Multiplicity result for non-homogeneous fractional Schrodinger--Kirchhoff-type equations in ℝ n

Abstract In this paper we consider the existence of multiple solutions for the non-homogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type M ⁢ ( ∫ ℝ n ∫ ℝ n | u ⁢ ( x ) - u ⁢ ( z ) | p | x - z | n + p ⁢ s ⁢ 𝑑 z ⁢ 𝑑 x ) ⁢ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u = f ⁢ ( x , u ) + g ⁢ ( x ) M\Bigg{(}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(z)|^{p}}{|x-{% z}|^{n+ps}}\,dz\,dx\Bigg{)}(-\Delta)_{p}^{s}u+V(x)|u|^{p-2}u=f(x,u)+g(x) in ℝ n {\mathbb{R}^{n}} , where (-Δ ) p s )_{p}^{s} is the fractional p-Laplacian operator with 0¡s¡1¡p¡ ∞ \infty , p s¡n, f : ℝ n \mathbb{R}^{n} × \times ℝ \mathbb{R} → \to ℝ \mathbb{R} is a continuous function, V : ℝ n \mathbb{R}^{n} → \to ℝ + \mathbb{R}^{+} is a potential function and g : ℝ n \mathbb{R}^{n} → \to ℝ \mathbb{R} is a perturbation term. Assuming that the potential V is bounded from bellow, that f(x,t) satisfies the Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, and that g(x) is sufficiently small in L p ′ L^{p^{\prime}} ( ℝ n \mathbb{R}^{n} ), we obtain some new criterion to guarantee that the equation above has at least two non-trivial solutions.


Introduction
The aim of this article is to study a Schrödinger-Kirchhoff-type equation with fractional p-Laplacian in ℝ n , M ℝ n ℝ n |u(x) − u(z)| p |x − z| n+ps dz dx (−∆) s p u + V(x)|u| p− u = f(x, u) + g(x), (1.1) where < s < < p < ∞, ps < n, and the operator (−∆) s p is the fractional p-Laplacian which may be defined along a function φ ∈ C ∞ (ℝ n ) as where x ∈ ℝ n , and B(x, ϵ) = {y ∈ ℝ n : |x − y| < ϵ}. We invite the reader to check [16, 19-21, 25, 35] and the references therein for further details on the fractional p-Laplace operator. The function g = g(x) can be viewed as a perturbation term. When p = and M = , equation (1.1) becomes the fractional Laplacian equation in ℝ n , which has been study by many researchers, see, for instance, [2, 6-8, 11, 34] and the references therein for some results.
In recent years, great attention has been paid on the study of problems involving the non-local fractional Laplacian or more general integro-differential operators. This type of operators arises in a quite natural way in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and game theory, as they are the typical outcome of stochastical stabilization of Lévy processes, see [3][4][5]23] and the references therein. The literature on fractional operators and their applications is very extensive, see, for example, [1, 2, 6-8, 11-15, 24, 27-40]. For a short introduction to the fractional Laplacian and the fractional Sobolev spaces, the reader is referred to [10]. Furthermore, research has been done in recent years for the regional fractional Laplacian, where the scope of the operator is restricted to a variable region near each point. We mention the work by Guan [17] and Guan and Ma [18] where they study these operators, their relation with stochastic processes and they develop an integration by parts formula. We also mention the work by Ishii and Nakamura [21], where they studied the Dirichlet problem for regional fractional Laplacian modeled on the p-Laplacian and the recent work by Felmer and Torres [12,13], where they considered the existence, symmetry properties and concentration phenomena of solutions of the non-linear Schrödinger equation with non-local regional diffusion. These regional operators present various interesting characteristics that make them very attractive from the point of view of mathematical theory of non-local operators.
On the other hand, Fiscella and Valdinoci in [15] first proposed a stationary Kirchhoff variational equation which models the non-local aspect of the tension arising from non-local measurements of the fractional length of the string. Indeed, problem (1.1) is a fractional version of a model, the so-called Kirchhoff equation, introduced by Kirchhoff in [22]. More precisely, Kirchhoff established a model given by the problem where ρ is the mass density, p is the initial tension, h is the area of the cross section, E is the Young modulus of the material and L is the length of the string, which extends the classical D'Alembert wave equation by considering the effects of the changes in the length of the strings during the vibrations. Note that non-local boundary problems like (1.2) can be used to model several physical and biological systems where u describes a process, which depend on the average of itself, such as the population density [9]. A parabolic version of problem (1.2) can be used to describe the growth and movement of a particular species. The movement, modeled by the integral term, is assumed to be dependent on the energy of the entire system with u being its population density. Alternatively, the movement of a particular species may be subject to the total population density within the domain (for instance, the spreading of bacteria) which gives rise to equations of the type It is worth pointing out that problem (1.2) received much attention only after Lions [26] proposed an abstract framework to it. For some motivation in the physical background for the fractional Kirchhoff model, we refer to [15,Appendix A]. Also, see for example [29][30][31]37] for non-degenerate Kirchhoff-type problems, and [1,36], for degenerate Kirchhoff-type problems in this direction. Very recently in [32], Pucci, Xiang and Zhang considered problem (1.1) under the following assumptions: For the Kirchhoff function M, suppose that: For the potential V, suppose that: For the nonlinearity f , suppose that: (f ) f : ℝ n × ℝ → ℝ is a Carathéodory function and there exist q, with θp < q < p * s , and a > such that |f(x, t)| ≤ a ( + |t| q− ) for a.e. x ∈ ℝ n and all t ∈ ℝ. (f ) There exists μ > θp such that , the authors first establish a Batsch-Wang-type compact embedding theorem for the fractional Sobolev spaces. Multiplicity results are then obtained by using the Ekeland's variational principle and the Mountain Pass Theorem. Also we mention the work by Xiang, Zhang and Ferrara [38], where they considered the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities and have obtained the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekeland's variational principle.
Inspired by these previous works, in this paper we consider problem (1.1) under some weaker assumptions. More precisely, we assume that the Kirchhoff function M satisfies (M )-(M ). Furthermore, regarding the potential V we only assume (V ) and for the functions f we suppose that: (H ) There exists some positive continuous function a : Before stating our main results, we introduce some useful notations. First of all, define the Gagliardo seminorm by where u : ℝ n → ℝ is a measurable function. Now, let the fractional Sobolev space be denoted as for any φ ∈ X s . From here on we set p ὔ = p/(p − ), the Hölder conjugate of p. Our main result in this paper is the following theorem. Then there exists a constant δ > such that problem (1.1) has at least two non-trivial solutions in X s , provided that ‖g‖ L p ὔ ≤ δ . Remark 1.1. We note that condition (V ) is used to establish some compact embedding theorems to guarantee that the (PS) condition holds, which is the essential step to obtain the existence of weak solutions of (1.1) via the Mountain Pass Theorem. In the present paper, we assume that V(x) is bounded from below but we could not obtain some compact embedding theorem. Therefore, one difficulty is to adapt some new technique to overcome this difficulty and test that the (PS) condition is verified, see Lemmas 3.1 and 3.2 below.
The remaining part of this paper is organized as follows. Some preliminary results are presented in Section 2. In Section3, we are devoted to accomplishing the proof of Theorem 1.1.

Preliminaries
To study the fractional problem (1.1), the so-called fractional Sobolev spaces W s,p (ℝ n ) with < s < are expedient. If < p < ∞, as usual, the norm is defined through We recall the Sobolev embedding theorem. Now we introduce some more notations and necessary definitions. Let B be a real Banach space. If I is a continuously Fréchet-differentiable functional defined on B, we write I ∈ C (B, ℝ). Recall that I ∈ C (B, ℝ) is said to satisfy the (PS) condition if every sequence {u n } n∈ℕ ⊂ B for which {I(u n )} n∈ℕ is bounded and I ὔ (u n ) → as n → ∞ possesses a convergent subsequence in B. Moreover, let B r be the open ball in B with radius r and centered at , and let ∂B r denote its boundary. Under the conditions of Theorem 1.1, we obtain the existence of the first weak solution of (1.1) using the following well-known Mountain Pass Theorem, see [33]. I(g(s)), As far as the second solution is concerned, we obtain it using the minimizing method, which is contained in a small ball centered at .

Proof of Theorem 1.1
The aim of this section is to establish the proof of Theorem 1.1. For this purpose, we are going to establish the corresponding variational framework to obtain solutions of (1.1). To this end, define the functional I : X s → ℝ by where If (M ) and (V ) hold, then J : X s → ℝ is of class C (X s ) and for all u, v ∈ X s . Moreover, J is weakly lower semi-continuous in X s , see [32]. Now we prove our key lemma.
Proof. From (H ) it is obvious that Now assume that u n ⇀ u in X s . Then, there is some constant K > such that for k ∈ ℕ. Furthermore, by Remark 2.1, given R > up to a subsequence, we may assume that: a.e in ℝ n and for all k ∈ ℕ. Thus, by the previous inequality and lim Hence, by the Lebesgue Dominated Convergence Theorem we have where μ ὔ + μ = μμ ὔ . On the other hand, from (H ), for any ϵ > there exists R > such that Consequently, in view of (2.1), (3.6)-(3.8), for k large enough, we have (3.9) Here, we apply the Young inequality Consequently, we obtain that which yields that Φ ὔ (u k ) → Φ ὔ (u) as u k ⇀ u, so Φ ὔ is compact.
Therefore, H is weakly continuous in X s .
To prove Theorem 1.1 we first consider some lemmas. Proof. Although the proof of this lemma is just the repetition of the process of [32, Lemma 6], we give the details for convenience of the reader. Assume that {u k } k∈ℕ ⊂ X s is a sequence such that {I(u k )} k∈ℕ is bounded and I ὔ (u k ) → as k → ∞. Then there exists a constant C > such that for every k ∈ ℕ, where (X s ) * is the dual space of X s . Firstly, we show that {u k } k∈ℕ is bounded. In fact, in view of (M )-(M ), (H ), (3.10) and using also that μ > θp ≥ p > , we obtain where K = S p ‖g‖ L p ὔ . Hence, {u k } k∈ℕ is bounded in X s . Then, the sequence {u k } k∈ℕ has a subsequence, again denoted by {u k } k∈ℕ , and there exists u ∈ X s such that u k ⇀ u weakly in X s which implies that ⟨I ὔ (u k ) − I ὔ (u), u k − u⟩ → as k → ∞. Let φ ∈ X s be fixed and denote by B φ the linear functional on X s defined by for all v ∈ X s . By the Hölder inequality, B φ is continuous and hence On the other hand, by Lemmas 3.3 and 3.4 and the Mountain Pass Theorem, there exists a sequence {u k } k ∈ X s such that, as k → ∞, I(u k ) → c > and I ὔ (u k ) → .
In view of the proof of Lemma 3.2, we know that there exists a critical point u ∈ X s of I. Moreover, Thus, u ̸ = and u ̸ = u .