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Infinitely many solutions for a fourth-order differential equation on a nonlinear elastic foundation
Boundary Value Problems volume 2013, Article number: 258 (2013)
In this paper, existence results of infinitely many solutions for a fourth-order differential equation with nonlinear boundary conditions are established. The proof is based on variational methods. Some recent results are improved and extended.
In this paper, we consider a beam equation with nonlinear boundary conditions of the type
where and are real functions. 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. The derivation of the model can be found in [1, 2].
Owing to the importance of fourth-order two-point boundary value problems in describing a large class of elastic deflection, there is a wide literature that deals with the existence and multiplicity results for such a problem with different boundary conditions (see, for instance, [3–8] and the references therein).
Motivated by the above works, in the present paper we study the existence of infinitely many solutions for problem (1.1) when the nonlinear term satisfies the superlinear condition and sublinear condition at the infinity on u, respectively. As far as we know, this case has never before been considered.
Now we state our main results.
1.1 The superlinear case
We give the following assumptions.
() g is odd and satisfies
() There exist constants and such that
() uniformly for , where .
() , as uniformly for .
() There exist constants , , and such that for every and with ,
Theorem 1.1 Assume that ()-() hold and F is even in u. Then problem (1.1) has infinitely many solutions.
Remark 1.1 There exist some functions satisfying ()-(), but not satisfying the well-known (AR)-condition,
for some .
For example, take . Then . Obviously, ()-() are satisfied. Note that
for L being large enough, which implies (). However, it is easy to see that f does not satisfy (AR)-condition.
1.2 The sublinear case
We make the following assumptions.
() g is odd and satisfies for any .
() There exist constants and such that
() for any .
() There are constants and with such that
() There exist constants and such that
Theorem 1.2 Assume that ()-() hold and F is even in u. Then problem (1.1) has infinitely many solutions.
Remark 1.2 The condition () implies that .
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
In this section, the following two theorems will be needed in our argument. Let E be a Banach space with the norm and with for any . Set , and , for . Consider the -functional defined by
(C1) maps bounded sets to bounded sets uniformly for . Furthermore, for all .
(C2) for all ; or as ; or
for all ; as .
For , define ,
Theorem 2.1 ([, Theorem 2.1])
Assume that (C1) and (C2) (or ) hold. If for all , then for all . Moreover, for a.e. , there exists a sequence such that , and as .
Theorem 2.2 ([, Theorem 2.2])
Suppose that (C1) holds. Furthermore, we assume that the following conditions hold:
(D1) ; as on any finite dimensional subspace of E.
(D2) There exist such that for all and as uniformly for .
Then there exist , such that , as . In particular, if has a convergent subsequence for every k, then has infinitely many nontrivial critical points satisfying as .
Now we begin describing the variational formulation of problem (1.1), which is based on the function space
where is the Sobolev space of all functions such that u and its distributional derivative are absolutely continuous and belongs to . Then E is a Hilbert space equipped with the inner product and the norm
where denotes the standard norm. In addition, E is compactly embedded in the spaces and , and therefore, there exist immersion constants such that
Next, we consider the functional defined by
where F, G are the primitives
Since f, g are continuous, we deduce that J is of class and its derivative is given by
for all . Then we can infer that is a critical point of J if and only if it is a (classical) solution of problem (1.1).
Now we define a class of functionals on E by
It is easy to know that for all and the critical points of correspond to the weak solutions of problem (1.1). We choose a completely orthonormal basis of E and define . Then , can be defined as that at the beginning of Section 2.
3 Proofs of Theorems 1.1 and 1.2
We will prove Theorem 1.1 by using Theorem 2.1. Firstly, we give the following three useful lemmas.
Lemma 3.1 Under the assumptions of Theorem 1.1, there exists large enough such that for all .
Proof Let , then there exists such that
Otherwise, for any positive integer n, there exists such that
for all k. Set , then and
for all k. Since , it follows from the compactness of the unit sphere of that there exists a subsequence, say , such that converges to some in . Hence, we have . By the equivalence of the norms on the finite-dimensional space , we have in , i.e.,
Thus there exist such that
In fact, if not, we have
for all positive integer n. This implies that
as , which gives a contradiction. Therefore, (3.4) holds.
and . By (3.2) and (3.4), we have
for all positive integer n. Let n be large enough such that and . Then we have
This implies that
for all large n, which is a contradiction with (3.3). Therefore, (3.1) holds.
For any , let . By condition (), for , there exists such that
Hence one has
for all with . It follows from ()-() and (3.1) that
for all with . Since , for large enough, we have . □
Lemma 3.2 Under the assumptions of Theorem 1.1, there exist , such that for all .
Proof Set . Then as . Indeed, it is clear that , so that , as . For every , there exists such that and . By the definition of , in E. Then it implies that in . Thus we have proved that . By (), we have
By (), for any , there exists such that
Therefore, there exists such that
Hence, for any , choose , by () and (3.5), we have
Let . Then, for any with , we have
uniformly for λ as . □
Lemma 3.3 Under the assumptions of Theorem 1.1, there exist as , such that , , where .
Proof It is easy to verify that (C1) and (C2) of Theorem 2.1 hold. By Lemmas 3.1, 3.2 and Theorem 2.1, we can obtain the result. □
Proof of Theorem 1.1 For the sake of notational simplicity, in what follows we always set for all . By Lemma 3.3, it suffices to prove that is bounded and possesses a strong convergent subsequence in E. If not, passing to a subsequence if necessary, we assume that as . In view of (), there exists such that
and combining (), we have
This implies that
Note that from (), . Let , then
By (), there exists such that
By (2.5), () and the Hölder inequality, one has
where is a constant independent of n. By (3.8) we obtain
combining this inequality with (3.6) and (3.7) yields that
as . Combining this with (3.9), we have
since . This is a contradiction. Therefore, is bounded in E. Without loss of generality, we may assume in E. Then in . Note that
Taking , we have , which means that in E and . Hence, has a critical point with . Consequently, we obtain infinitely many solutions since . □
Lemma 3.4 Under the assumptions of Theorem 1.2, there exists small enough such that and as uniformly for .
Proof For any , by using defined in Lemma 3.2, together with () and (), we have
for all with . Obviously, as . So and as uniformly for . □
Lemma 3.5 Under the assumptions of Theorem 1.2, there exists small enough such that for all .
Proof For any , by ()-() and the equivalence of the norms on the finite-dimensional space , we have
Since , for small enough, we can get for all . □
Proof of Theorem 1.2 It is easy to verify that (C1) and (D1) hold under the assumptions of Theorem 1.2. By Lemmas 3.4 and 3.5, the condition (D2) is also satisfied. Therefore, by Theorem 2.2 there exist , such that , as . In the following we show that is bounded. Indeed, note that
for some . Since , (3.10) yields that is bounded in E. By a standard argument, this yields a critical point of such that . Since as , we can obtain infinitely many critical points. □
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The author would like to express her sincere thanks to the referees for their helpful comments.
The author declares that she has no competing interests.
The author read and approved the final manuscript.