Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020View this Special Issue
Multiplicity Results for Variable-Order Nonlinear Fractional Magnetic Schrödinger Equation with Variable Growth
In this paper, we prove the multiplicity of nontrivial solutions for a class of fractional-order elliptic equation with magnetic field. Under appropriate assumptions, firstly, we prove that the system has at least two different solutions by applying the mountain pass theorem and Ekeland’s variational principle. Secondly, we prove that these two solutions converge to the two solutions of the limit problem. Finally, we prove the existence of infinitely many solutions for the system and its limit problems, respectively.
In this paper, we consider the multiplicity of nontrivial solutions of the following concave-convex elliptic equation involving variable-order nonlinear fractional magnetic Schrödinger equation: where; is a continuous function; is a bounded subset in with for all ; is the variable-order fractional magnetic Laplace operator; the potential with ; is a parameter; magnetic field with ; are two bounded nonnegative measurable function; ; and . In , the fractional magnetic Laplacian has been defined as for . In , the variable-order fractional magnetic Laplace is defined as follows: for each , along any . Inspired by them, we define the variable-order fractional magnetic Laplacian as follows: for each ,
Since is a function, magnetic field with , we see that operator is variable order fractional magnetic Laplace operator. Especially, when , reduce to the usual fractional magnetic Laplace operator. When reduce to the usual fractional Laplace operator. Very recently, for , , and ; in , under appropriate assumptions, the 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 , the authors obtained the existence, multiplicity, and concentration of nontrivial solutions for the following indefinite fractional elliptic equation by using Nehari manifold decomposition:
When and the authors in  give some sufficient conditions to ensure the existence of two different weak solutions and use the variational method and the mountain pass theorem to obtain the two weak solutions of problem (12) which converge to two solutions of its limit problems and the existence of infinitely many solutions to its limit problem:
For , in , the authors study the existence of solutions for the following equation on the whole space by using the method of Nehari manifold decomposition,and obtain some sufficient conditions for the existence of nontrivial solutions of the following equation:
In recent years, with the continuous deepening of research, the fractional magnetic problem has attracted extensive attention of researchers. More and more researchers have studied the solvability of the fractional magnetic problem (see [5–8]). We know that the fractional magnetic Laplacian operator is introduced in literature ; comes from magnetic Laplacian . However, as far as we know, up to now, few papers have studied the existence and multiplicity of solutions for the variable-order fractional magnetic Schrödinger equation. Therefore, motivated by the above literature, we are interested in the existence and multiplicity of solutions to problem (1) with variable growth. As far as we know, this is the first time to study the existence and multiplicity of nontrivial solutions for the variable-order nonlinear fractional magnetic Schrödinger equation with variable exponents.
It is worth noting that in this paper, we not only prove that there exist two different nontrivial solutions to problem (1), but we also show that the two nontrivial solutions of problem (1) converge to two solutions of the limit problem for problem (1). The novelty of this paper is that, compared with , we write the original fractional magnetic Schrödinger equation without variable growth to a variable-order fractional magnetic Schrödinger equation with variable growth. In addition, compared with , we write the variable order fractional Schrödinger equation to a variable order fractional magnetic Schrödinger equation.
Inspired by the above works, we assume that and are continuous functions satisfying the following:
(S2): is symmetric, that is, for all .
(V1): is a nonempty bounded domain and .
(V2): there exists a nonempty open domain such that , for all .
(V3): is a continuous function on and .
(V4): there exists a constant such that or all , where is the Hilbert space related to magnetic field (see Section 2).
For the variable exponents , we assume that and satisfy the following assumption:
(H1): for all .
(H2): for all .
In addition, we assume that , satisfy the following assumption:
(H3): are bounded nonnegative measurable function such that , on open interval and
Based on the hypothesis , we can give the following definition of weak solutions for problem (1).
Now, we will describe the first main result as follows.
Theorem 2. Assume that hold. Let . Then, the problem (1) allows at least two different solutions for all .
Theorem 3. Let and be two solutions obtained in Theorem 2. Then, and in as , where are two nontrivial solutions of the following problem:
Remark 4. In general, if is a continuous function, magnetic field with , then the variable-order fractional magnetic Laplacian can be defined as follows: for each , along any .
2. Preliminaries and Notations
In this section, we first give the definition of the variable exponential Lebesgue space. Secondly, we define variable-order fractional magnetic Sobolev spaces and prove the compact conditions between them. Finally, we give the variational setting for problem (1) and theorems that will be used later.
In this paper, we use to represent -dimensional Lebesgue measure of a measurable set . In addition, for each , we will use to represent the real part of and to represent the complex conjugate of . Let and be a nonempty set. A measurable function is called a variable exponent, and we define . If is finite, then the exponent is said to be bounded. The variable exponent Lebesgue space is with the Luxemburg norm then is a Banach space, and when is bounded, we have the following relations
For bounded exponent, the dual space can be identified with , where the conjugate exponent is defined by . If , then the variable exponent Lebesgue space is a separable and reflexive. In particular, with the scalar product
By Lemma 3.2.20 of  and , we know that in the variable exponent Lebesgue space, Hölder inequality is still valid. For all with , the following inequality holds:
Let be a nonempty open subset of , and let be a measurable function, and there exist two constants such that for all . Set
Equip with the norm
Especially, if , then the space is the usual fractional Sobolev space .
Lemma 8. Let be a smooth bounded subset of and let satisfying and satisfying . Then, there exists such that , for any . That is, the embedding is continuous. Moreover, is compact.
Proof. By Theorem 2.1 of , we know that is continuous and compact, there exists such that . then, for any , we have Hence, the embedding is continuous and compact.
Let be a continuous function and . For a function , define and the corresponding norm denoted by . We consider the space of measurable functions such that ; then, is a Hilbert space. Define as the closure of in ; then, is a Hilbert space. Especially, if , then the space is the variable-order fractional Sobolev space ; if and , then the space is the usual fractional Sobolev space .
In order to define weak solutions of problem (1), we introduce the functional space equipping with the scalar product which induces the following norm Hence, generalizes to the variable-order fractional Sobolev space (see ) and the magnetic framework the space introduced in . Next, we state and prove some properties of space , which will be useful in the sequel.
Lemma 6. There exists a constant , depending only on and , such that for any . Thus, is an equivalent norm of .
Proof. For any , by Lemma 3.1 in , we have the pointwise diamagnetic inequality from which we immediately have
Since is bounded, there exists such that and ; then, we have
Thus, we obtain where and .
Combining the above two aspects, we have which implies that is the equivalent norm of a norm .
Lemma 2.3. Let be a bounded subset of . Assume that is a continuous function satisfying and is a continuous function satisfying . If , then is continuous and is compact, that is, there exists such that
Proof. For any , we have where Since , we have
In view of which is bounded, there exists a compact set such that . By Lemma 2.2 of , we know that is locally bounded and is compact, Thus, we obtain
Next, we give the variational setting for problem (1). For , we need the following scalar product and norm:
Let be equipped with the norm (that is, in ). Obviously, for . Set . Moreover, for , we can get
For simplicity, we let . Therefore, by condition ,
Associated with problem (1), we consider the energy functional
In fact, one can verify that is well-defined of class in and for all . Therefore, if is a critical point of , then is a solution of problem (1).
Now we give the theorems that we need later.
Theorem 8 (see [2, 12]). Let be a real infinite dimensional Banach space and a functional satisfying the condition as well as the following three properties: (1), and there exists two constants such that for all with (2) is even(3)For all finite dimensional subspaces , there exists such that for all , where . Then, poses an unbounded sequence of critical values characterized by a minimax argument
Theorem 9 (see ). Let be a real Banach space and . Suppose satisfies the (PS) condition, which is even and bounded from below, and . If for any , there exists a -dimensional subspace of and such that , where , then at least one of the following conclusions holds: (i)There exists a sequence of critical points satisfying for all and as (ii)There exists such that for any , there exists a critical point such that and It is easy to verify that is a separable Hilbert space. Let , then . we define
Theorem 10 (see , fountain theorem). Suppose that satisfying . Assume that for every , there exist such that
(D2): as .
(D3): satisfies condition for every .
Then, has an unbounded sequence of critical values which have the form where .
Theorem 11 (see , dual fountain theorem). Suppose that satisfying . Assume that for every , there exist such that