Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems
© Gu and An; licensee Springer. 2013
Received: 8 November 2012
Accepted: 10 January 2013
Published: 6 February 2013
In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous second-order Hamiltonian systems by using the variant fountain theorem.
- (A)is measurable in t for all , continuously differentiable in u for a.e. and there exist and such that
for all and a.e. .
Here and in the sequel, and always denote the standard inner product and the norm in respectively.
There have been many investigations on the existence and multiplicity of periodic solutions for Hamiltonian systems via the variational methods (see [1–7] and the references therein). In , Zhang and Liu studied the asymptotically quadratic case of under the following assumptions:
(AQ3) uniformly for .
They obtained the existence of infinitely many periodic solutions of (1.1) provided is even in u (see Theorem 1.1 of ).
The subquadratic condition (AQ1) is widely used in the investigation of nonlinear differential equations. This condition was weakened by some researchers; see, for example,  of Jiang and Tang. This paper considers the case of , then . Motivated by  and , we replace (AQ1) with the following condition:
for and a.e. .
Meanwhile, we weaken the condition (AQ3) to () as follows:
Then our main result is the following theorem.
Theorem 1.1 Assume that (), (AQ2), () hold and is even in u. Then (1.1) possesses infinitely many solutions.
Remark The conditions (AQ1) and (AQ3) are stronger than () and (). Then Theorem 1.1 above is different from Theorem 1.1 of .
for all with and respectively. It is known that is compact.
Theorem 2.1 [, Theorem 2.2]
Assume that the functional defined above satisfies the following:
(T1) maps bounded sets to bounded sets uniformly for , and for all ;
(T2) for all , and as on any finite-dimensional subspace of E;
Particularly, if has a convergent subsequence for every k, then has infinitely many nontrivial critical points satisfying as .
for all and . Then for all . Let , . Note that , where Φ is the functional defined in (2.1).
3 Proof of Theorem 1.1
We firstly establish the following lemmas.
Lemma 3.1 Assume that () and () hold. Then for all and as on any finite-dimensional subspace of E.
Proof Since , by (2.4), it is obvious that for all .
where is the Lebesgue measure.
for any with . This implies as on Y. □
where and for all .
Proof Comparing this lemma with Lemma 2.7 of , we find that these two lemmas have the same condition (AQ2) which is the key in the proof of Lemma 2.7 of . We can prove our lemma by using the same method of , so the details are omitted. □
Now it is the time to prove our main result Theorem 1.1.
For the sake of notational simplicity, in the following we always set for all .
Step 1. We firstly prove that is bounded in E.
Thus, on a subset Ω of with positive measure.
which is a contradiction to (3.8).
Step 2. We prove that has a convergent subsequence in E.
for some .
In view of the compactness of and (3.9), the right-hand side of (3.10) converges strongly in E and hence in E. Together with (3.9), we have in E.
Now, from the last assertion of Theorem 2.1, we know that has infinitely many nontrivial critical points. The proof is completed. □
The authors thank the referee for his/her careful reading of the manuscript. The work is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (No. 61001139).
- Chen G, Ma S: Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum 0. J. Math. Anal. Appl. 2011, 379: 842-851. 10.1016/j.jmaa.2011.02.013MathSciNetView ArticleGoogle Scholar
- Ding Y, Lee C: Periodic solutions for Hamiltonian systems. SIAM J. Math. Anal. 2000, 32: 555-571. 10.1137/S0036141099358178MathSciNetView ArticleGoogle Scholar
- He X, Wu X: Periodic solutions for a class of nonautonomous second order Hamiltonian systems. J. Math. Anal. Appl. 2008, 341(2):1354-1364. 10.1016/j.jmaa.2007.11.028MathSciNetView ArticleGoogle Scholar
- Jiang Q, Tang C: Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 2007, 328: 380-389. 10.1016/j.jmaa.2006.05.064MathSciNetView ArticleGoogle Scholar
- Wang Z, Zhang J: Periodic solutions of a class of second order non-autonomous Hamiltonian systems. Nonlinear Anal. 2010, 72: 4480-4487. 10.1016/j.na.2010.02.023MathSciNetView ArticleGoogle Scholar
- Zhang Q, Liu C: Infinitely many periodic solutions for second-order Hamiltonian systems. J. Differ. Equ. 2011, 251: 816-833. 10.1016/j.jde.2011.05.021View ArticleGoogle Scholar
- Zou W: Multiple solutions for second-order Hamiltonian systems via computation of the critical groups. Nonlinear Anal. TMA 2001, 44: 975-989. 10.1016/S0362-546X(99)00324-7View ArticleGoogle Scholar
- Zou W: Variant fountain theorems and their applications. Manuscr. Math. 2001, 104: 343-358. 10.1007/s002290170032View ArticleGoogle Scholar