, and Mingqi Xiang * Combined e ects of Choquard and singular nonlinearities in fractional Kirchho problems


 The aim of this paper is to study the existence and multiplicity of solutions for a class of fractional Kirchho problems involving Choquard type nonlinearity and singular nonlinearity. Under suitable assumptions, two nonnegative and nontrivial solutions are obtained by using the Nehari manifold approach combined with the Hardy-Littlehood-Sobolev inequality.


Introduction
In this paper, we study the following Choquard-Kirchho problem involving singular nonlinearity: Here µ ∈ ( , N), p * µ,s = p · N−µ N−ps is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, λ, µ > are two parameters, < β < , and(−∆) s p is the fractional p−Laplacian which, up to a normalization constant, is de ned for any x ∈ R N as |φ(x) − φ(y)| p− (φ(x) − φ(y)) |x − y| N+ps dy for any φ ∈ C ∞ (R N ). Here Bε(x) denotes the ball in R N centered at x with radius ε > . For further details about the fractional Laplacian and its applications, we refer to [31]. Throughout our paper, the following assumptions will be satis ed: (f ) f : R N −→ R + such that f ∈ L q (R N ), where q = p * s p * s − +β with p * s = Np N−ps is the fractional critical Sobolev exponent; When s = and b = , the equation (1.1) covers the Choquard-Pekar equation which has come forth in quantum physics of a polaron at rest [39] and which describes the modeling of an electron ensnared in its own hole [26], see also [32]. Equation (1.1) is also related with the fractional Kircho model which was rst proposed by Fiscella and Valdinoci [11]. Indeed, the study of Kirchho -type problems, which arise in various models of physical and biological systems, have received more and more attention in recent years. Precisely, Kirchho in [24]  fractional Kirchho problems, we refer to [10, 19-23, 28-30, 49, 50] and the references cited there. In recent years, much attention has been focused on the existence and properties of nontrivial solutions for fractional Choquard equation involving fractional p−Laplacian,see for example [37,41,47,51]. In [37], Mukherjee and Sreenadh studied the following subcritical Choquard system involving fractional p−Laplacian and perturbations (−∆) s p u + a (x)u|u| p− = α(|x| −µ * |u| q )|u| q− u + β(|x| −µ * |v| q )|u| q− u + f (x) in R n , (−∆) s p v + a (x)v|v| p− = γ(|x| −µ * |v| q )|v| q− v + β(|x| −µ * |u| q )|v| q− v + f (x) in R n .
The authors proved that the system admites at least two solutions by means of Nehari manifold and minimax methods. Pucci, Xiang and Zhang in [41] discussed the following Schrödinger-Choquard-Kirchho type fractional p−Laplacian equations with upper critical exponents where p * µ,s = (pN − pµ/ )/(N − ps) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and The authors established the existence and asymptotic behavior of solutions for above problem in the cases of f satis es superlinear and sublinear nonlinearities, respectively. The main techniques used in the paper are the mountain pass lemma and Ekeland's variational principle. Recently, Yang et al. [51] considered the following problem where Ω is a bounded domain in R N with Lipschitz boundary, < p < ∞, < s < , The authors analyzed the minimizer of energy functional associated to the problem (1.2) on positive Nehari and sign-changing Nehari sets, and obtained the existence of positive and sign-changing solutions for the problem (1.2). Further discussions on Choquard equation can be found in the survey papers [33,36] and the references cited there.
On the other hand, critical and subcritical fractional problems involving singular nonlinearity have received more and more attention. In [2], Barrios etal. considered the existence of solutions for the problem where γ > , p > and M ∈ { , }. The existence of solutions were obtained by using the approximation method by Boccardo and Orsina. Canino etal. [6] extended the above problem to the fractional p-Laplacian and considered the following problem where γ > . The authors considered two cases: γ ∈ ( , ) and γ > . The approximation method by Boccardo and Orsina was used to get the existence of solutions. Moreover, the uniqueness and symmetry of solutions were also investigated. In [12], Fiscella and Mishra studied the following Kirchho problem involving singular and critical nonlinearities where < q < , L(u) is de ned as in (1.1) with p = and θ ∈ [ , * s / ), λ > , and g is a sign-changing function. The authors analyzed the bering map and gave the compactness property of the energy functional corresponding to the problem (1.3). The authors required b small enough in order to get some key estimates of the energy functional on the Nehari manifold. When λ is small enough, the authors obtained the existence of two positive solutions by using Nehari manifold method. For fractional Kirchho problems with singular nonlinearity, we also refer the interested readers to [44]. Very recently, Goel and Sreenadh [16] used the similar method as in [12] to discuss the following critical Choquard-Kirchho type problem where K(u) = − a + ϵ p ( Ω |∇u| dx) θ− ∆u with a > , p > N − (N ≥ ) and θ ∈ [ , * µ ). Here < µ < N, < q ≤ and λ is a positive parameter. The authors established the existence of two positive solutions for (1.4). They applied minimization argument on the Nehari sub-manifolds to obtain the rst solution. To get the second solution, the authors divided the proof into two cases: µ < min{ , N} and µ ≥ min{ , N}. Furthermore, do Ó, Giacomoni and Mishra [9] studied fractional Kirchho system with critical and concaveconvex nonlinearities.
In present paper, we are interested in the multiplicity of solutions for fractional Kirchho equations with Choquard and singular nonlinearities. Since the energy functional associated to (1.1) in general is not differentiable on D s,p (R N ), the usual critical point theory is not available. Inspired by [9,45,46], we shall use the Nehari manifold approach to get the existence of two solutions for (1.1). Clearly, equation (1.1) is di erent from the problems considered in the literature, since (1.1) deals with fractional p-Kirchho equations with Choquard type and singular nonlinearities. Thus, our equation and result are new. De nitely, we encounter some di culties in analyzing the berling map and discussing the existence of local minimizes on Nehari manifold. Our discussions are more elaborate than the papers in the literature.
To introduce the main result of this paper more precisely, we rst give the de nition of weak solutions. Here S > denotes the best constant of embedding from D s,p (R N ) to L p * s (R N ) and Cg(N, µ) > will be given by (2.2).
Our result is the following theorem. The rest of our paper is organized as follows. In Section 2, we recall some de nitions and preliminaries which will be used in our discussion. In Section 3, the properties of bering maps are analyzed. Furthermore, a compactness result is also given. In Section 4, two nontrivial and nonnegative solutions are obtained by applying the Nehari manifold approach.

Preliminaries
In this section, we recall some basic results on fractional Sobolev spaces and the Hardy-Littlehood-Sobolev inequalty. For the details, we refer to [8,38,42,43]. Firstly, we denote by D s,p (R N ) the usual fractional Sobolev space (Hardy-Littlehood-Sobolev inequality, see [27]) Assume that < r, t < ∞, < µ < N and Then there exists C(N, µ, r, t) > such that

In this case, the equality in (2.1) holds if and only if u = ch and
Note that, by Theorem 2.1, we know Hence, by the fractional Sobolev embedding theorem, if u ∈ D s,p (R N ) this occurs provided that < * Hence, p * µ,s is said to be the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
where C(N, µ) > is a suitable constant. Here and from now on, we shortly denote by · ν for the norm of Lebesgue space L ν (R N ). Further, by the fractional Sobolev inequality, we get and S > is given by

Fibering map analysis
First, we de ne the energy functional I λ : D s,p (R N ) −→ R corresponding to equation (1.1) as Here De ne the Nehari manifold Then u ∈ N λ if and only if Φ λ,u ( ) = . Thus, it is natural to divide N λ into three parts as and A simple calculation gives that Let h λ,u (t) = . Then we have Then there exists a unique tmax > such that h λ,u (tmax) = . Thus, we have ψ λ,u (tmax) = , ψ λ,u (t) is increasing on ( , tmax), decreasing on (tmax, +∞). In the following, we estimate ψ λ,u (tmax) as follows It follows from the condition (f ) and Hölder's inequality that By (3.6) and (2.2), we deduce that and C(λ) = if and only if When λ ∈ ( , Λ ), we have C(λ) > and ψ λ,u (tmax) > . Thus we can nd two zero points t + and t − of ψ λ,u (t) with t Proof. Arguing by contradiction, we assume that there exists u ∈ N λ \{ }. Then we have and , we obtain that T λ (u) = for all u ∈ N λ \{ }. It follows from the de nition of T and (2.2) that On the other hand, we deduce from (3.8) and (3.6) that which yields that (3.10) Combining (3.9) with (3.10), we obtain This contradicts the assumption λ < Λ . In conclusion, we prove that N λ = { } for all λ ∈ ( , Λ ).
Proof. For u ∈ N + λ . By (f ) and (3.6), we get For U ∈ N − λ . It follows from (2.2) that which leads to Clearly, A > A λ since λ ∈ ( , Λ ). Thus, the proof is complete. Proof. We only show the proof for the case u ∈ N + λ while the proof of the case u ∈ N − λ is similar. De ne This which means that Observe by (3.12) that Then we can choose su ciently small R ∈ ( ,R) such that The proof is completed.
Lemma 3.6. The functional I λ is coercive and bounded from below on N λ .
Proof. For u ∈ N λ , it follows from θp < q and (3.6) that Since p > − β, we know that I λ is coercive on N λ . De ne then G(t) attains its minimum at Thus Therefore, we get that I λ is bounded from below on N λ . Set By Lemma 3.2 and 3.4, we know that N + λ ∪ { } and N − λ are two closed sets in E. Using Ekeland's variational principle [1], we can extract a minimizing sequence(un)n ⊂ N + . In view of Lemma 3.6, we derive that the sequence (un)n is bounded in N λ with [un]s,p ≤ C for some C > . Thus, up to a subsequence still denoted by (un)n we may assume that there exists u ∈ D s,p (R N ) such that un u weakly in D s,p (R N ) un → u a.e. in R N . (3.14) Lemma 3.7. For λ ∈ ( , Λ ). Then there exists a constant C > such that the following conclusions hold: Proof. We only prove the case (i), while the proof of (ii) follows similarly. Since (un)n ⊂ N + λ , it su ces to prove that Arguing by contradiction, we assume that By f ∈ L p * s p * s − +β (R N ) and Vitali's convergence theorem, one can prove that Hence, there exists A > such that [un] p s,p −→ A as n −→ ∞. Consequently, we get By Lemma 3.1, for λ ∈ ( , Λ ), we have n (x)) q (u + n (y)) q |x − y| µ dxdy. (3.17) Since (un)n ⊂ N + λ , it follows from (3.16) that Substituting (3.16) and (3.18) into (3.17), we arrive at this together with q > p implies that which is impossible. Therefore, the proof is complete.
Dividing the above estimate by t > and letting t −→ + , we obtain By (3.20), we have Then by Lemma 3.7-(i) and the boundedness of the sequence (un)n, we obtain that (ζ n ( ), φ) is bounded from below for every φ ∈ D s,p (R N ) with φ ≥ . Next, we show that (ζ n ( ), φ) is bounded from above. Arguing by contradiction, we assume that (ζ n ( ), φ) = ∞. Since