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.


Introduction
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 N ≥ 1; sð·Þ: ℝ N × ℝ N ⟶ ð0, 1Þ is a continuous function; Ω is a bounded subset in ℝ N with N > 2sðx, yÞ for all ðx, yÞ ∈ Ω × Ω; ð−ΔÞ sð·Þ A is the variable-order fractional magnetic Laplace operator; the potential V λ ðxÞ = λV + ðxÞ − V − ðxÞ with V ± = max f±V, 0g; λ > 0 is a parameter; magnetic field A ∈ C 0,α ðℝ N , ℝ N Þ with α ∈ ð0, 1; f , g > 0 are two bounded nonnegative measurable function; p, q ∈ CðΩÞ; and u : ℝ N ⟶ ℂ. In [1], the fractional magnetic Laplacian has been defined as for x ∈ ℝ N . In [2], the variable-order fractional magnetic Laplace ð−ΔÞ sð·Þ is defined as follows: for each x ∈ ℝ N , along any φ ∈ C ∞ 0 ðΩÞ. Inspired by them, we define the variable-order fractional magnetic Laplacian ð−ΔÞ sð·Þ A as follows: for each x ∈ ℝ N , For sð·Þ = α, pðxÞ, qðxÞ ≡ constant, and A = 0; in [4], the authors obtained the existence, multiplicity, and concentration of nontrivial solutions for the following indefinite fractional elliptic equation by using Nehari manifold decomposition: When A = 0, V − ðxÞ = 0, and f ðxÞ, gðxÞ ≡ constant, the authors in [2] 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 sð·Þ = s, pðxÞ, qðxÞ ≡ constant, in [1], 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][6][7][8]). We know that the fractional magnetic Laplacian operator ð−ΔÞ s A is introduced in literature [9]; ð−ΔÞ s A comes from magnetic Laplacian ð∇−iAÞ 2s . 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 [1], 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 [2], we write the variable order fractional Schrödinger equation to a variable order fractional magnetic Schrödinger equation.
Inspired by the above works, we assume that s : ℝ N × ℝ N ⟶ ð0, 1Þ and V are continuous functions satisfying the following: (S 2 ): sð·Þ is symmetric, that is, sðx, yÞ = sðy, xÞ for all ðx, Ω. In addition, we assume that f , g satisfy the following assumption: ( Based on the hypothesis ðS 2 Þ, we can give the following definition of weak solutions for problem (1).
Journal of Function Spaces Definition 1. We say that u ∈ E λ is a weak solution of problem for any v ∈ E λ , where E λ will be defined in Section 2.
Now, we will describe the first main result as follows.

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 jΩj to represent n-dimensional Lebesgue measure of a measurable set Ω ⊂ ℝ N . In addition, for each a ∈ ℂ, we will use Ra to represent the real part of a and a to represent the complex conjugate of a. Let N ≥ 1 and Ω ⊂ ℝ N be a nonempty set. A measurable function r : Ω ⟶ ½1, ∞Þ is called a variable exponent, and we define r + = esssup x∈Ω rðxÞ, r − = essinf x∈Ω rðxÞ. If r + is finite, then the exponent r is said to be bounded. The variable exponent Lebesgue space is with the Luxemburg norm then L rðxÞ ðΩ, ℂÞ is a Banach space, and when r is bounded, we have the following relations For bounded exponent, the dual space ðL rðxÞ ðΩ, ℂÞÞ′ can be identified with L r ′ðxÞ ðΩ, ℂÞ, where the conjugate exponent r′ is defined by r′ = r/ðr − 1Þ. If 1 < r − ≤ r + < ∞, then the variable exponent Lebesgue space L rðxÞ ðΩ, ℂÞ is a separable and reflexive. In particular, with the scalar product By Lemma 3.2.20 of [10] and k·k L rðxÞ ðΩ,ℂÞ = kj·jk L rðxÞ ðΩ,ℝÞ , we know that in the variable exponent Lebesgue space, Hölder inequality is still valid. For all u ∈ L rðxÞ ðΩ, ℂÞ, v ∈ L r ′ ðxÞ ðΩ, ℂÞ with rðxÞ ∈ ð1, ∞Þ, the following inequality holds: 3 Journal of Function Spaces Let Ω be a nonempty open subset of ℝ N , and let sð·Þ: 1Þ be a measurable function, and there exist two constants 0 < s 0 < s 1 Equip H sð·Þ ðΩ, ℂÞ with the norm Hence, the embedding H sð·Þ ðΩ, ℂÞ°L rðxÞ ðΩ, ℂÞ is continuous and compact. In order to define weak solutions of problem (1), we introduce the functional space  [2]) and the magnetic framework the space introduced in [9]. Next, we state and prove some properties of space H sð·Þ 0,A ðΩ, ℂÞ, which will be useful in the sequel. Lemma 6. There exists a constant C 2 > 0, depending only on N, s 1 and Ω, such that from which we immediately have Since Ω is bounded, there exists r > 1/2 such that Ω ⊂ B r and jB r \ Ωj > 0; then, we have Thus, we obtain where C 1 = ð2rÞ N+2s 1 /jB r \ Ωj and C 2 2 = C 1 + 1.
Next, we give the variational setting for problem (1). For λ > 0, we need the following scalar product and norm: Let E = fu ∈ H sð·Þ 0,A ðΩ, ℂÞ: For simplicity, we let kuk 2 λ,V ≔ kuk 2 H sð·Þ 0,A ðΩ,ℂÞ + Ð Ω V λ u 2 dx. Therefore, by condition ðV 4 Þ, Associated with problem (1), we consider the energy functional Ψ λ : E λ ⟶ ℝ, In fact, one can verify that Ψ λ is well-defined of class C 1 in E λ and Journal of Function Spaces for all u, v ∈ E λ . Therefore, if u ∈ E λ is a critical point of Ψ λ , then u is a solution of problem (1). Now we give the theorems that we need later.
Theorem 8 (see [2,12]). Let X be a real infinite dimensional Banach space and I ∈ C ′ ðXÞ a functional satisfying the ðPSÞ c condition as well as the following three properties: (1) Ið0Þ = 0, and there exists two constants ρ, δ > 0 such that IðuÞ ≥ δ for all u ∈ X with kuk = ρ (2) I is even Then, I poses an unbounded sequence of critical values characterized by a minimax argument Theorem 9 (see [13]). Let X be a real Banach space and I ∈ C 1 ðX, ℝÞ. Suppose I satisfies the (PS) condition, which is even and bounded from below, and Ið0Þ = 0. If for any k ∈ N, there exists a k-dimensional subspace X k of X and ρ k > 0 such that sup X k ∩S ρ k I < 0, where S ρ k = fu ∈ X : kuk X = ρ k g, then at least one of the following conclusions holds: (i) There exists a sequence of critical points fu k g satisfying Iðu k Þ < 0 for all k and ku k k X ⟶ 0 as k ⟶ ∞ (ii) There exists r > 0 such that for any 0 < a < r, there exists a critical point u such that kuk X = a and I ðuÞ = 0 It is easy to verify that E λ is a separable Hilbert space. Let Theorem 10 (see [14], fountain theorem). Suppose that I ∈ C 1 ðE, ℝÞ satisfying Ið−uÞ = IðuÞ. Assume that for every k ∈ N, there exist r k > γ k > 0 such that (D 1 ): a k = max fIðuÞ: u ∈ Y k , kuk = r k g ≤ 0.
(D 7 ): I satisfies ðPSÞ * c condition for every c ∈ ½d k 0 , 0. Then, I has a sequence of negative critical values converging to 0.

Proof of Theorem 1
In this part, we first recall that definitions of functional Ψ λ satisfies the ðPSÞ c condition and ðPSÞ * c condition in E λ at the level c ∈ ℝ and use the usual mountain pass theorem (see [2]) to find a ðPSÞ c sequence in E λ . Second, we show that functional Ψ λ satisfies the ðPSÞ * c condition in E λ at the level c < c 0 . Finally, we give the proof of problem (1).
Definition 12 (see [2]). Let I ∈ C 1 ðE, ℝÞ and c ∈ ℝ. The functional I satisfies the ðPSÞ c condition if any sequence fu n g ⊂ E such that Iðu n Þ ⟶ c and I ′ðu n Þ ⟶ 0 as n ⟶ ∞ admits a strongly convergent subsequence in E.
Definition 13 (see [16]). Let I ∈ C 1 ðE, ℝÞ and c ∈ ℝ. The functional I satisfies the ðPSÞ * c condition (with respect to Y n ) if any sequence fu n g ⊂ E such that fu n g ∈ Y n , Iðu n Þ ⟶ c and I ′ j Y n ðu n Þ ⟶ 0 as n ⟶ ∞ admits a strongly convergent subsequence in E.
In order to obtain our main results by using the mountain pass theorem, we first prove that Ψ λ satisfies the mountain pass geometry (i) and (ii).
Proof. Assume that fu n g be a ðPSÞ * c sequence in E λ with c < c 0 ; then, fu n g ∈ Y n , Ψ λ ðu n Þ ⟶ c λ and Ψ λ ′j Y n ðu n Þ ⟶ 0 as n ⟶ ∞: It follows from (42) and (43) and the Hölder inequality that On the contrary, we suppose that fu n g is not bounded in E λ . Then, there exists a subsequence still denoted by 9 Journal of Function Spaces fu n g such that ku n k λ ⟶ ∞ as n ⟶ ∞. Then, it follows from (69) that which leads to a contradiction. Hence, fu n g is bounded in E λ for all λ > 0. Therefore, there exist a subsequence of fu n g still denoted by fu n g and u 0 in E λ such that u n ⇀ u 0 in E λ , u n ⟶ u 0 a:e:in Ω, where r ′ ðxÞ = rðxÞ/rðxÞ − 1. The next step is to show that u n ⟶ u 0 in E λ . By Lemma 7, we can get u n ⟶ u 0 in L rðxÞ ðΩ, ℂÞ. Thus, Making use of Hölder inequality, we can obtain