Variational Approach for the Variable-Order Fractional Magnetic Schrödinger Equation with Variable Growth and Steep Potential in RN∗

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.


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 N ≥ 1,sð·Þ: ℝ N × ℝ N → ð0, 1Þ, is a continuous function, 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, and the magnetic field is A ∈ C 0,α ðℝ N , ℝ N Þ with α ∈ ð0, 1,p, q ∈ Cðℝ N Þ 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 Laplace ð−ΔÞ sð·Þ is defined as for each x ∈ ℝ N , along any u ∈ C ∞ 0 ðΩÞ. Inspired by them, we define the variable-order fractional magnetic Laplacian ð−ΔÞ sð·Þ A as for each x ∈ ℝ N , 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 sð·Þ = 1, pðxÞ, qðxÞ ≡ constant, and A = 0, 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 sð·Þ, pðxÞ, qðxÞ ≡ constant, and A = 0, 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 [6][7][8][9][10][11][12] deal with the existence and multiplicity of fractional magnetic problems. Some papers see [8,[13][14][15][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 ℝ N . 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 ℝ N . 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 ℝ N but also our main results are based on the study for the decomposition of Nehari manifolds. On the one hand, rela-tive 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 pðxÞ to ð2, ∞Þ and the research range from the bounded region Ω to the whole space ℝ N . 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.
is nonempty and has a smooth boundary with for all λ > 0, where D sð·Þ, A ðℝ N , ℂÞ is the Hilbert space related to the magnetic field A (see Section 2).

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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 V λ satisfied the conditions ðV 1 Þ − ðV 3 Þ as the steep well potential.
(H 4 ) g ∈ L ∞ ðℝ N , ℂÞ and ∥g∥ ∞ ≔ ∥g∥ L ∞ ðℝ N ,ℂÞ > 0: (13) In what follows, it will always be assumed that the hypothesis ðS 2 Þ holds. Then, we will give the following definition of weak solutions for problem (1). Definition 1. We say that u ∈ X λ is a weak solution of equa- for any v ∈ X λ , where X λ will be given in Section 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][19][20] for details. Furthermore, we give the variational setting for equation (1) and some preliminary results. Denote If p + < ∞, then p is said to be bounded. If ð1/pðxÞÞ + ð1/ p′ðxÞÞ = 1, then p′ðxÞ = pðxÞ/pðxÞ − 1 is called the dual variable exponent of pðxÞ. The variable exponent Lebesgue space can be defined as with the norm then L pðxÞ ðℝ N , ℂÞ is a Banach space. When p is bounded, we have min ∥u∥ For bounded exponent, the dual space ðL pðxÞ ðℝ N , ℂÞÞ ′ can be identified with L p ′ ðxÞ ðℝ N , ℂÞ, where p ′ ðxÞ is called the dual variable exponent of pðxÞ. Especially, with the real scalar product hu, vi L 2 ðℝ N ,ℂÞ ≔ R Ð ℝ N u vdx, for all u, v ∈ L 2 ðℝ N , ℂÞ. By Lemma 11, 20 of [20] and ∥· 3 Advances in Mathematical Physics ∥ L pðxÞ ðℝ N ,ℂÞ = ∥ | · |∥ L pðxÞ ðℝ N ,ℝÞ , we know that in the variable exponent Lebesgue space, the H€ older inequality is still valid. For all u ∈ L pðxÞ ðℝ N , ℂÞ, v ∈ L p′ðxÞ ðℝ N , ℂÞ with pðxÞ ∈ ð1,∞Þ , the following inequality holds Define Equip D sð·Þ ðℝ N , ℂÞ with the inner product and the corresponding norm ∥u∥ 2 sð·Þ = hu, ui sð·Þ . Especially, if sð·Þ ≡ constant, then the space D sð·Þ ðℝ N , ℂÞ is the usual fractional Sobolev space D s ðℝ N , ℂÞ.
Through the above lemma, we know that D sð·Þ A ðℝ N , ℂÞ ↪D sð·Þ ðW, ℂÞ, and from Theorem 2.1 of [2], we know that for Ω be a bounded subset of ℝ N and p : Ω → ½1,∞Þ is continuous functions, D sð·Þ ðΩ, ℂÞ is continuously embedded into L pðxÞ ðΩ, ℂÞ, so we seek another method to prove the size relationship between Ð ℝ N juðxÞj pðxÞ dx, Remark 8 (see [6] Remark 9). There holds From the above inequality, we immediately obtain the embedding D sð·Þ A ðℝ N , ℂÞ↪L p ðℝ N , ℂÞ which is continuous.

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From the above inequality, it holds that which shows that X is continuously embedded into D sð·Þ A ð ℝ N , ℂÞ. Similarly, for all λ ≥ 1/kM 2 | fV + < kg | , there holds where θ = 1 − M 2 jfV + < kgj/M 2 jfV + < kgj. In addition, we have This together M 2 | fV + < kg | <1 yields that For the sake of notational simplicity, we let ∥u∥ 2 λ,V ≔ ½u 2 sð·Þ,A + Ð ℝ N V λ u 2 dx. Hence, by condition ðV 4 Þ, we have Related to equation (1), we think the functional Ψ λ : X λ → ℝ, In fact, we can easily verify that Ψ λ is well-defined of class C 1 in X λ and for all u, v ∈ X λ . Therefore, if u ∈ X λ is a critical point of Ψ λ , then u is a solution of equation (1). Since the energy functional Ψ λ is unbounded below on X λ , in order to overcome this problem, we use the Nehari manifold N λ = fu ∈ X λ \ f 0g: hΨ λ ′ ðuÞ, ui = 0g to study the energy functional. In addition, we also note that N λ contains every nonzero solution of equation (1). Especially, all critical points of must be located in N λ , and the local minimizers on N λ are usually critical points of Ψ λ .

Main Results
To start with, we can get an estimate of Ψ λ . Then, we will discuss some basic properties of N λ . Finally, we prove Theorem 2 and Theorem 3 using the variational methods.
Lemma 10. Ψ λ is coercive and bounded below on N λ . Furthermore, one has Proof. If u ∈ N λ , in view of (37), (40), and H€ older inequality, it gains 6 Advances in Mathematical Physics Therefore, Ψ λ is coercive and bounded below on N λ .
We know that N λ is linked to the behavior of the function of the form L u ðtÞ: t → Ψ λ ðtuÞ for t > 0. This map is called as the fibering map which can be traced back to basic works [1,22,23]. If u ∈ X λ , then After observation, we can get that and thus, for u ∈ X λ \ f0g and t > 0, L u ′ðtÞ = 0 if and only if t u ∈ N λ , i.e., positive critical points of L u correspond points on the Nehari manifold. Especially, L u ′ ð1Þ = 0 if and only if u ∈ N λ . We found that N λ can be divided into three parts corresponding local minimal, local maximum, and points of inflection. Based on the above, we can define For each u ∈ N λ , we can find that Now, we will deduce some results of N + λ , N 0 λ , and N − λ .

Lemma 11.
Assume u 0 is a local minimizer of Ψ λ on N λ and Proof. If u 0 is a local minimizer of Ψ λ on N λ , then u 0 is a solution of the optimization problem where KðuÞ = ∥u∥ 2 λ,V − Ð ℝ N f ðxÞjuj qðxÞ dx − Ð ℝ N gðxÞjuj pðxÞ dx. Consequently, by the theory of Lagrange multipliers, there exists ν ∈ ℝ such that Ψ λ ′ðu 0 Þ = νK ′ðu 0 Þ. Therefore, It follows from u 0 ∈ N λ that Thus, If u 0 ∉ N 0 λ , then hK ′ðu 0 Þ, u 0 i ≠ 0. In view of (50), it gains ν = 0. (1) ∀u ∈ N + λ ∪ N 0 λ , one has Proof. By the definitions of N + λ and N 0 λ , we can obtain It is easy to get that which implies that Proof. If the conclusion does not hold, then there exists λ ≥ 1/kM 2 | fV + < kg | , such that N 0 λ ≠ ∅. Then, for u ∈ N 0 λ , by (40), (48), and the H€ older 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
Proof. First, we assume fu n g is a ðPSÞ c sequence with c < C 0 . In view of Lemma 10, there exists a positive constantĈ related to λ such that ∥u n ∥ λ ≤Ĉ. Consequently, there is a sub-sequence which is still denote as fu n g and u 0 in X λ such that Besides, Ψ λ ′ ðu 0 Þ = 0: Let v n = u n − u 0 . Making use of the Vitali theorem, it holds that In fact, note that f ∈ L 2/2−qðxÞ ðℝ N , ℂÞ, for any 0 < ε < 1; then, there exists rðεÞ > 0 such that for ζ ∈ ℝ N and r > rðεÞ, For each Ω 0 ⊂ B r ðζÞ, one has It is easy to get that f f ðxÞjv n j qðxÞ g is a equi-integrable on B r ðζÞ. Besides, f ðxÞjv n j qðxÞ → 0, a.e., in B r ðζÞ. It follows from the Vitali theorem that

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Hence, there holds which implies that lim n→∞ Ð ℝ N f ðxÞjv n j qðxÞ dx = 0: Next, we assert that u n → u 0 in X λ . In fact, by (V 2 ), we obtain In light of the H€ older inequality with the Sobolev inequality, we have By Bre ′ is-Lieb Lemma, we have By applying a Bre ′ is-Lieb type result on variable exponent Lebesgue space (see [24]) and (H 3 )-(H 4 ), it is easy to obtain that Similarly, Then, overall, we can get that Ψ λ ðv n Þ = Ψ λ ðu n Þ − Ψ λ ðu 0 Þ + oð1Þ and Ψ λ ′ ðv n Þ = oð1Þ. Then, by virtue of (77) and Lemma 10, we get that where Suppose by contradiction that fv n g is not bounded in X λ . Then, there exists a subsequence still denoted by fv n g such that ∥v n ∥ λ → ∞ as n → ∞. Hence, by virtue of (87), we have which is contradictory since 1 < q − ≤ q + < 2 < p − . Thus, fv n g is bounded in X λ for all λ > λ * ≥ 1/kM 2 | fV + < kg | . That is, there exist a constant M 1 > 0 such that ∥v n ∥ λ ≤ M 1 . From