Multiplicity results for fractional Schr\"odinger-Kirchhoff systems involving critical nonlinearities

In this paper, we study certain critical Schr\"{o}dinger-Kirchhoff type systems involving the fractional $p$-Laplace operator on a bounded domain. More precisely, using the properties of the associated functional energy on the Nehari manifold sets and exploiting the analysis of the fibering map, we establish the multiplicity of solutions for such systems.


Introduction
In recent years, a lot of attention has been paid to problems involving fractional and nonlocal operators. These types of problems arise in applications in many fields, e.g., in materials science [9], phase transitions [5,40], water waves [17,18], minimal surfaces [14], and conservation laws [10]. For more applications of such problems in physical phenomena, probability, and finances, we refer interested readers to [13,15,48]. Due to their importance, there are many interesting works on existence and multiplicity of solutions for fractional and nonlocal problems either on bounded domains or on the entire space, see [1,3,4,6,24,25,35,37,38,39].
In the last decade, many scholars have paid extensive attention to the Kirchhofftype elliptic equations with critical exponents, see [21,26,34], for the bounded domains and [27,29,30] for the entire space. In particular, in [23], the authors considered the following Kirchhoff problem where t ≥ 0 and M (t) = a + bt for some a > 0 and b ≥ 0. Here, and in the rest of this paper, Ω will denote a bounded domain in R n with Lipschitz boundary ∂Ω. Under suitable conditions, and by using the truncation technique method combined with the Mountain pass theorem, the authors proved that for λ > 0 large enough, problem (1.1) has at least one nontrivial solution. Later, the fractional Kirchhoff-type problems were extensively studied by many authors using different methods, see [7,8,16,22,28,31,32,33,36,41,43,44,45,46]. In particular, by using the Nehari manifold method and the Symmetric mountain pass theorem, the authors in [44] investigated the multiplicity of solutions for some p-Kirchhoff system with Dirichlet boundary conditions.
Mingqi et al. [31] studied the following Schrödinger-Kirchhoff type system (1.2) where λ > 0, α + β = p * s := np n−sp , V : R n → [0, ∞) is a continuous function, the Kirchhoff function M : (0, ∞) → (0, ∞) is continuous, and H u and H v are Caratheodory functions. Under some suitable assumptions and by applying the Mountain pass theorem with Ekeland's variational principle, the authors obtained the existence and asymptotic behavior of solutions for system (1.2).
By the same methods as in [31], Fiscella et al. [22] studied the existence of solutions for a critical Hardy-Schrödinger-Kirchhoff type system involving the fractional p-Laplacian in R n . Using the Three critical points theorem, Azroul et al. [8] established the existence of three weak solutions for a fractional p-Kirchhoff-type system on a bounded domain with homogeneous Dirichlet boundary conditions. Recently, Benkirane et al. [7] have established the existence of three solutions for the (p, q)-Schrödinger-Kirchhoff type system in R n via the Three critical points theorem.
Motivated by the above-mentioned papers, we consider in this paper the following Schrödinger-Kirchhoff type system involving the fractional p-Laplacian and critical nonlinearities where . V1 and . V2 will be given later (see (1.6)), n > ps, 0 < s < 1 < q < p, λ is a positive parameter, the weight functions a 1 and a 2 are positive and bounded on Ω, and (−∆) s p is the fractional p-Laplace operator, defined by For more details about the fractional p-Laplacian operator and the basic properties of fractional Sobolev spaces, we refer the reader to [19]. Throughout this paper, the index i will denote integers 1 or 2, and we shall assume that the potential function V i : Ω → (0, ∞) is continuous and that there exists v i > 0 such that inf is a continuous function satisfying the following conditions Moreover, we shall assume that f, g ∈ C(Ω × R × R, [0, ∞[) are positively homogeneous functions of degree (q − 1), i.e., for all t > 0 and (x, Finally, we shall also assume that there exists a function H : , where H u (respectively, H v ) denotes the partial derivative of H with respect to u (respectively, v). We note that the primitive function H belongs to C 1 (Ω×R×R, R) and satisfies the following assumptions for all t > 0, (x, u, v) ∈Ω × R × R, and some constant γ > 0, (1.5) Before stating our main result, let us introduce some notations. For s ∈ (0, 1), we define the functional space which is endowed with the norm and Ω c = R n \ Ω. From now on, we shall denote by . q the norm on the Lebesgue space L q (Ω). It is well known that W s,p (Q), . W s,p (Q) is a uniformly convex Banach space. Next, L p (Ω, V i ) denotes the Lebesgue space of real-valued functions, with V i (x)|w| p ∈ L 1 (Ω), endowed with the following norm Let us denote by W s,p Vi (Q) the completion of C ∞ 0 (Q) with respect to the norm According to [ [2] for more details. Let S p,Vi be the best Sobolev constants for the embeddings from W s,p Vi (Q) into L p * s (Ω), which is given by For simplicity, in the rest of this paper, S will denote the following expression Next, we define the notion of solutions for problem (1.3).
The main result of this paper is the following.
This paper is organized as follows. In Section 2, we present some notations and preliminary results related to the Nehari manifold and fibering maps. In Section 3, we prove Theorem 1.1.

The Nehari manifold method and fibering maps analysis
This section collects some basic results on the Nehari manifold method and the fibering maps analysis which will be used in the forthcoming section, we refer the interested reader to [12,13,20], for more details. We begin by considering the Euler-Lagrange functional J λ : W → R, which is defined by We can easily verify that J λ ∈ C 1 (W, R), moreover its derivative J ′ λ from the space W into its dual space W ′ is given by 2) From the last equation, we can see that the critical points of the functional J λ are exactly the weak solutions for problem (1.3). Moreover, since the energy functional J λ is not bounded from below on W, we shall show that J λ is bounded from below on a suitable subset of W, which is known as the Nehari manifold and is defined by Hence, from (2.2), we see that elements of N λ correspond to nontrivial critical points which are solutions of problem (1.3).
It is useful to understand N λ in terms of the stationary points of the fibering maps ϕ u,v : (0, ∞) → R, defined by A simple calculation shows that for all t > 0, we have Now, in order to obtain a multiplicity of solutions, we divide N λ into three parts By the Lagrangian multipliers theorem, there exists δ ∈ R, such that Moreover, by (2.3) and the constraint β(u 0 , v 0 ) = 0, we have Since (u 0 , v 0 ) ∈ N 0 λ , we have ϕ ′′ u0,v0 (1) = 0. Thus, by (2.8) we get δ = 0. Consequently, by substitution of δ in (2.7), we obtain J ′ λ (u 0 , v 0 ) = 0. This completes the proof of Lemma 2.1.
In order to understand the Nehari manifold and fibering maps, let us define the function ψ u,v : (0, ∞) → R as follows Thus, it is easy to see that (tu, tv) ∈ N λ if and only if ψ u,v (t) = 0. (2.10) Moreover, by a direct computation, we get and (2.13) Now we shall prove the following crucial result.
Proof. We begin by noting that by (2.9), we have Now, if we combine equations (1.5) and (1.7) with the Hölder inequality, we obtain and where a = max( a 1 ∞ , a 2 ∞ ), A(u, v) = (u, v) p , and S is given by equation (1.8).
On the other hand, by combining equations (2.14), (2.15) with (H 2 ), we obtain where m is given by equation (2.12) and F u,v is defined for t > 0 by Since 1 < q < p < p * s , it is easy to see that lim t→0 + F u,v (t) < 0 and lim t→∞ F u,v (t) = −∞. So, by a simple calculation we can prove that F u,v attains its unique global maximum at Moreover, where λ * is given by (2.13). If we choose λ < λ * , then we get from (2.16) Hence, by a variation of ψ u,v (t) there exist unique t 1 < t max (u, v) and unique t 2 > t max (u, v), such that ψ ′ u,v (t 1 ) > 0 and ψ ′ u,v (t 2 ) < 0. Moreover ψ u,v (t 1 ) = 0 = ψ u,v (t 2 ). Finally, it follows from (2.10) and (2.11) that (t 1 u, t 1 v) ∈ N + λ and (t 2 u, t 2 v) ∈ N − λ . This completes the proof of Lemma 2.2. We can see from Lemma 2.2 that sets N + λ and N − λ are nonempty. In the following lemma we shall give a property related to N 0 λ . Lemma 2.3. Assume that condition (H 2 ) holds. Then for all λ ∈ (0, λ * ), we have N 0 λ = ∅. Proof. We shall argue by contradiction. Assume that there exists λ > 0 in (0, λ * ) such that N 0 λ = ∅. Let (u 0 , v 0 ) ∈ N 0 λ . Then invoking (H 2 ), (2.5) and (2.15), we have On the other hand, by (H 2 ), (2.6) and (2.14), one has Combining (2.20) and (2.21), we get Next, we define the function H on (0, ∞) by Since 1 < q < p < p * s , it follows that lim Moreover, by (H 2 ), (H 3 ) and (2.15), we have Since q < p and θp < p * s , it follows that J λ is coercive and bounded from below on N λ . This completes the proof of Lemma 2.4.

Proof of the main result
In this section, we shall prove the main result of this paper (Theorem 1.1). First, we need to prove two propositions.

By (H 3 ), it can be shown that lim
Since lim t→0 ω u,v (t) = 0 and lim t→∞ ω u,v (t) = −∞, it follows that ω u,v attains its global maximum at Moreover, from (2.14) and the fact that p * s > θp, we have sup Therefore, using the variations of the functions ζ u,v and ω u,v , we get Hence, there exists t 0 > 0, such that This completes the proof of Proposition 3.1. Proof. Let {(u k , v k )} be a Palais-Smale sequence for J λ at level c, that is