EXISTENCE RESULTS FOR ANISOTROPIC FRACTIONAL ( p 1 ( x, . ) , p 2 ( x, . )) -KIRCHHOFF TYPE PROBLEMS

In this paper, we investigate the existence and multiplicity of solutions for a class of fractional ( p 1 ( x, . ) , p 2 ( x, . )) -Kirchhoff type problems with Dirichlet boundary data of the following form

|x − y| N +sp i (x,y) dxdy ) (−∆) s p i (x,.) u(x) More precisely, by means of mountain pass theorem with Cerami condition, we show that the above problem has at least one nontrivial solution.Moreover, using Fountain theorem, we prove that ( P s

Introduction and statement of the main results
The study of differential equations involving p(x)-Laplacian operators have been a very interesting and exciting topic in recent years (see in particular the fascinating monograph [24] and the references therein for further details).This type of problems are extremely attractive because they can be used to model dynamical phenomena which arise from the study of electrorheological fluids or elastic mechanics.Problems with variable exponent growth conditions also appear in the modelling of stationary thermorheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes of filtration of an ideal barotropic gas through a porous medium.The detailed application backgrounds of the p(x)-Laplacian can be found in [4,19,27,36,40] and the references therein.
For problems involving different growth rates depending on the underlying domains, they also involve equations with (p(x), q(x))-growth conditions where several p(x)-Laplacian operators involved, interacting with one another.This (p(x), q(x))growth condition is a natural generalization of the anisotropic (p, q)-growth condition.In that context, the systems involving the (p(x), q(x))-Laplacian (or (p 1 (x), p 2 (x), • • • , p n (x))-Laplacian) can be good candidates for modeling phenomena which ask for distinct behavior of partial differential derivatives in various directions, for related problems we just mention [3,21,33,37].
On the other hand, in the recent years increasing attention has been paid to the study of pseudo-differential and nonlocal fractional operators (as (−∆) s , (−∆) s p and their generalizations) and related fractional differential equations.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 stochastic stabilization of Lévy processes, see for instance [5,18,31] and the references therein.For a selfcontained overview of the basic properties of fractional Sobolev spaces and fractional Laplacian (or fractional p-Laplacian operator), we refer the reader to [16,35] and to the references included.
It is therefore, a natural question is to see which results "survive" when the p(x)-Laplacian is replaced by the fractional p(x, .)-Laplacian.In a few last years, to our best knowledge, there have been some mathematicians extending the study of classical exponent variable case to include fractional case (see for instance [2, 6-11, 13, 14, 17, 23, 29, 32]), the authors established some definitions and basic properties about new fractional Sobolev spaces with variable exponents and obtained some existence results for nonlocal fractional problems.
Motivated by the papers mentioned above and the results introduced in [1,22] and the references therein, we aim to discuss the existence of a nontrivial solutions for a fractional (p 1 (x, .),p 1 (x, .))-Kirchhofftype problem with homogeneous Dirichlet boundary data of the following form (x,y)  |x − y| N +spi(x,y) dxdy (−∆) where: for any x ∈ Ω, and s ∈ (0, 1).
• Here, for i = 1, 2, the operator (−∆) s pi(x,.) is the fractional p i (x, .)-Laplaciandefined as follows with p.v. is a commonly used abbreviation in the principal value sense.
, is a Kirchhoff function with the following assumptions (K 0 ) : M i : R + −→ R + is a differentiable function and there exists (K 1 ) : For i = 1, 2, there exists α i ∈ (0, 1) such that Note that (K 1 ) implies that where c 1 is a positive constant and q ∈ C + (Ω) such that 1 < q − ⩽ q(x) < (p max ) * s (x) for any x ∈ Ω (see Notation 2.1 and Section 2).
Condition (f 2 ) originates in the study of L. Jeanjean [28] in the case p(x, y) ≡ 2 for the Laplacian equation.This condition is crucial to obtain the compactness condition of the Palais-Smale or Cerami type for an elliptic equation in the whole space R N .In that context, these results for superlinear problems in bounded domains have been initially investigated by Miyagaki and Souto [34], Motivated by this work, many authors studied the existence of nontrivial solutions for nonlinear elliptic problems under the following condition: In our study, we suppose that the nonlinearity f satisfies the condition (f 2 ) instead of the well-known Ambrosetti-Rabinowitz (AR) type condition: In fact, we can take θ = 1 But, in general, (AR) does not imply (f 2 ), (see [38,Example 3.4]).Now, it is worth mentioning that (−∆) s p(x,.) is a nonlocal pseudo-differential operator of elliptic type which can be seen as a generalization of the fractional p-Laplacian operator (−∆) s p in the constant exponent case (i.e., when p(x, y) = p = constant).On the other hand, we remark that the above expression is the fractional version of the well-known p(x)-Laplacian operator (where p(x) = p(x, x)) which is associated with the variable exponent Sobolev space.
For the Kirchhoff function M , a typical prototype is due to Kirchhoff in 1883, and it is given by and when M (t) > 0 for all t ⩾ 0, Kirchhoff problems are said to be nondegenerate and this happens for example if a > 0 and b ⩾ 0 in the model case (1.1).Otherwise, if M (0) = 0 and M (t) > 0 for all t > 0, the Kirchhoff problems are called degenerate and this occurs in the model case (1.1) when a = 0 and b > 0.
One typical feature of problem (P s Mi ) is the nonlocality, in the sense that the value of (−∆) s p(x,.)u(x) at any point x ∈ Ω depends not only on the values of u on Ω, but actually on the entire space R N .Moreover, the presence of the functions M i , i = 1, 2, which implies that the first equation in (P s Mi ) is no longer a pointwise equation, it is no longer a pointwise identity, Therefore, the Dirichlet datum is given in R N \ Ω (which is different from the classical case of the p(x)-Laplacian) and not simply on ∂Ω.Hence, it is often called nonlocal problem.This causes some mathematical difficulties which make the study of such a problem particularly interesting.
As far as we know, there is no work that deals with a nonlocal problem involving fractional (p 1 (x, .),p 2 (x, .))-Laplacianoperator except [23] in which the authors considered problem (P s Mi ) for the case M 1 = M 2 ≡ 1 and they established some existence results for the problem with indefinite weights in an appropriate space of functions by means of variational techniques and Ekeland's variational principle.Moreover, in [6], using mountain pass theorem, the authors studied the existence of weak solutions for a quasilinear elliptic system involving the fractional (p(x, .),q(x, .))-Laplacianoperators.Very recently, the authors in [12] studied the equation (−∆) s p(x,.)u(x) = f (x, u(x)) without assuming the (AR) type condition.Therefore, without this condition it becomes a very difficult task to get the compactness condition.That is why, to our best knowledge, the present studied anisotropic Kirchhoff type problem is the first contribution in this direction.The purpose of this work is to improve the results of the above-mentioned papers.So, using the weaker assumption (f 2 ) instead of (AR)-condition and some variant min-max theorem, we overcome these difficulties and we prove the existence and multiplicity of weak solutions for problem (P s Mi ).Hence, our main results can be stated as follows.Theorem 1.1.Assume that the assumptions (K 0 ), (K 1 ) and (f 0 )-(f 3 ) hold.If p + max < q − , then problem (P s Mi ) has at least one nontrivial solution.Theorem 1.2.Assume that (K 0 ), (K 1 ), and (f 0 )-(f 3 ) are satisfied.Moreover, we suppose that If q − > p + max , then problem (P s Mi ) has a sequence of weak solutions {±u k } ∞ k=1 such that J (±u k ) −→ +∞ as k → +∞.The rest of this paper is organized as follows: In section 2, we give some definitions and fundamental properties of generalized Lebesgue spaces and fractional Sobolev spaces with variable exponent.In section 3, we discuss the existence of nontrivial weak solutions of problem (P s Mi ) by means of mountain pass theorem with Cerami condition.Furthermore, using Fountain theorem, we show that problem (P s Mi ) has infinitely many (pairs) of solutions with unbounded energy.As a conclusion, we extend all our results directly to the fractional multi p(x, .)-Laplaciancase.Moreover, in order to illustrate our results, we consider a particular example of the Kirchhoff functions M i and the nonlinearity f .

Variatoinal setting and preliminary results
For the reader's convenience, we briefly review the definitions and list some useful properties of the generalized Lebesgue spaces.Furthermore, we recall some qualitative properties of the fractional Sobolev spaces with variable exponent and several important properties of fractional p(x, .)-Laplacianoperator.

Variable exponent Lebesgue spaces
In this subsection, we give some basic results of variable exponent Lebesgue spaces L q(•) (Ω).For more details, we refer the reader to [25,30] and the references therein.Consider the set For all q ∈ C + (Ω), we define For any q ∈ C + (Ω), we define the variable exponent Lebesgue space as This vector space endowed with the Luxemburg norm, which is defined by dx ⩽ 1 is a separable reflexive Banach space.
Let q ∈ C + (Ω) be the conjugate exponent of q, that is, Then we have the following Hölder-type inequality.
A very important role in manipulating the generalized Lebesgue spaces with variable exponent is played by the modular of the L q(•) (Ω) space, which is defined by

Fractional Sobolev spaces with variable exponent
In this subsection, we present some preliminary results and basic properties of fractional Sobolev spaces with variable exponent that were introduced in [10].For a deeper treatment on these spaces, we refer the reader to [9,14,29].
Let Ω be a Lipschitz bounded open set in R N .We denote by Q the set Let p : Q −→ (1, +∞) be a continuous bounded function, we assume that Throughout this paper, s is a fixed real number such that 0 < s < 1.
We would like to mention that a continuous and compact embedding theorem is proved in [29] under the assumption q(x) > p(x) = p(x, x).The authors in [9] give a slightly different version of continuous compact embedding theorem assuming that q(x) = p(x) = p(x, x), in this case the space E becomes y) |x − y| N +sp(x,y) dxdy < +∞, for some λ > 0 .
Then, there exists a constant C = C(N, s, p, r, Ω) > 0 such that, for any u ∈ W , That is, the space W is continuously embedded in L r(x) (Ω).Moreover, this embedding is compact.
In [10], we compared the spaces W and X, and we established the compact and continuous embedding of X into Lebesgue spaces with variable exponent.
Then, there exists a constant That is, the space X is continuously embedded in L r(x) (Ω).Moreover, this embedding is compact.
(i) The assertion (iii) in Theorem 2.2 remains true if we replace X by X 0 .
Let denote by L the operator associated to the (−∆ p(x,.) ) s defined as where ., .denotes the usual duality between X 0 and its dual space X * 0 .Lemma 2.3 ( [14]).Assume that assumptions (2.2) and (2.3) are satisfied.Then, the following assertions hold: (i) L is a bounded and strictly monotone operator.
(iii) L is a homeomorphism.
In order to facilitate the investigation of problem (P s Mi ), the following notations are required.
for any x ∈ Ω.
It is easy to see that pmax , pmin ∈ C + (Ω) and p max , p min ∈ C + (Q).For simplicity, we set E = W s,pmax(x,y) (Ω) and It is clear that E and E 0 are separable and reflexive Banach spaces under the norms For i = 1, 2, we denote by ρ 0 pi(x,y) the modular on W s,pi(x,y) 0 (Q) defined by Remark 2.2.Obviously, from Theorem 2.2, for any q ∈ C + (Ω) such that q(x) < (p max ) * s (x) for all x ∈ Ω, we have E → L q(x) (Ω), and this embedding is continuous and compact.Moreover, by Remark 2.1-(i), this result remains true if we replace E by E 0 .Now, we give the definition of the Cerami condition (C) which is introduced by Cerami in [20].Definition 2.2.Let X be a Banach space and J ∈ C 1 (X, R).Given c ∈ R, we say that Φ satisfies the Cerami c condition (we denote condition (C c )), if Note that condition (C) is weaker than the Palais-Smale condition.However, it was shown in [15] that from condition (C) it is possible to obtain a deformation lemma, which is fundamental in order to get some min-max theorems.More precisely, let us recall the following version of the mountain pass lemma with Cerami condition which will be used in the sequel.
Proposition 2.3.Assume that X is a separable Banach space, Φ ∈ C 1 (X, R) is an even functional satisfying the Cerami condition (C).Moreover, for each k = 1, 2, . . ., there exist R k > r k > 0 such that Then, Φ has a sequence of critical values which tends to +∞.

Proof of Existence and multiplicity results
By a weak solution for (P s Mi ), we mean a function u ∈ E 0 such that for all ϕ ∈ E 0 , where |x − y| N +spi(x,y) dxdy.In this case, the weak formulation (3.1) is the Euler-Lagrange equation of the energy functional J : E 0 −→ R defined by Standard arguments (see, for instance [9, Lemma 3.1]) and the continuity of M i , i = 1, 2, imply that J is well defined and J ∈ C 1 (E 0 , R).Moreover, for all u, ϕ ∈ E 0 , its Gateaux derivative is given by Thus, the weak solutions of (P s Mi ) coincide with the critical points of J .

Compactness Cerami condition for the functional J
In this subsection, we establish the following compactness result which plays the most important role in this chapter.
Proof.We first show that J satisfies the assertion (C 1 ) of Cerami condition (C c ) (see Definition 2.2).Indeed, for all c ∈ R, let {u n } ⊂ E 0 be a bounded sequence such that Since E 0 is a reflexive space, then without loss of generality, we can assume that u n u in E 0 , which implies that Thus, we have On the other hand, by (f 0 ) and Hölder's inequality in Lemma 2.1, we have where 1 q(x) + 1 q(x) = 1.Hence, as 1 < q − ⩽ q(x) < (p max ) * s (x) for all x ∈ Ω, we have that E 0 is compactly embedded in L q(x) (Ω).It follows that (3.4) Besides this, since 1 < p − i ⩽ pi (x) < (p max ) * s (x), i = 1, 2, for all x ∈ Ω, then E 0 is compactly embedded in L pi(x) (Ω), for i = 1, 2. So, again by Hölder's inequality in Lemma 2.1, we get Using ( K 1 ), for i = 1, 2, we can easily obtain that Hence, from (K 0 ), it follows that Since {u n } ⊂ E 0 and u ∈ E 0 , by Lemma 2.2, we deduce that M i σ pi(x,y) (u n ) and M i σ pi(x,y) (u) are bounded.Thence, by assumption (K 0 ), we get Now, since u n u in E 0 , using (3.2), we get Hence, by the same argument as before, we deduce that Combining (3.6) and (3.7), we conclude that From Lemma 2.3-(iii), L is a mapping of type (S + ), and since L is a sum of two operators of type (S + ).Then, by [26, Lemma 6.8-(b)], L is also of type (S + ).Hence L is a mapping of type (S + ).
Next, we show that J satisfies the assertion (C 2 ) of Cerami condition (C c ) (see Definition 2.2), we argue by contradiction.Indeed, we assume that there exists c ∈ R and {u n } ⊂ E 0 such that From (3.8), it is easy to see that Hence, for a subsequence of {ϕ n }, still denoted by {ϕ n }, and ϕ ∈ E 0 , we get ϕ n ϕ in E 0 , (3.10) where q is given in assumption (f 0 ).
• If ϕ = 0, as in [28], we can define a sequence {t n } ⊂ R such that If there is a number of t n satisfying the above equality, one choose one of them.Fix (3.11), we get From (f 0 ), we have Hence, by the continuity of t −→ F (., t), we have Therefore, Then, for n large enough, 2Ap ∈ (0, 1), using (K 0 ), Lemma 2.2, Proposition 2.1, and Remark 2.2, we obtain Since J (0) = 0 and J (u n ) −→ n→+∞ c, then From (3.9) and (f 2 ), we have Now, we consider the following function For i = 1, 2, using (K 1 ), we obtain This fact implies that dθ i (t) dt ⩾ 0 for all t ⩾ 0, i = 1, 2. Hence t → θ i (t) is increasing.
Thus, by (f 1 ), we get Using ( K 1 ), for i = 1, 2, we can easily deduce that where ci is a positive constant.Since J Proposition 2.1, and the continuous embedding of E 0 into L pi(x) (Ω), i = 1, 2, we deduce via the Fatou lemma that c1 + c2 (p − min ) By (3.17), we obtain a contradiction.

Existence of weak solution via mountain pass theorem
By means of mountain pas theorem with Cerami condition given in Proposition 2.2, we establish the first main result of this paper which is an existence theorem for problem (P s Mi ) as stated in Theorem 1.1.Proof of Theorem 1.1.By Lemma 3.1, J satisfies the Cerami condition (C c ) in E 0 .To apply Proposition 2.2, we will show that J possesses the mountain pass geometry.
• Firstly, we claim that there exist R, a > 0 such that J (u) ⩾ a for any u ∈ E 0 with u E0 = R. (3.19)Indeed, Since pi (x) < (p max ) * s (x) for any x ∈ Ω, from Remark 2.2, we have that E 0 embedded in L pi(x) (Ω), that is, there exist ci > 0, i = 1, 2, such that Moreover, as p + max , q(x) < (p max ) * s (x) for any x ∈ Ω, then there exist c 2 , c 3 > 0 such that .
• Secondly, we affirm that there exists u 0 ∈ E 0 \ B R6 (0) such that In fact, from (f 1 ), we choose a constant B > Let l > 1 be large enough, by the above inequality and (3.18), we have Hence, as Consequently, there exist l 0 > 1 and u 0 = l 0 ϕ 0 ∈ E 0 \ B R (0) such that (3.23) hold true.Hence, in the light of mountain pass theorem with Cerami condition (Proposition 2.2), we deduce that J has at least one nontrivial critical value, that is, problem (P s Mi ) has at least one nontrivial solution.This completes the proof.□

Infinitely many solutions for problem (P s M i )
In this subsection, we provide a multiplicity result for problem (P s Mi ).The main tools used here is the Fountain theorem with Cerami condition (see, Proposition 2.3).
Since E 0 is a separable and reflexive Banach space, from [39, Section 17, Theorems 2-3], there exist To establish the proof of the above result, we need the following auxiliary lemma.
Lemma 3.2.Let r ∈ C + (Ω) such that r(x) < (p max ) * s (x) for any x ∈ Ω, define Since E 0 is a reflexive space, so {u k } has a weakly convergent subsequence, which we still denoted by {u k }, we suppose that u k u.We claim that u = 0.In fact, for any e Now, we are ready to prove Theorem 1.2.
Proof of Theorem 1.2.We show that J verifies the assumptions of Fountain theorem given in Proposition 2.3.Indeed, from condition (f 4 ), J is an even function.By Lemma 3.1, J satisfies conditon (C c ) .Next, we verify that (Φ 1 ) and (Φ 2 ) in Theorem 2.3 are satisfied.
The assertion (Φ 1 ) is verified.Hence, for R k large enough (R k > r k ), from the above fact, we conclude that max {u∈Yk:∥u∥E 0 =R k } J (u) ≤ 0, which implies that the assertion (Φ 2 ) is verified.Consequently, by the Fountain theorem, we achieve the proof of Theorem 1.2.□

A multi fractional p(x, .)-Laplacian Kirchhoff type problem
We could extend all our results directly to the fractional multi p(x, .)-Laplaciancase by considering the fractional (p 1 (x, .),p 2 (x, .),• • • , p n (x, .))-Laplacianproblem of the following form ✠ As a particular case of Kirchhoff functions M i , we consider where α = min{α 1 , α 2 }.Note that the above function does not satisfy (AR).But it is easy to see that it is satisfies (f 0 )-(f 4 ).In this case problem (P n s ) becomes It is clear that • M i (t) ⩾ a i for all t ⩾ 0, i = 1, • • • , n.
If we take in (K 1 ), α i = 1 2 , for all i = 1, . . ., n.It follows that M i satisfies the assumptions (K 0 ) and (K 1 ).Therefore, the results obtained in Theorems 1.1 and 1.2 stay true for problem (P s,n K ).The problem and results are all new.