (2004)

5. Fan, XL, Zhang, QH: Existence of solutions for -Laplacian Dirichlet problem. Nonlinear Anal.. 52, 1843–1852 (2003). Publisher Full Text

6. Fan, XL, Wu, HQ, Wang, FZ: Hartman-type results for -Laplacian systems. Nonlinear Anal.. 52, 585–594 (2003). Publisher Full Text

7. Fan, XL, Zhang, QH, Zhao, D: Eigenvalues of -Laplacian Dirichlet problem. J. Math. Anal. Appl.. 302, 306–317 (2005). Publisher Full Text

8. Fu, YQ: The principle of concentration compactness in spaces and its application. Nonlinear Anal.. 71(5-6), 1876–1892 (2009). Publisher Full Text

9. El Hamidi, A: Existence results to elliptic systems with nonstandard growth conditions. J. Math. Anal. Appl.. 300, 30–42 (2004). Publisher Full Text

10. Hudzik, H: On generalized Orlicz-Sobolev space. Funct. Approx. Comment. Math.. 4, 37–51 (1976)

11. Kováčik, O, Rákosník, J: On spaces and . Czechoslov. Math. J.. 41, 592–618 (1991)

12. Růžička, M: Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin (2000)

13. Zhang, QH: Existence of positive solutions for elliptic systems with nonstandard -growth conditions via sub-supersolution method. Nonlinear Anal.. 67(4), 1055–1067 (2007). Publisher Full Text

14. Zhang, QH, Qiu, ZM, Liu, XP: Existence of solutions for a class of weighted -Laplacian system multi-point boundary value problems. J. Inequal. Appl.. 2008, (2008) Article ID 791762

15. Zhang, QH: Boundary blow-up solutions to -Laplacian equations with exponential nonlinearities. J. Inequal. Appl.. 2008, (2008) Article ID 279306

16. Zhikov, VV: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv.. 29, 33–36 (1987). Publisher Full Text

17. Bandle, CV, Kwong, MK: Semilinear elliptic problems in annular domains. Z. Angew. Math. Phys.. 40, 245–257 (1989). Publisher Full Text

18. Deimling, K: Nonlinear Functional Analysis, Springer, Berlin (1980)

19. Wang, HY: On the existence of positive solutions for semilinear elliptic equations in the annulus. J. Differ. Equ.. 109, 1–7 (1994). Publisher Full Text

20. Hu, L, Wang, LL: Multiple positive solutions of boundary value problems for systems of nonlinear second-order differential equations. J. Math. Anal. Appl.. 335, 1052–1060 (2007). Publisher Full Text

21. Liu, LS, Liu, BM, Wu, YH: Positive solutions of singular boundary value problems for nonlinear differential systems. Appl. Math. Comput.. 186, 1163–1172 (2007). Publisher Full Text

22. Liu, WW, Liu, LS, Wu, YH: Positive solutions of a singular boundary value problem for systems of second-order differential equations. Appl. Math. Comput.. 208, 511–519 (2009). Publisher Full Text

23. Wei, ZL, Zhang, MC: Positive solutions of singular sub-linear boundary value problems for fourth-order and second-order differential equation systems. Appl. Math. Comput.. 197, 135–148 (2008). Publisher Full Text

24. Yang, ZL: Positive solutions to a system of second-order nonlocal boundary value problems. Nonlinear Anal.. 62, 1251–1265 (2005). Publisher Full Text

25. Zhou, YM, Xu, Y: Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations. J. Math. Anal. Appl.. 320, 578–590 (2006). Publisher Full Text

26. Feng, MQ, Ji, DH, Ge, WG: Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math.. 222, 351–363 (2008). Publisher Full Text

27. Feng, MQ: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl. Math. Lett.. 24, 1419–1427 (2011). Publisher Full Text

28. Feng, MQ, Zhang, XM, Ge, WG: New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl.. 2011, (2011) Article ID 720702

29. Ma, HL: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Anal.. 68, 645–651 (2008). Publisher Full Text

30. Zhang, XM, Ge, WG: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput.. 219, 3553–3564 (2012). Publisher Full Text