Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 47, pp. 1–21. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu MULTIPLICITY OF HIGH ENERGY SOLUTIONS FOR FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS WITH CRITICAL FREQUENCY SIQI QU, XIAOMING HE Abstract. In this article we study the fractional Schrödinger-Poisson system ε2s(−∆)su+ V (x)u = φ|u|2 ∗ s−3u, x ∈ R3, (−∆)sφ = |u|2 ∗ s−1, x ∈ R3, where s ∈ (1/2, 1), ε > 0 is a parameter, 2∗s = 6/(3−2s) is the critical Sobolev exponent, V ∈ L 3 2s (R3) is a nonnegative function which may be zero in some region of R3. By means of variational methods, we present the number of high energy bound states with the topology of the zero set of V for small ε. 1. Introduction In the past decades, the nonlinear Schrödinger-Poisson system −∆u+ V (x)u+K(x)φu = f(x, u), x ∈ R3, −∆φ = K(x)u2, x ∈ R3, (1.1) has been the interesting object for many researcher. As a model describing the interaction of a charge particle with an electromagnetic field, it arises in mathe- matical physics context [5], and is usually known as Schrödinger-Poisson system. In the pioneering paper [5], Benci and Fortunato studied the eigenvalue problem for (1.1) in bounded domain Ω ⊂ R3 by using the variational methods. After that, the existence and multiplicity of solutions for Schrödinger-Poisson system (1.1) under variant assumptions on V,K and f , had been widely investigated by numerous authors and there have developed many effective methods to deal with equations or systems with nonlocal terms, we refer the interested readers to see [1, 2, 8, 24, 29, 37, 38] and the references therein. For the Schrödinger-Poisson system with a nonlocal critical term −∆u+ V (x)u−K(x)φ|u|3u = f(x, u), x ∈ R3, −∆φ = K(x)|u|5, x ∈ R3, (1.2) Liu [23] obtained the existence of positive solutions by using mountain pass theorem and the concentration-compactness principle. Li and He [25] studied the existence 2020 Mathematics Subject Classification. 35B35, 35B40, 35K57, 35Q92, 92C17. Key words and phrases. Fractional Schrödinger-Poisson system; high energy solution; critical Sobolev exponent. ©2022. This work is licensed under a CC BY 4.0 license. Submitted March 11, 2022. Published July 5, 2022. 1 2 S. QU, X. HE EJDE-2022/47 and multiplicity of positive solutions for (1.2) by using the variational methods. Az- zollini, d’Avenia, Vaira [3] studied the existence and nonexistence results of positive and sign-changing solutions for (1.2) on bounded domains. Li, Li and Shi [19, 20] considered positive solutions to the another Schrödinger-Poisson-type systems with critically growing nonlocal term −∆u+ bu+ qφ|u|3u = f(u), x ∈ R3, −∆φ = |u|5, x ∈ R3, (1.3) and obtained the existence of positive solutions to (1.3). Guo [15] obtained two positive bound state solutions of (1.3) with b = 0, q = −1, f(u) = λQ(x)|u|p−2u, a sublinear term, by decomposition the Nehari manifold and fine estimates. In the setting of the fractional Laplacian, system (1.1), or (1.2) becomes the fractional Schrödinger-Poisson type system. It is a fundamental equation in frac- tional quantum mechanics in the study of particles on stochastic fields modeled by Lévy processes [7, 17, 18]. By using a perturbation approach, Zhang, do Ó and Squassina [33] considered the existence and the asymptotical behaviors of positive solutions to the fractional Schrödinger-Poisson system (−∆)su+ V (x)u+K(x)φu = f(x, u), x ∈ R3, (−∆)tφ = K(x)u2, x ∈ R3. (1.4) with V (x) = 0 and K(x) = λ > 0, a parameter, and a general subcritical or crit- ical nonlinearity f . Teng [31] analyzed the existence of ground state solutions of (1.4) with K(x) = 1 and f(x, u) = µ|u|q−1u + |u|2∗s−2u, q ∈ (2, 2∗s), by combining Pohozaev-Nehari manifold, arguments of Brezis-Nirenberg type, the monotonicity trick and global compactness Lemma. Murcia and Siciliano [27] studied the semi- classical state of the system ε2s(−∆)su+ V (x)u+K(x)φu = f(u), x ∈ RN , εθ(−∆)α/2φ = γαu 2, x ∈ RN . (1.5) and established the multiplicity of positive solutions that concentrate on the min- ima of V (x) as ε → 0 by the Ljusternik-Schnirelmann category theory. Liu and Zhang [22] studied multiplicity and concentration of solutions of (1.5) when the nonlinearity f is critical growth. Recently, Yang, Yu and Zhao [32] considered the fractional Schrödinger-Poisson system with critical exponent ε2s(−∆)su+ V (x)u+ φu = f(u) + |u|2 ∗ s−2u, x ∈ R3, ε2t(−∆)tφ = u2, x ∈ R3, (1.6) where f(u) = λ|u|p−2u, λ > 0, 4s+2t s+t < p < 2∗s, and the potential V satisfies the following hypotheses introduced by del Pino and Felmer [11]: (A1) V ∈ C(R3,R) and infx∈R3 V (x) > 0. (A2) There exists a bounded open set Λ ⊂ R3 such that V0 := inf Λ V < min ∂Λ V and M = {x ∈ Λ : V (x) = V0} 6= ∅. Using penalization techniques and concentration-compactness principle, the authors obtained a positive ground state solution for ε > 0 small, and they showed that these ground state solutions concentrate around a local minimum of V as ε→ 0. Later, Ambrosio [4] obtained the multiplicity and concentration of positive solutions to EJDE-2022/47 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 3 (1.6) by the penalization techniques and Ljusternik-Schnirelmann theory. Recently, Chen, Li and Peng [10] obtained multiple higher e nergy solutions of (1.6) by a global compactness lemma and Lusternik-Schnirelman theory, where f(u) ≡ 0. For more related results for (1.6), we refer to [14, 34] and references therein. In the fractional scenario, (1.2) takes the form (−∆)su+ V (x)u−K(x)φ|u|2 ∗ s−3u = f(x, u), x ∈ R3, (−∆)sφ = K(x)|u|2 ∗ s−1, x ∈ R3, (1.7) and we note that there are only two papers that deals with fractional Schrödinger- Poisson system with nonlocal critical term after a bibliography review. Feng [12] proved the existence of mountain pass type solution of (1.7) and extended the main results of [23] to the fractional Lapalcian case. He [16] considered the fractional Schrödinger-Poisson system with doubly critical exponents (−∆)su+ V (x)u− φ|u|2 ∗ s−3u = |u|2 ∗ s−2u+ f(u), x ∈ R3, (−∆)sφ = |u|2 ∗ s−1, x ∈ R3, (1.8) and proved the existence of a mountain pass solution by employing the concentration- compactness principle and mountain pass theorem. The novelty of (1.7) and (1.8) is that the second equation is nonlocal critical growth and driven by nonlocal operators, which make the study of problem (1.7) and (1.8) more interesting and challenging. As we observed that, the previous results on the existence and multiplicity of solutions for systems (1.7) and (1.8), were mainly focused on the existence of ground state solutions under the condition (A1). The multiplicity of semiclassical states for (1.7) with both critical frequency (V (x) ≥ 0, 6≡ 0, x ∈ R3) and critical growth has not been considered before. The purpose of this paper is to fill this gap. Concretely speaking, we study the fractional Schrödinger-Poisson system ε2s(−∆)su+ V (x)u = φ|u|2 ∗ s−3u, x ∈ R3, (−∆)sφ = |u|2 ∗ s−1, x ∈ R3, (1.9) where the potential V (x) satisfies the assumption (A3) V ∈ L 3 2s (R3), V (x) ≥ 0 on R3, and the set M = {x ∈ R3 : V (x) = 0} is nonempty and bounded. Recall that, if Y is a closed subset of a topological space X, the Lusternik- Schnirelmann category catX(Y ) is the least number of closed and contractible sets inX, which cover Y . For any fixed τ > 0. DenoteMτ = {x ∈ RN : dist(x,M) ≤ τ}. Now, we state our main results. Theorem 1.1. Let (A3) be satisfied. Then, for any τ > 0, there exist ετ > 0 such that for any ε ∈ (0, ετ ), then system (1.9) has at least catMτ (M) high energy semiclassical states in Ds,2(R3). The proof of Theorem 1.1 is of variational. From the technical point of view, the appearance of the double non-localities from nonlocal critical convolution term and the nonlocal operator in system (1.9) make the bounded (PS) sequences could not converge. It is difficult for us to check the (PS)c condition since the nodal solutions of (1.9) do not possess the double energy characteristics, as we known the double energy property plays a key role in proving the main result in [10, 34]. To overcome 4 S. QU, X. HE EJDE-2022/47 these difficulties, we shall employ some idea from [13] and establish a new global compactness lemma in the fractional case, and some estimates become more subtle and delicate to be established. We shall construct two barycenter functions, and use Ljusternik-Schnirelmann category theory to obtain the desired results. As far as we know, the multiplicity of high energy solutions for system (1.9) has not been studied in the literature. This article is organized as follows: In Section 2 we give some preliminary results. In Section 3, we introduce the limit problem and prove some useful lemmas. In Section 4, we give a global compactness lemma which describes the behavior of Palais-Smale sequences, and we regain the compactness if the functional energy lies in a suitable interval. In Section 5, we first define two barycenter functions and present some estimations; after these preparations, we complete the proof of Theorem 1.1 by means of the Ljusternik-Schnirelmann category theory. 2. Preliminaries We recall that the fractional Sobolev space Ds,2(R3) is defined as Ds,2(R3) = {u ∈ L2∗s (R3) : ∫ R3 |(−∆)s/2u|2dx <∞}, with the norm [26, 28] ‖u‖2 := ∫ R3 |(−∆)s/2u|2dx = ∫∫ R6 |u(x)− u(y)|2 |x− y|3−2s dx dy. The embedding Ds,2(R3)→ L2∗s (R3) is continuous and there exists a best constant S > 0 such that S = inf u∈Ds,2(R3)\{0} ∫ R3 |(−∆)s/2u|2dx( ∫ R3 |u|2∗sdx )2/2∗s . (2.1) From the Lax-Milgram theorem, we have that, for given u ∈ Ds,2(R3), there exists a unique solution φ = φu ∈ Ds,2(R3) satisfying (−∆)sφu = |u|2∗s−1 in a weak sense. The function φu is represented by φu(x) = Cs ∫ R3 |u(y)|2∗s−1 |x− y|3−2s dy, x ∈ R3, where Cs = π− 3 2 2−2sΓ(3− 2s)(Γ(s))−1, see [16, 12] for instance. Then function φu has the following properties. Lemma 2.1. For each u ∈ Ds,2(R3), the function φu has the following properties: (i) φθu = θ2∗s−1φu for all θ > 0. (ii) For each u ∈ Ds,2(R3), one has ‖φu‖ ≤ S−1/2|u|2 ∗ s−1 2∗s and∫ R3 φu|u|2 ∗ s−1dx ≤ S−1/2 (∫ R3 |u|2 ∗ sdx )(2∗s−1)/2∗s ‖φu‖ ≤ S−1|u|2(2∗s−1) 2∗s . (iii) If un ⇀ u in Ds,2(R3), and un → u a.e. in R3, then φun ⇀ φu in Ds,2(R3); and φun − φun−u − φu → 0 in Ds,2(R3). (iv) If un ⇀ u in Ds,2(R3), and un → u a.e. in R3, then∫ R3 φun |un|2 ∗ s−1dx− ∫ R3 φun−u|un − u|2 ∗ s−1dx− ∫ R3 φu|u|2 ∗ s−1dx→ 0, (2.2) EJDE-2022/47 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 5 and φun |un|2 ∗ s−3un − φun−u|un − u|2 ∗ s−3(un − u)− φu|u|2 ∗ s−3u ⇀ 0, (2.3) in (Ds,2(R3))∗, where (Ds,2(R3))∗ is the dual space of Ds,2(R3). Proof. Item (i) is obvious from the definition of φu. (ii) For each u ∈ Ds,2(R3), we can deduce that∫ R3 |(−∆)s/2φu|2dx = ∫ R3 φu|u|2 ∗ s−1dx ≤ (∫ R3 |u|2 ∗ sdx )(2∗s−1)/2∗s (∫ R3 |φu|2 ∗ sdx )1/2∗s ≤ S−1/2 (∫ R3 |u|2 ∗ sdx )(2∗s−1)/2∗s ‖φu‖, (2.4) and so ‖φu‖ ≤ S−1/2 (∫ R3 |u|2 ∗ sdx )(2∗s−1)/2∗s = S−1/2|u|2 ∗ s−1 2∗s . Therefore,∫ R3 φu|u|2 ∗ s−1dx ≤ S−1/2 (∫ R3 |u|2 ∗ sdx )(2∗s−1)/2∗s ‖φu‖ ≤ S−1|u|2(2∗s−1) 2∗s . (iii) From the Sobolev embedding, one has un ⇀ u ∈ L2∗s (R3). Then |un|2 ∗ s−1 ⇀ |u|2∗s−1 in L 2∗s 2∗s−1 (R3). Using Brezis-Lieb lemma [6], we have that |un|2 ∗ s−1 − |un − u|2 ∗ s−1 − |u|2 ∗ s−1 → 0 in L 2∗s 2∗s−1 (R3). (2.5) Therefore, for any v ∈ Ds,2(R3) ↪→ L2∗s (R3), we obtain (φun , v) = ∫ R3 |un|2 ∗ s−1v dx→ ∫ R3 |u|2 ∗ s−1v dx = (φu, v), which reveals that φun ⇀ φu in Ds,2(R3). Since for every w ∈ Ds,2(R3). |〈φun − φvn − φu, w〉| = ∣∣ ∫ R3 w(|un|2 ∗ s−1 − |vn|2 ∗ s−1 − |u|2 ∗ s−1) ∣∣ ≤ |w|2∗s ||un| 2∗s−1 − |vn|2 ∗ s−1 − |u|2 ∗ s−1|2∗s/(2∗s−1), then φun − φun−u − φu → 0 in Ds,2(R3), which implies the assertion. (iv) Set vn = un − u, then vn ⇀ 0 in Ds,2(R3) ↪→ L2∗s (R3) and vn → 0 a.e. in R3. By item (iii) we have φvn ⇀ 0 in Ds,2(R3). Since un → u a.e. in R3, and un ⇀ u ∈ L2∗s (R3), we infer that |un|2 ∗ s−1 ⇀ |u|2∗s−1 in L 2∗s 2∗s−1 (R3), and so, (2.5) holds. Consequently, by the weak convergence of {un} and Hölder inequality, as n→∞, we obtain∫ R3 φun |un|2 ∗ s−1dx− ∫ R3 φvn |vn|2 ∗ s−1dx− ∫ R3 φu|u|2 ∗ s−1dx = ∫ R3 [φun − φvn − φu]|un|2 ∗ s−1dx+ ∫ R3 φvn [|un|2 ∗ s−1 − |un − u|2 ∗ s−1 − |u|2 ∗ s−1]dx + ∫ R3 φvn |u|2 ∗ s−1dx+ ∫ R3 φu[|un|2 ∗ s−1 − |u|2 ∗ s−1]dx→ 0, 6 S. QU, X. HE EJDE-2022/47 which implies (2.2). To prove (2.3), we have by item (iii) that φun ⇀ φu inDs,2(R3), which yields φun ⇀ φu in L2∗s (R3). On the other hand, by virtue of un → u a.e. in R3 and∫ R3 |φun |un|2 ∗ s−3un| 2∗s 2∗s−1 dx ≤ (∫ R3 |φun |2 ∗ sdx ) 1 2∗s−1 (∫ R3 |un|2 ∗ xdx ) 2∗s−2 2∗s−1 dx ≤ C. From [36, Proposition 5.4.7], we see that φun |un|2 ∗ s−3un ⇀ φu|u|2 ∗ s−3u, φvn |vn|2 ∗ s−3vn ⇀ 0 in L 2∗s 2∗s−1 (R3). Hence, for any ϕ ∈ Ds,2(R3), we have∫ R3 [φun |un|2 ∗ s−3un − φun−u|un − u|2 ∗ s−3(un − u)]ϕdx→ ∫ R3 φu|u|2 ∗ s−3uϕdx, and (2.3) follows. � Lemma 2.2. If N ≥ 3, and V (∈ L N 2s (RN ), then the functional H(u) = ∫ RN V (x)u2dx is weakly continuous in Ds,2(RN ). Proof. It is similar to the proof of [35, Lemma 2.13], we sketch it here for conve- nience. The functional H is well defined by the Sobolev space and Hölder inequal- ities. Assume that un ⇀ u weakly in Ds,2(RN ). Since the Sobolev embedding Ds,2(RN ) ↪→ L2∗s (RN ) is continuous, we have un ⇀ u weakly in L2∗s (RN ), and so, u2 n ⇀ u2 weakly in L N N−2s (RN ). Note that V ∈ L N 2s (RN ) = (L N N−2s (RN ))∗, thus,∫ RN V (x)u2dx→ ∫ RN V (x)u2dx as n→∞, which implies the conclusion. � Finally, we recall that the Hardy-Littlewood-Sobolev inequality. Proposition 2.3 ([21]). Let t, r > 1 and 0 < α < n with 1/t+ α/n+ 1/r = 2, f ∈ Lt(Rn) and h ∈ Lr(Rn). There exists a sharp constant C(t, n, α, r) independent of f, h such that ∫∫ R2n f(x)h(y) |x− y|α dx dy ≤ C(t, n, α, r)|f |t|h|r. (2.6) If t = r = 2N 2N−α , then C(t,N, α, r) = C(N,α) = πα/2 Γ(π2 − α 2 ) Γ(N − α 2 ) { Γ(π2 ) Γ(N) }−1+ α N . In this case there is equality in (2.6) if and only if f ≡ (constant)h and h(x) = A(γ2 + |x− a|2)− 2N−α 2 for some A ∈ C, 0 6= γ ∈ R and a ∈ RN . EJDE-2022/47 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 7 3. Limit problem To study (1.9), we need to introduce the limit equation. First, we introduce the system (−∆)su = φ|u|2 ∗ s−3u, x ∈ R3, (−∆)sφ = |u|2 ∗ s−1, x ∈ R3. (3.1) It is well known that the best embedding constant S is achieved at the function [30]: U(x) = β1 (1 + |x|2)(3−2s)/2 , β1 = (S3/(2s)Γ(3) π3/2Γ(3/2) ) 3−2s 6 , and U(x) is a ground state solution of the equation (−∆)su = |u|2 ∗ s−2u, x ∈ R3. (3.2) Moreover, by the invariance of scaling and translation, we known that the function Uδ,z0(x) := δ− 3−2s 2 U (x− z0 δ ) solves (3.2), and satisfies ‖Uδ,z0‖2 = |Uδ,z0 | 2∗s 2∗s = S 3 2s . (3.3) The following observation is useful in the energy estimation of the functionals below. Lemma 3.1. Assume that u, φ are positive solutions of (3.1), then u(x) = φ(x) = Uδ,z0(x) for some z0 ∈ R3, and δ > 0. Proof. Let u and φ be a pair of positive solution to (3.1). Then we have (−∆)s(u− φ) = (φ− u)|u|2 ∗ s−2, x ∈ R3. Multiplying this equation by u− φ, and integrating by part, we obtain∫ R3 |(−∆)s/2(u− φ)|2dx+ ∫ R3 (u− φ)2|u|2 ∗ s−2dx = 0. Whence, we can conclude u = φ. Furthermore, u satisfies (3.1), and the conclusion follows. � The functional of (1.9) is Iε(u) = ε2s 2 ∫ R3 |(−∆)s/2u|2dx+ 1 2 ∫ R3 V (x)u2dx− 1 2(2∗s − 1) ∫ R3 φu|u|2 ∗ s−1dx, and introduce the limit equation of (1.9) as ε2s(−∆)su = φ|u|2 ∗ s−3u, x ∈ R3, (−∆)sφ = |u|2 ∗ s−1, x ∈ R3. (3.4) whose energy functional is denoted by Jε(u) = ε2s 2 ∫ R3 |(−∆)s/2u|2dx− 1 2(2∗s − 1) ∫ R3 φu|u|2 ∗ s−1dx. By Lemma 3.1, it is easy to see that u = ε 3−2s 4 Uδ,x0 is the ground state of (3.4) and Jε(u) = ε 3+2s 2 2 ∫ R3 |(−∆)s/2Uδ,x0 |2dx− ε 3+2s 2 2(2∗s − 1) ∫ R3 φUδ,x0 |Uδ,x0 |2 ∗ s−1dx 8 S. QU, X. HE EJDE-2022/47 = ε 3+2s 2 (1 2 − 1 2(2∗s − 1) ) S 3 2s = 2s 3 + 2s ε 3+2s 2 S 3 2s . For each ε > 0, we define the Nehari manifolds of Iε, Jε as follows Nε = {u ∈ Ds,2(R3)\{0} : I ′ε(u)u = 0}, N∞ε = {u ∈ Ds,2(R3)\{0} : J ′ε(u)u = 0}, mε := inf u∈Nε Iε(u), m∞ε := inf u∈N∞ε Jε(u). Lemma 3.2. Let (A3) hold. Then mε = 2s 3+2sε 3+2s 2 S 3 2s . Proof. For any u ∈ Nε, by Lemma 2.1, we have ε2s ∫ R3 |(−∆)s/2u|2dx ≤ ε2s ∫ R3 |(−∆)s/2u|2dx+ ∫ R3 V (x)u2dx = ∫ R3 φu|u|2 ∗ s−1dx ≤ S−1|u|2(2∗s−1) 2∗s ≤ S−2∗s (∫ R3 |(−∆)s/2u|2dx )2∗s−1 , (3.5) which implies that∫ R3 φu|u|2 ∗ s−1dx = ‖u‖2 + ∫ R3 V (x)u2dx ≥ ‖u‖2 ≥ ε 3−2s 2 S 3 2s . (3.6) Hence, Iε(u) = Iε(u)− 1 2(2∗s − 1) I ′ε(u)u = 2s 3 + 2s ε2s‖u‖2 + 2s 3 + 2s ∫ R3 V (x)u2dx ≥ 2s 3 + 2s ε 3+2s 2 S 3 2s , (3.7) which implies that mε ≥ 2s 3 + 2s ε 3+2s 2 S 3 2s . Now, consider vn(x) = ε 3−2s 4 w(x − zn) ∈ N∞ε , where w is a positive solution of (3.1) centered at zero, and zn ∈ R3 satisfies |zn| → ∞. Let tn > 0 be such that tnvn ∈ Nε. Using vn ⇀ 0 in Ds,2(R3), and Lemma 2.2, we have∫ R3 V (x)v2 ndx→ 0. Then tn → 1. Hence, mε ≤ Iε(tnvn) = Iε(vn) + on(1) = Jε(vn) + on(1) = 2s 3 + 2s ε 3+2s 2 S 3 2s + on(1), which implies that mε = 2s 3+2sε 3+2s 2 S 3 2s . � Lemma 3.3. Let (A3) hold. Then mε = m∞ε holds, and mε is not attained. EJDE-2022/47 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 9 Proof. For u ∈ Ds,2(RN ), we define the function γ(t) = 〈I ′ε(tu), tu〉 = t2ε2s ∫ R3 |(−∆)s/2u|2dx+ t2 ∫ R3 V (x)u2dx− t2(2∗s−1) ∫ R3 φu|u|2 ∗ s−1dx = at2 − bt22∗α,s , (3.8) where a = ∫ R3 (ε2s|(−∆)s/2u|2 + V (x)u2)dx, b = ∫ R3 φu|u|2 ∗ s−1dx. It is easy to see that there exist unique t(u) > 0 and s(u) > 0 such that t(u)u ∈ Nε, s(u)u ∈ N∞ε , and Iε(t(u)u) = max t>0 Iε(tu), Jε(s(u)u) = max t>0 Jε(tu). For any u ∈ Nε, m∞ ≤ I∞(s(u)u) ≤ ε2s 2 ‖s(u)u‖2 + 1 2 ∫ R3 V (x)|s(u)u|2dx− |s(u)|2(2∗s−1) 2(2∗s − 1) ∫ R3 φu|u|2 ∗ s−1dx = Iε(s(u)u) ≤ Iε(u), (3.9) which implies that m∞ε ≤ mε. (3.10) Assume that w is a positive solution of (3.4) centered at origin, {zn} ⊂ RN satisfy- ing |zn| → ∞ as n→∞, wn(x) = ε 3−2s 4 w(x− zn) and tn = t(wn). It is clear that wn ⇀ 0 in Ds,2(RN ), and by Lemma 2.2, one has∫ R3 V (x)w2 ndx→ 0. (3.11) Now, we have that Iε(tnwn) = t2nε 2s 2 ‖wn‖2 + t2n 2 on(1)− |tn| 2(2∗s−1) 2(2∗s − 1) ∫ R3 φwn |wn|2 ∗ s−1dx. (3.12) Noting that wn ∈ N∞ε and tnwn ∈ Nε, we obtain ε2s‖wn‖2 = ∫ R3 φwn |wn|2 ∗ s−1dx, (3.13) ε2st2n‖wn‖2 + t2n ∫ R3 V (x)w2 ndx = t2(2∗s−1) ∫ R3 φwn |wn|2 ∗ s−1dx. (3.14) By (3.13) and (3.14) we have ε2s(t 2(2∗s−2) n − 1)‖wn‖2 = ∫ R3 V (x)w2 ndx = on(1), (3.15) which means that tn → 1 as n→∞. By (3.12), we see that limn→∞ Iε(un) = m∞ε . Therefore, we obtain mε ≤ m∞ε , and so mε = m∞ε . Next, we prove that mε cannot be achieved. Suppose by contradiction that, there exists some ũ ∈ Nε such that Iε(ũ) = mε = m∞ε . Then, from s(ũ)ũ ∈ N∞ε , we have m∞ε ≤ Jε(s(ũ)ũ) 10 S. QU, X. HE EJDE-2022/47 ≤ ε2s 2 ‖s(ũ)ũ‖2 + 1 2 ∫ RN V (x)|s(ũ)ũ|2dx− 1 2(2∗s − 1) ∫ RN φs(ũ)ũ|s(ũ)ũ|2 ∗ s−1dx = I(s(ũ)ũ) ≤ I(ũ) = m∞ε . We infer that∫ RN V (x)|s(ũ)ũ|2dx = 0, s(ũ) = 1⇒ ũ ≡ 0 on R3\M. Hence, ũ ∈ N∞ε , Jε(ũ) = m∞ε . Thus, by Lemma 3.1, ũ(x) = ε 3−2s 4 Uδ,z0(x) > 0,∀x ∈ R3, for some δ > 0, z0 ∈ R3, which leads to a contradicts to ũ(x) ≡ 0 on R3\M . � Corollary 3.4. Assume that {un} is a Palais-Smale sequence of Iε constrained on Nε, then {un} is a Palais-Smale sequence of Iε. If u is a critical point of Iε constrained on Nε, then u must be a critical point of Iε. Proof. Let {un} be a Palais-Smale sequence of Iε constrained on Nε, then there exists λn ∈ R such that on(1) = I ′ε(un)− λnQ′ε(un), where Qε(u) = I ′ε(u)u. Note that {un} is bounded in Ds,2(R3). Thus, one has on(1) = I ′ε(un)un − λnQ′ε(un)un = −λnQ′ε(un)un, On the other hand, since 2∗s > 2, then by (3.6), we derive that Q′ε(un)un = 2ε2s ∫ R3 |(−∆)s/2un|2dx+ 2 ∫ R3 V (x)u2 ndx− 2(2∗s − 1) ∫ R3 φun |un|2 ∗ s−1dx = [2(2− 2∗s)] ∫ R3 φun |un|2 ∗ s−1dx ≤ [2(2− 2∗s)]ε 3−2s 2 S 3 2s < 0. (3.16) Thus, λn → 0 as n → ∞. Moreover, by the boundedness of {un}, we assert that {Q′ε(un)} is bounded. Hence, I ′ε(un) → 0 as n → ∞. If u is a critical point of Iε constrained on Nε, then there exists λ ∈ R such that I ′ε(u) = λQ′ε(u). Therefore, 0 = Qε(u) = I ′ε(u)u− λQ′ε(u)u. By the same calculation as for (3.16), one finds that Q′ε(u)u ≤ [2(2− 2∗s)]ε 3−2s 2 S 3 2s . Thus, λ = 0, and then, I ′ε(u) = 0. � 4. Global compactness The following global compactness lemma plays a key role in proving the com- pactness of the (PS) sequences. Lemma 4.1. Under condition (A3), for each ε > 0, suppose that {un} ⊂ Ds,2(R3) is a Palais-Smale sequence of Iε at level c. Then, replacing un if necessary, with a subsequence, there exist a number k ∈ N, sequences of points x1 n, . . . , x k n ∈ R3 and radii r1 n, . . . , r k n such that: EJDE-2022/47 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 11 (1) u0 n ≡ un ⇀ u0 in Ds,2(R3); (2) ujn ≡ (uj−1 n − uj−1)rjn,xjn ⇀ uj in Ds,2(R3), j = 1, 2, . . . , k; (3) ‖un‖2 → ∑k j=0 ‖uj‖2; (4) Iε(un)→ Iε(u 0) + ∑k j=1 Jε(u j), as n → ∞ where u0 is a solution of (1.9) and uj , 1 ≤ j ≤ k, are the nontrivial solutions of (3.4). Moreover, we agree that in the case k = 0 the above holds without uj. Proof. Note that {un} is a (PS)c sequence for Iε; then, we can prove that {un} is bounded in Ds,2(R3). Without loss of generality, we may assume that un ⇀ u0 in Ds,2(R3) as n → ∞, un → u0 a.e. in R3. Moreover, I ′ε(u 0) = 0. In fact, for any ϕ ∈ C∞0 (R3), we obtain I ′ε(un)ϕ = ε2s ∫ R3 (−∆)s/2un(−∆)s/2ϕdx+ ∫ R3 V (x)unϕdx− ∫ R3 φun |un|2 ∗ s−3unϕ. From Lemma 2.1 we can see that φun ⇀ φu0 in Ds,2(R3) and so φun ⇀ φu0 in L2∗s (R3). Therefore,∫ R3 (φun − φu0)|u0|2 ∗ s−3u0ϕ→ 0 as n→∞. (4.1) Since un → u a.e. in R3 and by Hölder inequality, we obtain∫ R3 ∣∣φun(|un|2 ∗ s−3un − |u0|2 ∗ s−3u0) ∣∣ 2∗s 2∗s−1 dx ≤ C ( |φun | 2∗s 2∗s−1 2∗s |un| 2∗s (2 ∗ s−2) 2∗s−1 2∗s + |φun | 2∗s 2∗s−1 2∗s |u0| 2∗s (2 ∗ s−2) 2∗s−1 2∗s ) ≤ C, (4.2) and so φun(|un|2 ∗ s−3un − |u0|2∗s−3u) ⇀ 0 in L 2∗s 2∗s−1 (R3), and thus∫ R3 φun(|un|2 ∗ s−3un − |u0|2 ∗ s−3u0)ϕdx→ 0 as n→∞, (4.3) which together with (4.1) implies∫ R3 φun |un|2 ∗ s−3unϕdx→ ∫ R3 φu0 |u0|2 ∗ s−3uϕdx as n→∞. (4.4) By (4.4) and the weak convergence of un ⇀ u0 in Ds,2(R3), we have I ′ε(u 0)ϕ = lim n→∞ I ′ε(un)ϕ = 0, ∀ϕ ∈ C∞0 (R3). (4.5) Therefore, I ′ε(u 0) = 0 and u is a critical point of Iε by density of C∞0 (R3) in Ds,2(R3). Let v1 n(x) = un(x)− u0(x). By Brezis-Lieb Lemma [6] and Lemma 2.1 (iv), one can easily obtain ‖v1 n‖2 = ‖u1 n‖2 − ‖u0‖2 + on(1), (4.6) Iε(v 1 n) = Iε(u 1 n)− Iε(u0) + on(1), (4.7) I ′ε(v 1 n) = I ′ε(u 1 n)− I ′ε(u0) + on(1). (4.8) Note that v1 n ⇀ 0 in Ds,2(R3). Then, by Lemma 2.2,∫ R3 V (x)|v1 n|2dx = on(1), ∫ R3 V (x)v1 nϕdx = on(1)‖ϕ‖, 12 S. QU, X. HE EJDE-2022/47 for each ϕ ∈ Ds,2(R3). Therefore, Jε(v 1 n) = Iε(v 1 n) + on(1) = Iε(un)− Iε(u) + on(1), J ′ε(v 1 n) = I ′ε(v 1 n) + on(1) = on(1). (4.9) If v1 n → 0 in Ds,2(R3), then the proof is complete. If v1 n 9 0 in Ds,2(R3), then there exists ζ > 0 such that Jε(v 1 n) > ζ > 0. (4.10) We assert that there exist two sequences {rn} ⊂ R+ and {yn} ⊂ R3 such that gn = (v1 n)rn,yn ⇀ g 6= 0, in Ds,2(R3), (4.11) where (v1 n)rn,yn = r 3−2s 2 n v1 n(rnx+ yn). In fact, by (4.9) we have ε2s‖v1 n‖2 = ∫ R3 φv1n |v 1 n|2 ∗ s−1dx+ on(1), (4.12) Jε(v 1 n) = 2s 3 + 2s ∫ R3 φv1n |v 1 n|2 ∗ s−1dx+ on(1). (4.13) From (4.10), Lemma 2.1(ii) and the boundedness of {un}, we derive that 0 < d1 < |v1 n| 2∗s−1 2∗s < D1, (4.14) for some d1, D1 > 0. We introductive a Lévy concentration function Qn(r) := sup x∈R3 ∫ Br(z) |v1 n|2 ∗ sdx. Since Qn(0) = 0 and Qn(∞) > d 6 3+2s 1 , we can assume that there exist sequences {rn} ⊂ R+ and {yn} ⊂ R3 such that sup x∈R3 ∫ Br(z) |v1 n|2 ∗ sdx = ∫ Brn (yn) |v1 n|2 ∗ sdx = b, where 0 < b < min { d 6 3+2s 1 , ( S 2CsD1 ) 6 6s−3 } . We define gn = (v1 n)rn,yn , without loss of generality, we may suppose that gn ⇀ g in Ds,2(R3), and gn → g a.e. in R3. Direct calculations show that sup z∈R3 ∫ B1(z) |gn(x)|2 ∗ sdx = ∫ B1(0) |gn(x)|2 ∗ sdx = ∫ Brn (yn) |v1 n|2 ∗ sdx = b, (4.15) ‖v1 n‖2 = ‖gn‖2, |v1 n|2∗s = |gn|2∗s , (4.16)∫ R3 φgn |gn|2 ∗ s−1dx = ∫ R3 φv1n |v 1 n|2 ∗ s−1dx . (4.17) Based on the above two properties, we have Jε(gn) = Jε(v 1 n) = Jε(un)− Jε(u0) + on(1), (4.18) J ′ε(gn) = J ′ε(v 1 n) = on(1). (4.19) EJDE-2022/47 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 13 If g = 0, then gn → 0 in L2 loc(R3). Assume that ψ ∈ C∞0 (R3) satisfying suppψ ⊂ B1(y∗) for some y∗ ∈ R3, and ∇ψ(x)| ≤ C for x ∈ R3. Note that ‖ψgn‖2 = ∫∫ R6 |ψ(x)gn(x)− ψ(y)gn(y)|2 |x− y|3+2s dx dy = ∫∫ R6 (gn(x)− gn(y))(ψ2(x)gn(x)− ψ2(y)gn(y)) |x− y|3+2s dx dy + ∫∫ R6 |ψ(x)− ψ(y)|2gn(x)gn(y) |x− y|3+2s dx dy. (4.20) By a direct computation, we have∫∫ R6 |ψ(x)− ψ(y)|2gn(x)gn(y) |x− y|3+2s dx dy ≤ (∫∫ R6 |ψ(x)− ψ(y)|2g2 n(x) |x− y|3+2s dx dy )1/2 × (∫∫ R6 |ψ(x)− ψ(y)|2g2 n(x) |x− y|3+2s dx dy )1/2 . (4.21) We next show that∫∫ R6 |ψ(x)− ψ(y)|2g2 n(x) |x− y|3+2s dx dy = ∫∫ R6 |ψ(x)− ψ(y)|2g2 n(y) |x− y|3+2s dx dy = on(1). In fact, we have∫∫ R6 |ψ(x)− ψ(y)|2g2 n(x) |x− y|3+2s dx dy = ∫ B1(y∗) ∫ B1(y∗) |ψ(x)− ψ(y)|2g2 n(x) |x− y|3+2s dx dy + 2 ∫ B1(y∗) ∫ Bc1(y∗) |ψ(x)− ψ(y)|2g2 n(x) |x− y|3+2s dx dy ≤ ∫ B1(y∗) g2 n(x)dx ∫ B1(y∗) |∇ψ(y + θ(x− y))|2|x− y|2 |x− y|3+2s dy + C ∫ B1(y∗) g2 n(x)dx ∫ Bc1(y∗) 1 |x− y|3+2s dy ≤ C1 ∫ B1(y∗) g2 n(x)dx ∫ 2 0 r2 r1+2s dr + C ∫ B1(y∗) g2 n(x)dx ∫ ∞ 1 r2 r3+2s dr ≤ C2 ∫ B1(y∗) g2 n(x) dx→ 0 (4.22) as n → ∞, in view of gn → 0 in L2 loc(R3), where θ = θ(y) ∈ (0, 1). Similarly, we have ∫∫ R6 |ψ(x)− ψ(y)|2g2 n(y) |x− y|3+2s dx dy → 0, as n→∞. (4.23) By (4.20)–(4.23), we have ‖ψgn‖2 = ∫∫ R6 (gn(x)− gn(y))(ψ2(x)gn(x)− ψ2(y)gn(y)) |x− y|3+2s dx dy + on(1). (4.24) 14 S. QU, X. HE EJDE-2022/47 Furthermore, based on the above results and Proposition 2.3, we have ε2sS|ψgn|22∗s ≤ ε 2s‖ψgn‖2 = ε2s ∫∫ R6 (gn(x)− gn(y))(ψ2(x)gn(x)− ψ2(y)gn(y)) |x− y|3+2s dx dy + on(1) = ε2s ∫ R3 φgn |gn|2 ∗ s−1ψ2dx+ on(1) ≤ ε2sCs|gn| 2∗s−1 2∗s (∫ R3 (|gn|2 ∗ s−1ψ2) 6 3+2s dx ) 3+2s 6 + on(1) = ε2sCs|gn| 2∗s−1 2∗s (∫ R3 |gn| 6s−3 3−2s 6 3+2s |gnψ| 12 3+2s dx ) 3+2s 6 + on(1) ≤ ε2sCs|gn| 2∗s−1 2∗s (∫ R3 |ψgn|2 ∗ sdx ) 3−2s 3 (∫ B1(y∗) |gn|2 ∗ sdx ) 6s−3 6 + on(1) ≤ ε2sCsD1b 6s−3 6 (∫ R3 |ψgn|2 ∗ sdx ) 3−2s 3 + on(1) < 1 2 ε2sS|ψgn|22∗s + on(1), which implies that gn → 0 in L 2∗s loc(R3), contradicting with (4.15). Therefore, g 6= 0. By (4.9) and the weakly sequential continuity of I ′ε, we know J ′ε(g) = 0, hence the sequences {gn}, {r1 n}, {y1 n} are the wanted sequences. By iteration, we obtain sequences vjn = uj−1 n −uj−1, j ≥ 2, and the re-scaled functions ujn = (vjn)rjn,yjn ⇀ uj in Ds,2(R3), where uj is a nontrivial solution to (3.4). Furthermore, by (4.6),(4.9) and (4.18), we obtain ‖ujn‖2 = ‖vjn‖2 = ‖uj−1 n ‖2 − ‖uj−1‖2 + on(1) = . . . = ‖un‖2 − j−1∑ i=0 ‖ui‖2 + on(1), (4.25) Jε(u j n) = Jε(v j n) = Jε(u j−1 n )− Jε(uj−1) + on(1) = . . . = Iε(un)− Iε(u0)− j−1∑ i=1 Jε(u i). (4.26) Moreover, as 0 = J ′ε(u j)uj = ε2s‖uj‖2 − ∫ R3 φuj |uj |2 ∗ s−1dx ≥ ε2s‖uj‖2 − S−1|uj |2(2∗s−1) 2∗s ≥ ‖uj‖2[ε2s − S−2∗s (‖uj‖2)2∗s−2], (4.27) we have ‖uj‖2 ≥ ε 3−2s 2 S 3 2s , and the iteration must terminate at some index k ≥ 0. � Corollary 4.2. Let {un} be a (PS)c sequence for Iε with c ∈ (mε, 2 6s−3 4s mε). Then for each ε > 0, {un} is relatively compact in Ds,2(R3). EJDE-2022/47 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 15 Proof. From Lemma 4.1, it follows that there exist a number k ∈ N, a solution u0 of (1.9) and solutions u1, . . . , uk of (3.4), such that ‖un‖2 → k∑ j=0 ‖uj‖2; Iε(un)→ Iε(u 0) + k∑ j=1 Jε(u j). By Lemma 3.3, if u0 6= 0, then I(u0) > mε. On the other hand, for each nontrivial solution uj of (3.4), if uj is positive or negative, we have that Jε(u j) = 2s 3 + 2s ε 3+2s 2 S 3 2s = mε. If uj changes its sign, then, from the proof of [13, Proposition 3.2], we have for all t+, t− > 0, Jε(t +(uj)+) 4s 6s−3 + Jε(t −(uj)−) 4s 6s−3 ≤ Jε(uj) 4s 6s−3 . Fixing t+, t− > 0 such that J ′ε(t ±(uj)±)(t±(uj)±) = 0, it follows that Jε(t ±(uj)±) ≥ 2s 3 + 2s ε 3+2s 2 S 3 2s , which implies that Jε(u j) ≥ 2 6s−3 4s 2s 3 + 2s ε 3+2s 2 S 3 2s = 2 6s−3 4s mε. Since c ∈ (mε, 2 6s−3 4s mε), we must have k = 0, and so un → u0 in Ds,2(R3). � 5. Proof of Theorem 1.1 In this section, we are devoted to showing the multiplicity of high energy semi- classical states. For small τ > 0, we may choose ρ = ρ(τ) > 0 such that Mτ ⊂ Bρ(0). Let χ(x) = { x if |x| < ρ, ρx |x| if |x| ≥ ρ. (5.1) We define β : Nε → R3 and γ : Nε → R+ by β(u) = 1 ε 3−2s 2 S 3 2s ∫ R3 χ(x)|(−∆)s/2u|2dx, γ(u) = 1 ε 3−2s 2 S 3 2s ∫ R3 |χ(x)− β(u)|(−∆)s/2u|2dx. It is easy to see that for any Uδ,z ∈ Ds,2(R3), there exists a unique tδ,z in (0,+∞) such that Φδ,z(x) := tδ,zε 3−2s 4 Uδ,z(x) ∈ Nε. (5.2) We also introduce the set Λ = Λ(ρ, δ1, δ1) = {(x, δ) ∈ R3 × R : |x| < ρ/2, δ1 < δ < δ2}. (5.3) A direct computation yields that, for any fixed z ∈ R3,∫ R3 V (x)U2 δ,z(x)dx→ 0 as δ → 0. 16 S. QU, X. HE EJDE-2022/47 Then, for each ε > 0 and any fixed z ∈ R3, we see that limδ→0 tδ,z = 1. Hence, for every ε > 0, there exist δ1 = δ1(ε) and δ2 = δ2(ε) with δ1 < δ2 and δ1, δ2 → 0 as ε→ 0, such that sup{Iε(δ,z) : (z, δ) ∈ Λ} < ε 3+2s 2 ( 2s 3 + 2s S 3 2s + h(ε) ) , (5.4) where h(ε)→ 0 as ε→ 0. Lemma 5.1. limδ→0 γ(Φδ,z) = 0 uniformly for |z| ≤ ρ/2. Proof. For 0 < 2ξ < ρ, by tδ,z = 1 + oδ(1), we have γ(Φδ,z) = 1 S 3 2s ∫ R3 |χ(x)− β(Φδ,z)|(−∆)s/2Uδ,z|2dx+ oδ(1) = 1 S 3 2s ∫ R3\Bξ(z) |χ(x)− β(Φδ,z)|(−∆)s/2Uδ,z|2dx + 1 S 3 2s ∫ Bξ(z) |χ(x)− β(Φδ,z)|(−∆)s/2Uδ,z|2dx+ oδ(1) := I1 + I2 + oδ(1). Note that I1 = 1 S 3 2s ∫ R3\Bξ(0) |χ(x)− β(Φδ,0)|(−∆)s/2Uδ,0|2dx ≤ Cρ ∫ R3\Bξ(0) |(−∆)s/2Uδ,0|2dx→ 0 as δ → 0. For I2, we have β(Φδ,z) = β(ε 3−2s 4 Uδ,z) + oδ(1) = 1 S 3 2s ∫ R3 χ(x)|(−∆)s/2Uδ,z(x)|2dx+ oδ(1) = z + δ3+2s S 3 2s ∫ R3 [χ(δx+ z)− z]|(−∆)s/2U1,0|2dx+ oδ(1) = z + oδ(1). (5.5) Consequently, S 3 2s I2 ≤ ∫ Bξ(z) |χ(x)− χ(z)||(−∆)s/2Uδ,z|2dx + ∫ Bξ(z) |χ(z)− β(Φδ,z)||(−∆)s/2Uδ,z|2dx ≤ 2 ∫ Bξ(z) |x− z||(−∆)s/2Uδ,z|2dx+ 2ξS 3 2s + ∫ Bξ(z) |χ(z)− z||(−∆)s/2Uδ,z|2dx+ oδ(1) ≤ 4ξS 3 2s + oδ(1). where we have used [9, Lemma 2], which says χ(x)− χ(z) ≤ 2|x− z|+ 2ξ, x ∈ Bξ(z). Since ξ > 0 is arbitrary, we have limδ→0 γ(Φδ,z) = 0, uniformly for |z| ≤ ρ/2. � EJDE-2022/47 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 17 We now define a set Ñε ⊂ Nε by Ñε = { u ∈ Nε : 2s 3 + 2s ε 3+2s 2 S 3 2s < Iε(u) < ε 3+2s 2 ( 2s 3 + 2s S 3 2s + h(ε) ) , (β(u), γ(u)) ∈ Λ } , where Λ is given by (5.3). According to Lemma 5.1, we can modify δ1(ε) and δ2(ε) such that Ñε 6= ∅ for ε > 0 small. Lemma 5.2. limε→0 supu∈Ñε dist(β(u),Mτ ) = 0, for any τ > 0. Proof. Let un ∈ Ñεn be such that dist(β(un),Mτ ) = sup u∈Ñεn dist(β(u),Mτ ) + on(1), and assume that εn → 0. It suffices to find a sequence zn ∈Mτ such that β(un) = zn + on(1). (5.6) Since un ∈ Nεn , by (3.6) we see that ‖un‖2 ≥ ε 3−2s 2 n S 3 2s . Hence, 2s 3 + 2s ε 3+2s 2 n S 3 2s ≤ 2s 3 + 2s ε2s n ‖un‖2 ≤ 2s 3 + 2s ε2s n ‖un‖2 + 2s 3 + 2s ∫ R3 V (x)u2 ndx = Iεn(un)− 1 2(2∗s − 1) I ′εn(un)un < ε 3+2s 2 n ( 2s 3 + 2s S 3 2s + h(εn) ) . (5.7) Setting wn := ε 2s−3 4 un, we have from (5.7) that ‖wn‖2 → 2s 3 + 2s S 3 2s and ε−2s n ∫ R3 V (x)w2 ndx→ 0 as n→∞. (5.8) Using un ∈ Nεn again, we have∫ R3 |(−∆)s/2wn|2dx+ ε−2s n ∫ R3 V (x)w2 ndx = ∫ R3 φwn |wn|2 ∗ s−1dx, (5.9) which implies that {wn} is a PS sequence for I1. It follows from Lemmas 3.3 and 4.1 with ε = 1 that there exist a number k ∈ N, sequences of points x1 n, . . . , x k n ∈ R3 and radii r1 n, . . . , r k n such that: (1) w0 n ≡ wn ⇀ w0 in Ds,2(R3); (2) wjn ≡ (wj−1 n − wj−1)rjn,xjn ⇀ wj in Ds,2(R3), j = 1, 2, . . . , k; (3) ‖wn‖2 → ∑k j=0 ‖wj‖2; (4) I1(wn)→ I1(w0) + ∑k j=1 J1(wj), as n → ∞ where u0 is a solution of (1.9) and uj , 1 ≤ j ≤ k, are the nontrivial solutions of (3.4). If w0 6= 0, by Lemma 3.3 we know that I1(w0) > 2s 3 + 2s S 3 2s , 18 S. QU, X. HE EJDE-2022/47 which contradicts with the above conclusion (4) since J1(wj) > 2s 3+2sS 3 2s and I1(wn)→ 2s 3+2sS 3 2s . Therefore, w0 = 0. Moreover, using J1(wj) > 2s 3 + 2s S 3 2s and I1(wn)→ 2s 3 + 2s S 3 2s , again, we must have k = 1 and w1 is a ground state of (3.4) with I1(w1) = 2s 3+2sS 3 2s . So, there exist δ1 > 0 and z1 ∈ R3 such that w1 = Uδ1,z1 , and there exists (r1 n, x 1 n) ∈ R + ×R3 such that ‖(wn)r1n,x1 n − w1‖ → 0. Consequently, there exist a sequence of points {zn} ⊂ R3 and a sequence of {σn} ⊂ (0,+∞) such that ‖hn‖ := ‖wn − Uσn,zn‖ → 0, where zn = x1 n + r1 nz1 and σn = r1 nδ1. We claim that σn → 0 and {zn} is bounded. (5.10) Indeed, denoting Ψσn,zn = ε 3−2s 4 n Uσn,zn by hn → 0 in Ds,2(R3), one has β(un) = 1 ε 3−2s 2 S 3 2s ∫ R3 χ(x)|(−∆)s/2un|2dx = 1 S 3 2s ∫ R3 χ(x)|(−∆)s/2wn|2dx = 1 S 3 2s ∫ R3 χ(x)|(−∆)s/2Uσn,zn |2dx+ on(1) = 1 ε 3−2s 2 S 3 2s ∫ R3 χ(x)|(−∆)s/2Ψσn,zn |2dx+ on(1) = β(Ψσn,zn) + on(1). (5.11) From un ∈ Ñε we may assume β(Ψσn,yn) ⊂ Bρ/2(0). If σn → ∞, then we know that for each R > 0, by [28, Proposition 2.2], we have lim n→∞ 1 ε 3−2s 2 n ∫ BR(0) |(−∆)s/2Ψσn,zn |2dx = lim n→∞ ∫ BR(0) |(−∆)s/2Uσn,zn |2dx ≤ lim n→∞ ∫ BR(0) |∇Uσn,zn |2dx = 0. Using this fact and the definition of the mapping γ, we obtain γ(Ψσn,zn) = ε− 3−2s 2 S 3 2s ∫ R3 |χ(x)− β(Ψσn,zn)|(−∆)s/2Ψσn,zn |2dx ≥ ε− 3−2s 2 S 3 2s ∫ R3 |χ(x)||(−∆)s/2Ψσn,zn |2dx− β(Ψσn,zn) ≥ ρε− 3−2s 2 S 3 2s ∫ R3\BR(0) |(−∆)s/2Ψσn,zn |2dx− ρ 2 = ρε− 3−2s 2 S 3 2s ∫ R3 |(−∆)s/2Ψσn,zn |2dx− ρ 2 + on(1) = ρ 2 + on(1). (5.12) Since ‖hn‖ := ‖un−Ψσn,zn‖ → 0, one has γ(un) = γ(Ψσn,zn) + on(1). So, γ(un) > ρ 2 + on(1). However, from un ∈ Ñεn , we obtain δ1(εn) < γ(un) < δ2(εn), (5.13) EJDE-2022/47 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 19 where δi(εn) → 0, i = 1, 2 as n → ∞. This leads to a contradiction, and so, {σn} is bounded. Now we assume that σn → σ̄ ≥ 0 as n → ∞. If σ̄ > 0, then we must have that |zn| → ∞. Otherwise, Uσn,zn would converge strongly in Ds,2(R3), and so would wn. Consequently, I1 possesses nontrivial minimizer on N1, which is impossible by Lemma 3.3. Then, for every R > 0, the fact that limn→∞ |zn| = ∞ implies that lim n→∞ ∫ BR(0) |(−∆)s/2Uσn,zn |2dx = 0. Hence, one can similarly obtain the estimation (5.12), a contradiction to (5.13). The proof of the boundedness of the sequence {zn} is similar, and it is omitted here. Hence, (5.10) holds. Now, we may assume that zn → z∗ and σn → 0. By choosing subsequences of {σn} and {εn}, still denoted by {σn} and {εn} such that σni εni = oni(1) as ni → ∞, we may replace {σni} by {εni} and relabel {εni} by {εn}. Define vn(x) = ε 3−2s 4 n wn(εnx+ zn). Then vn → U1,0 in Ds,2(R3), and we obtain lim n→∞ ∫ R3 V (εnx+ zn)vn(x)2dx = lim n→∞ ε−2s n ∫ R3 V (x)wn(x)2dx = 0; which implies that ∫ R3 V (z∗)U2 1,0(x)dx = 0. Therefore, V (z∗) = 0 and z∗ ∈ M . Furthermore, zn ∈Mτ for large n. Then by (5.11), we obtain β(un) = 1 S 3 2s ∫ R3 χ(x)|(−∆)s/2Uσn,zn |2dx+ on(1) = 1 S 3 2s ∫ R3 [χ(εnx+ zn)− zn]|(−∆)s/2U1,0|2dx+ zn + on(1). Since εnx+ zn → z∗ ∈M , we deduce that β(un) = zn + on(1), hence the sequence {zn} is what we need. Consequently, (5.6) is true and we complete the proof. � Proof of Theorem 1.1. For any τ > 0, set small ε = ετ > 0. Then Φ : [δ1, δ2]×M → Ñε given by Φ(δ, z) = Φδ,z is well defined and by Lemma 5.2, we have β(Ñε) ⊂Mτ . From (5.5), we obtain β(Φδ,z) = z + oδ(1) uniformly in z ∈M. For δ ∈ [δ1, δ2], we denote β(Φδ,z) = z + µ(z) for z ∈ M , where |µ(z)| < τ/2 uniformly for z ∈ M . Define H(t, (δ, z)) := (δ, z + (1 − t)µ(z)). It is easy to see that H : [0, 1]× [δ1, δ2]×M → [δ1, δ2]×Mτ is continuous. Obviously, H(0, (δ, z)) = (δ, β(Φδ,z)), H(1, (δ, z)) = (δ, z). Therefore, Θ(δ, z) := (δ, β(Φδ,z))) : [δ1, δ2]×M → [δ1, δ2]×Mτ is homotopic to the inclusion mapping Id : [δ1, δ2]×M → [δ1, δ2]×Mτ . Thus, we have cat(Ñε) ≥ cat[δ1,δ2]×Mτ ([δ1, δ2]×M) = catMτ (M). By Corollaries 3.4 and 4.2, the functional Iε satisfies the (PS)c condition on Ñε. Hence, the Ljusternik-Schnirelman theory of critical points implies that Iε has at least catMτ (M) solutions. � Acknowledgments. This work was supported by the NSFC (12171497, 11771468, 11971027). 20 S. QU, X. 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Xie; Bound state solutions of Schrödinger-Poisson system with critical exponent, Discret. Contn. Dyn. Syst. 37 (2017), 605-625. [38] X. Zhong, C. Tang; Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in R3, Nonlinear Anal. RWA 39 (2018), 166-184. Siqi Qu College of Science, Minzu University of China, Beijing, 100081, China Email address: qusiqi78@gmail.com Xiaoming He College of Science, Minzu University of China, Beijing, 100081, China Email address: xmhe923@muc.edu.cn 1. Introduction 2. Preliminaries 3. Limit problem 4. Global compactness 5. Proof of Theorem ?? Acknowledgments References