A nonlinear boundary value problem for fourth-order elastic beam equations
© Song; licensee Springer. 2014
Received: 25 February 2014
Accepted: 25 July 2014
Published: 23 September 2014
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.
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 , .
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 –. 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 , , – and the references therein. More precisely, in , 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 .
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
- (a)For every and every ; the restriction of the functional
If ; then for each , the following alternative holds: either
(b1) possesses a global minimum, or
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.
the mapping is measurable for every ;
the mapping is continuous for almost every ;
- (c)for every there exists a function such that
for almost every .
holds for any .
3 Main results
where α, β are given by (A1), c is a positive constant, and , are given by (A3).
Letbe an-Carathéodory function and. Assume that
(A2) for all;
wherewhen, for everyproblem (1.1) has an unbounded sequence of weak solutions in E.
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.
and the conclusion is achieved. □
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 .
has an unbounded sequence of weak solutions for everyin E.
has infinitely many distinct weak solutions in E.
Letbe an-Carathéodory function and. Moreover, assume that (A2) and
(A1)′: for alland;
wherewhen, for everyproblem (1.1) has a sequence of weak solutions, which strongly converges to zero in E.
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. □
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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