Existence results for a two point boundary value problem involving a fourth-order equation

We study the existence of non-zero solutions for a fourth-order differential equation with nonlinear boundary conditions which models beams on elastic foundations. The approach is based on variational methods. Some applications are illustrated.


Introduction
In this paper, we consider the following fourth-order problem = µg(u(1)), where f : [0, 1] × lR → lR is an L 1 -Carathéodory function, g : lR → lR is a continuous function and λ, µ are positive parameters.The problem (P λ,µ ) 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 results for this kinds of problems has been extensively studied.In particular, by using a variational approach, the existence of three solutions for the problems (P λ,1 ) and (P λ,λ ) has been established respectively in [6] and in [4].Moreover, in [8] the author obtained the existence of at least two positive solutions for the problem (P 1,1 ).Finally, we point out that the problem (P λ,µ ) can be also studied by iterative methods (see for instance [7]) 2 G. Bonanno, A. Chinnì and S. A. Terzian and, for fourth order equations subject to conditions of different type, we refer, for instance, to [3,5] and references therein.
In this paper we will deal with the existence of one non-zero solution for the problem (P λ,µ ).Precisely, using a variational approach, under conditions involving the antiderivatives of f and g, we will obtain two precise intervals of the parameters λ and µ for which the problem (P λ,µ ) admits at least one non-zero classical solution (see Theorem 3.1).As a way of example, we present here a special case of our results.Theorem 1.1.Let f : lR → lR be a nonnegative continuous function.Then, for each λ ∈ 0, We explicitly observe that in Theorem 1.1, assumptions on the behavior of f , as for instance asymptotic conditions at zero or at infinity, are not requested, whereby f is a totally arbitrary function.
The paper is arranged as follows.In Section 2, we recall some basic definitions and our main tool (Theorem 2.2), which is a local minimum theorem established in [1].Finally, Section 3 is devoted to our main results.Precisely, under a suitable behaviour of f and for parameters µ small enough, the existence of a non-zero solution for (P λ,µ ) is obtained (Theorem 3.1) and a variant is highlighted (Theorem 3.3).Moreover, some consequences are pointed out (Corollaries 3.4 and 3.5) and a concrete example of application is given (Example 3.7).

Basic definitions and preliminary results
We consider the space where H 2 ([0, 1]) is the Sobolev space of all functions u : [0, 1] → lR 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 inner product ) is compact (see [6]) and it results for each u ∈ X.We consider the functionals Φ, Ψ λ,µ : X → lR defined by for each u ∈ X and for each λ, µ > 0 where F(x, ξ) := ξ 0 f (x, t) dt and G(ξ) := ξ 0 g(t) dt for each x ∈ [0, 1], ξ ∈ lR.By standard arguments, Φ is sequentially weakly lower semicontinuous and coercive.Moreover, Φ and Ψ λ,µ are in C 1 (X) and their Fréchet derivatives are respectively for each u, v ∈ X.In [6] the authors proved that Φ admits a continuous inverse on X * and Ψ is compact.In particular, in Lemma 2.1 of [6] it has been shown that, for each λ, µ > 0, the critical points of the functional are solutions for problem (P λ,µ ).
In order to obtain solutions for the problem (P λ,µ ), we make use of a recent critical point result, where a novel type of Palais-Smale condition is applied (see Theorem 3.1 of [1]).We recall it.Definition 2.1.Let Φ and Ψ two continuously Gâteaux differentiable functionals defined on a real Banach space X and fix r ∈ lR.The functional I = Φ − Ψ is said to verify the Palais-Smale condition cut off upper at r (in short (P.S.) [r] ) if any sequence {u n } n∈IN in X such that has a convergent subsequence.
Then, for each ,

Existence of one solution
Before introducing the main result, we define some notation.With α ≥ 0, we put Then, for each and for each g : lR → lR continuous, there exists η λ,g > 0, where such that for each µ ∈]0, η λ,g [ the problem (P λ,µ ) admits at least one non-zero solution u λ such that u λ ∞ , u λ ∞ < γ.
Denote by v the function of X defined by for which it results Taking into account that v(x) ∈ [0, δ] for each x ∈ 3 8 , 3  4 , condition ( f 2 ) ensures that F(x, δ) dx ≥ 0, which implies This ensures that (3.4) For each u : Now, taking into account it results 2 In all cases, taking into account (3.4) and (3.5), we have Moreover, we observe that from δ < γ, taking ( f 1 ) into account, we obtain 8π 4 2 In fact, arguing by a contradiction, if we assume δ < γ ≤ 8π 4 2 3 3 δ, we obtain and this is an absurd by ( f 1 ).Therefore, we have Φ( v) = 4π 4 δ 2 2 3 3 < γ 2 2 = r and the condition (a 1 ) of Theorem 2.2 is verified.
Moreover, since , Theorem 2.2 guarantees the existence of a local minimum point u λ for the functional I λ such that 0 < Φ(u λ ) < r and so u λ is a nontrivial classical solution of problem (P λ,µ ) such that u λ ∞ , u λ ∞ < γ.
Remark 3.2.We observe that in Theorem 3.1 we read By reversing the roles of λ and µ, we obtain the following result.

Corollary 3 . 4 .F
Assume that f : lR → lR is a continuous and non negative function such that ( f 1 ) lim sup t→0 +