Existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian and local nonlinearity

In this paper, we deal with the existence of nontrivial solutions for the following Kirchhofftype equation


Introduction
In this paper, we investigate the existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian and local nonlinearity. More precisely, we consider the following Kirchhoff-type equation where 0 < s < 1 < p < ∞, sp < N, λ > 0 is a real parameter, (−∆) s p is the fractional p-Laplacian operator which, up to a normalization constant, may be defined as |φ(x) − φ(y)| p−2 (φ(x) − φ(y)) |x − y| N+sp dy, ∀ x ∈ R N , along any φ ∈ C ∞ 0 (R N ), where B ε (x) := {y ∈ R N : |x − y| < ε}. In fact, many scholars have paid more attention to the fractional and nonlocal operators in recent years. This type of operator occurs naturally in many field of science, such as finance, continuum mechanics, free boundary obstacle problems, population dynamics, plasma physics and anomalous diffusion. For more details on this type of operator, we refer to [2,5,6] and the references therein. For the fractional Sobolev spaces and the study of the fractional Laplacian by using variational methods, we refer the readers to [9,20,27]. In order to simplify our statements, we first suppose on the potential function V that (V 1 ) V ∈ C(R N ) and there exists a constant V 0 > 0 such that inf x∈R N V(x) ≥ V 0 ; (V 2 ) there exists R > 0 such that lim |y|→∞ meas ({x ∈ B R (y) : V(x) ≤ d}) = 0 for any d ∈ R + .
Condition (V 2 ) was originally from [3], it can be used to solve the problem of the lack of compactness in the whole space R N . Moreover, the condition (V 2 ) is weaker than the coercivity condition lim |x|→∞ V(x) = ∞.
As we all know, Kirchhoff equations was first proposed by Kirchhoff in 1883 (see [15]), which was related to the celebrated D'Alembert wave equation for free vibrations of elastic strings. Here, ρ is the mass density, P 0 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. Bernstein [4] and Pohozaev [24] were the early scholars devoted to study Kirchhoff equations. After Lions [18] proposed an abstract framework for Kirchhoff problems, many scholars have studied Kirchhoff equations by using variational methods. For example, we refer to [12,19] for Kirchhoff equations involving subcritical nonlinearities; we also collect some articles, see [17,21] for Kirchhoff equations involving critical and supercritical nonlinearities.
Recently, Fiscella and Valdinoci [11] proposed a stationary Kirchhoff type variational model, which considered the nonlocal aspect of the tension arising from nonlocal measurements of fractional length of the string. More precisely, they studied the following fractional Kirchhoff type problem involving critical growth By combining a truncated technique with the mountain pass theorem, they obtained the existence of nontrivial solutions for the above equation when λ is large enough. Afterwards, many scholars have studied the existence and multiplicity of nontrivial solutions, ground state solutions, sign-changing solutions for fractional Kirchhoff  However, many scholars usually supposed that the nonlinearity f (u) satisfies the growth condition | f (u)| ≤ C(|u| + |u| q−1 ), q ∈ (p, p * s ) and the following conditions or the Ambrosetti-Rabinowitz condition (AR) there exists µ ∈ (p, p * s ) such that 0 < µF(u) ≤ f (u)u for all u ∈ R. Indeed, under the above conditions, it is easy to obtain the existence and multiplicity of solutions for fractional Kirchhoff type equations by using variational methods. For example, Cheng and Gao [7] established the existence of least energy sign-changing solutions when f (u) satisfies (S 1) and (S 2). Under the nonlinearity f (u) satisfies (AR) condition, Nyamoradi and Zaidan [22], they proved the existence of nontrivial solutions. In addition, some scholars studied the existence of solutions for fractional Kirchhoff type equations when f (u) satisfies the Berestycki-Lions type conditions, see for instance [14,33].
As far as we know, there are no papers dealing with the fractional Kirchhoff type equation with local nonlinearity, which the nonlinearity f (u) is superlinear just in a neighborhood of u = 0. There is no doubt that serious difficulties will be encountered, because there is no assumption about the function f (u) at infinity. This difficulty makes the study of fractional Kirchhoff type equations become more interesting and challenging. In order to overcome this difficulty, Li and Su [16] used a truncation argument due to Costa and Wang (see [8]) to get the existence and multiplicity of solutions for Kirchhoff type equations. In [13], Huang and Jia also obtained the existence of positive solutions for quasilinear Schrödinger equations via the truncation argument in [8].
Motivated by the above works, the purpose of this paper is to give existence results for Eq (1.1). To the best of our knowledge, the existence of solutions for fractional Kirchhoff equations with local nonlinearity has not been studied yet. Before stating our result, we consider M : , is supposed to satisfy the following conditions: (M 1 ) M ∈ C(R + 0 ) and there exists m 0 > 0 such that M(t) ≥ m 0 for any t ∈ R + 0 ; (M 2 ) the function t → M(t) is increasing; Moreover, we assume the following assumptions on f : In the following, let's state our result.
The energy functional associated with Eq (1.1) is given by Due to we assume that the nonlinearity f (x, u) is superlinear only in a neighborhood of u = 0, the energy functional may be not well defined. Therefore, we can not directly use the variational method to prove the existence of solutions. In order to prove Theorem 1.1, we will use a truncation argument, which came from [8,13,16]. More precisely, we first show that the existence of nontrivial solutions for the revised equation via the mountain pass theorem. By Moser iteration method and L ∞ -estimate, we can obtain solution of revised equation, which is the solution of the original Eq (1.1) when λ is sufficiently large. In addition, our assumptions on the nonlinearity f just in a neighborhood of the origin, which are greatly relax. The results of this paper are new and can enrich the previous ones in the literature. Remark 1.2. It is worth mentioning that this paper is the first time to assume the nonlinearity f (x, u) just in a neighborhood at u = 0 and discuss existence of solutions for fractional Kirchhoff-type equations. Therefore, our results are new, and enrich the previous ones in the literature. This paper is organized as follows. In Section 2, we recall some basic properties of the fractional Sobolev spaces and introduce a truncation argument. In Section 3, we give the proof of Theorem 1.1.

Preliminaries
In this section, we first recall some basic results of the fractional Sobolev spaces. Let 0 < s < 1 < p < ∞ be real numbers and N > sp. The fractional Sobolev space W s,p (R N ) is given by denotes the Lebesgue space of real-valued functions and equipped with the norm According to Lemma 10 in the Appendix of [25], we also know that X = (X, · X ) is a uniformly convex Banach space. Furthermore, X is a reflexive Banach space. The dual space of (X, · X ) is denoted by (X * , · X * ). Denote the best fractional Sobolev constant: are continuous. Thus, there exists a constant C r > 0 such that Moreover, if r ∈ [p, p * s ), then the embedding X → → L r (B R ) is compact for any R > 0. Lemma 2.2. ( [25]) Suppose that (V 1 ) and (V 2 ) are satisfied. Let r ∈ [p, p * s ) be a fixed exponent. If {u n } is a bounded sequence in X, then there exists u 0 ∈ X ∩ L r (R N ) such that up to a subsequence, u n → u 0 strongly in L r (R N ), as n → +∞, for all r ∈ [p, p * s ). Next, we show that the definitions of (PS ) c condition and the mountain pass theorem. Definition 2.3. Let E ∈ C 1 (X, R). We say the E satisfies the (PS ) c condition at level c ∈ R in X, if any (PS ) c sequence {u n } ⊂ X, that is, E(u n ) → c, E (u n ) → 0 in X * , as n → ∞, admits a convergent subsequence in X. Theorem 2.4. ( [32]) Let X be a real Banach space, suppose E ∈ C 1 (X, R) satisfies the (PS ) c condition with E(0) = 0. Moreover, (i) there exist , η > 0 such that E(u) ≥ η for all u ∈ X, with u X = , (ii) there exists e ∈ X satisfying < e X such that E(e) < 0. Define is a critical value of E(u).
In order to prove Theorem 1.1, we need to modify and extend f to a suitable f . The argument was developed in [8,13,16]. Based on this, we make the following truncation argument.
Finally, we introduced the well-known Simon inequality as follows, which will be used later. Lemma 2.6.( [28]) There exist constants c p , C p > 0 such that for any x, y ∈ R N , it holds 3. Proof the Theorem 1.1 In this section, we will complete the proof of Theorem 1.1. We first prove that the existence of nontrivial solutions for the modified equation. More precisely, we consider the following fractional Kirchhoff type equation where f is given by (2.5). The energy functional I λ : X → R associated with Eq (3.1) Under the assumptions of Theorem 1.1, by using the similar proof method in [25], we can know that I λ ∈ C 1 (X, R) and for any u, v ∈ X. Obviously, the critical points of the energy functional I λ are exactly the weak solutions of Eq (3.1). In the following, let us first verify that I λ has mountain pass geometry. Lemma 3.1. Suppose that (V 1 ), (V 2 ), (M 1 ) and ( f 1 )-( f 3 ) are satisfied. Then there exist ζ λ , ρ λ > 0 such that I λ (u) ≥ ζ λ for any u ∈ X, with u X = ρ λ .
For u ∈ C ∞ 0 (R N ), with u X = 1. According to k > 2p, (3.4) and (3.5), we get Consequently, we can takeū = tu ∈ X such that I λ (ū) < 0 for t sufficiently large. Now, similar to the proof of Lemma 6 in [25], we prove that I λ satisfies (PS ) c condition. Proof. Let {u n } n∈N ⊂ X be a (PS ) c sequence for the functional I λ at level c ∈ R, that is, (3.6) Let us first prove that {u n } is bounded in X. For this purpose, we assume that u n X → +∞, as n → ∞. By (M 1 )-(M 3 ), one has Hence, it follows from (M 1 ), (3.2), (3.3), (3.7) and Lemma 2.5 (ii) that as n → ∞. By k > 2p, we get a contradiction. Therefore, {u n } n∈N is bounded in X.
Next, we prove that there exists u ∈ X such that u n → u in X, as n → ∞. Combining Lemma 2.1-Lemma 2.2 and Theorem A.1 in [32], there exist a subsequence, still denoted by {u n } and a function u in X such that a.e. in R N , |u n | ≤ h r , a.e. in R N , for some h r ∈ L r (R N ), (3.8) as n → ∞. In view of Lemma 2.5 (i) and Hölder's inequality, we have Now, let u ∈ X be fixed and denote by B u the linear functional on X defined by From u n u in X and I λ (u n ) → 0 in X * , we obtain I λ (u n ) − I λ (u), u n − u → 0 as n → ∞. Therefore, it follows from (3.3) and (3.8)-(3.11) that Finally, we divided into two cases to prove u n − u X → 0, as n → ∞. To this aim, we first assume that p ≥ 2. In view of Lemma 2.6, (3.10) and (3.14), we have (1), (3.15) as n → ∞. By (V 1 ), (3.13) and Lemma 2.6, we can show that as n → ∞. Hence, combining (3.15) with (3.16), we have u n − u X → 0 as n → ∞.
In the end, we assume that 1 < p < 2. Let us introduce the following elementary inequality . Similarly, it follows from (3.8) that there exists a constant l * > 0 such that u n p,V ≤ l * for all n ∈ N. Therefore, we conclude from Hölder's inequality and (3.13) that . Consequently, we have u n − u X → 0, as n → ∞. This concludes the proof of Lemma 3.3.
Next, we prove that the existence of nontrivial solutions for Eq (3.1) via the mountain pass theorem. Proof. In view of (3.4), (3.5) and Theorem 3.4, we have for all sufficiently large λ, whereū is given by Lemma 3.2 and Θ > 0 is a constant.
Now, similar to the proof of Theorem 1.1 in [1], we use the Moser iteration technique to estimate u λ L ∞ (R N ) . Lemma 3.7. If u ∈ X is a nontrivial solution of Eq (3.1), then u ∈ L ∞ (R N ). In addition, there exists a constant Ξ > 0 such that where β > 1 and u L = min{u, L}. Hence, we conclude from the functions introduced above that By the definition of ξ(u), we have uu p(β−1) Considering ξ(u) as a test function in (3.1), we get (3.20) Now, we define w L := uu β−1 L . .

Conclusions
In this paper, we have considered a class of fractional Kirchhoff type equation. Under suitable assumptions on V and M, using a truncation argument and the mountain pass theorem, we have established the existence of nontrivial solutions. It is expected that the results proved in this paper may be starting point further research in this field.