Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 14, pp. 1–31. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND CONCENTRATION RESULTS FOR FRACTIONAL SCHRÖDINGER-POISSON SYSTEM VIA PENALIZATION METHOD ZHIPENG YANG, WEI ZHANG, FUKUN ZHAO Abstract. This article concerns the positive solutions for the fractional Schrödinger- Poisson system ε2s(−∆)su+ V (x)u+ φu = f(u) in R3, ε2t(−∆)tφ = u2 in R3, where ε > 0 is a small parameter, (−∆)α denotes the fractional Laplacian of orders α = s, t ∈ (3/4, 1), V ∈ C(R3,R) is the potential function and f : R→ R is continuous and subcritical. Under a local condition imposed on the potential function, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values. Moreover, we considered some properties of these positive solutions, such as concentration behavior and decay estimate. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory. 1. Introduction and statement of main results In recent years, a great deal of work has been done on the semiclassical standing waves for the fractional Schrödinger equation i~ ∂Ψ ∂t = ~2s(−∆)sΨ + Ṽ (x)Ψ− f(|Ψ|) in R3 × R, (1.1) where i is the imaginary unit, ~ is the Planck constant, Ṽ (x) = V (x) + E is the potential function with the constant E and f(exp(iθ)ξ) = exp(iθ)f(ξ) for θ, ξ ∈ R is a nonlinear function. Equations of type (1.1) were introduced by Laskin [21, 22], and come from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths. It also appeared in several areas such as optimization, finance, phase transitions, stratified materials, crystal dislocation, flame propagation, con- servation laws, materials science and water waves (see [8, 12]). An interesting Schrödinger equation appears when it describes quantum (non- relativistic) particles interacting with the electromagnetic field generated by the 2010 Mathematics Subject Classification. 49J35, 58E05. Key words and phrases. Penalization method; fractional Schrödinger-Poisson; Lusternik-Schnirelmann theory. c©2021 Texas State University. Submitted October 5, 2018. Published March 16, 2021. 1 2 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 motion, that is a fractional nonlinear Schrödinger-Poisson system (also called frac- tional Schrödinger-Maxwell system), i~ ∂Ψ ∂t = ~2s(−∆)sΨ + Ṽ (x)Ψ + φΨ− f(|Ψ|) in R3 × R, ~2t(−∆)tφ = |Ψ|2 in R3. (1.2) A solution of the form e−iEt/~u(x) is called a standing wave and (e−iEt/~u(x), φ(x)) is a solution of (1.2) if and only if (u(x), φ(x)) satisfies ε2s(−∆)su+ V (x)u+ φu = f(u) in R3, ε2t(−∆)tφ = u2 in R3. (1.3) For s = t = 1, Equation (1.3) gives back the classical Schrödinger-Poisson equa- tion −ε2∆u+ V (x)u+ φu = f(u) in R3, −ε2∆φ = u2 in R3, (1.4) which was proposed by Benci and Fortunato [6] in 1998 for a bounded domain, and is related to the Hartree equation [23]. Recently, to better simulate the interaction effect among many particles in quantum mechanics, a nonlinear version of the Schrödinger equation coupled with a Poisson equation have begun to receive much attention, we refer the interested readers to [2, 3, 4, 9, 14, 17, 35, 41] and the references therein. When s, t ∈ (0, 1) to the best of our knowledge, there are just a few references on the existence for the fractional Schrödinger-Poisson systems, maybe because the standard techniques developed for the local Laplacian do not work immediately. Teng [34] considered the fractional Schrödinger-Poisson system (1.2) with subcrit- ical and critical nonlinearity and he established the existence of ground state solu- tions. For other existence results we refer to [19, 24, 27, 38, 40] and the references therein. Assuming f(u) ∼ |u|p−2u(4 < p < 2∗s) and the global condition 0 < inf x∈R3 V (x) < lim inf |x|→∞ V (x) = V∞, firstly introduced by Rabinowitz [29], the authors in [25] obtained the multiplicity and concentration of positive solutions for the system ε2s(−∆)su+ V (x)u+ φu = f(u) + |u|2 ∗ s−2u in R3, ε2t(−∆)tφ = u2 in R3, (1.5) via the standard Nehari manifold method. The concentration behavior of ground state solutions for subcritical and critical cases with two competing positive poten- tials was obtained in [39, 37]. Different from [25, 37, 39], in this paper, we establish the existence and con- centration of positive solutions for the fractional Schrödinger-Poisson system (1.3) when the potential function V ∈ C(R3,R) satisfies the following conditions: (A1) There is constant V0 > 0 such that V0 = infx∈R3 V (x), (A2) there is a bounded domain Ω such that V0 < min ∂Ω V. EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 3 Without loss of generality, we assume that M = {x ∈ Ω : V (x) = V0} 6= ∅ and 0 ∈M. Moreover, we assume that the continuous function f vanishes on (−∞, 0) and satisfies (A3) f(u) = o(u3) as u→ 0. (A4) There is p ∈ (4, 2∗s) such that lim u→∞ f(u) up−1 = 0. (A5) There is θ ∈ (4, 2∗s) such that 0 < θF (u) = θ ∫ u 0 f(τ)dτ ≤ f(u)u, ∀u > 0. (A6) The function f(u)/u3 is increasing on (0,∞). Remark 1.1. In view of (A4) and (A5), we have 4 < 2∗s = 6 3−2s , which implies that s > 3 4 . Moreover, if s, t ∈ (3/4, 1), then we have that 2s+ 2t > 3. We remark that the nonlinearity f is only continuous, we can not use standard arguments on the Nehari manifold as in [25]. To overcome the nondifferentiability of the Nehari manifold, we shall use some variants of critical point theorems from Szulkin and Weth [33]. Now we state our main results as follows. Theorem 1.2. Assume that (A1)–(A6) are satisfied with s, t ∈ (3/4, 1). Then for each δ > 0 such that Mδ = {x ∈ R3 : distx,M) ≤ δ} ⊂ Ω, there exists εδ > 0 such that (1.3) has at least catMδ (M) positive solutions for any ε ∈ (0, εδ). Moreover, if u denotes one of these positive solutions and ηε ∈ R3 its global maximum, then lim ε→0 V (ηε) = V0, and there exists a constant C > 0 (independent of ε) such that u(x) ≤ Cε3+2s ε3+2s + |x− ηε|3+2s , ∀x ∈ R3. This article is organized as follows. In section 2, besides describing the functional setting to stu dy problem (1.3), we give some preliminary Lemmas which will be used later. In section 3, influenced by the work [10] and [32], we introduce a modified functional and show it satisfies the Palais-Smale condition. In section 4, we stu dy the autonomous problem associated. This stu dy allows us to show that the modified problem has multiple solutions. Finally, we show the critical point of the modified functional which satisfies the original problem, and investigate its concentration and polynomial decay behavior, which completes the proof Theorem 1.2. 2. Variational setting and preliminary results Throughout this paper, we denote ‖ · ‖Lr the usual norm of the space Lr(R3), 1 ≤ r < ∞, Br(x) denotes the open ball with center at x and radius r, C or Ci(i = 1, 2, · · · ) denote some positive constants may change from line to line. ⇀ and → mean the weak and strong convergence. 4 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 2.1. Functional space setting. Firstly, fractional Sobolev spaces are the conve- nient setting for our problem, so we sketch the fractional order Sobolev spaces, the complete introduction can be found in [11]. We recall that, for any α ∈ (0, 1), the fractional Sobolev space Hα(R3) = Wα,2(R3) is defined as follows: Hα(R3) = {u ∈ L2(R3) : ∫ R3 ( |ξ|2α|F(u)|2 + |F(u)|2 ) dξ <∞}, whose norm is defined as ‖u‖2Hα(R3) = ∫ R3 ( |ξ|2α|F(u)|2 + |F(u)|2 ) dξ, where F denotes the Fourier transform. We also define the homogeneous fractional Sobolev space Dα,2(R3) as the completion of C∞0 (R3) with respect to the norm ‖u‖Dα,2(R3) := (∫∫ R3×R3 |u(x)− u(y)|2 |x− y|3+2α dx dy )1/2 = [u]Hα(R3). The fractional Laplacian, (−∆)αu, of a smooth function u : R3 → R, is defined by F((−∆)αu)(ξ) = |ξ|2αF(u)(ξ), ξ ∈ R3. Also (−∆)αu can be equivalently represented [11] as (−∆)αu(x) = −1 2 C(α) ∫ R3 u(x+ y) + u(x− y)− 2u(x) |y|3+2α dy, ∀x ∈ R3, where C(α) = (∫ R3 (1− cosξ1) |ξ|3+2α dξ )−1 , ξ = (ξ1, ξ2, ξ3). Also, by the Plancherel formula in Fourier analysis, we have [u]2Hα(R3) = 2 C(α) ‖(−∆)α/2u‖22. As a consequence, the following three norms are equivalent on Hα(R3):(∫ R3 |u|2 dx+ ∫∫ R3×R3 |u(x)− u(y)|2 |x− y|3+2α dx dy )1/2 ,(∫ R3 (|ξ|2α|F(u)|2 + |F(u)|2)dξ )1/2 ,(∫ R3 |u|2 dx+ ‖(−∆)α/2u‖22 )1/2 . Making the change of variable x 7→ εx, we can rewrite system (1.3) as the equivalent system (−∆)su+ V (εx)u+ φu = f(u) in R3, (−∆)tφ = u2 in R3. (2.1) If u is a solution of (2.1), then v(x) := u(x/ε) is a solution of (1.3). Thus, to stu dy the system (1.3), it suffices to stu dy the system (2.1). In view of the potential V (x), we introduce the subspace Hε = { u ∈ Hs(R3) : ∫ R3 V (εx)u2 dx < +∞ } , EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 5 which is a Hilbert space equipped with the inner product (u, v)Hε = ∫ R3 (−∆)s/2u(−∆)s/2v dx+ ∫ R3 V (εx)uv dx, and the norm ‖u‖2Hε = (u, u) = ∫ R3 |(−∆)s/2u|2 dx+ ∫ R3 V (εx)u2 dx. For convenience, we denote ‖ ·‖Hε by ‖ ·‖ε. For the reader’s convenience, we review some useful result for this class of fractional Sobolev spaces. Lemma 2.1 ([11]). Let 0 < α < 1, then there exists a constant C = C(α) > 0, such that ‖u‖2 L2∗α (R3) ≤ C[u]2Hα(R3) for every u ∈ Hα(R3), where 2∗α = 6 3−2α is the fractional critical exponent. More- over, the embedding Hα(R3) ↪→ Lr(R3) is continuous for any r ∈ [2, 2∗α] and is locally compact whenever r ∈ [2, 2∗α). Lemma 2.2 ([30]). If {un} is bounded in Hα(R3) and for some R > 0, lim n→∞ sup y∈R3 ∫ BR(y) |un|2 dx = 0, then un → 0 in Lr(R3) for any 2 < r < 2∗α. Lemma 2.3 ([28]). Let u ∈ Ds,2(R3), ϕ ∈ C∞0 (R3) and for each r > 0, ϕr(x) = ϕ(xr ). Then uϕr → 0 in Ds,2(R3) as r → 0. If, in addition, ϕ ≡ 1 in a neighbourhood of the origin, then uϕr → u in Ds,2(R3) as r → +∞. 2.2. Reduction method. It is clear that system (2.1) is the Euler-Lagrange equa- tions of the functional J : Hε ×Dt,2(R3)→ R defined by J(u, φ) = 1 2 ‖u‖2ε − 1 4 ∫ R3 |(−∆)s/2φ|2 dx+ 1 2 ∫ R3 φu2 dx− ∫ R3 F (u) dx . (2.2) It is easy to see that J exhibits a strong indefiniteness, namely it is unbounded both from below and from above on infinitely dimensional subspaces. This indefi- niteness can be removed using the reduction method described in [6]. First of all, for each fixed u ∈ Hε, there exists a unique φtu ∈ Dt,2(R3) which is the solution of (−∆)tφ = u2 in R3. We can write an integral expression for φtu in the form φtu(x) = Ct ∫ R3 u2(y) |x− y|3−2t dy, ∀x ∈ R3, which is called t-Riesz potential (see [20]), where Ct = 1 π3/2 Γ(3− 2t) 22tΓ(s) . Then (2.1) can be reduced to the first equation with φ represented by the solution of the fractional Poisson equation. This is the basic strategy of solving (2.1). To 6 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 be more precise about the solution φ of the fractional Poisson equation, we collect some useful Lemmas. Lemma 2.4 ([34]). For each u ∈ Hs(R3) and 4s+ 2t ≥ 3, we have: (i) φtu ≥ 0; (ii) φtu : Hs(R3)→ Dt,2(R3) is continuous and maps bounded sets into bounded sets; (iii) ‖φtu‖2Dt,2(R3) = ∫ R3 φ t uu 2 dx ≤ S2 t ‖u‖4 12 3+2t ; (iv) if un ⇀ u in Hs(R3), then φtun ⇀ φtu in Dt,2(R3); (v) if un → u in Hs(R3), then φtun → φtu in Dt,2(R3) and ∫ R3 φ t unu 2 n dx →∫ R3 φ t uu 2 dx. We define N : Hs(R3)→ R by N(u) = ∫ R3 φtuu 2 dx . It is clearly that N(u(·+ y)) = N(u) for any y ∈ R3, u ∈ Hs(R3) and N is weakly lower semi-continuous in Hs(R3). Moreover, similarly to the well-know Brezis-Lieb Lemma ([7]), we have the next Lemma. Lemma 2.5 ([34]). Let un ⇀ u in Hs(R3) and un → u a.e. in R3 with 2s+2t > 3. Then (i) N(un − u) = N(un)−N(u) + o(1); (ii) N ′(un − u) = N ′(un)−N ′(u) + o(1), in (Hs(R3))−1. Putting φ = φtu into the first equation of (2.1), we obtain a nonlocal semilinear elliptic equation (−∆)su+ V (εx)u+ φtuu = f(u) in R3. The corresponding functional I : Hε → R is defined by I(u) = 1 2 ∫ R3 |(−∆)s/2u|2 dx+ 1 2 ∫ R3 V (εx)u2 dx+ 1 4 ∫ R3 φtuu 2 dx− ∫ R3 F (u) dx. Note that if 4s+ 2t ≥ 3, then 2 ≤ 12 3+2t ≤ 2∗s, and thus Hs(R3) ↪→ L 12 3+2t (R3). Then by Hölder inequality and Sobolev inequality, we have∫ R3 φtuu 2 dx ≤ (∫ R3 |u| 12 3+2t dx ) 3+2t 6 (∫ R3 |φtu|2 ∗ t dx )1/2∗t ≤ S−1/2 t (∫ R3 |u| 12 3+2t dx ) 3+2t 6 ‖φtu‖Dt,2 ≤ C‖u‖2ε‖φtu‖Dt,2 <∞. Therefore, the functional I is well-defined for every u ∈ Hε. Next, we consider critical points of I using a variational method. 3. Modified problem In this section, we adapt for our case an argument explored by the penalization method introduction by del Pino and Felmer [10]. In fact, from (A3) and (A6) we have lim u→0 f(u) u = 0 EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 7 and the map u 7→ f(u) u is increasing in (0,∞). Now we fix some notation. Let K > 2, a > 0 be such that f(a) a = V0 K where V0 is given by (A1). We define f̃(u) = { f(u) if u ≤ a, V0u/K if u > a, g(x, u) = χΩ(x)f(u) + ( 1− χΩ(x) ) f̃(u), where χ is characteristic function of a set Ω. From hypotheses (A3)–(A6) we have that g is a Carathéodory function and satisfies the following properties: lim u→0 g(x, u) u3 = 0 uniformly in x ∈ R3; (3.1) lim u→∞ g(x, u) up−1 = 0 uniformly in x ∈ R3; (3.2) 0 ≤ θG(x, u) := θ ∫ u 0 g(x, τ)dτ < g(x, u)u, ∀x ∈ Ω, ∀u > 0, 0 ≤ 2G(x, u) ≤ g(x, u)u ≤ V0 K t2, ∀x ∈ R3\Ω, ∀u > 0. (3.3) u 7→ g(x, u) u3 is increasing for u > 0. (3.4) Moreover, from definition of g, we have g(x, u) ≤ f(u), ∀u ∈ (0,+∞), ∀x ∈ R3, g(x, u) = 0, ∀u ∈ (−∞, 0), ∀x ∈ R3. Now, we stu dy the modified equation (−∆)su+ V (εx)u+ φu = g(εx, t) in R3, (−∆)tφ = u2 in R3. (3.5) Note that positive solutions of (3.5) with u(x) ≤ a for each x ∈ R3\Ωε are also positive solution of (1.3), where Ωε = {x ∈ R3 : εx ∈ Ω}. The energy functional associated with (3.5) is Iε(u) = 1 2 ∫ R3 ( |(−∆)s/2u|2 + V (εx)u2 ) dx+ 1 4 ∫ R3 φtuu 2 dx− ∫ R3 G(εx, u) dx which is of C1 class and whose derivative is given by 〈I ′ε(u), v〉 = ∫ R3 ( (−∆)s/2u(−∆)s/2v+V (εx)uv ) dx+ ∫ R3 φtuuv dx− ∫ R3 g(εx, u)v dx. for all v ∈ Hε and associated norm ‖ · ‖ε. Hence the critical points of Iε in Hε are weak solutions of problem (3.5). Now, we denote the Nehari manifold associated to Iε by Nε = {u ∈ Hε\{0} : 〈I ′ε(u), u〉 = 0}. Obviously, Nε contains all nontrivial critical points of Iε. But we do not know whether Nε is of class C1 under our assumptions and therefore we cannot use minimax theorems directly on Nε. To overcome this difficulty, we will adopt a technique developed in [32, 33] to show that Nε is still a topological manifold, 8 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 naturally homeomorphic to the unit sphere of Hε, and then we can consider a new minimax characterization of the corresponding critical value for Iε. For this we denote by H+ ε the subset of Hε given by H+ ε = {u ∈ Hε : | supp(u+) ∩ Ωε| > 0} and S+ ε = Sε ∩H+ ε , where Sε is the unit sphere of Hε. Lemma 3.1. The set H+ ε is open in Hε. Proof. Suppose by contradiction there are a sequence {un} ⊂ Hε\H+ ε and u ∈ H+ ε such that un → u in Hε. Hence | supp(u+ n ) ∩ Ωε| = 0 for all n ∈ N and u+ n (x) → u+(x) a.e. in x ∈ Ωε. So, u+(x) = lim n→∞ u+ n (x) = 0, a.e. in x ∈ Ωε. But, this contradicts the fact that u ∈ H+ ε . Therefore H+ ε is open. � From definition of S+ ε and Lemma 3.1 it follows that S+ ε is a incomplete C1,1- manifold of codimension 1, modeled on Hε and contained in the open H+ ε . Hence, Hε = TuS + ε ⊕ Ru for each u ∈ S+ ε , where TuS + ε = {v ∈ Hε : (u, v)ε = 0}. In the rest of this section, we show some Lemmas related to the function Iε and the set H+ ε . First, we show the functional Iε satisfying the mountain pass geometry. Lemma 3.2. The functional Iε satisfies the following conditions: (i) There exist α, ρ > 0 such that Iε(u) ≥ α with ‖u‖ε = ρ; (ii) there exists e ∈ Hε satisfying ‖e‖ε > ρ such that Iε(e) < 0. Proof. (i) For any u ∈ Hε\{0}, it follows from (3.1) and (3.2) that there exists C > 0 such that |g(εx, u)| ≤ |u|3 + C|u|p−1, for all x ∈ R3, u ∈ R, |G(εx, u)| ≤ 1 4 |u|4 + C|u|p, for all x ∈ R3, u ∈ R. By the Sobolev embedding Hε ↪→ Lr(R3) for r ∈ [2, 2∗s], we have Iε(u) = 1 2 ∫ R3 ( |(−∆)s/2u|2 + V (εx)u2 ) dx+ 1 4 ∫ R3 φtuu 2 dx− ∫ R3 G(εx, u) dx ≥ 1 2 ‖u‖2ε − C1‖u‖4ε − C2‖u‖pε = 1 2 ‖u‖2ε ( 1− C1‖u‖2 − ε− C2‖u‖2ε ) Therefore, we can choose positive constants α, ρ such that Iε(u) ≥ α with ‖u‖ε = ρ. (ii) For each u ∈ H+ ε and τ > 0 we have Iε(τu) = τ2 2 ∫ R3 ( |(−∆)s/2u|2 + V (εx)u2 ) dx+ τ4 4 ∫ R3 φtuu 2 dx− ∫ R3 G(εx, τu) dx ≤ τ2 2 ‖u‖2ε + τ4 4 ∫ R3 φtuu 2 dx− C1τ θ ∫ Ωε |u+|θ dx+ C2| supp(u+) ∩ Ωε|. Since θ ∈ (4, 2∗s), conclusion (ii) follows. � Since f is only continuous, the next two results are very important because they allow us to overcome the non-differentiability of Nε and the incompleteness of S+ ε . EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 9 Lemma 3.3. Assume that (A1)–(A6) are satisfied. Then the following properties hold: (1) For each u ∈ H+ ε , let hu : R+ → R be given by hu(τ) = Iε(τu). Then there exists a unique τu > 0 such that h′u(τ) > 0 in (0, τu) and h′u(τ) < 0 in (τu,∞); (2) there is a σ > 0 independent on u such that τu > σ for all u ∈ S+ ε . Moreover, for each compact set W ⊂ S+ ε there is CW > 0 such that τu ≤ CW for all u ∈ W; (3) The map m̂ε : H+ ε → Nε given by m̂ε(u) = τuu is continuous and mε := m̂ε|S+ ε is a homeomorphism between S+ ε and Nε. Moreover, m−1 ε (u) = u ‖u‖ε . Proof. (1) From Lemma 3.2, it is sufficient to note that, hu(0) = 0, hu(τ) > 0 when τ > 0 is small and hu(τ) < 0 when τ > 0 is large. Since hu ∈ C1(R+,R), there is τu > 0 global maximum point of hu and h′u(τu) = 0. Thus, I ′ε(τuu)(τuu) = 0 and τuu ∈ Nε. We see that τu > 0 is the unique positive number such that h′u(τu) = 0. Indeed, suppose by contradiction that there are τ1 > τ2 > 0 with h′u(τ1) = h′u(τ2) = 0. Then, for i = 1, 2 we have that τ2 i ‖u‖2ε + τ4 i ∫ R3 φtuu 2 dx = ∫ R3 g(εx, τiu)τiu dx. Therefore, ‖u‖2ε τ2 i + ∫ R3 φtuu 2 dx = ∫ R3 g(εx, τiu) (τiu)3 u4 dx which implies that( 1 τ2 1 − 1 τ2 2 ) ‖u‖2ε = ∫ R3 (g(εx, τ1u) (τ1u)3 − g(εx, τ2u) (τ2u)3 ) u4 dx, contrary to (3.4). Thus, (1) is proved. (2) Suppose u ∈ S+ ε , then from (3.1) and (3.2) and Sobolev embeddings we obtain τu ≤ ∫ R3 g(εx, τuu)u dx ≤ εC1τ 4 u + CεC2τ p u . From previous inequality we obtain σ > 0 independent on u, such that τu > σ. Finally, if W ⊂ S+ ε is compact, and suppose by contradiction that there is {un} ⊂ W such that τn := τun → ∞. Since W is compact, there is u ∈ W such that un → u in Hε. It follows from the arguments used in the proof of Lemma 3.2 that Iε(τnun)→ −∞. On the other hand, by vn := τnun ∈ Nε and (3.3) and (3.4) we deduce that Iε(vn) = Iε(vn)− 1 θ I ′ε(vn)vn ≥ θ − 2 2θ ‖vn‖2ε + (1 4 − 1 θ ) ∫ R3 φtuu 2 dx+ 1 θ ∫ R3\Ωε ( g(εx, vn)vn − θG(εx, vn) ) dx ≥ θ − 2 2θ ‖vn‖2ε − θ − 2 2θ ‖vn‖2ε K = ( 1− 1 K )θ − 2 2θ ‖vn‖2ε, 10 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 which yields a contradiction. Therefore (2) is proved. (3) First of all we observe that m̂ε, mε and m−1 ε are well defined. In fact, by (1), for each u ∈ H+ ε , there exists a unique τu > 0 such that τuu ∈ Nε, hence there is a unique m̂ε(u) = τuu ∈ Nε. On the other hand, if u ∈ Nε then u ∈ H+ ε . Otherwise, we have | supp(u+) ∩ Ωε| = 0 and by Lemma 2.3 and (3.3) we have 0 < ‖u‖2ε ≤ ∫ R3 g(εx, u)u dx ≤ 1 K ∫ R3\Ωε V (εx)u2 dx ≤ ‖u‖ 2 ε K which is impossible since K > 2 and u 6= 0. Therefore, m−1 ε (u) = u ‖u‖ε ∈ S + ε , is well defined and it is a continuous function. Since m−1 ε (mε(u)) = m−1 ε (tuu) = τuu τu‖u‖ε = u, ∀u ∈ S+ ε , we conclude that mε is a bijection. To prove m̂ε : H+ ε → Nε is continuous, let {un} ⊂ H+ ε and u ∈ H+ ε be such that un → u in Hε. By (a2), there is a τ0 > 0 up to a subsequence such that τn := τun → τ0. Since τnun ∈ Nε we obtain τ2 n‖un‖2ε + τ4 n ∫ R3 φtunu 2 n dx = ∫ R3 g(εx, τnun)τnun dx, ∀n ∈ N. By Lemma 2.4 and passing to the limit as n→∞, it follows that τ2 0 ‖u‖2ε + τ4 0 ∫ R3 φtuu 2 dx = ∫ R3 g(εx, τ0u)τ0u dx, which means that τ0u ∈ Nε and τu = τ0. This proves m̂ε(un)→ m̂ε(u) in H+ ε . So, m̂ε, mε are continuous functions and (3) is proved. � Now we define the functions Ψ̂ε : H+ ε → R and Ψε : S+ ε → R, by Ψ̂ε(u) = Iε(m̂ε(u)), Ψε := Ψ̂ε|S+ ε . The next result is a direct consequence of Lemma 3.3. The details can be seen in [33]. For the convenience of the reader, here we do a sketch of the proof. Lemma 3.4. Assume that (A1)–(A6) are satisfied. Then (1) Ψ̂ε ∈ C1(H+ ε ,R) and Ψ̂′ε(u)v = ‖m̂ε(u)‖ε ‖u‖ε I ′ε(m̂ε(u))v, ∀u ∈ H+ ε , ∀v ∈ Hε. (2) Ψε ∈ C1(S+ ε ,R) and Ψ′ε(u)v = ‖mε(u)‖εI ′ε(mε(u))v, ∀v ∈ TuS+ ε . (3) If {un} is a (PS)c sequence of Ψε, then {mε(un)} is a (PS)c sequence of Iε. If {un} ⊂ Nε is a bounded (PS)c sequence for Iε, then {m−1 ε (un)} is a (PS)c sequence of Ψε. (4) u is a critical point of Ψε if and only if, mε(u) is a critical point of Iε. Moreover, corresponding critical values coincide and inf S+ ε Ψε = inf Nε Iε. EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 11 Proof. (1) Let u ∈ H+ ε and v ∈ Hε. From definition of Ψ̂ε and tu and the mean value theorem, we obtain Ψ̂ε(u+ hv)− Ψ̂ε(u) = Iε ( τu+hv(u+ hv) ) − Iε(τuu) ≤ Iε ( τu+hv(u+ hv) ) − Iε(τu+hvu) = I ′ε ( τu+hv(u+ θhv) ) τu+hvhv, where |h| is small enough and θ ∈ (0, 1). Similarly, Ψ̂ε(u+ hv)− Ψ̂ε(u) ≥ Iε(τu(u+ hv))− Iε(τuu) = I ′ε(τu(u+ ςhv))τuhv, where ς ∈ (0, 1). Since the mapping u 7→ τu is continuous according to Lemma 3.3, we see combining these two inequalities that lim h→0 Ψ̂ε(u+ hv)− Ψ̂ε(u) h = τuI ′ ε(τuu)v = ‖m̂ε(u)‖ε ‖u‖ε I ′ε(m̂ε(u))v. Since Iε ∈ C1, it follows that the Gâteaux derivative of Ψ̂ε is bounded linear in v and continuous on u. From [36] we know that Ψ̂ε ∈ C1(H+ ε ,R) and Ψ̂′ε(u)v = ‖m̂ε(u)‖ε ‖u‖ε I ′ε(m̂ε(u))v, ∀u ∈ H+ ε , ∀v ∈ Hε. The item (1) is proved. (2) This item is a direct consequence of the item (1). (3) We first note that Hε = TuS + ε ⊕Ru for every u ∈ S+ ε and the linear projection P : Hε → TuS + ε defined by P (v + τu) = v is continuous, namely, there is C > 0 such that ‖v‖ε ≤ C‖v + τu‖ε, ∀u ∈ S+ ε , v ∈ TuS+ ε , τ ∈ R. (3.6) Moreover, by (1) we have ‖Ψ′ε‖ = sup v∈TuS+ ε ,‖v‖ε=1 Ψ′ε(u)v = ‖w‖ε sup v∈TuS+ ε ,‖v‖ε=1 I ′ε(w)v, (3.7) where w = mε(u). Since w ∈ Nε, we conclude that I ′ε(w)u = I ′ε(w) w ‖w‖ε = 0. (3.8) Hence, from (3.6) and (3.8) we have ‖Ψ′ε(u)‖ ≤ ‖w‖ε‖I ′ε(w)‖ ≤ C‖w‖ε sup v∈TuS+ ε \{0} I ′ε(w)v ‖v‖ε = C‖Ψ′ε(u)‖, which shows that ‖Ψ′ε(u)‖ ≤ ‖w‖ε‖I ′ε(w)‖ ≤ C‖Ψ′ε(u)‖, ∀u ∈ S+ ε . (3.9) Since w ∈ Nε, we have ‖w‖ ≥ γ > 0. Therefore, the inequality in (3.9) together with Iε(w) = Ψε(u) imply the item (3). (4) It follow from (3.9) that Ψ′ε(u) = 0 if and only if I ′ε(w) = 0. The remainder follows from definition of Ψε. � As in [33], we have the following variational characterization of the infimum of Iε over Nε: cε = inf u∈Nε Iε(u) = inf u∈H+ ε max τ>0 Iε(τu) = inf u∈S+ ε max τ>0 Iε(τu) > 0. 12 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 The main feature of the modified functional is that it satisfies the Palais-Smale condition, as we can see from the next results. Lemma 3.5. Let {un} be a (PS)c sequence for Iε with c > 0, then {un} is bounded in Hε. Proof. Since {un} be a (PS)c sequence for Iε, then there is C > 0 such that C + ‖un‖ε ≥ Iε(un)− 1 θ I ′ε(un)un. From (3.3) and (3.4) we have C + ‖un‖ε ≥ (1− 1 K ) θ − 2 2θ ‖un‖2ε. Therefore, {un} is bounded in Hε by the fact that K, θ > 2. � Lemma 3.6 ([16]). Let {un} be a (PS)c sequence for Iε, then for each ξ > 0, there is a number R = R(ξ) > 0 such that lim sup n→∞ ∫ R3\BR ( |(−4)s/2un|2 + V (εx)u2 n ) dx < ξ. Lemma 3.7. The functional Iε verifies the (PS)c condition with c > 0 in Hε. Proof. Let {un} ⊂ Hε be a (PS)c sequence for Iε. From Lemma 3.5 we know that {un} is bounded in Hε. Passing to a subsequence, we may assume that un ⇀ u in Hs(R3), un → u in Lrloc(R3), 2 ≤ r < 2∗s, un(x)→ u(x) a.e. in R3. (3.10) By Lemma 3.6 we have for any ξ > 0, there exists an R = R(ξ) > 0 such that lim sup n→∞ ∫ R3\BR ( |(−4)s/2un|2 + V (εx)u2 n ) dx < ξ. (3.11) and so ∫ R3\BR ( |(−4)s/2u|2 + V (εx)u2 ) dx < ξ. (3.12) By (3.10)-(3.12), and the Sobolev’s Imbedding Theorem, we have that for any r ∈ [2, 2∗s) and any ξ > 0, there exists a C > 0 such that∫ R3 |un − u|r dx = ∫ BR |un − u|r dx+ ∫ R3\BR |un − u|r dx ≤ on(1) + C(‖un‖Hε(R3\BR) + ‖u‖Hε(R3\BR)) ≤ Cξ. Hence. we have proved that un → u in Lr(R3), 2 ≤ r < 2∗s. (3.13) Observe that ‖un − u‖2ε = 〈I ′ε(un)− I ′ε(u), un − u〉+ ∫ R3 ( g(εx, un)− g(εx, u) ) (un − u) dx − ∫ R3 (φtunun − φ t uu)(un − u) dx. EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 13 It is clear that 〈I ′ε(un)− I ′ε(u), un − u〉 → 0 as n→ +∞. According to assumptions (3.1) and (3.2) and the Hölder inequality, we obtain∫ R3 ( g(εx, un)− g(εx, u) ) (un − u) dx ≤ ∫ R3 C(|un|+ |u|+ |un|p−1 + |u|p−1)|un − u| dx ≤ C(‖un‖L2 + ‖u‖L2)‖un − u‖L2 + C(‖un‖p−1 Lp + ‖u‖p−1 Lp )‖un − u‖Lp . Since un → u in Lr(R3) for all r ∈ [2, 2∗s), we have that∫ R3 ( g(εx, un)− g(εx, u) ) (un − u) dx→ 0 as n→ +∞. By Hölder inequality, the Sobolev inequality and Lemma 2.4 we have | ∫ R3 φtunun(un − u) dx| ≤ ‖φtun‖L2∗t ‖un‖L 3 t ‖un − u‖L2 ≤ C‖φtun‖Dt,2‖un‖L 3 t ‖un − u‖L2 ≤ C‖un‖2 L 12 3+2t ‖un‖ L 3 t ‖un − u‖L2 , where C > 0 is a constant and we have used the fact that 2s+ 2t ≥ 3. Again using un → u in Lr(R3) for any r ∈ [2, 2∗s), we have∫ R3 φtunun(un − u) dx→ 0 as n→∞, Similarly, we obtain ∫ R3 φtuu(un − u) dx→ 0 as n→∞. Thus ∫ R3 (φtunun − φ t uu)(un − u) dx→ 0 as n→∞, so that ‖un − u‖ε → 0 and consequently un → u in Hε. � Lemma 3.8. The functional Ψε satisfies the Palais-Smale condition in S+ ε . Proof. Let {un} ⊂ S+ ε be a (PS)c sequence for Ψε. Thus, Ψε(un)→ c and ‖Ψ′ε‖∗ → 0, where ‖·‖∗ is the norm in the dual space (TunS + ε )′. It follows from Lemma 3.4(3) that {mε(un)} is a (PS)c sequence for Iε in Hε. From Lemma 3.7 we see that there is a u ∈ S+ ε such that mε(un)→ mε(u) in Hε. From Lemma 3.3(3), it follows that un → u in S+ ε . � 4. Multiplicity of solutions for the modified problem 4.1. Autonomous problem. Since we are interesting in giving a multiplicity re- sult for the modified problem, we start by considering the limit problem associated to (3.5), namely, the problem (−∆)su+ V0u+ φu = f(u) in R3, (−∆)tφ = u2 in R3, (4.1) 14 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 which has the associated functional I0(u) = 1 2 ∫ R3 (|(−∆)s/2u|2 + V0u 2) dx+ 1 4 ∫ R3 φtuu 2 dx− ∫ R3 F (u) dx. The functional is well defined on the Hilbert space H0 = Hs(R3) with the inner product (u, v)0 = ∫ R3 (−∆)s/2u(−∆)s/2v dx+ ∫ R3 V0uv dx, and the norm ‖u‖20 = ∫ R3 |(−∆)s/2u|2 dx+ ∫ R3 V0u 2 dx. We denote the Nehari manifold associated to I0 by N0 = {u ∈ H0\{0} : 〈I ′0(u), u〉 = 0}, and the open subset H+ 0 = {u ∈ H0 : | supp(u+)| > 0}, and S+ 0 = S0 ∩H+ 0 , where S0 is the unit sphere of H0. As in section 3, S+ 0 is an incomplete C1,1-manifold of codimension 1, modeled on H0 and contained in the open H+ 0 . Thus, H0 = TuS + 0 ⊕ Ru for each u ∈ S+ 0 , where TuS + 0 = {v ∈ H0 : (u, v)0 = 0}. Next we have the following Lemmas, their proofs follow from a similar argument the one used in Lemmas 3.3 and 3.4. Lemma 4.1. Let V0 be given in (A1) and (A3)–(A6) be satisfied. Then the following properties hold: (1) For each u ∈ H+ 0 , let gu : R+ → R be given by gu(τ) = I0(τu). Then there exists a unique τu > 0 such that g′u(τ) > 0 in (0, τu) and g′u(τ) < 0 in (τu,∞). (2) There is a σ > 0 independent on u such that τu > σ for all u ∈ S+ 0 . Moreover, for each compact set W ⊂ S+ 0 there is CW > 0 such that τu ≤ CW for all u ∈ W. (3) The map m̂ : H+ 0 → N0 given by m̂(u) = τuu is continuous and m := m̂|S+ 0 is a homeomorphism between S+ 0 and N0. Moreover, m−1(u) = u ‖u‖0 . We define the applications Ψ̂0 : H+ 0 → R by Ψ̂0(u) = I0(m̂(u)), and Ψ0 : S+ 0 → R by Ψ0 := Ψ̂0|S+ 0 . Lemma 4.2. Let V0 be given in (A1) and (A3)–(A6) be satisfied. Then: (1) Ψ̂0 ∈ C1(H+ 0 ,R) and Ψ̂′0(u)v = ‖m̂(u)‖0 ‖u‖0 I ′0(m̂(u))v, ∀u ∈ H+ 0 ∀v ∈ H0. (2) Ψ0 ∈ C1(S+ 0 ,R) and Ψ′0(u)v = ‖m(u)‖0I ′0(m(u))v, ∀v ∈ TuS+ 0 . (3) If {un} is a (PS)c sequence of Ψ0, then {m(un)} is a (PS)c sequence of I0. If {un} ⊂ N0 is a bounded (PS)c sequence for I0, then {m−1(un)} is a (PS)c sequence of Ψ0. EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 15 (4) u is a critical point of Ψ0 if and only if, m(u) is a critical point of I0. Moreover, corresponding critical values coincide and inf S+ 0 Ψ0 = inf N0 I0. As in the previous section, we have the following variational characterization of the infimum of I0 over N0: cV0 = inf u∈N0 I0(u) = inf u∈H+ 0 max τ>0 I0(τu) = inf u∈S+ 0 max τ>0 I0(τu) > 0. The next Lemma allows us to assume that the weak limit of a (PS)c sequence is non-trivial. Lemma 4.3. Let {un} ⊂ H0 be a (PS)c sequence with c > 0 for I0 with un ⇀ 0. Then, only one of the following statements holds. (i) un → 0 in H0, or (ii) there exist a sequence {yn} ⊂ R3 and constants R, β > 0 such that lim inf n→+∞ ∫ BR(yn) u2 n dx ≥ β > 0. Proof. Suppose (ii) does not occur. Then, for any R > 0, we have lim n→+∞ sup y∈R3 ∫ BR(y) u2 n dx = 0. Since {un} is bounded in H0, by Lemma 2.2, we have un → 0 in Lr(R3) for r ∈ (2, 2∗s). Thus, ∫ R3 f(un)un dx = ∫ R3 F (un) dx→ 0 as n→ +∞. Recalling that I ′0(un)un → 0 and Lemma 2.4 we obtain ‖un‖20 = on(1). Therefore the conclusion follows. � From Lemma 4.3 we can see that, if u is the weak limit of a (PS)cV0 sequence {un} for the functional I0, then we can assume u 6= 0. Otherwise we would have un ⇀ 0 and once it doesn’t occur un → 0, we conclude from Lemma 4.3 that there exist {yn} ⊂ R3 and R, β > 0 such that lim inf n→+∞ ∫ BR(yn) u2 n dx ≥ β > 0. Then set vn(x) = un(x+ yn), making a change of variable, we can prove that {vn} ia also a (PS)cV0 sequence for the functional I0, it is bounded in H0 and there is v ∈ H0 such that vn → v in H0 with v 6= 0. In the next Lemma we obtain a positive ground state solution for the autonomous problem (4.1). Theorem 4.4. Problem (4.1) has a positive ground state solution. 16 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 Proof. Let {un} ⊂ H0 be a (PS)cV0 sequence for I0. Arguing as Lemma 3.5, we have that {un} is bounded in H0. Then, up to a subsequence, we have un ⇀ u in Hs(R3), un → u in Lrloc(R3), 2 ≤ r < 2∗s, un(x)→ u(x) a.e. in R3. Similar to the proof in Lemmas 3.2 and 3.7, we know that the functional I0 has the mountain pass geometry and satisfies the (PS)cV0 condition. So problem (4.1) has ground state solution from [36]. Next we prove that the solution u is positive, which is a ground state solution of the equation (−∆)su+ V0u+ φtuu = f(u) in R3. (4.2) Using u− = max−u, 0 as a test function in (4.2) we obtain∫ R3 (−∆) s 2u(−∆) s 2u− dx+ ∫ R3 V0|u−|2 dx+ ∫ R3 φtu(u−)2 dx = 0. (4.3) On the other hand,∫ R3 (−∆) s 2u(−∆) s 2u− dx = 1 2 C(s) ∫∫ R3×R3 (u(x)− u(y))(u−(x)− u−(y)) |x− y|3+2s dx dy ≥ 1 2 C(s) ∫ {u>0}×{u<0} (u(x)− u(y))(−u−(y)) |x− y|3+2s dx dy + 1 2 C(s) ∫ {u<0}×{u<0} (u−(x)− u−(y))2 |x− y|3+2s dx dy + 1 2 C(s) ∫ {u<0}×{u>0} (u(x)− u(y))u−(x) |x− y|3+2s dx dy ≥ 0. Thus, from (4.3) and Lemma 2.4 (i), it follows that u− = 0 and u ≥ 0. Moreover, if u(x0) = 0 for some x0 ∈ R3 and the regularity of solutions [18] we have that (−∆)su(x0) = 0 and by [11, Lemma 3.2], we have (−∆)su(x0) = −C(s) 2 ∫ R3 u(x0 + y) + u(x0 − y)− 2u(x0) |y|3+2s dy; therefore, ∫ R3 u(x0 + y) + u(x0 − y) |y|3+2s dy = 0, yielding u ≡ 0, a contradiction. Therefore, u is a positive solution of the system (4.1) and the proof is complete. � The next result is a compactness result on autonomous problem which we will use later. Lemma 4.5. Let {un} ⊂ N0 be a sequence such that I0(un) → cV0 . Then {un} has a convergent subsequence in H0. Proof. Since {un} ⊂ N0, it follows from Lemma 4.1(3), Lemma 4.2(4) and the definition of cV0 that vn = m−1(un) = un ‖un‖0 ∈ S+ 0 , ∀n ∈ N, EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 17 Ψ0(vn) = I0(un)→ cV0 = inf S+ 0 Ψ0. Although S+ 0 is incomplete, due to Lemma 4.1, we can still apply the Ekeland’s variational principle [13] to the functional Θ0 : H → R ∪ {∞}, defined by Θ0(u) = Ψ0(u) if u ∈ S+ 0 and Θ0(u) =∞ if u ∈ ∂S+ 0 , where H = S+ 0 is the complete metric space equipped with the metric d(u, v) := ‖u−v‖0. In fact, take ε = 1 k2 in Theorem 1.1 of [13], we have a subsequence {vnk} ⊂ {vn} such that cV0 ≤ Ψ(vnk) ≤ cV0 + 1 k2 . From [13, Theorem 1.1], for λ = 1/k, there exist a sequence {ṽk} ⊂ S+ 0 such that Θ0(ṽk) ≤ Θ0(vnk) < cV0 + 1 k2 and ‖vnk − ṽk‖0 ≤ 1 k . In particular, for any u ∈ S+ 0 we have Ψ0(u) > Ψ0(ṽk)− 1 k ‖u− ṽk‖0. Hence, similar the proof for [13, Theorem 3.1], we have that there exists λk ∈ R such that ‖Ψ̂′0(ṽk)− λkg′0(ṽk)‖0 ≤ 1 k , where g0(u) = ‖u‖20 − 1. Which means that λk = 1 ‖g′0(ṽk)‖20 〈Ψ̂′0(ṽk), g′0(ṽk)〉+ ok(1), g′0(ṽk) = ṽk. From Lemma 4.2(1), λk = 〈Ψ̂′0(ṽk), ṽk〉+ ok(1) = τṽk〈I ′0(tṽk ṽk), ṽk〉+ ok(1) = ok(1). Therefore, we can conclude there is a sequence {ṽn} ⊂ S+ 0 such that {ṽn} is a (PS)cV0 sequence for Ψ0 on S+ 0 and ‖un − ṽn‖0 = on(1). The remainder of the proof follows from Lemma 4.2, Theorem 4.4 and arguing as in the proof of Lemma 3.8. � 4.2. Technical results. In this section we will relate the number of positive solu- tions of (3.5) to the topology of the set M. For this, we consider δ > 0 such that Mδ ⊂ Ω and by Theorem 4.4, we can choose w ∈ N0 with I0(w) = cV0 . Let η be a smooth nonincreasing cut-off function defined in [0,+∞) such that η(t) = 1 if 0 ≤ t ≤ δ 2 and η(t) = 0 if t ≥ δ. For each y ∈M, let Ψε,y(x) = η(|εx− y|)w( εx− y ε ). Then for small ε > 0, one has Ψε,y ∈ Hε\{0} for all y ∈ M. In fact, using the change of variable z = x− y ε , one has∫ R3 V (εx)Ψ2 ε,y(x) dx = ∫ R3 V (εx)η2(|εx− y|)w2( εx− y ε ) dx = ∫ R3 V (εz + y)η2(|εz|)w2(z)dz 18 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 ≤ C ∫ R3 w2(z)dz < +∞. Moreover, using the change of variable x′ = x− y ε , z ′ = z − y ε , we have ‖(−∆)s/2Ψε,y‖22 = 1 2 C(s) ∫∫ R3×R3 ∣∣η(|εx− y|)w( εx−yε )− η(|εz − y|)w( εz−yε ) ∣∣2 |x− z|3+2s dxdz = 1 2 C(s) ∫∫ R3×R3 ∣∣η(|εx′|)w(x′)− η(|εz′|)w(z′) ∣∣2 |x′ − z′|3+2s dx′dz′ = ‖(−∆)s/2η(|εx|)w(x)‖22 = ‖(−∆)s/2ηεw‖22, where ηε(x) = η(|εx|). By Lemma 2.3, we see that ηεw ∈ Ds,2(R3) as ε → 0, and hence Ψε,y ∈ Ds,2(R3) for ε > 0 small. Hence Ψε,y ∈ Hε. Now we proof Ψε,y 6= 0. In fact, ∫ R3 Ψ2 ε,y(x) dx = ∫ R3 η2(|εx− y|)w2( εx− y ε ) dx = ∫ |εx−y|<δ η2(|εx− y|)w2( εx− y ε ) dx ≥ ∫ |z|≤δ/2ε η2(|εz|)w2(z)dz ≥ ∫ B0(δ/2ε) w2(z)dz → ∫ R3 w2(z)dz > 0 as ε → 0. Then Ψε,y 6= 0 for small ε > 0. Therefore, there exists unique τε > 0 such that max τ≥0 Iε(τΨε,y) = Iε(τεΨε,y) and τεΨε,y ∈ Nε. We introduce the map Φε :M→ Nε by setting Φε(y) = τεΨε,y. By construction, Φε(y) has a compact support for any y ∈M and Φε is a continuous map. Lemma 4.6. The functional Φε(y) satisfies lim ε→0 Iε(Φε(y)) = cV0 uniformly in y ∈M. Proof. Suppose that the result is false. Then, there exist some δ0 > 0, {yn} ⊂ M and εn → 0 such that |Iεn(Φεn(yn))− cV0 | ≥ δ0. (4.4) From the definition of τεn we have 0 < τ2 εn ∫ R3 |(−∆)s/2Ψεn,yn |2 dx+ τ2 εn ∫ R3 V (εnx)Ψ2 εn,yn dx ≤ τ2 εn ∫ R3 |(−∆)s/2Ψεn,yn |2 dx+ τ2 εn ∫ R3 V (εnx)Ψ2 εn,yn dx + τ4 εn ∫ R3 φtΨεn,ynΨ2 εn,yn dx = τεn ∫ R3 g(εnx, τεnΨεn,yn)Ψεn,yn dx. (4.5) EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 19 It follows from (4.5) that τεn 6→ 0; then τεn ≥ τ0 > 0 for some τ0 > 0. If τεn → +∞, by (A6) and the boundedness of Ψεn,yn , we obtain 1 τ2 εn (∫ R3 |(−∆)s/2Ψεn,yn |2 dx+ ∫ R3 V (εnx)Ψ2 εn,yn dx ) + ∫ R3 φtΨεn,ynΨ2 εn,yn dx = ∫ R3 g(εnx, τεnΨεn,yn) (τεnΨεn,yn)3 Ψ4 εn,yn dx→ +∞ as n → +∞. But the left side of the above inequality tends to ∫ R3 φ t ww 2 dx since τεn → +∞ as εn → 0, which is impossible. Hence, 0 < τ0 ≤ τεn ≤ C. Without loss of generality, we may assume that τεn → T > 0. Next we claim that T = 1. By Lemma 2.3 and Lebesgue’s theorem we have lim n→+∞ ‖Ψεn,yn‖2εn = ‖w‖20, lim n→+∞ ∫ R3 φtΨεn,yn |Ψεn,yn |2 dx = ∫ R3 φtw|w|2 dx, lim n→+∞ ∫ R3 f(Ψεn,yn)Ψεn,yn dx = ∫ R3 f(w)w dx, (4.6) Therefore, 1 T 2 ∫ R3 |(−∆)s/2w|2 dx+ 1 T 2 ∫ R3 V0w 2 dx+ ∫ R3 φtww 2 dx = ∫ R3 f(Tw) (Tw)3 w4 dx. (4.7) Since w is a ground state solution of (4.1), then∫ R3 |(−∆)s/2w|2 dx+ ∫ R3 V0w 2 dx+ ∫ R3 φtww 2 dx = ∫ R3 f(w)w dx. (4.8) Combining (4.7)-(4.8), we have (1− 1 T 2 ) (∫ R3 |(−∆)s/2w|2 dx+ ∫ R3 V0w 2 dx ) = ∫ R3 (f(w) w3 − f(Tw) (Tw)3 ) w4 dx. (4.9) By (3.4), we deduce that T = 1. It follows from (4.6), we have lim n→+∞ Iεn(Φεn(yn)) = IV0 (w) = cV0 , (4.10) which contradicts (4.4). This completes the proof. � Let ρ = ρ(δ) > 0 be such that Mδ ⊂ Bρ(0). Consider χ : R3 → R3 be defined as χ(x) = x for |x| ≤ ρ and χ(x) = ρx |x| for |x| ≥ ρ. Finally, let us consider the barycenter map βε : Nε → R3 given by βε(u) = ∫ R3 χ(εx)u2(x) dx∫ R3 u2(x) dx ∈ R3. Lemma 4.7. The functional βε satisfies limε→0 βε(Φε(y)) = y uniformly in y ∈M. Proof. Suppose by contradiction that there exist δ0 > 0, {yn} ⊂ M and εn → 0 such that |βεn(Φεn(yn))− yn| ≥ δ0. (4.11) 20 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 Using the change of variables z = εnx−yn εn and the definition of βε, we have βεn(Φεn(yn)) = yn + ∫ R3(χ(εnz + y)− yn)|η(|εnz|)w(z)|2 dx∫ R3 |η(|εnz|)w(z)|2 dx . Since {yn} ⊂ M ⊂ Bρ(0) and χ ∣∣ Bρ ≡ id, we conclude that |βεn(Φεn(yn))− yn| = on(1), which contradicts (4.11) and the desired conclusion holds. � Lemma 4.8. Let εn → 0 and {un} ⊂ Nεn be such that Iεn(un) → cV0 . Then, there exists a sequence {ỹn} ⊂ R3 such that vn(x) = un(x + ỹn) has a convergent subsequence in H0. Moreover, passing to a subsequence, yn := εnỹn → y0 ∈M. Proof. By Lemma 3.5, {un} is bounded in H0. Note that cV0 > 0, and since ‖un‖εn → 0 would imply Iεn(un)→ 0, we can argue as in the proof of Lemma 4.3 to obtain a sequence {ỹn} ⊂ R3 and constants R, β > 0 such that lim inf n→+∞ ∫ BR(ỹn) u2 n dx ≥ β > 0. Define vn(x) := un(x + ỹn), then {vn} is also bounded in H0 and up to a subse- quence, we have vn ⇀ v 6= 0 in H0. Let τn > 0 be such that ṽn := τnvn ∈ N0 and set yn = εnỹn. Because {un} ⊂ Nεn , we have cV0 ≤ I0(ṽn) ≤ 1 2 ∫ R3 |(−∆)s/2ṽn|2 dx+ 1 2 ∫ R3 V (εnx+ yn)ṽ2 n dx + 1 4 ∫ R3 φtṽn ṽ 2 n dx− ∫ R3 G(εnx+ yn, ṽn) dx = τ2 n 2 ∫ R3 |(−∆)s/2un|2 dx+ τ2 n 2 ∫ R3 V (εnx)u2 n dx + τ4 n 4 ∫ R3 φtunu 2 n dx− ∫ R3 G(εnx, τnun) dx = Iεn(τnun) ≤ Iεn(un) = cV0 + on(1). This implies limn→+∞ I0(ṽn) = cV0 . Because ṽn ∈ N0, we obtain {ṽn} is bounded in H0. It follows from the boundedness of {vn} in H0 that {τn} is bounded, without loss of generality, we may assume that τn → τ0 ≥ 0. If τ0 = 0, in view of the boundedness of {vn} in H0, we have ṽn = τnvn → 0 in H0. Hence I0(ṽn) → 0, which contradicts cV0 > 0. Thus, τ0 > 0 and the weak limit of {ṽn} is different from zero. Hence, up to a subsequence, we have ṽn ⇀ τ0v := ṽ 6= 0 in H0 by the uniqueness of the weak limit. From Lemma 4.5, we know that ṽn → ṽ in H0. Moreover, ṽ ∈ N0. Now, we show that {yn} is bounded in R3. Suppose that after passing to a subsequence, |yn| → +∞. Then, by Fatou’s Lemma and Lemma 2.4 we have cV0 = I0(ṽn) < lim inf n→∞ (1 2 ∫ R3 |(−∆)s/2ṽn|2 dx+ 1 2 ∫ R3 V (εnx+ yn)ṽ2 n dx EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 21 + 1 4 ∫ R3 φtṽn ṽ 2 n dx− ∫ R3 F (ṽn) dx ) = lim inf n→∞ (τ2 n 2 ∫ R3 |(−∆)s/2un|2 dx+ τ2 n 2 ∫ R3 V (εnx)u2 n dx + τ4 n 4 ∫ R3 φtunu 2 n dx− ∫ R3 F (τnun) dx ) ≤ lim inf n→∞ Iεn(τnun) ≤ lim inf n→∞ Iεn(un) = cV0 , which yields a contradiction. Hence, {yn} is bounded and up to a subsequence, yn → y0 in R3. If y0 /∈ M, then V0 < V (y0) and we obtain a contradiction by the same manner as above. So, y0 ∈M and the proof is complete. � Let h : R+ → R+ be a positive function satisfying h(ε)→ 0+ as ε→ 0+. Define the set Ñε = {u ∈ Nε : Iε(u) ≤ cV0 + h(ε)}. Given y ∈M, from Lemma 4.6 we conclude that h(ε) = supy∈M |Iε(Φε(y))−cV0 | → 0 as ε→ 0+. Thus, Φε(y) ∈ Ñε and Ñε 6= ∅ for ε > 0. Lemma 4.9. For any δ > 0, it holds that lim ε→0 sup u∈Ñε inf y∈Mδ |βε(u)− y| = 0. Proof. Let {εn} ⊂ R+ be such that εn → 0. By definition, there exists {un} ⊂ Ñεn such that inf y∈Mδ |βεn(un)− y| = sup u∈Ñεn inf y∈Mδ |βεn(u)− y|+ on(1). So, it suffices to find a sequence {yn} ⊂ Mδ satisfying lim n→+∞ |βεn(un)− yn| = 0. (4.12) Since un ∈ Ñεn ⊂ Nεn , we obtain cV0 ≤ cεn ≤ Iεn(un) ≤ cV0 + h(εn). It follows that Iεn(un)→ cV0 . Thus, we can invoke Lemma 4.8 to obtain a sequence {ỹn} ⊂ R3 such that yn = εnỹn ∈Mδ for n large enough. Then βεn(un) = yn + ∫ R3(χ(εnx+ yn)− yn)|un(x+ ỹn)|2 dx∫ R3 |un(x+ ỹn)|2 dx . For all x ∈ R3 fixed, since εnx + yn → y ∈ Mδ, we have that the sequence {yn} satisfies (4.12). This completes the proof. � 4.3. Multiplicity of solutions for (3.5). Next we prove our multiplicity result by presenting a relation between the topology ofM the number of solutions of the modified problem (3.5), we will apply the Ljusternik-Schnirelmann abstract result in [31, 33]. Theorem 4.10. Assume that conditions (A1)–(A6) hold. Then, given δ > 0 there is ε̂δ > 0 such that for any ε ∈ (0, ε̂δ), problem (3.5) has at least catMδ (M) positive solutions. 22 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 Proof. For y ∈M, set γε(y) = m−1 ε (Φε(y)). It follows from Lemma 3.4 and Lemma 4.6 that lim ε→0 Ψε(γε(y)) = lim ε→0 Iε(Φε(y)) = cV0 , (4.13) uniformly for in y ∈M. Let S̃+ ε = {w ∈ S+ ε : Ψε(w) ≤ cV0 + h(ε)}, where h is given in the definition of Ñε. From (4.13), we know that there is a number ε̂ such that S̃+ ε 6= ∅ for ε ∈ (0, ε̂). For a fixed δ > 0, by Lemmas 3.3, 4.6, 4.7 and 4.9, we know that there exists a ε̂ = ε̂δ > 0 such that for any ε ∈ (0, ε̂δ), the diagram M Φε−→ Ñε m−1 ε−→ S̃+ ε mε−→ Ñε βε−→Mδ is well defined. From Lemma 4.7, there is a function λ(ε, y) with |λ(ε, y)| < δ 2 uniformly in y ∈ M, for all ε ∈ (0, ε̂), such that βε(Φε(y)) := y + λ(ε, y) for all y ∈ M. Define H(t, y) = y + (1 − t)λ(ε, y). Then, H : [0, 1] × M → Mδ is continuous. Obviously, H(0, y) = βε(Φε(y)), H(1, y) = y for all y ∈ M. That is, H(t, y) is homotopy between βε ◦ Φε and the inclusion map id : M → Mδ. This fact and [5, Lemma 4.3] implies that catS̃+ ε γε(M) ≥ catMδ (M). On the other hand, using the definition of Ñε and choosing ε̂δ small if necessary, we see that Iε satisfies the (PS) condition in Ñε recalling Lemma 3.7. By Lemma 3.4 and 3.8, we obtain that Ψε satisfies the (PS) condition in S̃+ ε . Therefore, the standard Ljusternik-Schnirelmann theory provides at least catS̃+ ε γε(M) critical points of Ψε restricted to S̃+ ε . Using Lemma 3.7 again, we infer that Iε has at least catMδ (M) critical points. Using the same arguments contained in the proof Theorem 4.4, we see that the system (3.5) has at least catMδ (M) positive solutions. � 5. Proof of Theorem 1.2 In this section we will prove our main result. The idea is to show that the solutions obtained in Theorem 4.10 satisfy the following estimate uε(x) ≤ a, for all x ∈ Ωcε for ε small enough. This fact implies that these solutions are in fact solutions of the original problem (2.1). The key ingredient is the following result, whose proof uses an adaptation of the arguments found in [12], which are related to the Moser iteration method [26]. Lemma 5.1. Let εn → 0+ and un ∈ Ñεn be a solution of (3.5). Then up to a subsequence, vn = un(x + ỹn) satisfies that vn ∈ L∞(R3) and there exists C > 0 such that ‖vn‖L∞(R3) ≤ C, ∀n ∈ N, where {ỹn} is given in Lemma 4.8. Proof. We define h(εnx, vn) := g(εnx+ εnỹn, vn)− V (εnx+ εnỹn)vn − φtvnvn. EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 23 From Lemma 3.5, {vn} is bounded in Hε, and hence in Lr(R3) for any r ∈ [2, 2∗s]. So there exists some C > 0 such that ‖vn‖q ≤ C, ∀ n ∈ N. Since vn is a solution of (3.5), it follows that φtvn(x) = ∫ R3 v2 n(y) |x− y|3−2t dy = ∫ {|x−y|≤1} v2 n(y) |x− y|3−2t dy + ∫ {|x−y|>1} v2 n(y) |x− y|3−2t dy ≤ ∫ {|x−y|≤1} v2 n(y) |x− y|3−2t dy + ∫ {|x−y|>1} v2 n(y) dy ≤ (∫ {|x−y|≤1} 1 |x− y|(3−2t)q′ dy )1/q′(∫ {|x−y|≤1} v2q n (y) dy )1/q + C ≤ C , where q′(3− 2t) < 3, 2q ∈ [2, 2∗s], 1 q + 1 q′ = 1 since 2s+ 2t ≥ 3. Therefore, |h(εnx, vn)| ≤ C(|vn|+ |vn|p−1) ≤ C(1 + |vn|2 ∗ s−1) (5.1) for n large enough. For T > 0, we define H(t) =  0, if t ≤ 0, tβ , if 0 < t < T, βT β−1(t− T ) + T β , if t ≥ T, with β > 1 to be determined later. Since H is Lipschitz with constant L0 = βT β−1, we have [H(vn)]Ds,2 = (∫∫ R3×R3 |H(vn(x))−H(vn(y))|2 |x− y|3+2s dx dy )1/2 ≤ (∫∫ R3×R3 L2 0|vn(x)− vn(y)|2 |x− y|3+2s dx dy )1/2 = L0[vn]Ds,2 . Therefore, H(vn) ∈ Ds,2(R3). Moreover, by the definition of H, we know that H is a convex function; then (−∆)sH(vn) ≤ H ′(vn)(−∆)svn (5.2) in the weak sense. Thus, from H(vn) ∈ Ds,2(R3) and (5.1) and (5.2), we have ‖H(vn)‖22∗s ≤ C ∫ R3 |(−∆)s/2H(vn)|2 dx = C ∫ R3 H(vn)(−∆)sH(vn) dx ≤ C ∫ R3 H(vn)H ′(vn)(−∆)svn dx = C ∫ R3 H(vn)H ′(vn)h(εnx, vn) dx 24 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 ≤ C ∫ R3 H(vn)H ′(vn) dx+ C ∫ R3 H(vn)H ′(vn)v 2∗s−1 n dx. Using that H(vn)H ′(vn) ≤ βv2β−1 n and vnH ′(vn) ≤ βH(vn), we have(∫ R3 ( H(vn) )2∗s dx)2/2∗s = Cβ (∫ R3 v2β−1 n dx+ ∫ R3 ( H(vn) )2 v 2∗s−2 n dx ) , (5.3) where C is a positive constant that does not depend on β. Note that the last integral is well defined for T in the definition of H. Indeed∫ R3 ( H(vn) )2 v 2∗s−2 n dx = ∫ vn≤T ( H(vn) )2 v 2∗s−2 n dx+ ∫ vn>T ( H(vn) )2 v 2∗s−2 n dx ≤ T 2β−2 ∫ R3 v 2∗s n dx+ C ∫ R3 v 2∗s n dx <∞. We choose now β in (5.3) such that 2β − 1 = 2∗s, and we name it β1, that is β1 := 2∗s + 1 2 . (5.4) Let R̂ > 0 to be fixed later. Using last integral in (5.3) and applying the Holder’s inequality with exponents γ := 2∗s 2 and γ′ := 2∗s 2∗s−2 , we have∫ R3 ( H(vn) )2 v 2∗s−2 n dx = ∫ vn≤R̂ ( H(vn) )2 v 2∗s−2 n dx+ ∫ vn>R̂ ( H(vn) )2 v 2∗s−2 n dx ≤ ∫ vn≤R̂ ( H(vn) )2 vn R̂2∗s−1 dx+ (∫ R3 ( H(vn) )2∗s dx)2/2∗s (∫ vn>R̂ v 2∗s n dx ) 2∗s−2 2∗s . (5.5) By Lemma 4.8, we know that {vn} has a convergent subsequence in H0, therefore we can choose R̂ large enough so that(∫ vn>R̂ v 2∗s n dx ) 2∗s−2 2∗s ≤ 1 2Cβ1 , where C is the constant appearing in (5.3). Therefore, we can absorb the last term in (5.5) by the left hand side of (5.3) to obtain(∫ R3 ( H(vn) )2∗s dx)2/2∗s ≤ 2Cβ1 (∫ R3 v 2∗s n dx+ R̂2∗s−1 ∫ R3 ( H(vn) )2 vn dx ) . Now we use that H(vn) ≤ vβ1 n and we take T →∞, to obtain(∫ R3 v 2∗sβ1 n dx )2/2∗s ≤ 2Cβ1 (∫ R3 v 2∗s n dx+ R̂2∗s−1 ∫ R3 v 2∗s n dx ) , and therefore vn ∈ L2∗sβ1(R3). (5.6) Let us suppose now β > β1. Thus, using that H(vn) ≤ vβn in the right-hand side of (5.3) and letting T →∞ we obtain(∫ R3 v 2∗sβ n dx )2/2∗s ≤ Cβ (∫ R3 v2β−1 n dx+ R̂2∗s−1 ∫ R3 v 2β+2∗s−2 n dx ) . (5.7) EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 25 Set c0 := 2∗s(2∗s−1) 2(β−1) and c1 := 2β − 1 − c0. Note that since β > β1, we have 0 < c0 < 2∗s, c1 > 0. Hence, applying Young’s inequality with exponents γ := 2∗s/c0 and γ′ := 2∗s/2 ∗ s − c0, we have∫ R3 v2β−1 n dx ≤ c0 2∗s ∫ R3 v 2∗s n dx+ 2∗s 2∗s − c0 ∫ R3 v 2∗sc1 2∗s−c0 n dx ≤ ∫ R3 v 2∗s n dx+ ∫ R3 v 2β+2∗s−2 n dx ≤ C ( 1 + ∫ R3 v 2β+2∗s−2 n dx ) , with C > 0 independent of β. Plugging into (5.7),(∫ R3 v 2∗sβ n dx )2/2∗s ≤ Cβ ( 1 + ∫ R3 v 2β+2∗s−2 n dx ) , with C changing from line to line, but remaining independent of β. Therefore,( 1 + ∫ R3 v 2∗sβ n dx ) 1 2∗s (β−1) ≤ ( Cβ ) 1 2(β−1) ( 1 + ∫ R3 v 2β+2∗s−2 n dx ) 1 2(β−1) . (5.8) Repeating this argument we define a sequence βm,m ≥ 1 such that 2βm+1 + 2∗s − 2 = 2∗sβm. Thus, βm+1 − 1 = (2∗s 2 )m (β1 − 1). Replacing this in (5.8) one has( 1 + ∫ R3 v 2∗sβm+1 n dx ) 1 2∗s (βm+1−1) ≤ ( Cβm+1 ) 1 2(βm+1−1) ( 1 + ∫ R3 v 2∗sβm n dx ) 1 2∗s (βm−1) . Defining Cm+1 := Cβm+1 and Am := ( 1 + ∫ R3 v 2∗sβm n dx ) 1 2∗s (βm−1) , we conclude that there exists a constant C0 > 0 independent of m, such that Am ≤ m∏ k=1 C 1 2(βk−1) k A1 ≤ C0A1. Thus, ‖vn‖∞ ≤ C0A1 <∞, (5.9) uniformly in n ∈ N, thanks to (5.6). This completes the proof. � Lemma 5.2. As |x| → ∞, un(x)→ 0 uniformly in n. Proof. Note that un satisfies (−∆)sun + un = Υn, x ∈ R3, where Υn(x) = un(x)− V (εn(x+ ỹn))un(x)− φtunun(x) + g(εn(x+ ỹn), un), x ∈ R3. Putting Υ(x) = u(x)− V (y0)u(x)− φtuu(x) + g(y0, u), by Lemma 4.7, we see that Υn → Υ in Lq(R3), ∀ q ∈ [2,+∞), 26 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 and there exists a C > 0 such that ‖Υn‖L∞(R3) ≤ C, ∀ n ∈ N. From [15], we have that un(x) = G ∗Υn = ∫ R3 G(x− y)Υn(y) dy, where G is the Bessel Kernel G(x) = F−1( 1 1 + |ξ|2s ). From [15, Theorem 3.3] it is known that G is positive, radially symmetric and smooth in R3\{0}, there is C > 0 such that G(x) ≤ C/|x|3+2s, and G ∈ Lq(R3) for all q ∈ [1, 3 3−2s ). Now arguing as in the proof of [1, Lemma 2.6], we conclude that un(x)→ 0 as |x| → ∞, (5.10) uniformly in n ∈ N. � We are now ready to prove the main result of this article. Proof of Theorem 1.2. We fix a small δ > 0 such thatMδ ⊂ Ω. We first claim that there exists some ε̃δ > 0 such that for any ε ∈ (0, ε̃δ) and any solution uε ∈ Ñε of the problem (3.5), it holds ‖uε‖L∞(R3\Ωε) < a. (5.11) To prove the claim we argue by contradiction. Suppose that for some sequence εn → 0+ we can obtain un ∈ Ñεn such that I ′εn(un) = 0 and ‖un‖L∞(R3\Ωε) ≥ a. (5.12) As in the proof of Lemma 4.8, we have that Iεn(un) → cV0 and we can obtain a sequence {ỹn} ∈ R3 such that εnỹn → y0 ∈M. If we take r > 0 such that Br(y0) ⊂ B2r(y0) ⊂ Ω we have B r εn ( y0 εn ) = 1 εn Br(y0) ⊂ Ωεn . Moreover, for any z ∈ B r εn (ỹn), it holds |z − y0 εn | ≤ ||z − ỹn|+ |ỹn − y0 εn | < 1 εn (r + on(1)) < 2r εn , for n large. For these values of n we have that B r εn (ỹn) ⊂ Ωεn ; that is, R3\Ωεn ⊂ R3\B r εn (ỹn). On the other hand, it follows from Lemma 5.2 that there is R > 0 such that un(x) < a for |x| ≥ R and all n ∈ N. Then it follows that vn(x− ỹn) < a for x ∈ BcR(ỹn), n ∈ N. Thus, there exists n0 ∈ N such that for any n ≥ n0 and r εn > R, it holds R3\Ωεn ⊂ R3\B r εn (ỹn) ⊂ R3\BR(ỹn). Then, it holds un(x) < a ∀x ∈ R3\Ωεn , which contradicts (5.12) and the claim holds. Let ε̂δ be given by Theorem 4.10, and let εδ := min{ε̂δ, ε̃δ}. We will prove the theorem for this choice of εδ. Let ε ∈ (0, εδ) be fixed. By using Theorem 4.10 we obtain catMδ (M) nontrivial solutions of problem (3.5). If u ∈ Hε is one of EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 27 these solutions, we have that u ∈ Ñε, and we can use (5.11) and the definition of g to conclude that g(·, u) = f(u). Hence, u is also a solution of (2.1). An easy calculation shows that ω(x) = u(xε ) is a solution of the original problem (1.3). Then (1.3) has at least catMδ (M) nontrivial solutions. Now we consider εn → 0+ and take a sequence un ∈ Hεn of solutions of (3.5) as above. To study the behavior of the maximum points of un, we first notice that, by the definition of g and (3.1), (3.2), there exists 0 < γ < a such that g(εnx, u)u ≤ V0 K u2, for all x ∈ R3, u ≤ γ. (5.13) Using a similar discussion as above, we obtain R > 0 and {ỹn} ⊂ R3 such that ‖un‖L∞(BcR(ỹn)) < γ. (5.14) Up to a subsequence, we may assume that ‖un‖L∞(BR(ỹn)) ≥ γ. (5.15) Indeed, if this is not the case, we have ‖un‖L∞ < γ, and therefore from I ′εn(un) = 0 and (5.13) it follows that ‖un‖2εn ≤ ∫ R3 g(εnx, un)un dx ≤ V0 K ∫ R3 u2 n dx. (5.16) The above expression implies that ‖un‖εn → 0 as n→∞, which leads to a contra- diction. Thus, (5.15) holds. By using (5.14) and (5.15) we conclude that the maximal points pn ∈ R3 of un belong to BR(ỹn). Hence, pn = ỹn + qn for some qn ∈ BR(0). Recalling that the associated solution of (1.3) is of the form ωn(x) = un(xε ), we conclude that the maximum point ηε of vn is ηε := εnỹn + εnqn. Since {qn} ⊂ BR(0) is bounded and εnỹn → y0 ∈M, we obtain lim n→∞ V (ηε) = V (y0) = V0. Next we estimate the decay properties of ωn. For this, we first claim that there exist C > 0 such that un(x) ≤ C 1 + |x|3+2s , ∀x ∈ R3. Indeed, we can use [15, Lemma 4.2 ], according to which there exists a continuous function ω̄ such that 0 < ω̄(x) ≤ C 1 + |x|3+2s , (5.17) (−∆)sω̄ + V0 2 ω̄ = 0 in R3\BR̄(0) (5.18) 28 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 for some suitable R̄ > 0. Thanks to (5.10) we have that un → 0 as |x| → ∞ uniformly in n. Therefore, for some large R1 > 0, we obtain (−∆)sun + V0 2 un = (−∆)sun + V (εn(x+ ỹn))un − ( V (εn(x+ ỹn))− V0 2 ) un = −φsvnun + g(εn(x+ ỹn), un)− ( V (εn(x+ yn))− V0 2 ) un ≤ g(εn(x+ ỹn), un)− V0 2 un ≤ V0 K un − V0 2 un ≤ 0, (5.19) for x ∈ R3\BR1 (0). Now we take R2 := max{R̄, R1} and set b := inf BR2 (0) ω̄ > 0, zn := (m+ 1)ω̄ − bvn, (5.20) where m := supn∈N ‖un‖L∞ < ∞. We next show that zn ≥ 0 in R3. For this we suppose by contradiction that there is a sequence {xjn} such that inf x∈R3 zn(x) = lim j→∞ zn(xjn) < 0, (5.21) Notice that lim |x|→∞ ω̄(x) = lim |x|→∞ un(x) = 0, and so lim |x|→∞ zn(x) = 0, uniformly in n ∈ N. Consequently, the sequence {xjn} is bounded and therefore, up to a subsequence, we may suppose that xjn → x∗n as j → ∞ for some x∗n ∈ R3. Hence (5.21) becomes inf x∈R3 zn(x) = zn(x∗n) < 0. (5.22) From the minimality property of x∗n, we have (−∆)szn(x∗n) = −C(s) 2 ∫ R3 zn(x∗n + y) + zn(x∗n − y)− 2zn(x∗n) |y|3+2s dy ≤ 0. (5.23) By (5.20), we obtain zn(x) ≥ mb+ ω̄ −mb > 0, in BR2(0). Therefore, combining this with (5.22), we see that x∗n ∈ R3 \BR2 (0). (5.24) From (5.17)-(5.19), we conclude that (−∆)szn + V0 2 zn ≥ 0, in R3 \BR2(0). (5.25) Thinks to (5.24), we can evaluate (5.25) at the point x∗n, and recalling (5.22), (5.23), we conclude that 0 ≤ (−∆)szn(x∗n) + V0 2 zn(x∗n) < 0, EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 29 this is a contradiction, so zn(x) ≥ 0 in R3. That is to say, un ≤ (m+ 1)b−1ω̄ with (5.17) imply that un(x) ≤ C 1 + |x|3+2s , ∀x ∈ R3. Therefore, ωn(x) = un( x εn − ỹn) ≤ C 1 + | xεn − ỹn| 3+2s = Cε3+2s n ε3+2s n + |x− εnỹn|3+2s = Cε3+2s n ε3+2s n + |x− ηεn |3+2s , ∀ x ∈ R3. Thus, the proof is complete. � Acknowledgments. This research was supported by the National Natural Science Foundation of China (No. 11771385). We would like to thank the anonymous referee for his/her careful readings of our manuscript and the useful comments made for its improvement. References [1] C. Alves, O.H. Miyagaki; Existence and concentration of solution for a class of fractional elliptic equation in RN via penalization method, Calc. Var. Partial Differential Equations, 55 (2016), 1-19. [2] A. Ambrosetti; On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257–274. [3] A. Azzollini; Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), 1746–1763. [4] A. Azzollini, P. d’Avenia, A. Pomponio; On the Schrödinger-Maxwell equations under the effect of a general nonlinear termk, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779–791. [5] V. Benci, G. Cerami; Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29–48. [6] V. Benci, D. Fortunato; An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283–293. [7] H. Brézis, E. Lieb; A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490. [8] C. Bucur, E. Valdinoci; Nonlocal diffusion and applications, volume 20 of Lecture Notes of the Unione Matematica Italiana, Springer; Unione Matematica Italiana, Bologna, 2016. [9] G. Cerami, G. Vaira; Positive solutions for some non-autonomous Schrödinger-Poisson sys- tems, J. Differential Equations, 248 (2010). 521–543. [10] M. del Pino, P. L. Felmer; Local mountain passes for semilinear elliptic problems in un- bounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121–137. [11] E. Di Nezza, G. Palatucci, E. Valdinoci; Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. [12] S. Dipierro, M. Medina, E. Valdinoci; Fractional elliptic problems with critical growth in the whole of Rn, volume 15 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2017. [13] I. Ekeland; On the variational principle, J. Math. Anal. Appl., 47 (1974), 324–353. [14] H. Fan; Multiple positive solutions for Schrödinger-Poisson systems involving concave-convex nonlinearities, Electron. J. Differential Equations, 2019 no. 86 (2019), 1-19. [15] P. Felmer, A. Quaas, J. Tan; Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237–1262. 30 Z. YANG, W. ZHANG, F. ZHAO EJDE-2021/14 [16] X. He, W. Zou; Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016), 1–39. [17] Y. Jiang, H. Zhou; Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582–608. [18] T. Jin, Y. Li, J. Xiong; On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111–1171. [19] T. Jin and Z. Yang; The fractional Schrödinger-Poisson systems with infinitely many solu- tions, J. Korean Math. Soc., 57 (2020), 489–506. [20] N. Landkof; Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. [21] N. Laskin; Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298–305. [22] N. Laskin; Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108. [23] P.-L. Lions; Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33–97. [24] W. Liu; Existence of multi-bump solutions for the fractional Schrödinger-Poisson system, J. Math. Phys., 57 (2016),091502. [25] Z. Liu, J. Zhang; Multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson systems with critical growth. ESAIM Control Optim. Calc. Var. , 23 (2017), 1515–1542. [26] J. Moser; A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457–468. [27] E. G. Murcia, G. Siciliano; Positive semiclassical states for a fractional Schrödinger-Poisson system, Differential Integral Equations, 30 (2017), 231–258. [28] G. Palatucci, A. Pisante; Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equa- tions, 50 (2014), 799–829. [29] P. H. Rabinowitz; On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43(1992), 270–291. [30] S. Secchi; Ground state solutions for nonlinear fractional Schrödinger equations in RN , J. Math. Phys., 54 (2013), 1–17. [31] M. Struwe; Variational methods. Springer-Verlag, Berlin, 1990. [32] A. Szulkin, T. Weth; Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802–3822. [33] A. Szulkin, T. Weth; The method of Nehari manifold. Int. Press, Somerville, MA, 2010. [34] K. Teng; Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261(2016), 3061–3106. [35] W. Wang, Q. Li; Existence and concentration of positive ground states for Schrödinger- Poisson equations with competing potential functions, Electron. J. Differential Equations, 2020, no. 78 (2020), 1-19. [36] M. Willem; Minimax theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. [37] Z. Yang, Y. Yu, F. Zhao; The concentration behavior of ground state solutions for a critical fractional Schrödinger-Poisson system, Math. Nachr., 292 (2019), 1837–1868. [38] Z. Yang, Y. Yu, F. Zhao; Concentration behavior of ground state solutions for a frac- tional Schrödinger-Poisson system involving critical exponent, Commun. Contemp. Math., 21 (2019),1–46 [39] Y. Yu, F. Zhao, L. Zhao; The concentration behavior of ground state solutions for a fractional Schrödinger-Poisson system, Calc. Var. Partial Differential Equations, 56 (2017), 1-25. [40] J. Zhang, J. M. do ó, M. Squassina; Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16(2016), 15–30. [41] L. Zhao, H. Liu, F. Zhao; Existence and concentration of solutions for the Schrödinger- Poisson equations with steep well potential, J. Differential Equations, 255 (2013), 1–23. Zhipeng Yang Department of Mathematics, Yunnan Normal University, Kunming 650500, China Email address: yangzhipeng326@163.com EJDE-2021/14 FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS 31 Wei Zhang School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China Email address: weizyn@163.com Fukun Zhao (corresponding author) Department of Mathematics, Yunnan Normal University, Kunming 650500, China Email address: fukunzhao@163.com 1. Introduction and statement of main results 2. Variational setting and preliminary results 2.1. Functional space setting 2.2. Reduction method 3. Modified problem 4. Multiplicity of solutions for the modified problem 4.1. Autonomous problem 4.2. Technical results 4.3. Multiplicity of solutions for (3.5) 5. Proof of Theorem 1.2 Acknowledgments References