Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities

1 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 [5,39], water waves [16,17], minimal surfaces [13], and conservation laws [10]. For more applications of such problems in physical phenomena, probability, and finances, we refer interested readers to [12,14,47]. Due to their importance, there are many interesting works on the existence and multiplicity of solutions for fractional and nonlocal problems either on bounded domains or on the entire space, see [1,3,4,6,23,24,34,36–38].

In the last decade, many scholars have paid extensive attention to Kirchhoff-type elliptic equations with critical exponents, see [20,25,33], for the bounded domains and [26,28,29] for the entire space. In particular, in [22], the authors considered the following Kirchhoff problem:

where t ≥ 0 and M ( t ) = a + b t for some a > 0 and b ≥ 0 . Here, and in the rest of this article, 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 [7,8,15,21,27,30–32,35,40,42–45]. In particular, by using the Nehari manifold method and the symmetric mountain pass theorem, Xiang et al. [43] investigated the multiplicity of solutions for some p -Kirchhoff system with Dirichlet boundary conditions.

Mingqi et al. [30] studied the following Schrödinger-Kirchhoff-type system:

where λ > 0 , α + β = p s ∗ ≔ n p n − s p , 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 [30], Fiscella et al. [21] 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, Azroul 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 articles, we consider in this article the following Schrödinger-Kirchhoff-type system involving the fractional p -Laplacian and critical nonlinearities:

where ∥ . ∥ V 1 and ∥ . ∥ V 2 will be given later (see (1.6)), n > p s , 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 ( − Δ ) p s is the fractional p -Laplace operator, defined as follows:

where B ε ( x ) = { y ∈ R n : ∣ x − y ∣ < ε } . For more details about the fractional p -Laplacian operator and the basic properties of fractional Sobolev spaces, we refer the reader to [18].

Throughout this article, 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 Ω V i ≥ v i . In addition, we shall assume that M i : ( 0 , ∞ ) → ( 0 , ∞ ) is a continuous function satisfying the following conditions:

( H 1 ) lim t → ∞ t 1 − p s ∗ p M i ( t ) = 0 .

( H 2 ) There exists m i > 0 such that for all t > 0 , we have M i ( t ) ≥ m i .

( H 3 ) There exists θ i ∈ 1 , p s ∗ p such that for all t > 0 , we have M i ( t ) t ≤ θ i M ^ i ( t ) , where M ^ i ( t ) = ∫ 0 t M i ( s ) d s .

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 , u , v ) ∈ Ω × R × R , we have

Finally, we shall also assume that there exists a function H : Ω ¯ × R × R → R satisfying

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 :

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

where Q = R 2 n \ ( Ω c × Ω c ) 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 V i s , p ( Q ) the completion of C 0 ∞ ( Q ) with respect to the norm

According to [18, (Theorem 6.7]), the embedding W V i s , p ( Q ) ↪ L ν ( Ω ) is continuous for any ν ∈ [ p , p s ∗ ] . Namely, there exists a positive constant C ν such that

Moreover, by [46, Lemma 2.1], the embedding from W V i s , p ( Q ) into L ν ( Ω ) , is compact for any ν ∈ [ 1 , p s ∗ ) .

Let W = W V 1 s , p ( Q ) × W V 2 s , p ( Q ) be equipped with the norm ‖ ( u , v ) ‖ = ( ‖ u ‖ V 1 p + ‖ v ‖ V 2 p ) 1 p . Then, ( W , ‖ . ‖ ) is a reflexive Banach space. The interested reader can refer to [2] for more details. Let S p , V i be the best Sobolev constants for the embeddings from W V i s , p ( Q ) into L p s ∗ ( Ω ) , which is given as follows:

For simplicity, in the rest of this article, S will denote the following expression:

Next, we define the notion of solutions for problem (1.3).

The following theorem is the main result of this article.

This article 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.

2 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 [11,12,19] for more details. We begin by considering the Euler-Lagrange functional J λ : W → R , which is defined as follows:

where

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 as follows:

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 as follows:

It is clear that ( u , v ) ∈ N λ if and only if

Hence, from (2.2), we see that the 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 , is defined as follows:

A simple calculation shows that for all t > 0 , we have

and

It is easy to see that for all t > 0 , we have

So, ( t u , t v ) ∈ N λ if and only if φ u , v ′ ( t ) = 0 . In the special case, when t = 1 , we obtain ( u , v ) ∈ N λ , if and only if φ u , v ′ ( 1 ) = 0 . On the other hand, from (2.3), we obtain

Now, in order to obtain a multiplicity of solutions, we divide N λ into three parts as follows:

In order to understand the Nehari manifold and fibering maps, let us define the function ψ u , v : ( 0 , ∞ ) → R as follows:

We note that t q − 1 ψ u , v ( t ) = φ u , v ′ ( t ) . Thus, it is easy to see that ( t u , t v ) ∈ N λ if and only if

Moreover, by a direct computation, we obtain

Therefore,

Hence, ( t u , t v ) ∈ N λ + , ( respectively , ( t u , t v ) ∈ N λ − ) if and only if ψ u , v ( t ) = 0 and ψ u , v ′ ( t ) > 0 (respectively, ψ u , v ( t ) = 0 , and ψ u , v ′ ( t ) < 0 ). Put

and

Now we shall prove the following crucial result.