Multiplicity of solutions for a class of fractional $p(x,\cdot)$-Kirchhoff type problems without the Ambrosetti-Rabinowitz condition

We are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{\Omega\times \Omega} \ \displaystyle{\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y) \ |x-y|^{N+p(x,y)s}}} \ dx \, dy\right)(-\Delta)^{s}_{p(x,\cdot)}u = f(x,u), \quad x\in \Omega, \\ \\ u= 0, \quad x\in \partial\Omega, \end{array}\right.$$ where $\Omega\subset\mathbb{R}^{N}$, $N\geq 2$ is a bounded smooth domain, $s\in(0,1),$ $p: \overline{\Omega}\times \overline{\Omega} \rightarrow (1, \infty)$, $(-\Delta)^{s}_{p(x,\cdot)}$ denotes the $p(x,\cdot)$-fractional Laplace operator, $M: [0,\infty) \to [0, \infty),$ and $f: \Omega \times \mathbb{R} \to \mathbb{R}$ are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo-Benci-Fortunato (Nonlinear Anal. 7:9 (1983), 981-1012), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti-Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.

Throughout this paper, we shall assume that M : R + 0 := [0, +∞) → R + 0 is a continuous function satisfying the following conditions: It is worth pointing out that condition (M 2 ) was originally used to establish multiplicity of solutions for a class of higher order p(x)-Kirchhoff equations [11].
In recent years, a lot of attention has been given to problems involving fractional and nonlocal operators. This type of operators arises in a natural way in many different applications, e.g., image processing, quantum mechanics, elastic mechanics, electrorheological fluids (see [8,15,16,34] and the references therein).
In their pioneering paper, Bahrouni and Rădulescu [6] studied qualitative properties of the fractional Sobolev space W s,q(x),p(x,y) ( ), where is a smooth bounded domain. Their results have been applied in the variational analysis of a class of nonlocal fractional problems with several variable exponents.
Recently, by means of approximation and energy methods, Zhang and Zhang [38] have established the existence and uniqueness of nonnegative renormalized solutions for such problems. When s = 1, the operator degrades to integer order. It has been extensively studied in the literature; see for example [9,10,18,19,21] and the references therein. In particular, when p(x, ·) is a constant, this operator is reduced to the classical fractional p-Laplacian operator.
For studies concerning this operator, we refer to [31,32,[39][40][41]. We emphasize that, unless the functions p(x, ·) and q(x) are constants, the space W s,q(x),p(x,y) ( ) does not coincide with the Sobolev space W s,p(x) ( ) when s is a natural number; see [16,23]. However, because of various applications in physics and in mathematical finance, the study of nonlocal problems in such spaces is still very interesting.
On the other hand, a lot of interest has in recent years been devoted to the study of Kirchhoff-type problems. More precisely, in 1883 Kirchhoff [24] established a model given by the following equation: a generalization of the well-known D' Alembert wave equation for free vibrations of elastic strings, where ρ, p 0 , λ, E, L are constants which represent some physical meanings, respectively.
In the study of problem (1.1), the following Ambrosetti-Rabinowitz condition given in [3] has been widely used: (AR): There exists a constant μ > p + such that Clearly, if the (AR) condition holds, then where c 1 , c 2 are two positive constants. It is well known that (AR) condition is very important for ensuring the boundedness of the Palais-Smale sequence. When the nonlinear term f satisfies the (AR) condition, many results have been obtained by using the critical point theory and variational methods; see for example [1, 2, 4, 5, 12-14, 17, 20, 28, 29, 36, 37]. In particular, Ali et al. [1] and Azroul et al. [5] have established the existence of nontrivial weak solutions for a class of fractional p(x, ·)-Kirchhoff-type problems by using the mountain pass theorem of Ambrosetti and Rabinowitz, direct variational approach, and Ekeland's variational principle.
Since the (AR) condition implies condition (1.5), one cannot deal with problem (1.1) by using the mountain pass theorem directly if f (x, t) is p + -asymptotically linear at ∞, i.e.
where l is a constant. For this reason, in recent years some authors have studied problem (1.1) by trying to omit the condition (AR); see for example [18,22,27]. Not having the (AR) condition brings great difficulties, so it is natural to consider if this kind of fractional problems have corresponding results even if the nonlinearity does not satisfy the (AR) condition. In fact, in the absence of Kirchhoff 's interference, Lee et al. [26] have obtained infinitely many solutions to a fractional p(x)-Laplacian equation without assuming the (AR) condition, by using the fountain theorem and the dual fountain theorem.
Inspired by the above work, we consider in this paper the fractional p(x, ·)-Kirchhofftype problem without the (AR) condition. Our situation is different from [1,5] since our Kirchhoff function M belongs to a larger class of functions, whereas the nonlinear term f is p + -asymptotically linear at ∞.
More precisely, let us assume that f satisfies the following global conditions: , for all x ∈ , and a number 0 > 0 such that, for each λ ∈ (0, 0 ), > 0, there exists C > 0 such that (F 3 ): the following is uniformly satisfied on : where γ is given by (M 1 ); (F 5 ): the following holds: A simple computation proves that the function does not satisfy the (AR) condition. However, it is easy to see that f (x, t) in (1.7) satisfies conditions (F 1 )-(F 5 ).
We can now state the definition of (weak) solutions for problem (1.1) (see Sect. 2 for details): The main result of our paper is the following theorem.

Theorem 1.1 Let q(x), p(x, y) be continuous variable functions such that sp
and that M : R + 0 → R + 0 is a continuous function satisfying conditions (M 1 ) and (M 2 ). Then there exists > 0 such that, for each λ ∈ (0, ), problem (1.1) has a sequence {u n } n of nontrivial solutions.
The paper is organized as follows. In Sect. 2, we shall introduce the necessary properties of variable exponent Lebesgue spaces and fractional Sobolev spaces with variable exponent. In Sect. 3, we shall verify the Cerami compactness condition. Finally, in Sect. 4, we shall prove Theorem 1.1 by means of a version of the mountain pass theorem.

Fractional Sobolev spaces with variable exponent
For a smooth bounded domain in R N , we consider a continuous function p : × → (1, ∞). We assume that p is symmetric, that is, We also introduce a continuous function q : → R such that We first give some basic properties of variable exponent Lebesgue spaces. Set Given r ∈ C + ( ), we define the variable exponent Lebesgue space as and this space is endowed with the Luxemburg norm, Then (L r(x) ( ), | · | r(x) ) is a separable reflexive Banach space; see [25, Theorem 2.5 and Corollaries 2.7 and 2.12].
We denote our workspace E 0 = W s,q(x),p(x,y) 0 ( ), the closure of C ∞ 0 ( ) in E. Then E 0 is a reflexive Banach space with the norm A thorough variational analysis of the problems with variable exponents has been developed in the monograph by Rădulescu and Repovš [33]. The following result provides a compact embedding into variable exponent Lebesgue spaces.
Assume that τ : − → (1, ∞) is a continuous function such that Then there exists a constant C = C(N, s, p, q, r, ) such that, for every u ∈ W s,q(x),p(x,y) , That is, the space W s,q(x),p(x,y) ( ) is continuously embeddable in L τ (x) ( ). Moreover, this embedding is compact. In addition, if u ∈ W s,q(x),p(x,y) 0 , the following inequality holds: Theorem 2.2 (see [6]) For all u, v ∈ E 0 , we consider the operator I : E 0 → E * 0 such that Then the following properties hold: (1) I is a bounded and strictly monotone operator.
(2) I is a mapping of type (S + ), that is, if u n u ∈ E 0 and lim sup n→∞ I(u n )(u nu) ≤ 0, then u n → u ∈ E 0 .

The cerami compactness condition
Let us consider the Euler-Lagrange functional associated to problem (1.1), defined by for all w ∈ E 0 . Therefore critical points of J λ are weak solutions of problem (1.1). In order to prove our main result (Theorem 1.1), we recall the definition of the Cerami compactness condition [30]. Definition 3. 1 We say that J λ satisfies the Cerami compactness condition at the level c ∈ R ((Ce) c condition for short), if every sequence {u n } n ⊂ E 0 , i.e., J λ (u n ) → c and J λ (u n ) E 0 * 1 + u n E 0 → 0, as n → ∞, admits a strongly convergent subsequence in E 0 . If J λ satisfies the (Ce) c condition for any c ∈ R then we say that J λ satisfies the Cerami compactness condition.

Claim 3.1 Under assumptions of Theorem
First, we prove that the sequence {u n } n is bounded in E 0 . To this end, we argue by contradiction. So suppose that u n E 0 → ∞, as n → ∞. We define the sequence {v n } n by v n = u n u n E 0 , n ∈ N.
It is clear that {v n } n ⊂ E 0 and v n E 0 = 1 for all n ∈ N. Passing, if necessary, to a subsequence, we may assume that This means that u n (x) = v n (x) u n E 0 → +∞ a.e. on , as n → ∞.
Moreover, it follows by condition (F 3 ) and Fatou's lemma that, for each x ∈ , Now, since u n E 0 > 1, it follows by (3.1), (3.3) and (3.6) that for all n ∈ N. We can now conclude that From (3.5) and (3.7) we obtain which is a contradiction. Therefore | | = 0 and v(x) = 0 a.e. on .
It follows from (M 1 ), (M 2 ), (F 4 ) and v n → v = 0 in L p -( ) that which means that This is a contradiction. As a consequence, we can conclude that Cerami sequence {u n } n is indeed bounded. This completes the proof of Claim 3.1.
We now complete the verification of the Cerami compactness condition (Ce) c for J λ .

Claim 3.2
The functional J λ satisfies condition (Ce) c in E 0 .
Proof Let {u n } n be a (Ce) c sequence for J λ in E 0 . Claim 3.1 asserts that {u n } n is bounded in E 0 . By Theorem 2.1, the embedding Since E 0 is a reflexive Banach space, passing, if necessary, to a subsequence, still denoted by {u n } n , there exists u ∈ E 0 such that Now, by condition (F 2 ), It follows from (3.8), (3.10) and Proposition 2.1 that Therefore we can infer from (3.9) and (3.11) that M σ p(x,y) (u n ) × × |u n (x)u n (y)| p(x,y)-2 (u n (x)u n (y))((u n (x)u(x)) -(u n (y)u(y))) |x -y| N+p(x,y)s dx dy → 0.
Since {u n } n is bounded in E 0 , using (M 2 ), we can conclude that the sequence of positive real numbers {M(σ p(x,y) (u n ))} is bounded from below by some positive number for n large enough. Invoking Theorem 2.2, we can deduce that u n → u strongly in E 0 . This completes the proof of Claim 3.2.

Proof of Theorem 1.1
To prove Theorem 1.1, we shall use the following symmetric mountain pass theorem.
Let us first verify that functional J λ satisfies the mountain pass geometry.
Consequently, since u E 0 > 1, all norms on the finite-dimensional space W are equivalent, so there is C W > 0 such that Let R = R(W ) > 0. Then for all u ∈ W with u E 0 ≥ R we obtain