Abstract
By using an infinitely many critical points theorem, we study the existence of infinitely many solutions for a fourth-order nonlinear boundary value problem, depending on two real parameters. No symmetric condition on the nonlinear term is assumed. Some recent results are improved and extended.
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1 Introduction
In this paper, we consider a beam equation with nonlinear boundary conditions of the type:
where λ, μ are two positive parameters, f, h are two -Carathéodory functions, and is real function. This kind of problem arises in the study of deflections of elastic beams on nonlinear elastic foundations. The problem has the following physical description: a thin flexible elastic beam of length 1 is clamped at its left end and resting on an elastic device at its right end , which is given by g. Then the problem models the static equilibrium of the beam under a load, along its length, characterized by f and h. The derivation of the model can be found in [1], [2].
Fourth-order boundary value problems modeling bending equilibria of elastic beams have been considered in several papers. Most of them are concerned with nonlinear equations with null boundary conditions; see [3]–[6]. When the boundary conditions are nonzero or nonlinear, fourth-order equations can model beams resting on elastic bearings located in their extremities; see for instance [1], [2], [7]–[11] and the references therein. More precisely, in [10], using variant fountain theorems, the author obtains the existence of infinitely many solutions for problem (1.1) with and under the symmetric condition and some other suitable assumptions of the nonlinear term f.
Motivated by the above works, in the present paper we establish some multiplicity results for problem (1.1) under rather different assumptions on the functions f, h and g. It is worth noticing that in our results neither the symmetric nor the monotonic condition on the nonlinear term is assumed. We require that f has a suitable oscillating behavior either at infinity or at zero. In the first case, we obtain an unbounded sequence of solutions (Theorem 3.1); in the second case, we obtain a sequence of nonzero solutions strongly converging at zero (Theorem 3.4), which improve and extend the results in [10].
The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proofs of our main results.
2 Variational setting and preliminaries
We prove our results applying the following smooth version of Theorem 2.1 of Bonanno and Bisci [12], which is a more precise version of Ricceri’s variational principle [13], Lemma 2.5].
Theorem 2.1
Let E be a reflexive real Banach space, letbe two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous and coercive, and Ψ is sequentially weakly upper semicontinuous. For every, let
Then the following properties hold:
-
(a)
For every and every ; the restriction of the functional
toadmits a global minimum, which is a critical point (local minimum) ofin E.
-
(b)
If ; then for each , the following alternative holds: either
(b1) possesses a global minimum, or
(b2) there is a sequenceof critical points (local minima) ofsuch that
-
(c)
If ; then for each , the following alternative holds: either
(c1) there is a global minimum of Φ which is a local minimum of, or
(c2) there is a sequenceof pairwise distinct critical points (local minima) ofthat converges weakly to a global minimum of Φ.
Let E be the Hilbert space
with the inner product and norm
where is the Sobolev space of all functions such that u and its distributional derivative are absolutely continuous and belongs to , and denotes the standard norm. In addition, E is compactly embedded in the spaces and , and therefore, there exist immersion constants , such that
We recall that is an -Carathéodory function if
-
(a)
the mapping is measurable for every ;
-
(b)
the mapping is continuous for almost every ;
-
(c)
for every there exists a function such that
for almost every .
Define the functions as follows:
for all , and . Thus we define the functional by
Definition 2.1
We say that a function is a weak solution of problem (1.1) if
holds for any .
3 Main results
In this section we establish the main abstract results of this paper. Let
and
where α, β are given by (A1), c is a positive constant, and , are given by (A3).
Theorem 3.1
Letbe an-Carathéodory function and. Assume that
(A1) there exist constantsandsuch that
(A2) for all;
(A3) there exist two functionsandsatisfying
and
such that
Then, for everyand for any-Carathéodory function, whose potentialfor all, is a nonnegative function satisfying the condition
if we put
wherewhen, for everyproblem (1.1) has an unbounded sequence of weak solutions in E.
Proof
Obviously, it follows from (A3) that . Fix . Since , we have
Now fix and set
Let the functionals be defined by
where . Put
Using the property of f, h and the continuity of g, we obtain and for any , we have
and
So, with standard arguments, we deduce that the critical points of the functional are the weak solutions of problem (1.1) and so they are classical. We first observe that the functionals Φ and Ψ satisfy the regularity assumptions of Theorem 2.1.
First of all, we show that . Let be a sequence of positive numbers such that and
Set for all . Then, for all with , taking (2.2) into account, one has . Note that . Then, for all ,
Since , from the assumption (A3) and the condition (3.2), we have
and combining the assumption , we obtain
This implies that
Let be fixed. We claim that the functional is unbounded from below. Since
there exist a sequence of positive numbers and such that and
for each large enough. For all we define by
From the condition (A3), it is easy to verify that . For any , one has
On the other hand, by (A2) and since H is nonnegative, from the definition of Ψ, we infer
for every large enough. Since , and , we have
Then the functional is unbounded from below, and it follows that has no global minimum. Therefore, by Theorem 2.1(b), there exists a sequence of critical points of such that
and the conclusion is achieved. □
Remark 3.1
Indeed, it is not difficult to find such functions and satisfying the condition (A3). For example, let and . We can choose
and
Remark 3.2
Under the conditions and , from Theorem 3.1 we see that for every and for each , problem (1.1) admits a sequence of classical solutions which is unbounded in E. Moreover, if , the result holds for every and .
Corollary 3.2
Letbe an-Carathéodory function and. Suppose that hypotheses (A1)-(A2) hold. Moreover, the condition (A3) is satisfied if (3.1) is replaced by
Then, for any-Carathéodory function, whose potentialfor all, is a nonnegative function satisfying the condition (3.2), if we put
wherewhen, the problem
has an unbounded sequence of weak solutions for everyin E.
Corollary 3.3
Under the assumptions of Corollary 3.2, for any nonnegative continuous function, the problem
has infinitely many distinct weak solutions in E.
Now, let
and
Theorem 3.4
Letbe an-Carathéodory function and. Moreover, assume that (A2) and
(A1)′: for alland;
(A3)′: there exist two functionsandsatisfying
and
such that
are satisfied. Then, for everyand for any-Carathéodory function, whose potentialfor all, is a nonnegative function satisfying the condition
if we put
wherewhen, for everyproblem (1.1) has a sequence of weak solutions, which strongly converges to zero in E.
Proof
It follows from (A3)′ that . Fix . Since , we have
Now fix and set
We take Φ, Ψ, and as in the proof of Theorem 3.1. Now, as has been pointed out before, the functionals Φ and Ψ satisfy the regularity assumptions required in Theorem 2.1. As first step, we will prove that . Let be a sequence of positive numbers such that and
By the fact that and the definition of δ, we have . Set for all . Then, for all with , taking (2.2) into account, one has . Note that . Then, for all ,
It follows from (A1)′ that . Then we have
From , we obtain
which implies that
Let be fixed. We claim that the functional does not have a local minimum at zero. Since
there exist a sequence of positive numbers and such that and
for each large enough. For all , let be defined by (3.3) with the above . Note that . Then, since for all and , we obtain
for every large enough. Thus, since
we see that zero is not a local minimum of . This, together with the fact that zero is the only global minimum of Φ, we deduce that the energy functional does not have a local minimum at the unique global minimum of Φ. Therefore, by Theorem 2.1(c), there exists a sequence of critical points of , which converges weakly to zero. In view of the fact that the embedding is compact, we know that the critical points converge strongly to zero, and the proof is complete. □
Remark 3.3
Applications similar to Corollaries 3.2 and 3.3 can also be made to Theorem 3.4. Now we give an example illustrating Theorem 3.4. Consider the problem
where . Obviously, . Let and , and choose
and
By calculating, we have . Thus, and . Furthermore, the conditions (A2) and (A3)′ are satisfied. Let . Then (A1)′ holds. Therefore, by Theorem 3.4, we find that problem (3.5) has a sequence of weak solutions which strongly converges to zero in E for all .
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Song, Y. A nonlinear boundary value problem for fourth-order elastic beam equations. Bound Value Probl 2014, 191 (2014). https://doi.org/10.1186/s13661-014-0191-6
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DOI: https://doi.org/10.1186/s13661-014-0191-6