{'fraction_non_alphanumeric': 0.08035335689045936, 'fraction_numerical': 0.03371024734982332, 'mean_word_length': 3.3501383338456807, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 21, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Fractional Schr\"odinger-Poisson systems with a general subcritical or critical nonlinearity

We consider a fractional Schr\"{o}dinger-Poisson system with a general nonlinearity in subcritical and critical case. The Ambrosetti-Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.


Introduction and main result
We are concerned with the fractional nonlinear Schrödinger-Poisson system where λ > 0, (−∆) α is the fractional Laplacian operator for α = s, t ∈ (0, 1). The fractional Schrödinger equation was introduced by Laskin [28] and arose in fractional quantum mechanics in the study of particles on stochastic fields modeled by Lévy processes. The operator (−∆) α can be seen as the infinitesimal generators of Lévy stable diffusion processes [2]. If λ = 0, the system (1.1) reduces to the nonlinear fractional scalar field equation This equation is related to the standing waves for fractional scalar field equation which is a physically relevant generalization of the classical NLS. For power type nonlinearities the fractional Schrödinger equation was derived by Laskin [28] by replacing the Brownian motion in the path integral approach with the so called Lévy flights, see e.g. [29]. So, the equation we want to study presents first of all as a perturbation of a physically meaningful equation. Also, in [21,22] the author obtained deep results about uniqueness and non-degeneracy of ground states for (1.2) in case g(u) = |u| p−2 u − u for subcritical p. See also [33] where the soliton dynamics for (1.3) with an external potential was investigated. In [24], the author studies the evolution equation associated with the one dimensional system In this case the diffusion is fractional only in the Poisson equation. Our system is more general and contain this as a particular case. If K α (x) = |x| α−N , in [20] the following equation is studied √ −∆u + u = (K 2 * |u| 2 )u, u ∈ H 1/2 (R 3 ), u > 0, and in [19] it is shown that the dynamical evolution of boson stars is described by the nonlinear evolution equation for a field ψ : [0, T ) × R 3 → C (see also [23]). The square root of the Laplacian also appears in the semi-relativistic Schrödinger-Poisson-Slater systems, [6]. See also the model studied in [15].
Observe that, taking formally s = t = 1, then system (1.1) reduces to the classical Schrödinger-Poisson system It describes systems of identically charged particles interacting each other in the case where magnetic effects can be neglected [7]. In recent years, the Schrödinger-Possion system (1.5) has been widely studied by many researchers. Here we would like to cite some related results, for example, positive solutions [10], ground state solutions [3], semi-classical states [16], sign-changing solutions [25]. See also [1] and the references therein. In [4], Azzollini, d'Avenia and Pomponio were concerned with (1.5) under the Berestycki-Lions conditions (H2)-(H4) with s = 1. The authors proved that (1.5) admits a positive radial solution if λ > 0 small enough. For the critical case, we refer to [36] and a recent work [38] of the authors of the present work.
1.1. Main results. In the present paper, we are mainly concerned with the positive solutions of (1.1). First, we consider the subcritical case with the Berestycki-Lions conditions. Precisely, we assume the following hypotheses on g: Our first result can read as Theorem 1.1. Suppose that g satisfies (H1)-(H4) and 2t + 4s ≥ 3.
u is a radial ground state solution of (1.2).
. For the details, see Section 2 below.
In [38], without the Ambrosetti-Rabinowtiz condition, the authors of the present work considered the existence and concentration of positive solutions to (1.1) in the critical case for s = t = 1.
It is natural to wonder if similar results can hold for the critical fractional case. This is just our second goal of the present paper. In the critical case, we assume the following hypotheses on g: H4) ′ there exists µ > 0 and q < 2 * s such that g(τ ) − bτ 2 * s −1 + aτ ≥ µτ q−1 for all τ > 0. Our second result can read as follows.
Remark 1.5. In the case s = 1, the hypotheses (H2) ′ -(H4) ′ were introduced in [37] (see also [5]) to obtain the ground state of the scalar field equation −∆u = g(u) in R N . In [34], X. Shang and J. Zhang considered the fractional problem (1.2) in the critical case (see also [35]). With the help of the monotonicity of τ → g(τ )/τ , the ground state solutions were obtained by using the Nehari approach. To the best of our knowledge, there are few results in the literature about the ground state of the critical fractional problem (1.2) with a general nonlinearity, particularly without the Ambrosetti-Rabinowtiz condition and the monotonicity of g(τ )/τ . Theorem 1.4 seems to be the first result in this direction.
Remark 1.6. Without loss generality, from now on, we assume that a = b = µ = 1.
In the rest of the paper, we use the perturbation approach to prove Theorem 1.1 and 1.4. Similar argument also can be found in [38].
The paper is organized as follows.
In Section 2 we introduce the functional framework and some preliminary results.
In Section 3 we construct the min-max level.
In Section 4, we use a perturbation argument to complete the proof of Theorem 1.1.
In Section 5, we give the proof of Theorem 1.4.

Preliminaries and functional setting
where F is the Fourier transform, i.e., i is the image unit. If φ is smooth enough, it can be computed by the following singular integral where c α is a normalization constant and P.V. stands the principal value. For any α ∈ (0, 1), we consider the fractional order Sobolev space and the inner product It is easy to know the inner products on H s (R 3 ) are equivalent(see [34]). The homogeneous Sobolev space D α,2 (R 3 ) is defined by and the inner product For the further introduction on the Fractional order Sobolev space, we refer to [17]. Let }. Now, we introduce the following Sobolev embedding theorems. Lemma 2.1 (see [27]). For any α ∈ (0, 1), Lemma 2.2 (see [12,17]). For any α ∈ (0, 1), 2.2. The variational setting. Now, we study the variational setting of (1.1). By Lemma 2.1, Then, for u ∈ H s (R 3 ), by Lemma 2.2 the linear operator P : D t,2 (R 3 ) → R defined by is well defined on D t,2 (R 3 ) and is continuous. Thus, it follows from the Lax-Milgram theorem that there exists a unique φ t . Formula (2.1) is called the t-Riesz potential. Substituting (2.1) into (1.1), we can rewrite (1.1) in the following equivalent form Obviously, the critical points of Γ λ are the weak solutions of (2.2). Definition 2.3. ( then we summarize some properties of φ t u , T (u), which will be used later. Lemma 2.4. If t, s ∈ (0, 1) and 2t + 4s ≥ 3, then for any u ∈ H s (R 3 ), we have is continuous and maps bounded sets into bounded sets.
The proof is similar as that in [31], so we omit the details here.
3. The subcritical case 3.1. The modified problem. It follows from Lemma 2.4 that Γ λ is well defined on H s (R 3 ) and is of class C 1 . Since we are concerned with the positive solutions of (2.2), similar as that in [8](see also [13]), we modify our problem first. Without loss generality, we assume that where ξ is given in (H4). Let τ 0 = inf{τ > ξ : g(τ ) = 0} ∈ [ξ, ∞], defineg : R → R, is a solution of (2.2) where g is replaced byg, then by the maximum principle [11] we get that u is positive and u(x) ≤ τ 0 for any x ∈ R 3 , i.e., u is a solution of the original problem (2.2) with g. Thus, from now on, we can replace g byg, but still use the same notation g. In addition, for τ > 0, let and for any ε > 0, there exists C ε > 0 such that Let G i (u) = u 0 g i (τ ) dτ, i = 1, 2, then by (3.1) and (3.2) for any ε > 0 there exists C ε > 0 such that 3.2. The limit problem. In the following, we will find the solutions of (2.2) by seeking the critical points of Γ λ . If λ = 0, problem (2.2) becomes which is referred as the limit problem of (2.2). We define an energy functional for the limiting problems (3.4) by In [13], X. Chang and Z.-Q. Wang proved that, with the same assumptions on g in Theorem 1.1, there exists a positive ground state solution U ∈ H 3 r (R 3 ) of (3.4). Moreover, each such solution U of (3.4) satisfies the Pohozǎev identity Let S be the set of positive radial ground state solutions U of (3.4), then S = φ and we have the following compactness result, which plays a crucial role in the proof of Theorem 1.1.
Proposition 3.1. Under the assumptions in Theorem 1.1, S is compact in H s r (R 3 ). As shown in [9], for general s ∈ (0, 1), we do not have a similar radial lemma in H s r (R 3 ). So the Strauss's compactness lemma(see [8]) is not applicable here. Before we prove Proposition 3.1, we start with the following compactness lemma, which is a special case of [13, Lemma 2.4.] Lemma 3.2 (see [13]). Assume Q ∈ C(R, R) satisfy and there exists a bounded sequence Then, up to a subsequence, we have Q(u n ) → v strongly in L 1 (R 3 ) as n → ∞.
Proof of Proposition 3.1 Let {u n } ∞ n=1 ⊂ S and denote by E the least energy of (3.4), then for any n, u n satisfies L(u n ) = E and the Pohozǎev identity (4.5), which implies that Obviously, { (−∆) s/2 u n 2 } is bounded. It follows from Lemma 2.2 that { u n 2 * s } is bounded. By (3.3), as we can see in [8], { u n 2 } is bounded, which yields that {u n } is bounded in H s r (R 3 ). Without loss generality, we can assume that there exists u 0 ∈ H s r (R 3 ) such that u n → u 0 weakly in H s r (R 3 ), strongly in L q (R 3 ) for q ∈ (2, 2 * s ) and u n (x) → u 0 (x) a.e. x ∈ R 3 . In the following, we adopt some ideas in [8] to prove that u n → u 0 strongly in H s r (R 3 ). For u ∈ H s (R 3 ), let then, we know u n is a minimizer of the following constrained minimizing problem By (3.1) and Lemma 3.2, we get that Then by the Fatou's Lemma, V (u 0 ) ≥ 3−2s 2s E, which implies that u 0 ≡ 0. Meanwhile, it is easy to know that J(u 0 ) ≤ E. Similar as that in [8], we know that u 0 satisfies J(u 0 ) = E and V (u 0 ) = 3−2s 2s E. It yields that By the Fatou's Lemma, we know u n 2 → u 0 2 as n → ∞. Thus, u n → u 0 strongly in H s r (R 3 ). The proof is completed.
In the following, we will find a solution u ∈ S d of problem (2.2) for sufficiently small λ > 0 and some 0 < d < 1. The following proposition is crucial to obtain a suitable (PS)-sequence for Γ λ and plays a key role in our proof.
Then for d small enough, there is u 0 ∈ S, up to a subsequence, such that u λ i → u 0 in H s r (R 3 ). Proof. For convenience, we write λ for λ i . Since u λ ∈ S d and S is compact, we know {u λ } is bounded in H s r (R 3 ). Then by Lemma 2.4 we see that lim It follows from [13,Lemma 3.3] that, there is u 0 ∈ H s r (R 3 ), up to a subsequence, such that u λ → u 0 strongly in H s r (R 3 ). Obviously, 0 ∈ S d for d small. This implies that u 0 ≡ 0, L(u 0 ) ≤ E and L ′ (u 0 ) = 0. Thus, L(u 0 ) = E, i.e., u 0 ∈ S. The proof is completed. By Proposition 3.5, for small d ∈ (0, 1) such that there exist ω > 0, λ 0 > 0 such that and λ ∈ (0, λ 0 ).
Proof of Theorem 1.1 concluded. It follows from Proposition 3.7 that there exists λ 0 > 0 such that for λ ∈ (0, λ 0 ), there exists {u n } ∈ Γ D λ λ ∩ S d with Γ ′ λ (u n ) → 0 as n → ∞. Noting that S is compact in H s r (R 3 ), we get that {u n } is bounded in H s r (R 3 ). Assume that u n → u λ weakly in H s r (R 3 ), then Γ ′ λ (u λ ) = 0. It follows from the compactness of S that u λ ∈ S d and u n − u λ s ≤ 3d for n large. So u λ ≡ 0 for small d > 0. By Lemma 2.4, Noting that G 2 (τ ) ≥ m 2 τ 2 for any τ ∈ R, it follows from (3.3) that for some C > 0, Then by Lemma 2.2, for small d > 0, it is easy to verify that Γ λ (u n − u λ ) ≥ 0 for large n. So Thus u λ is a nontrivial solution of (2.2). Finally, by Proposition 3.5 we can get the asymptotic behavior of u λ as λ → 0 + . The proof is completed.

The critical case
In this section, we consider the Schrödinger-Possion system (1.1) in the critical case. First, we establish the existence of ground state solutions to the fractional scalar field equation (1.2) with a general critical nonlinear term. Then by the perturbation argument, we seek the solutions of (1.1) in some neighborhood of the ground states to (1.2). 4.1. The limit problem. In this subsection, we use the constraint variational approach to seek the ground state solutions of (1.2). Similar argument also can be found in [8,18,37]. Let We recall that U is said to be a ground state solution of (1.2) if and only if I(U ) = m 0 , where m 0 := inf{I(u) : u ∈ H s (R 3 ) \ {0} is a solution of (1.2)} and The existence of ground state is reduced to look at the constraint minimization problem   Proof of Theorem 4.1. The proof follows the lines of that in [37]. For the completeness, we give the details here.
Next, obviously, M ∈ (0, +∞). For small ε > 0, V (v ε ) > 0, then If p ≥ 1, then (1 + t) p ≤ 1 + p(1 + t) 1+p t, for all t ≥ −1. It follows from (4.3) that for small ε > 0, which yields 1 Step 2. M can be achieved. Noting that g is odd and the Fractional Polya-Szegö inequality [30], without loss of generality, we can assume that there exists a positive minimizing sequence {u n } ⊂ H s r (R 3 ) such that V (u n ) = 1 and T (u n ) → M as n → ∞. By Lemma 2.2, it is easy to know that {u n } is bounded in H s r (R 3 ). By Lemma 2.1, we can assume that u n → u 0 weakly in H s (R 3 ), strongly in L q (R 3 ) and a.e. in R 3 . Set v n = u n − u 0 , then T (u n ) = T (v n ) + T (u 0 ) + o(1), and where o(1) → 0 as n → ∞. Let f (s) = g(s) − s 2 * s −1 + s, then it follows from Lemma 3.2 that Next, we prove u 0 is the minimizer for M . Set which is again a contradiction. Then we conclude λ 0 = 1, i.e., M is achieved by u 0 .
The proof is completed.
Remark 4.3. Furthermore, similar as that in [13], if we assume that g ∈ C 1 (R, R) additionally, U satisfies the Pohozǎev identity Similar as that in [26,37], U is also a mountain pass solution.
Let S 1 be the set of positive radial ground state solutions U of (1.2), then similar to Step 2 in the proof of Theorem 4.1, we have the following compactness result.