This paper intend to study the following critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields in :
Under suitable assumptions, together with the concentration compactness principle and variational method, we prove that the existence and multiplicity of semiclassical solutions for above problem as .
This paper deals with the existence and multiplicity of solutions for the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields in :
where is a positive parameter, , , is the usual Sobolev critical exponent, is an electric potential, and and are called the magnetic operator and magnetic potential, respectively. According to d’Avenia and Squassina in , the fractional operator , which up to normalization constants, can be defined on smooth functions as follows
and magnetic potential is given by
Throughout the paper, the electric potential , Kirchhoff function , and satisfy the following assumptions:
, , and there is such that the set has finite Lebesgue measure.
The Kirchhoff function is continuous, and there exists such that . There exists satisfying for all , where .
( ) and uniformly in as . ( ) There exist and such that . ( ) There exist , , and such that , and for all , where .
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 . 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 , up to correcting the operator by the factor , it follows that converges to as . 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.  studied some existence results of Schrödinger-Kirchhoff type equation involving the fractional -Laplacian and the magnetic operator:
where satisfies the subcritical growth condition. For the critical growth case, the authors in  first considered the following fractional Schrödinger equations:
They obtained the existence of ground state solution by using variational methods. Subsequently, Liang et al.  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. .
On the other hand, for case 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  first studied the local and global well-posedness of a fractional Schrödinger-Poisson system in one dimension. Zhang et al. in  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 , 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  concerned with the existence of multibump solutions for the fractional Schrödinger-Poisson system through the Lyapunov-Schmidt reduction method. Chen et al. in  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  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  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.
Let and hold. If satisfies and , then the following statements hold:
For any , there is such that if , then problem (1.1) has at least one solution satisfying(1.5)(1.6)
Moreover, in E as .
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 condition at special levels . 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).
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 , fractional Sobolev space is defined by
where and denotes the so-called Gagliardo semi-norm, that is,
and is endowed with the norm
For the reader’s convenience, we will use the following embedding theorem, see Lemma 3.5 in .
The space is continuously embedded in for all . Furthermore, the space is continuously compact embedded in for all and any compact set .
Next, we have the following diamagnetic inequality, and its proof can be found in the study by d’Avenia and Squassina .
Let , then . That is,
From Proposition 3.6 in , for all , we have
For problem (1.1), we will use the Banach space defined by
with the norm
By the assumption , we know that the embedding is continuous. Note that the norm is equivalent to the norm defined by
for each .
Obviously, for each , there is such that
where . Hereafter, we shortly denote by the norm of Lebesgue space with .
Now, let such that , we can see that
Then, by (2.2), we have
for , where
Then, by the Lax-Milgram theorem, there exists a unique such that such that
Therefore, we obtain the following -Riesz formula:
We note that the aforementioned integral is convergent at infinity since . Next we collect some properties of , which will be used in this paper. The following proposition can be proved by using similar arguments as [27,28].
Assume that holds, for any , we have
is continuous and maps bounded sets into bounded sets;
if in E, then in ;
for any and ;
for all . Moreover,
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:
for . Now, let us consider the Euler-Lagrange functional associated with (1.1), defined by
It is clear that is of class under the assumptions (see ). Moreover, for all , the Fréchet derivative of is given by
Thus, the weak solutions of (1.1) coincide with the critical points of .
The main result of this section is the following compactness result.
Let and hold. If satisfies and , then for any , satisfies condition, for all , where , that is, any -sequence has a strongly convergent subsequence in E.
Let be a sequence for , we first claim that is bounded in . In fact, by and in , it follows from and that
We know that is bounded in from . Furthermore, we can obtain by passing to the limit in (3.4). Hence, by diamagnetic inequality, is bounded in . 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
where is the best Sobolev constant, i.e.,
, and are Dirac measures at and , and are constants. Moreover, we have
In the following, we shall prove that
Now, we suppose on the contrary that . Then, we can construct a smooth cut-off function, take such that ; in , in . For any , define , where . It is not difficult to see that is bounded in . Then , which implies
It is easy to verify that
Note that the Hölder inequality yields
Lemma 3.4 in  gives that
Due to the fact that has the subcritical growth and has the compact support, we have that
where . This is an obvious contradiction. Hence, .
Next, we prove that . Suppose on the contrary that . To obtain the possible concentration of mass at infinity, we similarly define a cut off function such that on and on . We can verify that is bounded in , and hence, , and this implies that as
As mentioned earlier, we have
Similar to the proof of Lemma 3.4 in , we can show that
By (3.17), we obtain . Thus, we have
where . Thus,
From the Brézis–Lieb lemma, we obtain
By the weak lower semicontinuity of the norm, condition , and the Brézis–Lieb lemma, we have
Here, we use the fact that . This fact implies that strongly converges to in . Hence, the proof is complete.□
4 Proof of Theorem 1.1
To prove Theorem 1.1, let , and we first prove that functional has the mountain pass geometry.
Let and hold. If satisfies and , then
there exist two positive constants such that if and if , where ;
for any finite dimensional subspace ,
From condition , we can take and there exists such that
This fact implies that the conclusion in Lemma 4.1 holds true since .
Now we verify condition of Lemma 4.1. We note that implies that
Thus, for all , we have
Since all norms in a finite-dimensional space are equivalent, and since , we obtain that in Lemma 4.1 is valid. This completes the proof.□
Next, we will prove that satisfies on the special finite-dimensional subspace. To do this, by assumption , we choose such that . Without loss of generality, we can assume from now on that .
Let be defined by
From , we can obtain for all .
On the other hand, from Lemma 3.5 in , we know that
Thus, for any , one can choose with and supp so that