Abstract

In this paper, we show the existence of solutions for an indefinite fractional Schrödinger equation driven by the variable-order fractional magnetic Laplace operator involving variable exponents and steep potential. By using the decomposition of the Nehari manifold and variational method, we obtain the existence results of nontrivial solutions to the equation under suitable conditions.

1. Introduction

In this paper, we investigate the existence of solutions of the following concave-convex fractional elliptic equation driven by the variable-order fractional magnetic Laplace operator involving variable exponents: where , is a continuous function, is the variable-order fractional magnetic Laplace operator, the potential with , is a parameter, and the magnetic field is with and . In [1], the fractional magnetic Laplacian has been defined as for . In [2], the variable-order fractional Laplace is defined as for each , along any . Inspired by them, we define the variable-order fractional magnetic Laplacian as for each ,

Since is a function, magnetic field with , we see that operator is a variable-order fractional magnetic Laplace operator. Especially, when reduces to the usual fractional magnetic Laplace operator. When reduces to the usual fractional Laplace operator. Very recently, when and authors in [2] are given some sufficient conditions to ensure the existence of two different weak solutions, and used the variational method and the mountain pass theorem to obtain the two weak solutions of problem (5) which converge to two solutions of its limit problems, and the existence of infinitely many solutions to its limit problem:

In addition, authors studied the multiplicity and concentration of solutions for a Hamiltonian system driven by the fractional Laplace operator with variable-order derivative in [3]. For , , , and , in [4], authors obtained the multiplicity and concentration of the positive solution of the following indefinite semilinear elliptic equations involving concave-convex nonlinearities by the variational method:

For , , , and , in [5], under appropriate assumptions, Peng et al. obtained the existence, multiplicity, and concentration of nontrivial solutions for the following indefinite fractional elliptic equation by using the Nehari manifold decomposition:

In [1], by using the Nehari manifold decomposition, authors studied the concave-convex elliptic equation involving the fractional order nonlinear Schrödinger equation:

Some sufficient conditions for the existence of nontrivial solutions of equation (8) are obtained. Nevertheless, only a few papers see [612] deal with the existence and multiplicity of fractional magnetic problems. Some papers see [8, 1316] deal with the solvability of Kirchhoff problems. Inspired by above, we are interested in the existence and multiplicity of solutions to problem (1) with variable growth and steep potential in . As far as we know, this is the first time to study the multiplicity of nontrivial solutions of the indefinite fractional elliptic equation driven by the variable-order fractional magnetic Laplace operator with variable exponents and steep potential in . This result was improved in the recent paper [1].

It is worth mentioning that in this paper, we not only obtain the existence and multiplicity results of nontrivial solutions of the variable-order fractional magnetic Schrodinger equation with variable growth and steep well potential in but also our main results are based on the study for the decomposition of Nehari manifolds. On the one hand, relative to [1], we extend the exponent to variable exponent, thus introducing the variable exponent Lebesgue space. In addition, compared with [2], we extend the range of to and the research range from the bounded region to the whole space . On the other hand, if we want to find the nontrivial solution of the equation (1) by the variational method, we need some geometry, such as a mountain structure and a link structure. However, the energy functional of equation (1) does not have the mountain structure. In order to overcome this obstacle, we seek another method, the Nehari manifold. By decomposing the Nehari manifold into three parts, we obtain the existence of nontrivial solutions of each part.

Inspired by the above works, we assume

() There exist two constants such that for all .

() is symmetric, that is, for all .

() is a continuous function on and .

() There exists such that the set is a nonempty and has finite measure. In addition, , where is the Lebesgue measure and is the best Sobolev constant (see Lemma 9).

() is nonempty and has a smooth boundary with .

() There exists a constant such that for all , where is the Hilbert space related to the magnetic field (see Section 2).

() where

To the best of our knowledge, this type of hypothesis is the first introduced by Bartsch and Wang in [17]. In addition, we recall the potential satisfied the conditions as the steep well potential.

Concerning and , we suppose

() A measurable function satisfy

() A measurable function satisfy

() and , where will be given in Section 2.

() and (13)

In what follows, it will always be assumed that the hypothesis holds. Then, we will give the following definition of weak solutions for problem (1).

Definition 1. We say that is a weak solution of equation (1), if for any , where will be given in Section 2.
Our main results are as follows.

Theorem 2. Under ()–(), ()–(), and (), there exists a nonempty open set such that in Then, equation (1) allows at least a nontrivial solution for all .

Theorem 3. Suppose that (), (), ()–(), and ()–() are satisfied. Then, there exists such that for every , equation (1) has at least two nontrivial solutions.

Remark 4. Generally speaking, if is a continuous function, magnetic field with , then the variable-order fractional magnetic Laplacian can be defined as for each given , along any .

2. Preliminaries and Notations

For the reader’s convenience, we first review some necessary definitions that we are later going to use of variable exponent Lebesgue spaces. We refer the reader to [2, 3, 1820] for details. Furthermore, we give the variational setting for equation (1) and some preliminary results.

Denote

If , then is said to be bounded. If , then is called the dual variable exponent of . The variable exponent Lebesgue space can be defined as with the norm then is a Banach space. When is bounded, we have

For bounded exponent, the dual space can be identified with , where is called the dual variable exponent of . Especially, with the real scalar product , for all . By Lemma 11, 20 of [20] and , we know that in the variable exponent Lebesgue space, the Hlder inequality is still valid. For all with , the following inequality holds

Define

Equip with the inner product and the corresponding norm . Especially, if , then the space is the usual fractional Sobolev space .

Lemma 5 (see [3] Lemma 5). Let , , if ; if . The embedding are continuous.
For each function , set and the corresponding norm is defined as Set be the space of measurable functions such that ; then, is a Hilbert space. If we let as the closure of in , then is a Hilbert space.

Lemma 6. For each compact subset , the embedding is continuous.

Proof. Fixed any compact subset , for any , we have where Since , we have By Lemma 6 of [21], we know that is locally bounded, and is compact, Thus, we obtain By (24)-(27), we can easily get that which implies that the embedding is continuously embedded into .

Through the above lemma, we know that , and from Theorem 2.1 of [2], we know that for be a bounded subset of and is continuous functions, is continuously embedded into , so we seek another method to prove the size relationship between , , and .

Lemma 7 (see [6] Lemma 10). For every , it holds . More precisely, °Í

Remark 8 (see [6] Remark 9). There holds

Lemma 9. Let , where if; if. is continuously embedded into . Moreover, if , then can be continuously embedded into ; that is, there exists a constant such that

Proof. By Lemma 7, we know that for every , it holds . By Lemma 5, we know that for is continuous. In light of Remark 8, one has From the above inequality, we immediately obtain the embedding which is continuous.

For , define

Set be equipped with the inner product (i.e., in ). Obviously, for . Set . Combining condition and fractional Sobolev inequality, we could get which shows that

From the above inequality, it holds that which shows that is continuously embedded into . Similarly, for all , there holds where . In addition, we have

This together yields that

For the sake of notational simplicity, we let . Hence, by condition , we have

Related to equation (1), we think the functional

In fact, we can easily verify that is well-defined of class in and for all . Therefore, if is a critical point of , then is a solution of equation (1). Since the energy functional is unbounded below on , in order to overcome this problem, we use the Nehari manifold to study the energy functional. In addition, we also note that contains every nonzero solution of equation (1). Especially, all critical points of must be located in , and the local minimizers on are usually critical points of .

3. Main Results

To start with, we can get an estimate of . Then, we will discuss some basic properties of . Finally, we prove Theorem 2 and Theorem 3 using the variational methods.

Lemma 10. is coercive and bounded below on . Furthermore, one has

Proof. If , in view of (37), (40), and Hlder inequality, it gains Therefore, is coercive and bounded below on .

We know that is linked to the behavior of the function of the form for . This map is called as the fibering map which can be traced back to basic works [1, 22, 23]. If , then

After observation, we can get that and thus, for and if and only if , i.e., positive critical points of correspond points on the Nehari manifold. Especially, if and only if . We found that can be divided into three parts corresponding local minimal, local maximum, and points of inflection. Based on the above, we can define

For each , we can find that

Now, we will deduce some results of , and .

Lemma 11. Assume is a local minimizer of on and , then in .

Proof. If is a local minimizer of on , then is a solution of the optimization problem where . Consequently, by the theory of Lagrange multipliers, there exists such that . Therefore, It follows from that Thus, If , then . In view of (50), it gains .

Lemma 12. (1), one has (2), one has

Proof. By the definitions of and , we can obtain It is easy to get that . It follows from the definition of that which implies that .

Lemma 13. Let the condition (), (), and ()-() are satisfied. Then, for all , one has .

Proof. If the conclusion does not hold, then there exists , such that . Then, for , by (40), (48), and the Hlder inequality, we have This means that Thus, we have From (48), we seem to easily get that which implies that Combining (39) and (40) with the Sobolev inequality, we have This means that Hence, combining (57) and (61), we have which is a contradictive with (). Therefore, for all , one has

By Lemma 13, , we can easily get that and define

Furthermore, we derive the following results.

Lemma 14. Under the condition (), (), and ()-(). Then, for all , there exists such that . Particularly, .

Proof. Our proof is decoupled in the following two steps:

Step 1. We claim that there exist such that . Indeed, let , where small enough. It is follows from (48) that This shows that This yields at once that Similarly, From (67), we can easily get that Consequently, it derives from (67) and (68) that Hence, .

Step 2. We assert that there exist such that . In fact, let . From (48), we seem to easily get that which implies that Combining (39), (40), and (71) with Sobolev inequality, we have and so It follows from (44) that Consequently, if , then for some .

We note that if and satisfy the hypotheses in Theorem 3, we can choose , such that have and which are independent of that satisfy for all and which shows for all .

Lemma 15. Assume that the conditions ()-() and ()-() hold, then there exists such that satisfies the condition in for all and .

Proof. First, we assume is a sequence with . In view of Lemma 10, there exists a positive constant related to such that . Consequently, there is a subsequence which is still denote as and in such that Besides, Let . Making use of the Vitali theorem, it holds that In fact, note that , for any ; then, there exists such that for and , For each , one has It is easy to get that is a equi-integrable on . Besides, , a.e., in . It follows from the Vitali theorem that Hence, there holds which implies that
Next, we assert that in . In fact, by (), we obtain In light of the Hlder inequality with the Sobolev inequality, we have By Bris-Lieb Lemma, we have By applying a Bris-Lieb type result on variable exponent Lebesgue space (see [24]) and ()-(), it is easy to obtain that Similarly, Then, overall, we can get that and . Then, by virtue of (77) and Lemma 10, we get that where Suppose by contradiction that is not bounded in . Then, there exists a subsequence still denoted by such that as . Hence, by virtue of (87), we have which is contradictory since . Thus, is bounded in for all . That is, there exist a constant such that . From (83), we can get that Together with (77)-(90), there holds We find that there exists large enough such that for all . It follows from (91) that in for all .

Theorem 16. Assume that ()-() and ()-() hold, then there exists such that for every , has a minimizer in satisfying that (1) is a nontrivial solution of equation (1).

Proof. Combining Lemma 14 and the Ekeland variational principle in [25], there exists such that is a sequence for . Furthermore, using Lemma 10, we can get that is bounded in . Consequently, there exists a subsequence of (we still denote as ) and in such that Besides, In view of Lemma 15, we know that in and . In other words, is a solution of equation (1).
Now, we will check that . On the contrary, by combining (40), (), the Egoroff theorem and the Hlder inequality, there holds as which shows that This is contradictive with . Hence, ; that is, is a nontrivial solution of equation (1).

Proof of Theorem 17. The result of Theorem 2 is immediately available from Theorem 20.

Theorem 18. Assume that the conditions (), (), and ()-() are satisfied, then there exists such that for every , has a minimizer in satisfying that (1) is a nontrivial solution of equation (1).

Proof. According to Lemma 14 and the Ekeland variational principle in [25], there exists such that is a sequence for . Furthermore, using Lemma 10, we can get that is bounded in . Consequently, there exists a subsequence of (we still denote as ) and in such that Besides, In view of Lemma 15, we know that in and . In other words, is a solution of equation (1).
Now, we will check that . Suppose the contrary, combining (39), (40), Egoroff theorem, and (), there holds as which shows that This is contradictive with . Hence, ; that is, is a nontrivial solution of equation (1).

Proof of Theorem 19. It derives from Theorem 20, Theorem 22, and Lemma 14 that equation (1) has two nontrivial solutions and such that and with .

Theorem 20. Assume that (), (), and ()-() hold, then for all , has a minimizer in satisfying that (1) is a nontrivial solution of equation (1).

Proof. Combining Lemma 14 and the Ekeland variational principle in [25], there exists such that is a sequence for . Furthermore, using Lemma 10, we can get that is bounded in . Consequently, there exists a subsequence of (we still denote as ) and in such that Besides, In view of Lemma 15, we know that in and . In other words, is a solution of equation (1).
Now, we will check that . On the contrary, by combining (40), (), the Egoroff theorem, and the Hlder inequality, there holds as which shows that This is contradictive with . Hence, ; that is, is a nontrivial solution of equation (1).

Proof of Theorem 21. The result of Theorem 2 is immediately available from Theorem 20.
We note that if and satisfy the hypotheses in Theorem 3, we can choose , such that have and which are independent of that satisfy for all and which shows for all .

Theorem 22. Assume that the conditions (), (), and ()-() are satisfied, then for all , has a minimizer in satisfying that (1) is a nontrivial solution of equation (1).

Proof. According to Lemma 14 and the Ekeland variational principle in [25], there exists such that is a sequence for . Furthermore, using Lemma 10, we can get that is bounded in . Consequently, there exists a subsequence of (we still denote as ) and in such that Besides, In view of Lemma 15, we know that in and . In other words, is a solution of equation (1).
Now, we will check that . Suppose the contrary, combining (39), (40), Egoroff theorem, and (), there holds as which shows that This is contradictive with . Hence, ; that is, is a nontrivial solution of equation (1).

Proof of Theorem 23. It derives from Theorem 20, Theorem 22, and Lemma 14 that equation (1) has two nontrivial solutions and such that and with .

Data Availability

Not applicable data and material.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

Acknowledgments

The authors expresses their gratitude to the referees for their valuable comments and suggestions which have led to a significant improvement on the presentation and quality of this paper. This work is supported by the Natural Science Foundation of Yunnan Province under grants 2018FE001(-136) and 2017zzx199, the National Natural Science Foundation of People’s Republic of China under grants 11961078 and 11561072, the Yunnan Province, Young Academic and Technical Leaders Program (2015HB010), and the Natural Science Foundation of Yunnan Province under grant 2016FB011.