On the critical fractional Schrödinger - Kirchho ﬀ - Poisson equations with electromagnetic ﬁ elds

: This paper intend to study the following critical fractional Schrödinger - Kirchho ﬀ - Poisson equations with electromagnetic ﬁ elds in (cid:2) 3

Throughout the paper, the electric potential V , Kirchhoff function M, and f satisfy the following assumptions: , and there is > b 0 such that the set has finite Lebesgue measure. (ℱ) (f 1 ) ( ) ∈ × f C , 3 and ( ) First, our motivation to study problem (1.1) mainly comes from the application of the fractional magnetic operator. We note that the equation with fractional magnetic operator often arises as a model for various physical phenomena, in particular in the study of the infinitesimal generators of Lévy stable diffusion processes [2]. Also, the number of literature on nonlocal operators and their applications has been studied, and hence, we refer interested readers to [3][4][5][6][7]. To further research this kind of equation by variational methods, many scholars have established the basic properties of fractional Sobolev spaces, readers are referred to [8,9].
Next, we note that some works that appeared in recent years concerning the follwing magnetic Schrödinger equation without Poisson term: where the magnetic operator in (1.2) is given by As stated in the study by Squassina and Volzone [10], up to correcting the operator by the factor ( Thus, up to normalization, the nonlocal case can be seen as an approximation of the local one. Recently, many researchers have paid attention to the equations with fractional magnetic operator. In particular, Mingqi et al. [11] studied some existence results of Schrödinger-Kirchhoff type equation involving the fractional p-Laplacian and the magnetic operator: where f satisfies the subcritical growth condition. For the critical growth case, the authors in [12] first considered the following fractional Schrödinger equations: They obtained the existence of ground state solution u ε by using variational methods. Subsequently, Liang et al. [13] proved the existence and multiplicity of solutions to a class of Schrödinger-Kirchhoff type equation in the non-degenerate case. We draw the attention of the reader to the degenerate case involving the magnetic operator in the study by Liang et al. [14]. On the other hand, for case ≡ A 0 in problem (1.1), there have been numerous articles dedicated to the study of the fractional Schrödinger-Poisson system as it appears in an interesting physical context. For example, Giammetta in [15] first studied the local and global well-posedness of a fractional Schrödinger-Poisson system in one dimension. Zhang et al. in [16] obtained the existence of radial ground state solution to the fractional Schrödinger-Poisson system with a general subcritical or critical nonlinearity by using the perturbation approach. In [17], Murcia and Siciliano proved that the number of positive solutions for a class of doubly singularly perturbed fractional Schrödinger-Poisson system via the Ljusternick-Schnirelmann category. Liu in [18] concerned with the existence of multibump solutions for the fractional Schrödinger-Poisson system through the Lyapunov-Schmidt reduction method. Chen et al. in [19] admitted the existence of the Nehari-type ground state solutions for fractional Schrödinger-Poisson system by using the non-Nehari manifold approach. For more related results, we can cite the recent works [20][21][22][23][24] and the references therein.
Once we turn our attention to the Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields, we immediately see that the literature is relatively scarce. In this case, we can cite the recent works [25,26]. We call attention to Ambrosio in [27] proved that the multiplicity and concentration results for a class of fractional Schrödinger-Poisson type equation with magnetic field and subcritical growth. For the critical growth case, Ambrosio in [28] also obtained the multiplicity and concentration of nontrivial solutions to the fractional Schrödinger-Poisson equation with the magnetic field. However, to the best of our knowledge, semiclassical solutions to fractional magnetic Schrödinger-Poisson equations problem (1.1) have not ever been considered until now.
Inspired by the previously mentioned works, our main objective is to study the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields. The proof of these assertions is given by means of concentration compactness principle and variational method. For this purpose, we will use some minimax arguments. Moreover, due to the appearance of the critical term, the Sobolev embedding does not possess compactness. To this end, we need some technical estimations.
We are now in a position to state the existence result as follows.
, which satisfy the estimates (1.5) and (1.6). Moreover, The main feature of our consequence in the present paper is to establish the multiplicity result for problem (1.1) under the critical growth condition. There is no doubt that we encounter serious difficulties because of the lack of compactness. To overcome the challenge, we use the concentration-compactness principles for fractional Sobolev spaces according to [29][30][31] to prove the ( ) PS c condition at special levels c. On the other hand, we need to develop new techniques to construct sufficiently small minimax levels.
The rest of our paper is organized as follows. In Section 2, we briefly review some properties of the Sobolev spaces with fractional order. In Section 3, we prove the Palais-Smale condition at some special energy levels by using the concentration-compactness principles for fractional Sobolev spaces. Section 4 deals with the existence and multiplicity result for problem (1.1).

Preliminaries
In this section, we briefly review the definitions and list some basic properties of the Lebesgue spaces, which we use throughout this article.
For any is endowed with the norm For the reader's convenience, we will use the following embedding theorem, see Lemma 3.5 in [1].
Next, we have the following diamagnetic inequality, and its proof can be found in the study by d'Avenia and Squassina [1]. Moreover, For problem (1.1), we will use the Banach space E defined by Then, by the Lax-Milgram theorem, there exists a unique | | ψ u t such that Therefore, we obtain the following t-Riesz formula: We note that the aforementioned integral is convergent at infinity since | | ( ) . Next we collect some properties of | | ψ u t , which will be used in this paper. The following proposition can be proved by using similar arguments as [27,28]. is continuous and maps bounded sets into bounded sets; 3 Behavior of (PS) sequences In this section, to overcome the lack of compactness caused by the critical exponents, we intend to employ the second concentration-compactness principle, see [29][30][31] for more details. Moreover, to obtain the solution of problem (1.1), we will use the following equivalent form:

(3.2)
It is clear that ε is of class ( ) C E, 1 under the assumptions ( ) (see [32]). Moreover, for all ∈ u v E , , the Fréchet derivative of ε is given by

(3.3)
Thus, the weak solutions of (1.1) coincide with the critical points of ε . The main result of this section is the following compactness result. Proof. Let ( ) u n n be a ( ) PS cλ sequence for ε , we first claim that ( ) u n n is bounded in E. In fact, by ( ) → u c ε n

(3.4)
We know that { } u n n is bounded in E from { } / < < * σ μ max 2 , 4 2 s . Furthermore, we can obtain ≥ c 0 by passing to the limit in (3.4). Hence, by diamagnetic inequality, {| |} u n n is bounded in ( ) H s 3 . Then, by using the fractional version of concentration compactness principle in the fractional Sobolev space (see [29][30][31]), up to a subsequence, we have In the following, we shall prove that Now, we suppose on the contrary that ≠ ∅ J . Then, we can construct a smooth cut-off function, (3.14) Due to the fact that f has the subcritical growth and ϕ ε has the compact support, we have that Since ϕ ρ has compact support, so that, letting → ∞ n in . This is an obvious contradiction. Hence, = ∅ J .
Next, we prove that = ∞ ν 0. Suppose on the contrary that > ∞ ν 0. To obtain the possible concentration of mass at infinity, we similarly define a cut off function ( ) We can verify that { } u ϕ n R n is bounded in E, and hence,