On the $p$-fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity

In this article, we deal with the following $p$-fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: $$ M\left([u]_{s,A}^{p}\right)(-\Delta)_{p, A}^{s} u+V(x)|u|^{p-2} u=\lambda\left(\int_{\mathbb{R}^{N}} \frac{|u|^{p_{\mu, s}^{*}}}{|x-y|^{\mu}} \mathrm{d}y\right)|u|^{p_{\mu, s}^{*}-2} u+k|u|^{q-2}u,\ x \in \mathbb{R}^{N},$$ where $0<s<1<p$, $ps<N$, $p<q<2p^{*}_{s,\mu}$, $0<\mu<N$, $\lambda$ and $k$ are some positive parameters, $p^{*}_{s,\mu}=\frac{pN-p\frac{\mu}{2}}{N-ps}$ is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions $V$, $M$ satisfy the suitable conditions. By proving the compactness results with the help of the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.


Introduction
In this article, we intend to study the following p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity in |x − y| µ dy |u| p * µ,s −2 u + k|u| q−2 u, x ∈ R N , (1.1) where 0 < s < 1 < p, ps < N , p < q < 2p * s,µ , 0 < µ < N , λ and k are some positive parameters, ) is an electric potential, A ∈ C(R N , R N ) is a magnetic potential, and V , M satisfy the following assumptions (V ) V : R N → R is a continuous function and has critical frequency, that is, V (0) = min x∈R N V (x) = 0.Moreover, the set V τ 0 = x ∈ R N : V (x) < τ 0 has finite Lebesgue measure for some τ 0 > 0.
(M ) (m 1 ) The Kirchhoff function M : R + 0 → R + is a continuous and nondecreasing.In addition, there exists a positive constant m 0 > 0 such that M (t) ≥ m 0 for all t ∈ R + 0 ; (m 2 ) For some σ ∈ (p/q, 1] , we have M (t) ≥ σM (t)t for all t ≥ 0, where M (t) = t 0 M (s)ds.When p = 2, we know that the fractional operator (−∆) s A , which up to normalization constants, can be defined on smooth functions u as dy, x ∈ R N , see d'Avenia and Squassina [1].There already exist several papers dedicated to the study of the Choquard equation, this problem can be used to describe many physical models [2,3].Recently, d'Avenia and Squassina [1] considered the following fractional Choquard equation of the form and the existence of ground state solutions was obtained by using the Mountain pass theorem and the Ekeland variational principle.For more results on problems with the Hardy-Littlewood-Sobolev nonlinearity without the magnetic operator case, see [4][5][6][7][8][9].For the case p = 2, Iannizzotto et al. [10] investigated the following fractional p-Laplacian equation (1. 3) The existence and multiple solutions for problem (1.3) was proved by using the Morse theory.Xiang et al. [11] dealt with a class of Kirchhoff-type problems driven by nonlocal elliptic integro-differential operators, and two existence theorems were obtained with the help of the variational method.Souza [12] studied a class of nonhomogeneous fractional quasilinear equations in R N with exponential growth of the form By using a suitable Trudinger-Moser inequality for fractional Sobolev spaces, they established the existence of weak solutions for problem (1.4).In particular, Nyamoradi and Razani [13] considered a class of new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity.The existence of infinitely many solutions was obtained by using the concentration compactness principle and Krasnoselskii's genus theory.For more recent advances on this kind of problems, we refer the readers to [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].
On the other hand, one of the main features of problem (1.1) is the presence of the magnetic field operator A. When A = 0, some authors have studied the following equation which has appeared in recent years, where the magnetic operator in equation (1.5) is given by Squassina and Volzone [29] proved that up to correcting the operator by the factor (1 − s), it follows that (−∆) s A u converges to −(∇u − iA) 2 u as s → 1.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 problems with fractional magnetic operator.In particular, Mingqi et al. [30] proved some existence results for the following Schrödinger-Kirchhoff type equation involving the magnetic operator where f satisfies the subcritical growth condition.For the critical growth case, Binlin et al. [31] considered the following fractional Schrödinger-equation with critical frequency and critical growth The existence of ground state solution tending to trivial solution as ε → 0 was obtained by using variational method.Furthermore, Song and Shi [32] were concerned with a class of the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields, under suitable additional assumptions, the existence of infinite solutions was obtained by using the variational method.More results about fractional equations involving the Hardy-Littlewood-Sobolev and critical nonlinear and can be found in [33][34][35][36].Inspired by the above works, in this paper, we are interested in the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity.As far as we know, there have not been any results for problem (1.1) yet.We note that there are many difficulties in dealing with such problems due to the presence of the electromagnetic field and critical nonlinearity.In order to overcome these difficulties, we shall adopt the concentration-compactness principles and some new techniques to prove the (P S) c condition.Moreover, we shall use the variational methods in order to establish the existence and multiplicity of solutions for problem (1.1).Here are our main results.
Theorem 1.1.Suppose that conditions (V ) and (M ) are satisfied.Then there exists λ * > 0 such that if λ > λ * > 0, then there exists at least one solution u λ of problem (1.1) and u λ → 0 as λ → ∞.Theorem 1.2.Suppose that conditions (V ) and (M ) are satisfied.Then for any m ∈ N, there exists This paper is organized as follows.In Section 2, we present the working space and the necessary preliminaries.In Section 3, we apply the principle of concentration compactness to prove that the (P S) c condition holds.In Section 4, we check that the mountain pass geometry is established.In Section 5, we use the critical point theory and some subtle estimates to prove our main results.

Preliminaries
In this section we shall give the relevant notations and some useful auxiliary lemmas.For other background information we refer to Papageorgiou et al. [37].Let where s ∈ (0, 1) and The norm of the fractional Sobolev space is given by .
In order to study our problem (1.1), we shall use the following subspace of W s,p A (R, C) defined by with the norm The condition (V ) implies that E ֒→ W s,p A (R N , C) is continuous.Next, we state the well known Hardy-Littlewood-Sobolev inequality and the diamagnetic inequality which will be used frequently.

The Palais-Smale condition
First, we define the set In order to prove the compactness condition, we introduce the following fractional version of the concentration compactness principle.Lemma 3.1.(see Xiang and Zhang [39]) Assume that there exist bounded non-negative measures ω, ζ and ν on R N , and at most countable set {x i } i∈I ∈ Ω\{0} such that Then there exist a countable sequence of points {x i } ⊂ R N and families of positive numbers {ν i : i ∈ I}, where I is at most countable.Furthermore, we have where is the Dirac mass of mass 1 concentrated at {x i } ⊂ R N .
Lemma 3.2.(see Xiang and Zhang [39]) Let {u n } n ⊂ W s,p R N be a bounded sequence such that and define Then the quantities ω ∞ ,ζ ∞ and ν ∞ are well defined and satisfy In addition, the following inequality holds In order to prove the main results, we define the energy functional of problem (1.1) as follows Under hypothetical conditions (V ) and (M ), a simple test as in Willem [40], yields that J λ ∈ C 1 (E, R) and its critical points are weak solutions of problem (1.1), if where and v ∈ E.
Next, we state and prove the following lemma.
Lemma 3.3.Assume that conditions (V ) and (M ) hold.Then any (P S) c sequence {u n } n for J λ is bounded in E and c ≥ 0.
Proof.Suppose that {u n } n ⊂ E is a (P S) sequence for J λ .Then we have and It follows from (3.6), (3.7) and (M ) that This fact implies that {u n } n is bounded in E. We also obtain c ≥ 0 from (3.8).Now, we can show that the following compactness condition holds.
Lemma 3.4.Assume that conditions (V ) and (M ) hold.Then J λ (u) satisfies (P S) c condition, for all sp < N < sp 2 and Proof.Let {u n } n be a (P S) c sequence for J λ .Then, by Lemma 3.3, we know that the sequence {u n } n is bounded in E.Moreover, we know that there exists a subsequence, still denoted by {u n }, such that u n ⇀ u weakly in E.Moreover, we have Now, by the concentration-compactness principle, we may assume that there exist bounded non-negative measures ω, ζ and ν on R N , and an at most countable set {x i } i∈I ∈ Ω\{0} such that Now, there exists a countable sequence of points {x i } ⊂ R N and families of positive numbers {ν i : i ∈ I}, We can also get In the sequel, we shall prove that Suppose to the contrary, that I = ∅.Then we can define a smooth cut-off function such that φ ∈ where By the Hölder inequality, we know that This is an obvious contradiction to the choice of c.This completes the proof of (3.11).Next, we shall prove the concentration at infinity.To this end, set φ R ∈ C ∞ 0 (R N ) for R > 0, and R .Invoking Theorem 2.4 of Xiang and Zhang [39], we define By Lemma 3.2, we have Moreover, (3.17) Similar discussion as above yields Furthermore, proceeding as in the proof of (3.14), we can get ν ∞ = 0. Thus Moreover, it is easy to see that By (3.20), (3.21) and the Hölder inequality, we have It follows from u n → u a.e in R N and the Fatou lemma that and We note that (d 1 ) and is nondecreasing for t ≥ 0. Thus, by Then u n E → u E .We note that E is a reflexive Banach space, thus u n → u strongly converges in E. This completes the proof of Lemma 3.4.

Auxiliary results
First, we shall prove that functional J λ has a mountain path structure.
Lemma 4.1.Let the conditions (V ) and (M ) hold.Then where F is a finite-dimensional subspace of E.
Proof.It follows from the Hardy-Littlewood-Sobolev inequality that there exists C(N, µ) > 0 such that By virtue of (V ) and (M ), we get Since p * s,µ , q > p, we know that the conclusion (C 1 ) of Lemma 4.1 holds.
In order to prove the conclusion (C 2 ) of Lemma 4.1, we note that it follows from the condition (m 2 ) that Note that all norms are equivalent in a finite-dimensional space.Then the above fact together with p < p σ < 2p * s,µ implies that the conclusion (C 2 ) of Lemma 4.1 holds.
Invoking Binlin et al. [31, Theorem 3.2], we have inf and define So, we have Since q > p/σ, we can find Using the above analysis, we can prove the following conclusions.
Lemma 4.3.Let the conditions (V ) and (M ) hold.Then for each κ > 0, there exists λ κ > 0 such that for any 0 < λ κ < λ, and e λ ∈ E we have that e λ > ̺ λ , J λ (t e λ ) ≤ 0 and for C > 0. Similar to the previous discussion, we have and we can get the following estimate max for any ζ → 0 and C > 0. From (4.11), we get the following lemma.

Proofs of main results
In the section, we shall prove the existence and multiplicity of solutions for problem (1.1).

pN −p µ 2 N
−ps is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, V ∈ C(R N , R + 0