New class of sixth-order nonhomogeneous $p(x)$-Kirchhoff problems with sign-changing weight functions

We prove the existence of multiple solutions for the following sixth-order $p(x)$-Kirchhoff-type problem: $-M(\int_\Omega \frac{1}{p(x)}|\nabla \Delta u|^{p(x)}dx)\Delta^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) \ \ \mbox{on} \ \Omega$ and $ \ u=\Delta u=\Delta^2 u=0 \ \ \mbox{on} \ \partial\Omega,$ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N>3$, $\Delta_{p(x)}^3u = \operatorname{div}\Big(\Delta(|\nabla \Delta u|^{p(x)-2}\nabla \Delta u)\Big)$ is the $p(x)$-triharmonic operator, $p,q,r \in C(\overline\Omega)$, $10$, $\lambda>0$, $g: \Omega \times \mathbb{R} \to \mathbb{R}$ is a nonnegative continuous function while $f,h : \Omega \times \mathbb{R} \to \mathbb{R}$ are sign-changing continuous functions in $\Omega$. To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order $p(x)$-Kirchhoff type problems with sign changing Kirchhoff functions.


Introduction
Let Ω ⊂ R N be a smooth bounded domain and N > 3. This paper deals mainly with the following sixth-order p(x)-Kirchhoff-type problem in Ω, u = ∆u = ∆ 2 u = 0, on ∂Ω, where p, q, r ∈ C(Ω), 1 < p(x) < N 3 for all x ∈ Ω, M(s) = a−bs γ , a, b, γ > 0, λ > 0, g : Ω×R → R is a nonnegative continuous function, f, h : Ω × R → R are assumed to be continuous functions which may change sign in Ω, and p(x) u := div ∆(|∇∆u| p(x)−2 ∇∆u) is the p(x)-triharmonic operator which is not homogeneous and is related to the variable exponent Lebesgue space L p(x) (Ω) and the variable exponent Sobolev space W 1,p(x) (Ω). These facts imply some difficulties. For example, some classical theories and methods, including the Lagrange multiplier theorem and the theory of Sobolev spaces cannot be applied.
Such problems are called nonlocal problems because of the presence of the function M, which implies that the equation contains an integral over Ω, and is no longer pointwise identity. This causes some mathematical difficulties which make the study of such a problem particularly interesting. We call (1.1) a sixth order Kirchhoff type equation because it is related to the stationary analog of the equation where ρ, p 0 , h, E, L are constants which represent some physical meanings respectively. Eq. (1.2) extends the classical D'Alembert wave equation by considering the effects of the changes in the length of the strings during the vibrations. This kind of nonlocal problem also appears in other fields, for example in nonlinear elasticity theory and in modelling electrorheological fluids [39,40] and from the study of electromagnetism and elastic mechanics [25,43], and raises many difficult mathematical problems. After this pioneering models, many other applications of differential operators with variable exponents have appeared in a large range of fields, such as image restoration and image processing [8,29]. We refer the reader to [1,21,34] for an overview of references on this subject.
Throughout this paper, unless otherwise stated, we shall always assume that exponent p(x) is continuous on Ω with and p * (x) denotes the critical variable exponent related to p(x), defined for all x ∈ Ω by the . In the following, we denote by [W 3,p(·) 0 (Ω)] ′ the dual space (Ω) and q * (x) = p * (x) p * (x) − 1 the conjugate exponent of p * (x). Now, we introduce some conditions for problem (1.1) as follows: where η, µ are small positive numbers. .

Furthermore, there exists non-empty open domain
In recent years, great attention has been paid to the study of Kirchhoff problems. This brought new mathematical difficulties that made the study of Kirchhoff type equations particularly interesting. A typical prototype for M, due to Kirchhoff in 1883, is given by In particularly, Chen-Kuo-Wu in [7] studied the following semilinear boundary problem and proved by the Nehari manifold and fibering maps, the existence of multiple positive where Ω is a smooth bounded domain in R N , with 1 < q < 2 < p < 2 * = 2N N − 2 and the parameters a, b, λ > 0. The functions f (x), g(x) ∈ C(Ω) may change sign on Ω.
The study of Kirchhoff type equations has already been extended to the case involving the p-Laplacian operator of the following form (1.4) Chen-Huang-Liu [6] studied the nonhomogeneous case of (1.4) (that is h( and h(x) are continuous functions which may change sign on Ω. The parameters p, q, r satisfy 1 < q < p(k + 1) < r < p * = N p N − p . Using the Mountain pass theorem and Ekeland's variational principle, they showed that problem (1.4) has at least two positive solutions when λ is small enough.
For h(x) ≡ 0, Huang-Chen-Xiu [27] studied problem (1.4) where M(s) = a + bs k , 1 < q < p < r < p * , and proved that the problem has at least one positive solution when r > p(k + 1) and the functions f (x), g(x) are nonnegative. Motivated by [27], Li-Mei-Zhang [30] considered M(s) = a + bs, 1 < q < p < r ≤ p * and they proved the existence of multiple nontrivial nonnegative solutions by using the Nehari manifold when the weight functions f (x), g(x) change their signs (see also [22]).
However, many papers generalized the constant case to include the p(x)-Laplacian operator, e.g., in [9], using variational methods, we investigated a nonlocal p(x)-Laplacian Dirichlet problem and we showed via the mountain pass theorem combined with the Ekeland variational principle the existence of at least two distinct, non-trivial weak solutions in the case that where λ is a parameter and a(x), b(x), α(x) and β(x) satisfy suitable hypotheses and under some suitable conditions on M.
Recently, Hamdani et al. in [24] studied (1.5) when M(s) = a − bs and K(x, u) = λ|u| p(x)−2 u + g(x, u), where λ is a real parameter, a, b > 0 are constants and g is a continuous function satisfies the classical (AR) condition.
then the authors proved the existence and multiplicity results via the Mountain Pass theorem and the Fountain theorem. Also for further studies on this subject, we refer the reader to [5,10,11,13,15,31,35,42].
The problems involving p(x)−biharmonic operators have been widely investigated. For example, via the Mountain pass theorem, El Amrouss et al. [18] obtained the existence and multiplicity of solutions for a class of p(x)−biharmonic equation of the form where Ω is a bounded domain in R N , with smooth boundary ∂Ω, λ ≤ 0 and f satisfies the (AR) condition. On the other hand, similar variational methods are also used to study the p(x)-biharmonic operator. For example, see [2,3,4,12,32,33,44] and [14,23,26] for a general Kirchhoff problems with or without the (AR) condition. However, in literature the only result involving a sixth-order problem like (1.1) by using variational methods can be found in [38]. Motivated by the above papers, we consider the problem (1.1) with the new general nonlocal term M(s) = a − bs γ , a, b, γ > 0 and with sign-changing weight functions which presents interesting difficulties to discus the existence of multiple solutions. Before stating our results, we give the definition of weak solutions for problem (1.1). For this purpose, we denote by X the space (Ω) and define the norm . X of X by the formula It is well known that if 1 < p − ≤ p + < ∞ then (X, . X ) is a separable and reflexive Banach space. Moreover, u X and |∇∆u| p(·) are two equivalent norms on X, see [19,20,28].
The main results of this paper are the following: for all x ∈ Ω and assume that the conditions (H 1 ) and (H ′ 2 ) hold. Then there exists λ > 0 such that, for each λ ∈ (0, λ), problem (1.1) admits at least two nontrivial weak solutions in X.
The paper is organized as follows. In Section 2, we give the notations and recall some useful lemmas concerning the variable exponent Lebesgue and Sobolev spaces. In Section 3, we give some lemmas which are important for the proofs of our main results. In Section 4, we prove Theorem 1.1 (we omit the proof of Theorem 1.2 since it is very similar).

Variable exponent Lebesgue and Sobolev spaces
For the convenience of the reader, we recall in this section some results concerning spaces L p(·) (Ω) and W r,p(·) (Ω) which we call generalized Lebesgue-Sobolev spaces. Denote For any p(x) ∈ C + (Ω), we introduce the variable exponent Lebesgue space endowed with the so-called Luxemburg norm which is a separable and reflexive Banach space. A thorough variational analysis of the problems with variable exponents has been developed in the monograph by Rȃdulescu and Repovš [37] (we refer the reader also to [16,28]).
In the light of the variational structure of (1.1), we look for critical points of the associated Euler-Lagrange functional J : X → R defined as Note that J is a C 1 (X, R) functional and for any v ∈ X. Thus, critical points of J are weak solutions of (1.1).

Some Lemmas
In order to prove our main result -Theorem 1.1 -we need to apply the Mountain pass theorem and the Ekeland variational principle. We first prove the following lemmas. . Then there exist λ, δ, ρ, α > 0 such that for λ ∈ (0, λ) and |h| p * (·) p * (·)−1 < δ, we have J(u) ≥ α for all u ∈ X with u X = ρ. Moreover, there exists e ∈ X with e X > ρ, such that J(e) < 0.
Step 1. We first prove that {u n } is bounded in X. Arguing by contradiction, if {u n } is unbounded in X, up to a subsequence, we may assume that u n X → ∞ as n → ∞. Let θ be a fixed positive constant such that Then according to the conditions (H 1 ) and (H 2 ), for n large enough, we have From (3.5), it follows that which is a contradiction since u n X → ∞ as n → ∞. So, {u n } is bounded in X and the first assertion is proved.
Step 2. Now, we prove that {u n } has a convergent subsequence in X. Indeed, by Proposition 2.2, the embedding X ֒→ L s(·) (Ω) is compact, where 1 ≤ s(x) < p(x) * . Since X is a reflexive Banach space, passing if necessary, to a subsequence, there exists u ∈ X such that u n ⇀ u in X, u n → u in L s(·) (Ω), u n (x) → u(x), a.e. in Ω.
(3.6) From (2.2), we find that Meanwhile, by Hölder's inequality and (3.6) we estimate where We can easily verify that So, thanks to (3.6) we can deduce that |u n − u| θ 1 (·) → 0 as n → ∞. (3.9) Combining this and the fact that {u n } is bounded in X, we infer from (3.8) and (3.9) that Similarly, we obtain So, from (3.10) and (3.11), we can deduce that (3.7) implies Since {u n } is bounded in X, passing to a subsequence, if necessary, we may assume that when n → ∞ then If t 0 = 0 then {u n } converges strongly to u = 0 in X and the proof is finished. Otherwise, we need to consider the following two cases: → 0 is not true and no subsequence of → 0} converges to zero. Therefore, there exists δ > 0 such that We define Then It follows that To complete the argument we need the following lemma.
By the fundamental lemma of the variational method (see [41]) it follows that u = 0. Thus Hence, we can deduce that This is a contradiction since J(u n ) → c < γa γ+1 γ is not true and similarly to Case 1, we have that So, it follows from the two cases above that Ω |∇∆u n | p(x)−2 ∇∆u n (∇∆u n − ∇∆u) dx → 0.
Applying (S + ) mapping theory (see [14] for r = 3), we can now deduce that u n X → u X as n → ∞, which means that J satisfies the (PS ) c condition. This completes the proof.