On the multiplicity of solutions for a kind of fourth-order equation depending on two real parameters

where λ,μ ∈ ]0,∞[, g : R → R is a continuous function and f : [0, 1] × R → R is an L1Caratéodory function. Problem (1.1) describes the static equilibrium of a flexible elastic beam of length 1 when, along its length, a load f is added to cause deformation. Precisely, conditions u(0) = u′(0) = 0 mean that the left end of the beam is fixed and conditions u′′(1) = 0, u′′′(1) = μg(u(1)) mean that the right end of the beam is attached to a bearing device, given by the function g . Existence and multiplicity of solutions for fourth-order boundary value problems have been discussed by several authors in the last decades; see for example [1, 4, 5, 9–13, 16, 17, 19–21] and the references therein. In particular, Yang et al. [21] used Ricceri’s variational principle [18] to establish the existence of at least two classical solutions generated from g for problem (1.1) with μ = 1.


Introduction
In this paper, we consider the following fourth-order problem: mean that the right end of the beam is attached to a bearing device, given by the function g.
Existence and multiplicity of solutions for fourth-order boundary value problems have been discussed by several authors in the last decades; see for example [1, 4, 5, 9-13, 16, 17, 19-21] and the references therein.
In particular, Yang et al. [21] used Ricceri's variational principle [18] to establish the existence of at least two classical solutions generated from g for problem (1.1) with μ = 1.
The authors in [9], using a multiplicity result by Cabada and Iannizzotto [8], ensured the existence of at least two nontrivial classical solutions for the problem where the functions f : [0, 1] × R → R and g : R → R are continuous and λ ≥ 0 is a real parameter.
Bonanno et al. [4], by means of an abstract critical points result of Bonanno [2], studied the existence of at least one nonzero classical solution for problem (1.1).
In [12], by using a smooth version of [7, Theorem 2.1], Heidarkhani et al. established the existence of infinitely many generalized solutions for the following perturbed fourth-order problem: where λ > 0, μ ≥ 0 are two parameters, f , g are two L 2 -Caratéodory functions, and p, h are Lipschitz continuous functions such that p(0) = h(0) = 0. Also in [11], the present authors obtained sufficient conditions to guarantee that problem (1.1) has infinitely many classical solutions.
More recently, Heidarkhani and Gharehgazlouei [13], using an immediate consequence of [3,Theorem 3.3], ensured the existence of at least three generalized solutions for the problem where λ > 0, μ ≥ 0 are two parameters, f : [0, 1] × R → R is an L 1 -Caratéodory function, g : R → R is a nonnegative continuous function, and h : R → R is a Lipschitz continuous function such that h(0) = 0.
Motivated by the above works, the aim of the present paper is to offer the existence of three solutions for fourth-order problem (1.1) by using two kinds of critical point theorems obtained in [3,6].
For completeness, we cite the recent and nice works [14,15] as general references on the subject treated in this paper.

Abstract setting
In order to study problem (1.1), the variational setting is the space where H 2 ([0, 1]) is the Sobolev space of all function u : [0, 1] → R such that u and its distributional derivative u are absolutely continuous and u belongs to L 2 ([0, 1]). X is a Hilbert space with the inner product and the corresponding norm We observe that the norm · on X is equivalent to the usual norm It is well known that the embedding X → C 1 ([0, 1]) is compact and for all u ∈ X (see [21]). We say that u ∈ X is a weak solution of problem (1.1) whenever Our main tools are critical point theorems that we recall here in a convenient form. The first result has been obtained in [6], and it is a more precise version of Theorem 3.2 of [3]. The second one has been established in [3]. Theorem 3.6]) Let X be a reflexive real Banach space; : X → R be a coercive, continuously Gâteaux differentiable, and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X * ; : X → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Assume that there exist r > 0 and x ∈ X, with r < (x), such that (a 1 ) Then, for each λ ∈ r , the functionalλ has at least three distinct critical points in X.

Lemma 2.2 ([3, Theorem 3.3])
Let X be a reflexive real Banach space; : X → R be a convex, coercive, and continuously Gâteaux differentiable functional whose derivative admits a continuous inverse on X * ; : X → R be a continuously Gâteaux differentiable functional whose derivative is compact such that Assume that there are two positive constants r 1 , r 2 and x ∈ X, with 2r 1 < (x) < r 2 2 , such that and for every x 1 , x 2 ∈ X, which are local minima for the functionalλ and such that ( We use the following notations: Corresponding to f , g, we introduce the functions F, G as follows: for all x ∈ [0, 1] and ξ ∈ R. Also, for each θ and η of positive real numbers, define

Main results
In this section, we present our main result on the existence of at least three weak solutions for problem (1.1). In order to introduce our result, we fix θ , η > 0 such that where we read r/0 = +∞. For instance, δ = +∞ when lim sup |ξ |→+∞ With the above notations we are able to prove the following multiplicity property.
It is well known that is a differentiable functional whose differential at the point u ∈ X is for every v ∈ X. Moreover, in [21], the authors proved that is strongly continuous on X, which implies that is a compact operator. Furthermore, by standard arguments, is coercive and continuously differentiable whose differential at the point u ∈ X is for each v ∈ X. Also, in [21] it is proved that admits a continuous inverse on X * . Moreover, is sequentially weakly lower semicontinuous. One can show that the weak solutions of problem (1.1) are exactly the solutions of the equation (u)λ (u) = 0.
Fix λ ∈ and put r = θ 2 2 . Then, for u ∈ X with (u) ≤ r, Therefore, From this, if G θ = 0, it is clear that while if G θ > 0, it turns out to be true bearing in mind that Denote by w the function of X defined by x ∈ [ 3 4 , 1].
Proof Without loss of generality, we can assume f (x, t) ≥ 0 for every (x, t) ∈ [0, 1] × R. Fix λ, μ, and g as in the conclusion and take and as in the proof of Theorem 3.1. Arguing as in the proof of Theorem 3.1, we observe that the regularity assumptions of Lemma 2.2 on and are satisfied. Then, our aim is to verify (b 1 ) and (b 2 ).
This implies that (b 1 ) and (b 2 ) of Lemma 2.2 are verified. Finally, we verify that assumption (b 3 ) of Lemma 2.2 holds. Let u 1 and u 2 be two local minima forλ . Then, u 1 and u 2 are critical points forλ , and so they are weak solutions for problem (1.1). We claim that the weak solutions obtained are nonnegative. Indeed, if u 0 is a weak solution of problem (1.1), then one has for all v ∈ X. Arguing by a contradiction, assume that the set A := {x ∈ [0, 1] : u 0 (x) < 0} is nonempty and of positive measure. Put v 0 := min{0, u 0 }. Clearly, v 0 ∈ X. So, taking into account that u 0 is a weak solution and by choosing v = v 0 , from our sign assumptions on the data, one has