Existence and Symmetry of Solutions for a Class of Fractional Schrödinger–Poisson Systems

: In this paper, we investigate a class of Schrödinger–Poisson systems with critical growth. By the principle of concentration compactness and variational methods, we prove that the system has radially symmetric solutions, which improve the related results on this topic.


Introduction
In recent years, fractional equations or systems have been studied extensively by researchers due to their various applications in various fields, such as obstacle problems, electrical circuits, quantum mechanics, and phase transitions; see [1][2][3][4][5][6][7] and their references. It is particularly important to mention that Laskin in [7] established the following timedependent Schrödinger equation involving a fractional Laplacian when he expanded the Feynman path integral, from Brownian-like to Lévy-like quantum mechanical paths i ∂ψ ∂t = (−∆) α ψ + (V(x) + κ)ψ − g(x, t), (x, t) ∈ R 3 × R. (1) where κ > 0 is a constant and (−∆) α = F −1 (|ξ| 2α F u) is the fractional Laplacian of order α, and F denotes the usual Fourier transform in R 3 . The fractional Schrödinger equation is a fundamental equation in the fractional quantum mechanics when investigating the quantum particles on stochastic fields modeled, and it has been getting a lot of attention from researchers; see [8][9][10][11][12][13] and their references. For example, Li et al. in [13] studied the following form of fractional Schrödinger equations where α ∈ (0, 1), 2 * α = 2N N−2α is the fractional critical Sobolev exponent, λ > 0 is a parameter, V(x) is potential function with lim |x|→+∞ V(x) = 0, i.e., V(x) is vanishing at infinity. Using the variational method, they obtained the existence of a positive solution for (2).
When dealing with the quantum particle in three-dimensional space interacting with an unknown electromagnetic field, Benci and Fortunato in [14] first introduced the following classical Schrödinger-Poisson system −∆u + V(x)u + φu = f (x, u), x ∈ R 3 , −∆φ = u 2 , x ∈ R 3 , field can be described by coupling the nonlinear Schrödinger-Poisson system. Recently, system (3) has been widely investigated because it has a strong physical meaning; see [15][16][17] and their references. For example, Azzollini and Pomponio in [15] considered system (3) by variational methods; they established the existence of a ground state solution when potential V(x) is a positive constant or non-constant. Fractional Schrödinger-Poisson systems have received lots of attention in recent years, and many of the works have studied the existence of solutions of it; see [18][19][20][21][22] and their references. As far as we know, few studies have considered the existence of solutions for the fractional Schrödinger-Poisson system with critical growth. Gu et al. in [19] only studied the existence of a positive solution by variational methods, and there are no relevant articles that consider the existence of radially symmetric solutions of the fractional Schrödinger-Poisson system with critical growth. We tried to deal with this problem and obtained novel existence results by using new analytical methods, which are different from the related conclusions on this topic.
Motivated by above results, in this paper, we mainly study the following fractional Schrödinger-Poisson system where 0 < α ≤ t < 1, 4α + 2t > 3, 2 * α = 6 3−2α . We assume V(x) and K(x) are radially symmetric, i.e., V(x) = V(|x|) and K(x) = K(|x|) for any x ∈ R 3 , and satisfies the following assumptions and there exists a constant K 0 such that 0 ≤ ∇K(x) · x ≤ K 0 for any x ∈ R 3 ; (K2) Positive constants K 1 and K 1 exist such that K 1 ≤ K(x) ≤ K 2 for any x ∈ R 3 ; We make the following assumptions for the function f .
Now we state the main results of this paper. There exists λ 1 > 0 such that for any 0 < λ < λ 1 , problem (4) has a nontrivial radially symmetric solution.
This paper is organized as follows. In Section 2, we introduce the preliminaries of fractional Sobolev space and some lemmas. In Section 3, we give the proofs of our main result.

Preliminaries
In this section, we present a short review of fractional Sobolev spaces and some lemmas, which we used to prove our main result. The complete introduction of fractional Sobolev spaces can be found in [23]. For the properties of the function φ s u , see [18]. Throughout this paper, we denote · p as the usual norm of L p (R 3 ), C i (i = 1, 2, · · · ) or C denotes the positive constants.
For α ∈ (0, 1), the fractional Sobolev space H α (R 3 ) is defined by (1 + |ξ| 2α )|F u(ξ)| 2 dξ < +∞ . and the norm is and norm For the second equation in problem (4), it has an unique solution φ t u (see [18]). Substituting φ t u in (4), we have the following fractional Schrödinger equation whose solutions can be obtained by seeking critical points of the energy functional I λ : In the following, the symmetric mountain pass lemma is presented and used to prove our main result.

Lemma 2 ([25]
). Let X be an infinite-dimensional Banach space with X = M N, where M is a finite-dimensional subspace of X. Assume that I ∈ C 1 (X, R) is a functional that satisfies the (PS)-condition and the following properties: (A1) I is even for all u ∈ X and I(0) = 0. (A2) There exist two constant , σ > 0 such that I(u) ≥ σ.
In the following, we prove that the functional I λ satisfies the mountain pass geometry. and Therefore, by Lemma 1, (7) and Hölder's inequality, we have which implies that there exist two positive constants β, ρ > 0 such that [18]) and (8), for t > 0 and any u ∈ H α (R 3 ), we have Lemma 4. Assume that (f1)-(f4) are satisfied. Then, any (PS)-sequence for I λ is bounded.
Proof. Let {u n } ⊂ be a (PS)-sequence. By (f3), for θ > 4, we have Then the quantities µ ∞ and ν ∞ are well defined and satisfy lim sup , where x i , ν i are from Lemma 5 and µ ∞ , ν ∞ are given in Lemma 6, S α is the best In the following lemma, we show that the functional I λ satisfies the (PS)-condition.

By (6) and the Young inequality, one has
Then and G ε,n (x) → 0 a.e. on R 3 . By the Lebesgue dominated convergence theorem, we have By the arbitrariness of ε, we have Next we will verify that u n → u 0 in L 2 * α (R 3 ). We show that there exists λ * > 0 such that ν i = 0 and ν ∞ = 0 for 0 < λ < λ * and i ∈ J. We argue by contradiction. Assume that i 0 ∈ J exists such that ν i 0 > 0 or ν ∞ > 0, by Lemma 7, we have Let and By (f3), we obtain (11) and (12), we know that which means that By (6), we have t f (t) ≤ ε|t| 2 + C ε |t| p .
Choose λ large enough to satisfy λ > C ε , then lim sup By the proposition 3.6 in [23], we have Therefore, for any v ∈ H α (R 3 ), we have By (17), we have It is easy to know that By the Hölder inequality, one has By the following equality (see (3.7) in [27]) (14) and (15), we obtain that From (10) and (13), one has which contradicts 0 < λ < λ * . Therefore, for any i ∈ J, ν i = 0 and ν ∞ = 0. By Lemma 6, we know lim sup Thus, Since u n u 0 weakly in H α (R 3 ), then by (18), we have which implies that u n → u 0 in H α (R 3 ).

Lemma 9.
Let Ω be every finite subspace of H α (R 3 ). Then, for any u ∈ Ω, it holds Proof. Arguing by contradiction, there exists C 6 > 0 such that I(u n ) ≥ −C 6 for any n ∈ N.

Conclusions
In this paper, we study a class of fractional Schrödinger-Poisson systems with critical growth (4). Problem (4) comes from the interaction of a charge particle with an electromagnetic field in three-dimensional space. By using the principle of concentration compactness, we established a compactness result about functional I λ . Applying the mountain pass theorem, a nontrivial radially symmetric solution was obtained. Infinitely nontrivial radially symmetric solutions are presented by the symmetric mountain pass lemma. Therefore, our results improve the related conclusions on this topic.