Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials

where s, t ∈ (0, 1), 3 < 4s < 3 + 2t, q ∈ (1, 2s/2) are real numbers, (−∆)s stands for the fractional Laplacian operator, 2s := 6 3−2s is the fractional critical Sobolev exponent, K, V and h are non-negative potentials and V , h may be vanish at in nity. f is a C1-function satisfying suitable growth assumptions. We show that the above fractional Schrödinger-Poisson system has a positive and a sign-changing least energy solution via variational methods.


Introduction
In this paper, we study the existence of positive and sign-changing least energy solutions for the following fractional Schrödinger-Poisson system where s, t ∈ ( , ), < s < + t, q ∈ ( , * s / ) are real numbers, * s := − s is the fractional critical Sobolev exponent. (−∆) s is the fractional Laplacian operator de ned as where P.V. is a commonly used abbreviation for in the principal value sense, and Cs is a suitable normalization constant. We notice that, when s = t = and V(x) ≡ , Ghergu and Singh [10] considered problem (1.1) with Choquard nonlinearity, −∆u + u + K(x)ϕ|u| q− u = Iα * |u| p |u| p− u, in R −∆ϕ = K(x)|u| q , in R , (1.2) where Iα(x) = |x| −( −α) , α ∈ ( , ), is the Ruiz potential. The authors discussed the existence of a ground state solution by using a variational approach. They also used a Pohozaev type identity to derive conditions in terms of p, q, N, α and K for which no solutions exist. Formally, the system −∆u + V(x)u + K(x)ϕ|u| q− u = |u| p− u, in R −∆ϕ = K(x)|u| q , in R , (1.3) can be seen as a formal limit of (1.2) as α → . The classical nonlinear Schrödinger equation is used as an approximation to the Hartree-Fock model of a quantum many body-system of electrons under the presence of an external potential V ext . In such a setting, (1.4) and its stationary counterpart bear the name of Schrödinger-Poisson-Slater, Schrödinger-Poisson-Xα, or Maxwell-Schrödinger-Poisson equation. The convolution term in (1.4) represents the Coulombic repulsion between the electrons. The local term |u| p− u was introduced by Slater as a local approximation of the exchange potential in the Hartree-Fock model. We refer to [6,19,28] for more applied background for (1.4). When s = t = and q = , system (1.1) reduces to the following Schrödinger-Poisson system which plays a great role in looking for solitary waves of the nonlinear stationary Schrödinger equations interacting the electrostatic eld. It also appears in physics such as quantum mechanics models [22], semiconductor theory [5] and so on. The nonlinearity f represents the particles interacting with each other. The term Kϕu represents the interaction with the electric eld. We refer the interesting readers to [2] and the references therein for more details about the mathematical and physical backgrounds. Recently, there are many studies on the existence of solutions to (1.5) by using the methods of nonlinear analysis. In [47], the authors established the existence of a positive solution of (1.5) with a critical nonlinearity K(x)|u| * − u + µQ(x)|u| q− u, by using the concentration-compactness principle of P. L. Lions and methods of Brezis and Nirenberg. Later, He and Zou [12] studied the existence and concentration behavior of ground state solutions for (1.5) with a critical nonlinearity and double parameters perturbation, namely (1. 6) Under some suitable conditions on the nonlinearity f and the potential V , they proved that for ε small, (1.6) has a ground state solution concentrating around global minimum of the potential V in the semi-classical limit. For more results on the multiple nontrivial solutions, in nitely many nontrivial solutions, ground state solutions, positive solutions, semiclassical state solutions and sign-changing solutions for (1.5) under various conditions on V , K and f , we refer to [4, 8, 14, 15, 23, 25, 29-31, 38, 39, 41, 46] and references therein. In recent years, there is an increasing interest in the studying for the fractional Schrödinger-Poisson system (1.1) with q = , i.e., where s, t ∈ ( , ). For example, in [34], Teng considered (1. The author used the method of Pohozaev-Nehari manifold and global compactness Lemma to obtain a ground state solutions. In [33], Teng studied the existence of ground states to (1.7) with subcritical growth, i.e., f (x, u) = |u| p− u, by using a monotonicity trick and global compactness principle.
In [21], Murcia and Siciliano considered the following semiclassical problem and established the multiplicity of solutions for small ε via Ljusternik-Schnirelmann category theory, where g is subcritical at in nity. In [43], the authors obtained the multiplicity of solution for (1.8) when g is critical growth. For more results on the concentration behavior of semiclassical solutions for (1.8), we refer to [18,42,44]. For the existence results on ground state solutions, sign-changing solutions for (1.7), we refer to [17,21,37,45] and the references therein. When q ≠ , we only note that in [36], Teng and Agarwal studied the existence and non-existence of ground state states via variational method, for (1.1) with the subcritical Choquard nonlinearity (Iα * |u| p )|u| p− u, that is, The assumption on V in [32] was formulated as follows: Under some suitable conditions on p, q and K, the authors proved the existence of a nonnegative ground states to (1.9), by using the hypotheses (V ), (V ) and a minimization argument; and the existence of bound states by employing the hypotheses (V ), (V ) and a linking theorem.
In light of the above cited works, the main purpose of this paper is to study the existence of (signchanging) solutions of (1.1) under the simultaneous presence of nonlinearities having critical growth and potentials V , h are permitted to vanishing asymptotically as |x| → ∞, without any symmetry assumptions made on V , h. To the best of our knowledge, there is not any result for the existence of positive and signchanging least energy solutions for the fractional Schrödinger-Poisson system (1.1) in the existing literature.
In order to state the main results, we introduce the assumptions on the potentials K, V and h. The nonnegative function K satis es following condition: The functions V , h : R → R, are continuous and we say that (V , h) ∈ H if the following conditions hold true: (Vh ) V(x), h(x) > for all x ∈ R and h(x) ∈ L ∞ (R ); (Vh ) If {An} n∈N ⊂ R is a sequence of Borel sets such that the Lebesgue measure m(An) is less than or equal to R, for all n ∈ N and some R > , we have For what concerns the nonlinearity f , we assume that f ful lls the following conditions: |t| p− < +∞ if (Vh ) holds, for some p ∈ ( q, * s ); (f ) f (t) |t| q− is strictly increasing function of t ∈ R\{ }; (f ) there exist µ, ν ∈ ( q, * s ) and λ > , such that F(t) ≥ λt µ , ∀t ≥ , and lim |t|→+∞ f (t) |t| ν− = .
Remark 1.1. From the assumption about function h, there exists x ∈ R such that h(x ) > , without loss of generality, we may assume that x = . So, by the continuous of h, there exists small r > such that min |x|≤r h(x) := h > .
The main results of this paper can be stated as follows. < q < − s , and assume that (K) holds, (V , h) ∈ H, and f satis es either  [9] where they obtained the existence of nontrivial solutions for this single fractional Schrödinger equation using the hypotheses (Vh ) − (Vh ). When K(x) ≢ , owing to the nonlocal property of (−∆) s and the nonlocal term K(x)ϕ t u (see Sect. 2) the study of (1.1) becomes more involved and harder to handle, the usual method for nding sign-changing solutions is not applicable directly. In addition, in the critical case the invariance by dilations of R has to be considered. While dealing with problem (1.1), alike the local case: s = t = , we have to overcome the well known double lack of compactness caused by the critical nonlinearity and the unbounded domain R , which prevents us from using the variational methods in a standard way. To recover the compactness of Palais-Smale sequences, we need to exploit hypotheses (Vh ) − (Vh ) to obtain the Sobolev compactness embedding for the weighted space, see Proposition 2.3 below, and some new delicate estimates concerning the nonlocal term and the critical nonlinearity. We note that in [47], the authors avoided the lack of compactness of the Sobolev embedding by looking for solutions of (1.5) in the subspace of radial functions of H (R ), which is usually denoted by H r (R ), since in this case the embedding H r (R ) → L s (R ) ( < s < ) is compact. For the results on the classical Schrödinger equations with vanishing potentials, we refer to [1,3] and the references therein.
This paper is organized as follows. In Section 2 we present some facts about the involved fractional Sobolev spaces and technical results, which are useful to prove the main results. In Section 3, we show the existence of positive solution for problem (1.1) and complete the proof of Theorem 1.1. In Section 4, we show the existence of sign-changing solutions of problem (1.1), and prove Theorem 1.2.

Notations:
-B R (y) denotes the ball centered at y with radius R; B R := B R ( ).
-L p (Ω), ≤ p ≤ ∞, Ω ⊆ R , are the usual spaces with norm denoted by · L p (Ω) ; if Ω = R , we simply write · p . -Positive constants whose exact values are not important in the relevant arguments, and that may vary from line to line, are generally denoted by C or C i , where i ∈ N.

Variational setting and preliminary lemmas
In this section we outline the variational framework for studying problem (1.1) and present some preliminary results, which are useful for the proof of the main results in Sections 3, 4.
We rst x some notations as follows. The homogeneous fractional Sobolev space D s, (R )(s ∈ ( , )) is de ned by which is the completion of C ∞ (R ) under the norm The embedding D s, (R ) → L * s (R ) is continuous and the best constant Ss can be de ned as We de ne the work space for (1.1) by We recall the following well-known Hardy-Littlewood-Sobolev inequality as follows:

There exists a sharp constant C(t, µ, r), independent of f and h, such thaẗ
Now, we are going to reduce system (1.1) to a single equation. To this aim, for any u ∈ H s (R ), we de ne the map Lu : D t, (R ) → R as Using the Hölder's inequality, (K) and the Sobolev embedding inequality, we get Here we have used the fact < q/( Hence, Lu is a linear and continuous map on D t, (R ). Therefore, by the Lax-Milgram theorem, there exists a unique and the representation formula holds which is called t-Riesz potential, which can be rewritten as In the sequel, we often omit the constant C t for convenience. Substituting ϕ t u in (1.1), we have the following single fractional Schrödinger equation So, the energy functional I : H → R associated with (2.4), It is easy to see that I is of class C and for any critical point of I is a weak solution of (2.4). We shall seek the ground state solution of problem (2.4) on the Nehari manifold and de ne the following minimization problem: We call u is a least energy sign-changing solution to problem (2.4) if u is a solution of problem (2.5) with u ± ≠ and for u ± ≠ , a direct computation yields that So the methods to obtain sign-changing solutions of the local problems and to estimate the energy of the sign-changing solutions seem not suitable for our nonlocal one (2.4). In order to get a sign-changing solution for problem (2.4), we will consider the following minimization problem: Finally, we introduce the weighted Banach space: equipped with the norm: and lim Such a solution is nonnegative and has the following representation: Proof. (i) follows from the de nition of ϕ t u and (H). (ii) First of all, we claim that (2.10) In fact, using Proposition 2.1, we have For any δ > , using Young inequality, one has Then Here we have used the fact < q/( . Hence, by the Lebesgue dominated convergence theorem, one has lim n→∞ˆR G δ,n (x)dx = . Consequently, This shows that lim sup and then claim (2.10) holds. Similarly, we can deduce that (2.12) So, it follow from (2.11)-(2.12) that The proof of the second equality in item (ii) is similar, so we omit it and this completes the proof of Proposition 2.4.
Since hu is a continuous function and exploiting (f ), there is α ′ u > which is global maximum of hu with α ′ u u ∈ N. Next we prove that α ′ u is the unique critical point of hu. Assume by contradiction that there are α > α > critical points of hu. Then we have From which, taking into account (f ), we deduce which leads a contradiction.
In a standard way (see [40]), we can prove that I satis es the following mountain-pass geometrical structure.
(iii) Taking a sequence {un} ⊂ H such that un → u in H, then u ± n → u ± in H. By item (i), there exist (αu n , βu n ) and (αu , βu) such that αu n u + n + βu n u − n ∈ M and αu u + + βu u − ∈ M. Then and (2.32) It is obvious that αu n , βu n ≥ C for some constant C > . Next we claim that {αu n } and {βu n } are bounded in R + . In fact, without loss of generality, we suppose that αu n → ∞ as n → ∞, by u ± n → u ± ≠ in H and dividing the above (2.31) by α q un , we infer that βu n α q− un → +∞ or βu n αu n → +∞ as n → ∞, and thus βu n → +∞ as n → ∞.
(iv) We argue by contradiction. If u + n → in H and there exists M such that αu n ≤ M. By either (f ) or ( f ), (f ) and the Sobolev embedding inequality, we have that = I ′ (αu n un + + βu n un − ), αu n un + α un as n large enough, which contradicts the fact that αu n u + n + βu n u − n ∈ M. Hence αu n → ∞. Similarly, we can prove that βu n → ∞ if u − n → in H.

Existence of positive solution for (1.1)
In this section, we are going to show the existence of positive solution for problem (1.1), and prove Theorem 1.1. We rst present the following lemma implies that I satis es the local (PS)c-condition. Then, by (f ) and h(x) > in R , we get which implies {un} is bounded in H. Thus, there exists a subsequence of {un}, still denoted by {un} such that From (3.1) and (3.2), it is easy to check that u is a critical point of I. Therefore, I ′ (u), u = . Then, by h(x) > , (f ) and < q < * s / , we have that Let vn = un − u and so vn in H. In view of (3.2), Proposition 2.4 (ii)(a) and the Brezis-Lieb Lemma [7], we obtain that Similarly, we have that Proof. For any u ∈ N, we have I ′ (u), u = . Then, by either (f ) or ( f ), (f ) and the Sobolev embedding inequality, we have that which implies that there exists some constant C > such that u ≥ C. Therefore, we have which implies that c ≥ q− q C > . Now, we prove c < s S s s . We de ne (see [26]) By a simple computation, we havê (3.8) By Lemma 2.5, we know that there exists a unique αε > such that αε uε ∈ N. Thus, by the de nition of c , we have c ≤ I(αε uε). To complete the proof it su ces to prove that Since max α≥ I(αuε) = I(αε uε) ≥ c > , there exists A > such that αε ≥ A > . Moreover, since I(αuε) → −∞ as α → ∞ and I(αε uε) ≥ c > , we get that there exists A > such that αε ≤ A , and so < A ≤ αε ≤ A . Notice that First, we claim that I ≤ s S s s + O(ε − s ). (3.10) In fact, we de ne It is easy to see that g(α) attains its maximum at Therefore, by (3.6) and (3.7), we deduce that ). (3.12) By virtue of µ > q > s − s , we see that − s > − µ( − s) . Therefore, to nish the proof, it is enough to prove that lim (3.13) In fact, by K(x) ∈ L ∞ (R ) and Proposition 2.2, we havê Hence, (3.13) holds and then we see that (3.9) holds and this completes the proof.
Proof of Theorem 1.1. By Lemma 2.6, Lemma 3.1 and Lemma 3.2, we know that the functional I satis es the mountain-pass geometrical structure and the (PS)c -condition. Hence, the functional I has a critical value c > . That is, there exists a non-trivial u ∈ H such that I(u ) = c and I ′ (u ) = , which implies that u is the non-trivial ground state solution of problem (1.1). Next, we prove that u is positive. It is easy to see that all the above calculations can be repeated word by word, replacing I(u) by the functional By using u − as a test function, we obtain Using the same argument as Proposition 5.1 in [35], we have u ∈ L ∞ (R ). By standard argument to the proof of Proposition 4.4 in [32], using Proposition 2.9 in [27] twice, we have that u ∈ C ,α (R ) for some α ∈ ( , ) for s > . Furthermore, if u (x) = for some x ∈ R , then (−∆) s u (x ) = . Since u ∈ C ,α (R ), by Lemma 3.2 [22] and u (x ) = , we have It implies u ≡ , which is a contradiction.
Let G(u, v) be the functional de ned in H by    We know that either v ± n = u ± n − u ± or v ± n = u ∓ n − u ∓ for n large enough. Without loss of generality, assume that v ± n = u ± n − u ± , then from v ± n in H we see that It follows from I ′ (u) = and (f ) that