Existence Results for Perturbed Fourth-order Kirchhoff Type Elliptic Problems with Singular Term

where ∆pu := ∆(|∆u| ∆u) and, ∆pu := div(|∇u| ∇u) denote the p-biharmonic operator and the p-Laplacian operator, respectively, 1 < p < N2 , Ω ⊆ R N is an open bounded domain containing the origin in R , the boundary ∂Ω is smooth, and M : [0, +∞[→ R is a continuous function such that there are two positive constants m0 and m1 with m0 ≤ M(t) ≤ m1 for all t ≥ 0, and λ > 0, μ ≥ 0 and f : Ω × R → R is L-Carathéodory function. Biharmonic equations can describe the static form change of a beam or the sport of a rigid body. For example, this type of equation furnishes a model for studying traveling wave in suspension bridges (see [21]). Due to this, many researchers have discussed the existence of at least one solution, or multiple solutions, or even infinitely many solutions for fourth-order boundary value problems by using lower and upper solution methods, Morse theory, the mountain-pass theorem, constrained minimization and concentration-compactness principle, fixed-point theorems and degree theory, and variational methods and critical point theory, and we refer the reader to [1,5,7,8,9,10,13,16,17,18,19,20,23,24,25,26,27,30,34,36] and references therein. On the other hand, singular elliptic problems have been intensively studied in recent years, see for example, [3,14,15,22,28,29,31,35,37] and the references. Stationary problems involving singular nonlinearities, as well as the associated evolution equations, describe naturally several physical phenomena and applied economical models. For instance, nonlinear singular boundary value problems arise in the context


Introduction
The purpose of this paper is to establish the existence of multiple solutions for the following perturbed nonlocal fourth-order problem of Kirchhoff-type under Navier boundary condition where ∆ 2 p u := ∆(|∆u| p−2 ∆u) and, ∆ p u := div(|∇u| p−2 ∇u) denote the p-biharmonic operator and the p-Laplacian operator, respectively, 1 < p < N 2 , Ω ⊆ R N is an open bounded domain containing the origin in R N , the boundary ∂Ω is smooth, and 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, and λ > 0, µ ≥ 0 and f : Ω × R → R is L 1 -Carathéodory function.
Biharmonic equations can describe the static form change of a beam or the sport of a rigid body. For example, this type of equation furnishes a model for studying traveling wave in suspension bridges (see [21]). Due to this, many researchers have discussed the existence of at least one solution, or multiple solutions, or even infinitely many solutions for fourth-order boundary value problems by using lower and upper solution methods, Morse theory, the mountain-pass theorem, constrained minimization and concentration-compactness principle, fixed-point theorems and degree theory, and variational methods and critical point theory, and we refer the reader to [1,5,7,8,9,10,13,16,17,18,19,20,23,24,25,26,27,30,34,36] and references therein.
On the other hand, singular elliptic problems have been intensively studied in recent years, see for example, [3,14,15,22,28,29,31,35,37] and the references. Stationary problems involving singular nonlinearities, as well as the associated evolution equations, describe naturally several physical phenomena and applied economical models. For instance, nonlinear singular boundary value problems arise in the context Li in [22] considered the fourth order elliptic problem with Navier boundary conditions 4) and proved that, the problem (1.4) admits at least two distinct solutions. Our goal of this work is to show the existence three solutions and two solutions for the following p-harmonic equation: where Ω is bounded domaine in R N (N ≥ 5) containing the origin and with smooth boundary ∂Ω, 1 < p < N/2, and f : Ω × R −→ R is a carathéodory function such that for some non-negative constants a 1 , a 2 and q ∈]1, p * [, where Recall that a function f : is measurable for every y ∈ R; (C2) the function y −→ f (x, y) is continuous for a.e. x ∈ Ω.
The plan of the paper is as follows: Section 2 contains some preliminary lemmas. In Section 3, using of three critical points theorems obtained in [6] which we recall in the next section (Theorems 2.4) we ensure the existence of at least three weak solutions for the problem (1.1). Finally Section 4 contains our main results and their proofs to obtain the existence of at least two weak solutions for the problem (1.1).

Preliminaries
Here and in the sequel, X will denote the space W 2,p (Ω) W 1,p 0 (Ω). By the Hardy-Rellich inequality (see [11]), we know that where the best constant is Obviously, for any µ ∈ [0, H), this norm is equivalent to Ω (|∆u| p + |∇u| p )dx see, for instance, [33]. Fixing q ∈ [1, p * [, again from the Sobolev embedding theorem, there exists a positive constant c q such that u L q (Ω) ≤ c q u (∀u ∈ X) (2.4) and, in the particular, the embedding X ֒→ L q (Ω) is compact. Let us define the functionals Φ, Ψ : X −→ R by In this article, we assume that the following condition holds, Throughout the paper, denote  (1) Ψ ∈ C 1 (X, R) and for u, v in X, we have Then the Nemytskii operator properties implies that Ψ is a C 1 operator in L q (Ω). Since there is a continuous embedding of X into L q (Ω), the function Ψ is also C 1 in X and (2) It is enough to show that Ψ ′ is strongly continuous in X. Let {u n } ⊂ X be a sequence such that u n ⇀ u. Since, the embedding of X into L q (Ω) is compact, there exists a subsequence, noted also {u n }, such that u n → u in L q (Ω). According to the Krasnoselski's theorem, the Nemytskii operator Existence Results for Elliptic Problems with Singular Term . Then by Holder's inequality and embedding of X into L q (Ω), we have This completes the proof.
then T is coercive. Consequently, thanks to a Minty-Browder theorem [38], the operator T is surjection. For any x, y ∈ R N , we have the following elementary inequalities from which we can get the strictly monotonicity of T : where ., . denotes the usual inner product in R N , for every x, y ∈ R N . Indeed, for 1 < p < 2, it is easy to see that and for, p ≥ 2, we also observe that which means that T is strictly monotone. Thus T is injective and admits an inverse mapping. T −1 is continuous. Indeed, let {f n } be a sequence of X * such that f n −→ f in X * . Let u n and u in X such that By the coercivity of T , the sequence {u n } is bounded in the reflexive space X. This means that there exist a subsequence that we call again {u n }, such that u n ⇀û in X which implies Now we prove that T is a mapping of type (S + ), it follows that u n →û in X. (2.11) Indeed let u n ⇀ u in X and lim sup n→+∞ T (u n ) − T (u), u n − u ≤ 0. Since T is strictly monotone, then for every v ∈ X. Then u n → u in X (see Theorem 3.1 of [12]). So, T is a mapping of (S + ) type. On the other hand since T is the Fréchet derivative, it follows that T is continuous, thus from (2.11) we have, Hence, taking into account that T is an injection, we have u =û. This completes the proof.
To prove our main result in section 3, we use a three critical point theorem of [6]. We recall it in a convenient form.
Theorem 2.4 ( [6, Theorem 2.6]). Let X be a reflexive real Banach space, Φ : X → R be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X * , Ψ : X → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, such that Then, for each λ ∈ Λ r the functional Φ − λΨ has at least three distinct critical points in X.
Other toll is the following abstract result.  We say that a function u ∈ X is a (weak) solution of the problem (1.1) if

Existence of three weak solutions
In this section, we formulate our main results on the existence of at least three weak solutions for the problem (1.1).
Fix x 0 ∈ Ω and pick 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 where Γ denotes the Gamma function, and (3.1) We present our first existence result as follows. We recall that c q is the constant of the embedding W 1,p 0 ∩ W 2,p ֒→ L q (Ω) for each q ∈]p, p * [, and c 1 stands for c q with q=1.  .2)). Also let f : Ω × R −→ R is a Carathéodory function, satisfying condition (f 1 ). Moreover, assume that Then, for every λ ∈ Λ := Proof. In order to apply Theorem 2.4 to our problem, We introduce the functionals Φ, Ψ : X → R for each u ∈ X, as follows Now we show that the functionals Φ and Ψ satisfy the required conditions. We easily observe that Φ(0) = Ψ(0) = 0. By proposition 2.1 we know that Ψ is a differentiable functional whose differential at the point u ∈ X is the functional Ψ ′ (u) ∈ X * , given by for every v ∈ X, and Ψ ′ : X → X * is a compact operator. Moreover it is well known thatΨ is sequentially weakly upper semicontinuous, and Φ is continuously differentiable whose differential at the point u ∈ X is Φ ′ (u) ∈ X * , given by for every v ∈ X, while Proposition 2.3 gives that its Gâteaux derivative admits a continuous inverse on X * . Furthermore, Φ is sequentially weakly lower semicontinuous. Clearly, the weak solutions of the problem (1.1) are exactly the solutions of the equation It is easy to see that

5)
Existence Results for Elliptic Problems with Singular Term Due to (f 3 ), one has that so, thanks to (3.5) we get . On the other hand, due to (2.9), we get for every u ∈ X and Φ(u) < r. Now, from (2.4) and by using (3.8), one has for every u ∈ X such that Φ(u) < r. Hence and so condition (i) of Theorem 2.4 is verified. Now we prove that I λ is coercive. From (2.4) one has and so, for each u ∈ X with u ≥ max{1, 1 cγ }, from (f 4 ) and (2.9) we have where meas(Ω) denotes the Lebesgue measure of the open set Ω. Since γ < p, coercivity of I λ is obtained. Then, taking into account the fact that the weak solutions of the problem (1.1) are exactly critical points of the functional I λ , and we have the desired conclusion.

Remark 3.2.
We observe that, if f (x, 0) = 0, then by Theorem 3.1, we obtain the existence of at least three non-zero weak solutions. Example 3.3. The following function verifies the assumptions requested in Theorem 3.1. Let r > 1 be a real number and 1 < γ < p < q < p * . We consider the function f : Ω × R → R defined as where α : Ω → R be a Borel, bounded and positive function. condition (f 1 ) is easily verified. Taking into account that

Existence of two weak solutions
In this section, our goal is to obtain the existence of two distinct weak solutions for the problem (1.1).  .2)). Also let f : Ω × R → R be a Carathéodory function such that (f 1 ) holds. Moreover, assume that (f 5 ) there exist θ > p and t 0 > 0 such that for each x ∈ Ω and |t| ≥ t 0 .
Then, for each λ ∈]0, λ * [, the problem (1.1) admits at least two distinct weak solutions, where Proof. Our aim is to apply Theorem 2.5 to problem (1.1) in the case r = 1 to the space X = W 1,p 0 (Ω) ∩ W 2,p (Ω) and to the functional Φ, Ψ : X → R defined in the proof of Theorem 3.1. First we prove that I λ = Φ − λΨ satisfies (PS)-condition for every λ > 0. Namely, we will prove that any sequence {u n } ⊂ X satisfying I λ (u n ) → c, and contains a convergent subsequence. Due to (4.1), we can actually assume there is a constant C such that for every n. By (4.2) we can write Which of course implies that {u n } is bounded in X. By the Eberlian-Smulyan theorem, passing to a subsequence if necessary, we can assume that u n ⇀ u in X and u n → u in L q (Ω), so I Since Φ ′ verifies (S + ) condition, we have u n → u in X and so I λ satisfies (PS)-condition. From (f 5 ), by standard computations, there is a positive constant C such that for all x ∈ Ω and |t| > t 0 . In fact, setting a(x) := min |ζ|=t0 F (x, ζ) and Let g(t) = −(e −t + te −t ), then g(0) = −1 and g ′ (t) = te −t ≥ 0 for all t ≥ 0. Thus for all t ≥ 0 we havẽ Hence the condition (f 6 ) is satisfied. In view of Theorem 4.1, we have the following corollary. If we put θ = 3 and t 0 = 2 then (f 5 ) is satisfied too.