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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 620438, 10 pages
http://dx.doi.org/10.1155/2010/620438
Research Article

Infinitely Many Periodic Solutions for Nonautonomous Sublinear Second-Order Hamiltonian Systems

1Department of Mathematics, ZunYi Normal College, ZunYi 563002, China
2Department of Mathematics, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 6 February 2010; Accepted 7 June 2010

Academic Editor: Jean Mawhin

Copyright © 2010 Peng Zhang and Chun-Lei Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Two sequences of distinct periodic solutions for second-order Hamiltonian systems with sublinear nonlinearity are obtained by using the minimax methods. One sequence of solutions is local minimum points of functional, and the other is minimax type critical points of functional. We do not assume any symmetry condition on nonlinearity.

1. Introduction and Main Result

We are interested in the following second order Hamiltonian systems: where and satisfies the following assumption. is measurable in for each and continuously differentiable in for a.e. for and there exist such that

for all and .

Then the corresponding functional on given by

is continuously differentiable and weakly lower semicontinuous on , where

is a Hilbert space with the norm defined by

for (see [1]). Moreover,

for all . It is well known that the solutions of problem (1.1) correspond to the critical points of .

There are large number of papers that deal with multiplicity results for this problem. Infinitely many solutions for problem (1.1) are obtained in [24], where the symmetry assumption on the nonlinearity has played an important role. In recent years, many authors have paid much attention to weaken the symmetry condition, and some existence results on periodic and subharmonic solutions have been obtained without the symmetry condition (see [57]). Particularly, Ma and Zhang [6] got the existence of a sequence of distinct periodic solutions under some superquadratic and asymptotic quadratic cases. Faraci and Livrea [7] studied the existence of infinitely many periodic solutions under the assumption that is a suitable oscillating behaviour either at infinity or at zero.

In this paper, we suppose that the nonlinearity is sublinear, that is, there exist and such that for all and . We establish some multiplicity results for problem (1.1) under different assumptions on the potential . Roughly speaking, we assume that has a suitable oscillating behaviour at infinity. Two sequences of distinct periodic solutions are obtained by using the minimax methods. One sequence of solutions is local minimum points of functional, and the other is minimax type critical points of functional. In particular, we do not assume any symmetry condition at all.

Our main result is the following theorem.

Theorem 1.1. Suppose that satisfies assumptions (A) and (1.7). Assume that Then,(i)there exists a sequence of periodic solutions which are minimax type critical points of functional , and , as ;(ii)there exists another sequence of periodic solutions which are local minimum points of functional , and , as .

2. Proof of Theorems

For , let and . Then one has

Lemma 2.1. Let be the subspace of given by Suppose that (1.7) holds. Then as in .

Proof. It follows from (1.7) and Sobolev's inequality that for all in . By Wirtinger's inequality, the norm is an equivalent norm on . Hence the lemma follows from the equivalence and the above inequality.

Lemma 2.2. Suppose that (1.7) and (1.8) hold. Then there exists positive real sequence such that

The proof of this lemma is similar to the following lemma.

Lemma 2.3. Suppose that (1.7) and (1.9) hold. Then there exists positive real sequence such that where .

Proof. For any , let , where . It follows from (1.7) and Sobolev's inequality that for all . Hence we have for all . As if and only if , then the lemma follows from (1.9) and the above inequality.

Now we give the proof Theorem 1.1.

Proof of Theorem 1.1. Let be a ball in with radius . Define We claim that each intersects the hyperplane . In fact, let be the projection of onto , defined by For , define Then is a homotopy of with Moreover, for all . By homotopy invariance and normalization of the degree, we have which means that . Thus intersects the hyperplane .
By Lemma 2.1, the functional is coercive on . There is a constant such that Hence . In view of Lemma 2.2, for all large values of , For such , there exists a sequence in such that Applying Theorem  4.3 and Corollary  4.3 in [1], there exists a sequence in such that as .
Now, let us prove that the sequence is bounded in . For any large enough , we can find , such that For fixed , by Lemma 2.3, we can choose such that and cannot intersect the hyperplane . Let , where and . Then we have for large enough. Besides, by Sobolev's inequality and (1.7), it is obvious that As is an equivalent norm in , it follows that is bounded. Hence, is bounded. Also is bounded in .
We assume that hence as . Moreover, an easy computation shows that so as . Then, it is not difficult to obtain , as .
Now we have Thus, is critical point and is critical value of functional . For any , if , intersects the hyperplane . It follows that This inequality and Lemma 2.3 deduce that The first result of Theorem 1.1 is obtained.
For fixed , define the subset of by For , we have It follows that is bounded below on .Define and let be a minimizing sequence in , that is, From (2.29), is bounded in . Then, there is a subsequence, we also denoted by , such that The case that is a convex closed subset of implies that .As is weakly lower semicontinuous, we have Since , Suppose that is in the interior of , then is a local minimum of functional . In fact, let . For large , from Lemmas 2.2 and 2.3, we have , which means that is not on the boundary of .
Finally, as is a minimum of in , It follows from Lemma 2.2 that Therefore Theorem 1.1 is proved.

Acknowledgments

This work is supported by NNSF of China under Grant 10771173 and Natural Science Foundation of Education of Guizhou Province under Grant 2008067.

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