Multiplicity of Solutions for Variable-order Fractional Kirchhoff Problem With Singular Term

. In this paper, we consider a class of singular variable-order fractional Kirchhoﬀ problem of the form:

In recent years, much attention has been devoted to nonlinear elliptic equations with singularities because of their wide applications to physical models such as non-Newtonian fluids, boundary layer phenomena for viscous fluids, chemical heterogeneous, etc, see for example [11,14,21,28].Fractional Kirchhoff problems, p-fractional Kirchhoff problems and the variableorder fractional Kirchhoff problems with singularities has been developed very quickly and the investigation for the existence and multiplicity results attracted a considerable attention of researchers.We refer interested readers to [9,15,16,17,18,19,20,23,24,25,27].It is well known that, compared with the fractional Kirchhoff problem , the variable-order fractional Kirchhoff problem possesses more complicated nonlinear properties.This causes many problems and some classical theories and methods, such as the Lagrange multiplier Theorem, will fail in this new situation.We point out that, this kind of problems have been the subject of a large literature and many results have been obtained.For the details, we refer to some articles [12,13,22,26].M.Xiang, D. Hu, B. Zhang and Y. Wang in [26], studied the following variable-order fractional Kirchhoff problem: By using the Z 2 -mountain pass theorem combined with the theory of the generalized Lebesgue Sobolev spaces, the authors prove that problem (1.2) admits at least two distinct nonnegative solutions.
Compared with the fractional Kirchhoff problem without singularities, the fractional Kirchhoff problem with singularities possesses more complicated nonlinear properties.This causes many problems and some classical theories and methods, such as the mountain pass Theorem, Z 2 -mountain pass theorem, will fail in this new situation.More recently, A. Ghanmi, M. Kratou, K. Saoudi and D. Repovs in [9], considered in the following nonlocal p-Kirchhoff equation with singularities.
where Ω ⊂ R N , is a bounded domain, (−∆) s p is the nonlocal fractional p-Laplace operator, [u] s(.) is the Gagliardo seminorm, λ is a non-negative parameter ,N > sp, p > 1 and 0 < s < 1 < σ < p * s , with p * s = N p N −ps .By using variational techniques with a truncation argument, the authors prove the existence and the multiplicity of positive solutions to the problem (1.3).
Inspired by the papers above mentioned , we studied the variable-order singular fractional Kirchhoff problem (1.1).
(H 3 ) a is a positive function with: The main result of this paper is the following.Theorem 1.1.Under assumptions (H 1 )−(H 3 ), there exists λ 0 > 0 such that for all λ ∈ (0, λ 0 ), the problem (1.1) has at least two nonnegative solutions.
Remark 1.1.To our knowledge, we note that this is the first paper that studies the multiplicity of solutions for fractional Kirchhoff problems with singular term and variable-order with Nehari manifold approach.
The context of the paper is organized as follows.In Section 2, we recall some basic facts about the weighted variable exponent Lesbesgue and Sobolev spaces.In Section 3, we prove our main result.
For any µ ∈ C + (Ω) , we write, Define the variable exponent Lebesgue space by With the norm is a Banach space, separable and reflexive if and only if 1 The modular on the space L µ(x) (Ω) is defined by the mapping and it satisfies the following proposition.Proposition 2.2.(See [7,8]) For all u ∈ L µ(x) (Ω), we have, (1) Let Ω be a nonempty open subset of R N and s(.
be a measurable function such that, there exist two constants 0 < s 0 < s 1 < 1 and for all (x, y If we equip H s(.) (Ω) with the norm , then (H s(.) (Ω), ||u|| Ω ) became a Banach space, (See [26]).Let H s(.) 0 be the linear space of Lebesgue measurable functions from R N to R such that any function u = 0 ∈ R N \ Ω and belongs to L 2 (Ω) satisfies 0 (Ω) can be continuously into L p (Ω).Now, we give a compact embedding Theorem.
Theorem 2.1.(See [26]) Let Ω ⊂ R N be a smooth bounded domain.Assume that s and q are functions satisfying (H 1 ) and (H 2 ) respectively.Then, there exists C q = C(N, q + , s + , s − ) > 0, such that for any u ∈ H s(.) (Ω), we have this embedding is compact and then,

Proof of Theorem 1.1
In this section, we will prove the main result of this paper.Denote X the space H s(.) 0 (Ω).We associated to the problem (1.1) the functional Ψ λ defined by |u| q(x) dx.
(1) From (H 1 )−(H 3 ), Ψ λ is differentiable but not C 1 (X, R), due to the singular term and for all u, v ∈ X, we have where Under hypothesis (H 2 ), 2σ < q − ≤ q(x), then the functional Ψ λ is not bounded on X, but in the next we will prove that the functional Ψ λ is bounded on Nehari manifold N , defined by where the function φ λ,u : [0, ∞[→ R, defined by: φ λ,u (t) = Ψ λ (tu), and Then, we have Lemma 3.1.Under assumptions (H 1 ) − (H 3 ), Ψ λ is coercive and bounded from below on N .
Proof.From Hölder inequality and hypothesis (H 3 ) we have, .
(1) Let u ∈ N , by (3.1), we have (2) Since Ψ λ is bounded from below on N , then Ψ λ is bounded from below on N + , N − and N 0 .
(2) Ψ λ is bounded from below on N + and N − .
We note In the next, we will prove that Ψ λ has a nonnegative minimizer in N + and has a nonnegative minimizer in N − .
Proof.Ψ λ is bounded on N , particularly on N + , then, there exists a minimizing sequence {u n } on N + , such that, Ψ λ (u n ) → m + λ .From Lemma 3.1, we concluded that {u n } is bounded on reflexive space X.So, there exist {u n } and u 1 in X such that Next, we will prove that u n → u 1 in X.We suppose that u n u 1 in X, then by Brezis Lemma (See [3]), we have Now, by dominated convergence theorem and (3.2), we get For u n ∈ N + , we have By (3.5), we have From Lemma 3.4, there exists t u 1 > 0 such that t u 1 u 1 ∈ N + and φ ′ λ,u 1 (t u 1 ) = 0.By dominated convergence theorem, we get Clearly t u 1 u 1 is a minimizer of since t u 1 u 1 ∈ N + , this yields a contradiction.Thus, u n → u 1 in X.Using (3.5) and lemma 3.4, we have Nextly, we will prove that , and we get If we suppose that Then φ ′ λ,|u 1 | (1) < 0. So, by lemma 3.4, there exists t |u 1 | > 0 such that Clearly which is absurd.Thus Proposition 3.2.If λ ∈ (0, λ 0 ), there exists a nonnegative minimizer u 2 ∈ N − of Ψ λ .
Proof.Ψ λ is bounded below of N and in N − .Hence, there exists a bounded minimizing sequence {u n } ⊂ N − , such that, Ψ λ (u n ) → m − λ .Since Ψ λ is coercive on N , then {u n } is bounded on reflexive space X.So, there exist {u n } and u 2 in X such that Next, we prove that u n → u 2 in X. Arguing by contradiction, we assume that u n u 2 in X.Then For u n ∈ N − , we have From Lemma 3.4, there exists t u 2 > 0 such that t u 2 u 2 ∈ N − and φ ′ λ,u 2 (t u 2 ) = 0.By dominated convergence theorem and since u n ∈ N − , we get Also, we observe that the function Z(t) = Ψ λ (tu n ) for any t > 0 attains its maximum at t = 1.Then which is absurd.Then (u n ) → X.Moreover, by lemma 3.4 and 3.2, we deduce that u 2 ∈ N − .Finaly, u 2 is a minimizer of Ψ λ on X.Similar to Proposition 3.1, we get that |u 2 | is also a minimizer of Ψ λ on X.
Since the functional Ψ λ is not C 1 , then the existence of minimizer of Ψ λ on N + or on N − does not imply the existence of critical points of Ψ λ .So we will use a technical lemma which based on the implicit function theorem.Lemma 3.5.Assume that (H 1 )−(H 3 ) and u ∈ N ± .Then there exist ǫ > 0, and continuous function α : Proof.The proof will be give for u ∈ N + .Let f : X × R → R a function defined by f (v, t) = φ We conclude that u 1 ∈ N + is a nontrivial weak solution to the problem (1.1).Now, the prove of u 2 ∈ N − , is a weak solution of problem (1.1) follows immediately by using Lemma 3.5, Proposition 3.2 and the same reasoning as the previous one.Finally, since N + ∩ N − = ∅, we conclude that problem (1.1) has at least two nontrivial solutions.
Data Availability Statement: No data were produced for this paper.Conflicts of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest.Funding: No funding.Contribution The authors R. Chammem , A. Sahbani and A. Saidani wrote and revised this paper.