Multiple solutions for nonlocal elliptic problems driven by p(x)-biharmonic operator

In this article, we study the existence of at least three distinct weak solutions for nonlocal elliptic problems involving $ p(x) $-biharmonic operator. The results are obtained by means of variational methods. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results.


Introduction
In this paper we study the existence of at least three distinct weak solutions for the following problem        T (u) = λ f (x, u(x)), in Ω, where , is an open bounded domain with smooth boundary, ∆ 2 p(x) u is the operator defined as ∆(|∆u| p(x)−2 ∆u) and is called the p(x)-biharmonic which is a generalization of the p-biharmonic, p(x) ∈ C(Ω), ρ(x) ∈ L ∞ (Ω), M : [0, +∞) → R is a continuous function such that there are two positive constants m 0 and m 1 with m 0 ≤ M(t) ≤ m 1 for all t ≥ 0, N 2 < p − := ess inf x∈Ω p(x) ≤ p + := ess sup x∈Ω p(x) < ∞, λ > 0 and f : Ω × R → R is an L 1 -Carathéodory function. The Kirchhoff equation refers back to Kirchhoff [18] in 1883 in the study on the oscillations of stretched strings and plates, suggested as an extended version of the classical D'Alembert's wave equation by taking into account the effects of the changes in the length of the string during the vibrations. Kirchhoff's equation like problem (P f λ ) model several physical and biological systems where u describes a process which depend on the average of itself. Lions in [23] has proposed an abstract framework for the Kirchhoff-type equations. After the work by Lions, various problems of Kirchhoff-type have been widely investigated, we refer the reader to the papers [7,24,27] and the references therein.
The main interest in studying problem (P f λ ) is given by the presence of the variable exponent p(·). Problems involving such kind of growth conditions benefited by a special attention in the last decade since they can model with sufficient accuracy phenomena arising in different branches of science. Two important models where operators involving variable exponents were considered come from the study of electrorheological fluids [8,28] and elastic mechanics [34].
Fourth-order equations have various applications in areas of applied mathematics and physics such as micro-electro-mechanical systems, phase field models of multi-phase systems, thin film theory, thin plate theory, surface diffusion on solids, interface dynamics, flow in Hele-Shaw cells (see [4,6,26]). The fourth-order equation can also describe the static form change of beam or the sport of rigid body. In [22], Lazer and Mckenna have pointed out that this type of nonlinearity furnishes a model to study travelling waves in suspension bridges. Numerous authors investigated the existence and multiplicity of solutions for the problems involving p(x)-biharmonic operators. We refer to [10,12,16,19,21,30,31]. In the last decade, Kirchhoff type equations involving the p(x)-Laplacian have been investigated, for instance see [3, 9, 13-15, 17, 25].
In this paper, we are interested to discuss the existence of at least three distinct weak solutions for problem (P f λ ). No asymptotic condition at infinity is required on the nonlinear term. In Theorem 3.1 we establish the existence of at least three distinct weak solutions for problem (P f λ ). Theorem 3.3 is a consequence of Theorem 3.1. As a consequence of Theorem 3.3, we obtain Theorem 3.4 for the autonomous case. We present example 3.5 to illustrate Theorem 3.4.
Then for each λ ∈ 1 β(r 1 , r 2 ) , the functional Φ − λΨ admits three distinct critical points We refer the interested reader to the papers [2,11,20] in which Theorem 2.1 has been successfully used to ensure the existence of at least three solutions for boundary value problems.
Let Ω be a bounded domain of R N , denote: We can introduce the norm on L p(x) (Ω) by: Let X be the generalized Lebesgue-Sobolev space W m,p(x) (Ω) defined by putting W m,p(x) (Ω) as which is equipped with the norm: γ is the multi-index and |γ| is the order.
A bounded operator T : X → R is said to be compact if T (B X ) has compact closure in R.
, Ω ⊂ R is a bounded region, then X → C(Ω) is a compact embedding.
According to 2.3, for each u ∈ X, there exists a constant c > 0 that depends on p(·), N, Ω: (c) for every ε > 0 there exists a function l ε ∈ L 1 (Ω) such that for a.e. x ∈ Ω, Corresponding to the functions f and M, we introduce the functions F : Ω × R → R and M : [0, +∞) → R, respectively, as follows and For our convenience, set Proposition 2.6. Let T = Φ : X → X * be the operator defined by for every u, v ∈ X. Then T admits a continuous inverse on X * .
Proof. For any u ∈ X \ {0}, since p − > 1, it follows that the map T is coercive. Since T is the Fréchet derivative of Φ, it follows that T is continuous and bounded. Using the elementary inequality [29] |x which means that T is strictly monotone. Thus T is injective. Consequently, thanks to Minty-Browder theorem [33], the operator T is a surjection and admits an inverse mapping. Thus it is sufficient to show that T −1 is continuous. For this, let (v n ) ∞ n=1 be a sequence in X * such that v n → v in X * . Let u n and u in X such that By the coercivity of T , we conclude that the sequence (u n ) is bounded in the reflexive space X. For a subsequence, we have u n →ũ in X, which implies Therefore, by the continuity of T , we have Moreover, since T is an injection, we conclude that u =ũ.

Main results
Fix x 0 ∈ Ω and choose s > 0 such that B(x 0 , s) ⊂ Ω, where B(x 0 , s) denotes the ball with center at x 0 and radius of s. Put r N−1 dr, Γ denotes the Gamma function, and L := Θ 1 + Θ 2 + Θ 3 .
Theorem 3.1. Assume that there exist positive constants θ 1 , θ 2 , θ 3 and η ≥ 1 with Then for every problem (P f λ ) has at least three weak solutions u 1 , u 2 and u 3 such that Proof. Our goal is to apply Theorem 2.1 to the problem (P f λ ). We consider the auxiliary problem If a weak solution of the problem (Pf λ ) satisfies the condition −θ 3 ≤ u(x) ≤ θ 3 for every x ∈ Ω, then, clearly it turns to be also a weak solution of (P f λ ). Therefore, it is enough to show that our conclusion holds for (P f λ ). We define functionals Φ and Ψ as given in (2.3) and (2.4), respectively. Let us prove that the functionals Φ and Ψ satisfy the required conditions in Theorem 2.1. It is well known that Ψ is a differentiable functional whose differential at the point u ∈ X is for every v ∈ X, as well as it is sequentially weakly upper semicontinuous. Recalling (2.1), we have for all u ∈ X with u > 1, which implies Φ is coercive. Moreover, Φ is continuously differentiable whose differential at the point u ∈ X is for every v ∈ X, while Proposition 2.6 gives that Φ admits a continuous inverse on X * . Furthermore, Φ is sequentially weakly lower semicontinuous. Therefore, we observe that the regularity assumptions on Φ and Ψ, as requested of Theorem 2.1, are verified. Define w by setting It is easy to see that w ∈ X and, and It is easy to see that w ∈ X and, in particular, since In particular, one has On the other hand, bearing (A 1 ) in mind, from the definition of Ψ, we infer F(x, η)dx. Choose and θ 2 < θ 3 , we achieve r 1 < Φ(w) < r 2 and r 3 > 0. For all u ∈ X with Φ(u) < r 1 , taking (2.1) and (2.2) into account, one has u ≤ max (p + r 1 ) So, thanks to the embedding X → C 0 (Ω), one has u ∞ < θ 1 . From the definition of r 1 , it follows that Hence, one has As above, we can obtain that and sup Therefore, since 0 ∈ Φ −1 (−∞, r 1 ) and Φ(0) = Ψ(0) = 0, one has On the other hand, for each u ∈ Φ −1 (−∞, r 1 ) one has Due to (A 2 ) we get α(r 1 , r 2 , r 3 ) < β(r 1 , r 2 ).
Remark 3.2. If f is non-negative, then the weak solution ensured in Theorem 3.1 is non-negative. Indeed, let u 0 be the weak solution of the problem (P f λ ) ensured in Theorem 3.1, then u 0 is nonnegative. Arguing by a contradiction, assume that the set A = {x ∈ Ω : u 0 (x) < 0} is non-empty and of positive measure. Putv(x) = min {0, u 0 (x)} for all x ∈ Ω. Clearly,v ∈ X and one has for everyv ∈ X. Thus we have i.e., u 0 (A) ≤ 0 which contradicts with this fact that u 0 is a non-trivial weak solution. Hence, the set A is empty, and u 0 is positive.
Then for every problem (P f λ ) has at least three weak solutions u 1 , u 2 and u 3 such that Proof. Choose θ 2 = F(x, t)dx Moreover, since θ 1 < η p + p − , from (A 4 ) we have F(x, η)dx Hence, from (A 4 ), (3.2) and (3.3), it is easy to observe that the assumption (A 2 ) of Theorem 3.1 is satisfied, and it follows the conclusion.
The following result is a consequence of Theorem 3.3.
Then for every λ > λ where problem (P f λ ) possesses at least four distinct non-trivial solutions.