Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 9256192, 14 pages
https://doi.org/10.1155/2018/9256192

Research Article

Infinitely Many Solutions for a Class of Fractional Impulsive Coupled Systems with -Laplacian

Junping Xie1,2 and Xingyong Zhang3,4

¹School of Geosciences and Info-Physics, Central South University, Changsha, Hunan 410083, China
²Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring, Central South University, Ministry of Education, Changsha, China
³Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, China
⁴School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Correspondence should be addressed to Xingyong Zhang; zhangxingyong1@163.com

Received 23 January 2018; Accepted 28 March 2018; Published 8 May 2018

Academic Editor: Antonio Iannizzotto

Copyright © 2018 Junping Xie and Xingyong Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By using the symmetric mountain pass lemma, we investigate the problem of existence of infinitely many solutions for a class of fractional impulsive coupled systems with -Laplacian, which possesses mixed type nonlinearities, and the nonlinearities do not need to satisfy the well-known Ambrosetti-Rabinowitz condition.

1. Introduction and Main Results

In this paper, we are concerned with existence of infinitely many solutions for the following fractional impulsive differential system with -Laplacian:

where , with , with , and , and (or ) denotes the right Riemann-Liouville fractional derivative of order (or ), (or ) is the left Caputo fractional derivative of order (or ), , are continuously differentiable, , , and

where

, , and satisfies and the following assumptions.

is measurable in for each , continuously differentiable in for a.e. , and there exist and such that

for all and a.e. .

It is well known that critical point theory is a very important and effective tool to investigate the existence and multiplicity of various solutions for partial differential equations, ordinary differential equations, Hamiltonian systems, difference equations, and so on. Lots of important and interesting results have been established (see, e.g., [1–11] and reference therein). In 2011, Jiao and Zhou [12] first used critical point theory to investigate the existence of solutions for a class of fractional boundary value problems. Since then, critical point theory has also become an effective tool to obtain the existence and multiplicity results of solutions for various fractional differential equations (see, e.g., [13–17] and reference therein). Particularly, in [14], Zhao et al. considered existence of solutions for the following fractional coupled differential system with a parameter:

where is a parameter, , and . By using a critical point theorem in [18], they obtained system (5) which has at least three weak solutions. In [13], Li et al. investigated a class of fractional coupled differential systems with a parameter:

where is a parameter, , and . By using the least action principle and symmetric mountain pass theorem, they obtained system (6) which has at least one solution under asymptotically quadratic case and has infinitely many solutions under superquadratic case. For the superquadratic case, they assumed the following well-known Ambrosetti-Rabinowitz (AR) condition.

There are constants , such that

for all and .

Over the past ten years, integer order impulsive differential equations with different boundary value conditions have been investigated deeply via variational methods (e.g., see [19–26] and reference therein). Recently, Bonanno et al. [27] and Rodrìguez-López and Tersian [28] were concerned with the following second-order impulsive fractional differential equation:

where , and are two parameters, , , and . By using variational methods, they obtained some existence results about one or three solutions of (8). Subsequently, in [29], Nyamoradi and Rodríguez-López investigated the existence and multiplicity of solutions for (8) with . They obtained some existence results about one or infinitely many solutions of (8) by using the least action principle, the mountain pass theorem, and the symmetric mountain pass theorem. In [30],Y. Zhao and Y. Zhao investigated the existence and multiplicity of solutions for a class of perturbed fractional differential system with impulsive effects and one parameter, and they obtained system that has at least one or two nontrivial solutions by using two abstract critical point theorems due to [31]. In [32], Heidarkhani et al. investigated the multiplicity of solutions for a class of perturbed fractional differential system with impulsive effects and two parameters, and they obtained that system has infinitely many solutions by using the smooth version of an abstract critical point theorem due to [33]. In [34], Zhao et al. investigated the existence of solution for (8) with . By using the Morse theory and local linking argument, they obtained that equation has at least one nontrivial solution.

In [35], Zhao and Tang investigated the following impulsive fractional differential equations with -Laplacian:

where and . By using the mountain pass theorem, a critical point theorem in [36], and symmetric mountain pass theorem, they obtained two multiplicity results of solutions for (9). In detail, they obtained the following theorems.

Theorem A (see [35]). Suppose the following conditions hold.
There exists a constant such that for any , , where is defined in Section 2.
There exists a constant such that for all and , where .
There exist constants such that and , where

There exist constants such that for all , .
Then (9) has at least two weak solutions.

Theorem B (see [35]). Suppose – hold and and are odd about , where . Then (9) has infinitely many weak solutions.

Motivated by [12–14, 35], in this paper, we investigate the existence of infinitely many solutions for system (1). Obviously, system (1) is more general and complex than system (5), system (6), and (9). We present some techniques in [35], which were applied to fractional -Laplacian impulsive differential equation and can also be applied to fractional -Laplacian impulsive differential system, and present some more relaxed superquadratic conditions for nonlinearities than those in [35]. It is remarkable that the fractional coupled -Laplacian differential systems are different from the fractional -Laplacian differential equations. One stark difference is that the solutions of system (1) are the combination of and but not of (5), which causes the fact that system (1) number is possibly more than that of (9) and, hence, it is impossible that system (1) reduces to system (9). Moreover, since, in general, and we present more relaxed superquadratic conditions, it is difficult to prove the boundness of Cerami sequence (see the definition in Section 2 below) and we have to develop some techniques on inequalities. When , system (1) becomes the following integer order -Laplacian impulsive differential system:

There have been some results on existence and multiplicity of solutions for integer order -Laplacian impulsive differential systems with different boundary value conditions (see, e.g., [4, 37, 38]). However, system (11) which has Dirichlet boundary value is different from those systems in [4, 37, 38] and our assumptions on are more relaxed than the well-known (AR) condition. Hence, our results are still new for integer order -Laplacian impulsive differential systems. Next, we state our results.

Theorem 1. Suppose that the following conditions hold.
, .
is even in and for a.e. .
There exist constants , , such that

for a.e. and all with .
There exists a positive constant such that

for a.e. and all with .
There exists positive constants such that

for a.e. and all with .
There exist , , such that

for a.e. and all with .

There exist constants and such that

for a.e. and all with .
There exist constants such that

for all with , .
There exist constants such that

for all with , .
There exist constants such that

for all with , .
There exist constants such that

for all with , .
, is even in , .
, is even in , .
Then system (1) has an unbounded sequence of weak solutions.

Theorem 2. Suppose that , , , –, –, , , , , and the following conditions hold.
There exist positive constants with

and such that

for all with , , and such that

for all with , , where

Then system (1) has an unbounded sequence of weak solutions.

It is easy to prove that the following condition implies that and hold.

there are constants , such that

for a.e. and all with .

Indeed, obviously, (AR)′ implies that holds with and . Moreover, by the proof of Theorem 1.2 in [4], (AR)′ and imply that there exist positive constants such that

for all and a.e. , and so it is easy to see that holds. Then by Theorems 1 and 2, we have the following corollaries.

Corollary 3. Suppose that , , , –, , –, , and hold. Then system (1) has an unbounded sequence of weak solutions.

Corollary 4. Assume that , , , –, , , , , , , and hold. Then system (1) has an unbounded sequence of weak solutions.

Remark 5. There exist examples satisfying Theorems 1 and 2. For example, let , , , , , , and

for all and a.e. , where and .

With similar proofs of Theorems 1 and 2, we can obtain the corresponding theorems for the following -Laplacian system:

where for all and a.e. .

Theorem 6. Suppose that the following conditions hold.
is measurable in for each , continuously differential in for a.e. , and there exist and such that

for all and a.e. .
.
is even in and for a.e. .
There exist , , such that

for a.e. and all with .
There exists a positive constant such that

for a.e. and all with .
There exist positive constants such that

for a.e. and all with .
There exist , such that

for a.e. and all with .

There exist constants and such that

for a.e. and all with .
There exist constants such that

for all with , .
There exist constants such that

for all with , .
, is even in , .
Then system (29) has an unbounded sequence of weak solutions.

Theorem 7. Suppose that , , , –, –, , , and the following condition holds.
There exist positive constants with

and such that

for all with , .
Then system (29) has an unbounded sequence of weak solutions.

Corollary 8. Suppose that , , , –, , and – and the following condition holds.
There are constants , such that

for a.e. and all with .
Then system (29) has an unbounded sequence of weak solutions.

Corollary 9. Assume that , , , –, , , , , and hold. Then system (29) has an unbounded sequence of weak solutions.

Remark 10. Corollaries 8 and 9 are still different from Theorem B. Indeed, if , , and for a.e. and all , system (29) reduces to (9). However, it is easy to see that and (or ) are different from . There exist examples satisfying and but not satisfying and . For example, let and for all , . Then . It is easy to see that , satisfy and . Set , . Obviously, , do not satisfy . Moreover, there exist examples satisfying – but not satisfying . For example, let

for a.e. . Finally, one can also establish some results which are similar to Theorem A for system (1) and system (29) by combining those assumptions and arguments of Theorems 1 and 2 with those ideas proving Theorem A.

2. Preliminaries

In this section, we recall some known definitions and lemmas about fractional derivatives. For more details, the readers can see [12, 39–42].

Let and

Definition 11 (see [40, 42]). Let and . and denote the left and right Riemann-Liouville fractional derivatives of order for function , respectively, which are defined by

Definition 12 (see [40, 42]). Let and . and denote the left and right Caputo fractional derivatives of order for function , respectively, which are defined by

Remark 13 (see [40, 42]). When , and .

Let

with the norm , and, for ,

with the norm .

For and , we define as the closure of , with respect to the norm:

Then by Proposition 3.1 in [12], is separable and reflexive Banach space, and if , then , and . Moreover, by Remark 3.1 in [12], , .

Proposition 14 (see [12]). Assume that and . For all ,

where . Moreover, if , then

where and .

By Proposition 14, it is easy to obtain that

Proposition 15 (see [12]). Assume that and , and the sequence converges weakly to in . Then in .

Assume that . Let . On , define the norm:

for all .

Similar to Definitions 2.4 and 2.5 in [35], we also present the following two definitions.

Definition 16. Let

If satisfies the first equation of (1) for a.e. and the second equation of (1) for a.e. , , , , and exist and satisfy the impulsive conditions of (1), and boundary conditions and , then we call a classical solution of (1).

Definition 17. For any , if the following two equalities

hold then the vector function is called a weak solution of (1).

For , we define the functional by

where

It follows from , the continuity of and and Theorem 5.41 in [42], that and are continuously differentiable and so and

Hence, the critical point of is a weak solution of (1). Similar to the arguments of Propositions .5 and .6 in [35], it is easy to obtain that is a classical solution of (1) if is a weak solution of (1).

Assume that is a real Banach space and . For any sequence , if is bounded and as , then we call a Palais-Smale sequence. If any Palais-Smale sequence has a convergent subsequence, then we call which satisfies Palais-Smale condition.

Similar to the proofs in [39], we will also use the following symmetric mountain pass theorem to prove our main results.

Lemma 18 (see [2]). Let be an infinite dimensional Banach space and let be even and satisfy Palais-Smale condition, and . If , where is finite dimensional, and satisfies the following, then possesses an unbounded sequence of critical values.

(i)There are constants such that .(ii)For each finite dimensional subspace , there is such that on .

Remark 19. As shown in [43], a deformation lemma can be proved with replacing Palais-Smale condition with Cerami condition, which implies that Lemma 2.1 in [2] is true under Cerami condition. We say that satisfies Cerami condition; that is, for every sequence , has a convergent subsequence if is bounded and as , where with the norm is the dual space of .

3. Proofs of Theorems

Lemma 20. Assume that , , , , , , and hold. Then satisfies Cerami condition.

Proof. For any sequence , suppose that there is a positive constant such that

By and assumption , there exist positive constants , , and such that

It follows from that

for all with and a.e. . Assumption and (60) imply that there exists a positive constant such that

for all and a.e. .
Assume that . Then, for any with and all ,

and, for any with and all ,

Then, (62) and (63) imply that

for all . Similarly, if , we have

for all . Combining (64) and (65), we have

for all . Moreover, for and , we have

for all , and

for all with . It follows from assumptions (59), (61), (66), (67), and (68) that there exist positive constants such that

for all with and a.e. . By (69) and assumption , there exist positive constants such that

for all and a.e. . By , , and , there exists positive constant such that

for all and a.e. . Moreover, by , , and , there exist positive constants and such that

for all . By (58) and (70), we have

which implies that there exists a positive constant such that . Then and . By (60) and , there exists such that

for all with and a.e. . Let for all and a.e. , , , and . By (58), , , (61), (72), (73), (74), (75), and (51), there exist positive constants and such that

Since , , , , , and , the boundness of and implies that and are bounded. Going if necessary to a subsequence, assume that in and in . Then, by Proposition 15, we obtain and , and so and . Similar to the arguments of Lemma 3.1 in [35], we can obtain that and . Therefore, and as , which shows that as .

Lemma 21. Assume that , , , , and hold. There are constants such that .

Proof. Choosing /, . Then, for all , we have and then, by (51), and, similarly, . Note that and . Hence, it follows from , , , , and (66) that

Since , (77) implies that there exists a sufficiently large constant such that for all with .

Lemma 22. Assume that , , , and hold. There are constants such that .

Proof. By (77) and , it is easy to obtain that

Then (22) and (78) imply that there exists a sufficiently large constant such that for all with .

Lemma 23. For each finite dimensional subspace , there is such that on .

Proof. For each given finite dimensional space , we claim that there exists such that on . Indeed, obviously, for any , can be rewritten by , where , , and and are finite dimensional ones. So there exist positive constants such that

Similar to the proof of Theorem 1.1 in [4], by and assumption , there exists positive constant such that

for all and a.e. . By , there exist

and positive constant such that

for all with and a.e. . Then by (82) and assumption , there exist positive constants and such that

for all and a.e. . By and , there exist positive constants such that

for all and a.e. . It follows from (83), (80), (84), and (79) that

Thus, (81) and the above inequality imply that there exists a sufficiently large constant such that if .

Proof of Theorems 1 and 2. By , , and , we have and is even. Similar to the proof of Lemma 4.4 in [44], we can choose , , and define and . Then . Note that . Thus by Lemma 21 (or Lemma 22) and Lemma 23 it is easy to see that (i) and (ii) of Lemma 18 hold. Lemma 20 implies that satisfies Cerami condition. So, by Lemma 18 and Remark 19, there exists a sequence such that and then it is easy to see that as .

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (no. 11226135 and no. 11301235).

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