Nontrivial Solutions for Time Fractional Nonlinear Schrödinger-Kirchhoff Type Equations

Abstract

We study the existence of solutions for time fractional Schrödinger-Kirchhoff type equation involving left and right Liouville-Weyl fractional derivatives via variational methods.

1. Introduction

In recent years, there has been a great interest in studying problems involving fractional Schrödinger equations [1–5], Kirchhoff type equations [6–8], fractional Navier-Stokes equations [9, 10], and fractional ordinary differential equations and Hamiltonian systems [11–17], and so forth. For further details and applications, we refer the reader to [18, 19] and the references cited therein.

On the other hand, the integer-order Schrödinger-Kirchhoff type equations have also been investigated by many authors; for example, see [20–23]. In fact, Schrödinger-Kirchhoff type equations play an important role in modelling several physical and biological systems. However, to the best of our knowledge, the existence of solutions to the time fractional Schrödinger-Kirchhoff type equations has yet to be addressed.

The objective of the present paper is to study time fractional Schrödinger-Kirchhoff type equation of the formwhere , and , respectively, denote left and right Liouville-Weyl fractional derivatives of order on , are constants, is parameter, , , and is a potential function.

The rest of the paper is organized as follows. Section 2 contains preliminary concepts of fractional calculus and fractional Sobolev space, while some important lemmas, which are needed in the proof of main results, are obtained in Section 3. We present our main results in Section 4.

2. Preliminaries

In this section, we recall important definitions and concepts of fractional calculus and then prove certain results about fractional Sobolev space related to our study of the problem at hand.

Definition 1 (see [24]). The left and right Liouville-Weyl fractional integrals of order on are defined byrespectively, where .
The left and right Liouville-Weyl fractional derivatives of order on are defined byrespectively, where .
The definitions (3) may be written in an alternative form as follows:

Also, we define the Fourier transform of as

For any , we define the seminorm and norm, respectively, as [16]and let the space denote the completion of with respect to the norm .

Next, for , we give the relationship between classical fractional Sobolev space and , where is defined by with the normand seminorm Observe that the spaces and are equal and have equivalent norms (see [16]).

Therefore, we define

Let The space is a reflexive and separable Hilbert space with the inner productand the corresponding norm Define the space with the norm

Lemma 2. is a uniformly convex Banach space.

Proof. is obviously Banach space. Now, we can prove that is uniformly convex. To this end, let and with and . Using the following inequality: we getwhich implies that . Hence, taking such that , we have . Therefore, is uniformly convex.

In the sequel, we need the following assumptions., ;there exists such that, for any , there exists such that ; and there exist constants and such that as uniformly in ;there exist such that as uniformly in ; for all ; and there exists such that there exist , ( is defined in Remark 6), , and small constants such that

Lemma 3. Assume that (V1) holds. Then the embeddings are continuous. In particular, there exists a constant such thatMoreover, if (V1) and (V2) hold, then the embedding is compact.

Proof. Clearly, the chain of embeddings is continuous and consequently one can obtain (23). Also in view of (V1), (V2), and following the method of proof similar to that of Lemma  2.2 in [15], the embedding is compact.

Lemma 4. Let . Then and there exists a constant such that

Proof. The proof is similar to that of Theorem  2.1 in [16], so we omit it.

Also by Lemma 4, there is a constant such that

Remark 5. If with , then it follows by Lemma 4 that for all as

Remark 6. From Remark 5 and Lemma 3, it is easy to verify that the imbedding of in is also compact for . Hence, for all , the imbedding of in is continuous and compact, which together with Lemma 4 implies that there exists such that

Lemma 7. Assume that (V1) and (V3) hold. Then the embedding is continuous and compact for .

Proof. By (V3) and Hölder’s inequality, we have for some positive constant . So Lemma 4 implies that for some positive constant . Hence, by Remark 6, we can get continuous embeddings into for . Now, we will show that the embedding is compact for . Let such that and such that . In view of (V3), given , for large enough, one can obtain Then, On the other hand, by Sobolev’s theorem (see, e.g., [25]) which implies that uniformly on , there is such that for all . Thus in . So, for , we have and consequently, in for .

Definition 8. Let be a Banach space, . One says that satisfies the Palais-Smale (PS) condition if any sequence for which is bounded and as possesses a convergent subsequence.

In order to establish the main results, we need the following known Theorems.

Theorem 9 (see [26, Theorem  2.2]). Let be a real Banach space and satisfies condition. Suppose and (i)there are constants such that ;(ii)there is an such that .Then possesses a critical value . Moreover can be characterized as where

Theorem 10 (see [26, Theorem  9.12]). Let be an infinite dimensional Banach space and let be even, satisfying (PS) condition, and . If , where is finite dimensional and satisfies the following conditions:there exist constants such that ;for any finite dimensional subspace , there is such that on ,then possesses an unbounded sequence of critical values.

3. Some Lemmas

Recall that is said to be a weak solution of problem (1) ifand the energy functional is given by the formulawhere

In view of assumptions (V1) and (F1), the functional is of class and by similar method in Theorem  4.1 in [27] and the definition of Gâteaux derivative, one can get

Lemma 11. Assume that (V) and (F1)–(F3) hold. Then satisfies the (PS) condition.

Proof. Let be a sequence such that is bounded and as . Then there exits such that and . So, by (F3), (23), and the fact that , we get Hence, is bounded in .
So, passing onto subsequence if necessary, thanks to Lemma 3, we haveWe will prove that Let be fixed and denote by the linear functional on defined by and set for all . In view of the Hölder inequality and definition of , we have Since in and as in , therefore as . Now, using (F1) and Hölder inequality, we obtain which, in view of (39), yieldsSince a.e. in , it follows by Fatou’s lemma thatNoting that is a nondecreasing function for , we get Now, in view of as , (46), and (47), one hasThen, from (48)-(49), we get Hence, we obtain . As is a reflexive Banach space (see Lemma 2), it is isomorphic to a locally uniformly convex space. So the weak convergence and norm convergence imply strong convergence. This completes the proof.

Let be a total orthonormal basis of and define , ,

Lemma 12. Assume that (V1) holds. Then, for ,

Proof. The proof is similar to that of Lemma  3.8 in [28]. So it is omitted.

In view of Lemma 12, we can choose an integer such thatwhere is a constant given in condition (F1). Let and set and . Hence and are subspaces of , and .

Lemma 13. Suppose that (V1), (V2), and (F1) are satisfied. Then there exist constants such that .

Proof. In view of (V2), (53), and definition of the space , we haveTherefore, from (23), (55), and (F1) and for large enough value of , we get Since , there exist constants such that .

Lemma 14. Assume that (F1) and (F4) are satisfied. Then, for any finite dimensional subspace , there is such that on .

Proof. Since all the norms in the finite dimensional space are equivalent, there exists a constant such that From (F1) and (F4), for any , there exists a constant such that Thus for all . Consequently, there is a large such that on . Therefore, the proof is completed.

4. Existence of Weak Solutions

In this section, we present our main results.

Theorem 15. Assume that (V1), (V2), (F1), (F3), (F4), and (F5) hold. Then problem (1) has infinitely many nontrivial weak solutions whenever is sufficiently large.

Proof. We know that , and it is even by (F5). Let and and be as defined in Section 2. By Lemmas 11, 13, and 14, it follows that satisfies all the condition of the Theorem 10. Therefore, problem (1) has infinitely many nontrivial weak solutions whenever is sufficiently large.

Theorem 16. Assume that (V1), (V2), (F1), (F2), (F3), and (F4) hold. Then problem (1) has at least one nontrivial weak solution when .

Proof. We complete the proof in three steps.
Step  1. Clearly and satisfies the (PS) condition by Lemma 11.
Step  2. It will be shown that there exist constants such that satisfies condition (i) of Theorem 9. For any , by (F1) and (F2), there exists a constant such thatThus, by (23) and (60), for small , we get for all , where . So it suffices to choose so that Step  3. It remains to prove that there exists an such that and , where is defined in Step  2. Let us consider for all . Take . By (F1) and (F4), for any , there is a constant such that So we have as . Thus, there is a point such that . By Theorem 9, possesses a critical value given by where Hence there is such that and ; that is, problem (1) has a nontrivial weak solution in .

Theorem 17. Assume that (V1), (V3), (F5), (F6), and (F7) hold. Then problem (1) has infinitely many nontrivial weak solutions for .

Proof. One can obtain the proof by employing the method of proof for Theorem 15 and using Lemma 7.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project was supported by National Natural Science Foundation of China (11671339).