Negative Energy Solutions for a New Fractional 
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Introduction and the Main Results
In this article, we investigate the existence and multiplicity of solutions for the Kirchhoff type problem involving the fractional pðxÞ-Laplacian operator. More precisely, we consider the following problem ðP λ Þ: where Ω is a bounded domain in ℝ N with Lipschitz boundary and a ≥ b > 0 are constants, s ∈ ð0, 1Þ and N > spðx, yÞ for all ðx, yÞ ∈ Ω × Ω. λ is a given positive parameter, and f is a continuous function. The operator Lu = ð−ΔÞ s pð·Þ u is the fractional pðxÞ-Laplacian operator, which is defined by Lu ≔ P:V: The study for this problem and the related knowledge on fractional Sobolev spaces with variable exponent please refer to [11,12] and the references therein.
In the recent years, the fractional calculus and the related problems are studied widely, see for example, the literature [13,14]. The elementary knowledge of fractional derivative and integral can be found in [15,16]. Especially, the Caputo derivative is discussed in [17]. These references will be helpful for our research later.
The following Kirchhoff equation was introduced by Kirchhoff in [18]: where ρ, p 0 , h, E, L are constants that have physical meaning. We call the problem (P λ ) a new problem of Kirchhoff type because it contains the new Kirchhoff term: which makes the problem P λ more interesting, meaningful, and difficult. Up to now, the study on the Kirchhoff type problems involving pðxÞ-Laplacian operator and the fractional pðxÞ-Laplacian operator is very active, see [19][20][21][22][23].
Many results concerning the existence and multiplicity of solutions have been appeared. Especially, [24] proved the existence of two weak solutions by using the variational methods in Orlicz-Sobolev spaces, [25] studied a class of Kirchhoff nonlocal fractional equations, and obtained the existence of three solutions. We also mention that [26] discussed a class of p-Kirchhoff equations via the fountain theorem and dual fountain theorem, and [27] studied the existence of nonnegative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator. It is well known that the (AR) condition plays an important role in verifying the Palais-Smale condition. However, there are a lot of functions not satisfying this condition. Hence, many people pay attention to find the new reasonable conditions instead of the (AR) condition, see [28][29][30][31] and the references therein. Motivated by these work, we use Ekeland's variational principle and dual fountain theorem to study the existence and multiplicity of negative energy solutions for a new fractional pðxÞ-Kirchhoff problem without the (AR) condition. Our results generalize the related work in two ways. Firstly, we deal with the problem ðP λ Þ in the fractional framework and [32] consider only the integer framework. Secondly, compare to [33], we add a new Kirchhoff function and consider the case of variable exponents.
Throughout this paper, the nonlinearity f ðx, tÞ: Ω × ℝ ⟶ ℝ is a Carathe ′ odory function satisfying (f1): There exists a positive constant C such that for any φ ∈ X 0 , where X 0 will be introduced in Section 2.
It is well known that a weak solution for problem ðP λ Þ is a critical point of the following energy functional I defined on X 0 by for all u ∈ X 0 . Moreover, we have for any u, φ ∈ X 0 . Under our assumptions, I is well defined in X 0 and I ∈ C 1 ðX 0 , ℝÞ. We say that a weak solution u for problem ðP λ Þ is a negative energy solution if the energy IðuÞ < 0.
In order to reduce our statements, we define the function space and q − , q + , p − , p + , r − will be introduced in Section 2. The main results of this paper are as follows.

Journal of Function Spaces
Theorem 2. Assume that the function f satisfies (f1)-(f4) and Then, there exists a λ * > 0 such that for any 0 < λ < λ * , the problem ðP λ Þ has at least one solution u 0 with negative energy.

Theorem 3.
Assume that the function f satisfies (f1)-(f5) and Then, for any λ > 0, problem ðP λ Þ has infinitely many solutions fu n g in X 0 with negative energy converging to 0.
The rest of this paper is organized as follows. In Section 2, some basic properties of the variable exponent fractional Sobolev spaces and Lebesgue spaces are given. In Section 3, it is proved that the functional satisfies the Cerami compactness condition in certain energy levels. In Section 4, by using Ekeland's variational principle, we give the proof of Theorem 2. Finally, in Section 5, we prove Theorem 3 by using the dual fountain theorem.
Throughout this paper, for simplicity, we use letters C i ði = 1, 2, ⋯, NÞ to denote positive constants in different cases, and we will specify them whenever it is necessary.

Preliminary Results
In this section, we recall some preliminary results of generalized Lebesgue spaces L qðxÞ ðΩÞ with variable exponent and generalized fractional Sobolev spaces W s,hðxÞ,pðx,yÞ ðΩÞ which will be used later. The readers can consult [34][35][36] and the references therein for more details. Let s ∈ ð0, 1Þ and Ω be a bounded domain in ℝ N with Lipschitz boundary, and qðxÞ, hðxÞ, pðx, yÞ be continuous functions satisfying We assume that pðx, yÞ is symmetric; that is, pðx, yÞ = pðy, xÞ for all ðx, yÞ ∈ Ω × Ω, such that For any qðxÞ ∈ C + ð ΩÞ, we introduce the variable exponent Lebesgue space as endowed with the so-called Luxemburg norm Define a mapping ρ : L qðxÞ ðΩÞ ⟶ ℝ by Lemma 4 (See [37]). The space ðL qðxÞ ðΩÞ, juj qðxÞ Þ is separable, uniformly convex, and reflexive and its conjugate space is ðL q′ðxÞ ðΩÞ, juj q ′ ðxÞ Þ, where q ′ ðxÞ is the conjugate function of qðxÞ, i.e., for all u ∈ L qðxÞ ðΩÞ, v ∈ L q′ðxÞ ðΩÞ, and the Hölder type inequality holds.
Lemma 5 (see [38]). Suppose that u n , u ∈ L qðxÞ ðΩÞ. Then, the following properties hold The fractional Sobolev spaces with variable exponent X = W s,hðxÞ,pðx,yÞ ðΩÞ are defined by

Journal of Function Spaces
Define the corresponding variable exponent norm and set Then, the (X, k·k X ) becomes an uniformly convex and reflexive Banach spaces (see [39]). Let X 0 = W s,hðxÞ,pðx,yÞ 0 and then (X 0 , k·k X 0 ) is also an uniformly convex and reflexive Banach spaces. X * 0 is the dual spaces of X 0 .
We state Ekeland's variational principle and dual fountain theorem which will be used in the proofs of Theorems 2 and 3.

Journal of Function Spaces
Then, I has a sequence of negative critical values converging to 0.

Cerami Compactness Condition
We discuss the compactness properties of our energy functional I related to the ðCeÞ c condition and ðCeÞ * c condition.
Definition 11. Let I ∈ C 1 ðX, ℝÞ, and we say that I satisfies the Cerami condition at the level c ∈ ℝ ( ðCeÞ c for short), if any sequence fu n g ⊂ X 0 with possesses a convergent subsequence in X 0 .

Definition 12.
Let I ∈ C 1 ðX, ℝÞ, and we say that I satisfies ðCeÞ * c condition at the level c ∈ ℝ ( ðCeÞ * c for short), if any sequence fu − n j g n j ⊂ X 0 , namely, u − n j ∈ A n j , with possesses a convergent subsequence in X 0 . Assume that fu n g is a bounded sequence in X 0 . By Theorem 6, there exists u ∈ X 0 such that u n ⇀ u in X 0 , u n ⟶ u a:e:in Ω, u n ⟶ u in L q x ð Þ Ω ð Þ: ð38Þ Lemma 13. Let u n , u, v ∈ X 0 such that (38) holds. Then, passing to a subsequence, the following properties hold Proof.
(i) By Hölder's inequality (Lemma 4),Theorem 6, and (38), we have and thus lim n⟶∞ ð Ω u n j j q x ð Þ−2 u n u n − u ð Þdx = 0: ð41Þ (ii) By virtue of conditions (f1) and (f2), for any ε ∈ ð0, 1Þ, there exists C ε > 0 such that combining with Hölder's inequality, Theorem 6 and (38),and it follows that which implies that The following lemma about the ðCeÞ c condition will play an important role in the proof of our main results.

Lemma 14.
Let the function f satisfy (f1), (f3), and (f4), then the energy functional I satisfies the ðCeÞ c condition, where precisely c < a 2 /2b. Proof. Step 1. We prove that fu n g is bounded in X 0 . Let fu n g be a ðCeÞ c sequence, Iðu n Þ ⟶ c, ð1 + ku n k X 0 ÞI ′ ðu n Þ ⟶ 0, which implies that where oð1Þ ⟶ 0 as n ⟶ ∞: We claim that fu n g is a bounded sequence. Suppose to the contrary that Denote ω n ≔ u n /ku n k X 0 , then ω n ∈ X 0 with kω n k X 0 = 1. Up to subsequences, for some ω ∈ X 0 , we get ω n ⇀ ω in X 0 , ω n ⟶ ωa:e: in Ω, ω n ⟶ ω in L qðxÞ ðΩÞ: There are only two cases need to be discussed ω = 0 and ω ≠ 0:
Step 2. We prove that fu n g has a convergent subsequence in X 0 . Since ð1+∥u n ∥ÞI ′ðu n Þ ⟶ 0, asn ⟶ ∞, fu n g is bounded in X 0 , we have therefore, we obtain and we deduce from Lemma 13 that Since fu n g is bounded in X 0 , passing to a subsequence, we may assume (i) If t 0 = 0, then u n strongly converges to u = 0 in X 0 is not true. Because fu n g is bounded in X 0 , so, is bounded.
Proof. Consider a sequence fu − n j g n j ⊂ X 0 such that u − n j ∈ A n j , Iðu − n j Þ ⟶ c, I ′ ðu − n j Þj A n j ′ ⟶ 0, in X * 0 , as n j ⟶ ∞: By the same method used in the proof of Lemma 14, we can prove that fu − n j g n j has a strongly convergent subsequence in X 0 .
The details are omitted.
Remark 16. Since the ðCeÞ c and ðCeÞ * c conditions hold for the energy c < a 2 /2b, we discuss the negative energy solutions for the problem ðP λ Þ.

Proof of Theorem 2
We firstly prove two lemmas.