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Jianwen Zhou, Bianxiang Zhou, Liping Tian, Yanning Wang, "Variational Approach for the Variable-Order Fractional Magnetic Schrödinger Equation with Variable Growth and Steep Potential in ", Advances in Mathematical Physics, vol. 2020, Article ID 1320635, 15 pages, 2020. https://doi.org/10.1155/2020/1320635
Variational Approach for the Variable-Order Fractional Magnetic Schrödinger Equation with Variable Growth and Steep Potential in
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.
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 , the fractional magnetic Laplacian has been defined as for . In , 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  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 . For , , , and , in , 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 , 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 , 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 [6–12] deal with the existence and multiplicity of fractional magnetic problems. Some papers see [8, 13–16] 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 .
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 , we extend the exponent to variable exponent, thus introducing the variable exponent Lebesgue space. In addition, compared with , 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).
To the best of our knowledge, this type of hypothesis is the first introduced by Bartsch and Wang in . 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).
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, 18–20] for details. Furthermore, we give the variational setting for equation (1) and some preliminary results.
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  and , we know that in the variable exponent Lebesgue space, the Hlder inequality is still valid. For all with , the following inequality holds
Equip with the inner product and the corresponding norm . Especially, if , then the space is the usual fractional Sobolev space .
Lemma 5 (see  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 , 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 , 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 .
Remark 8 (see  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