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

and Applied Analysis 3 Our main result is the following theorem. Theorem 1.1. Suppose that F t, x satisfies assumptions (A) and 1.7 . Assume that lim sup r→ ∞ inf x∈RN,|x| r |x|−2α ∫T 0 F t, x dt ∞, 1.8 lim inf R→ ∞ sup x∈RN,|x| R |x|−2α ∫T 0 F t, x dt −∞. 1.9


Introduction and Main Result
We are interested in the following second order Hamiltonian systems: ü t ∇F t, u t 0 a.e.t ∈ 0, T , where T > 0 and F : 0, T × R N → R satisfies the following assumption.
A F t, x is measurable in t for each x ∈ R N and continuously differentiable in x for a.e.t ∈ 0, T , F t, 0 0 for a.e.t ∈ 0, T , and there exist a ∈ C R , R , b ∈ L 1 0, T; R such that for all x ∈ R N and a.e.t ∈ 0, T .
Then the corresponding functional ϕ on H 1 T given by is continuously differentiable and weakly lower semicontinuous on H 1 T , where is a Hilbert space with the norm defined by It is well known that the solutions of problem 1.1 correspond to the critical points of ϕ.
There are large number of papers that deal with multiplicity results for this problem.Infinitely many solutions for problem 1.1 are obtained in 2-4 , where the symmetry assumption on the nonlinearity F 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 5-7 .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 F t, x is a suitable oscillating behaviour either at infinity or at zero.
In this paper, we suppose that the nonlinearity ∇F t, x is sublinear, that is, there exist f, g ∈ L 1 0, T; R and α ∈ 0, 1 such that for all x ∈ R N and a.e.t ∈ 0, T .We establish some multiplicity results for problem 1.1 under different assumptions on the potential F. Roughly speaking, we assume that F 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 F t, x satisfies assumptions (A) and 1.7 .Assume that Then, i there exists a sequence of periodic solutions {u n } which are minimax type critical points of functional ϕ, and ϕ u n → ∞, as n → ∞; ii there exists another sequence of periodic solutions {u * m } which are local minimum points of functional ϕ, and ϕ u * m → −∞, as m → ∞.

Proof of Theorems
For Suppose that 1.7 holds.Then Proof.It follows from 1.7 and Sobolev's inequality that for all u in H 1 T .By Wirtinger's inequality, the norm is an equivalent norm on H 1 T .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 {a n } 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 {b m } such that where Proof.For any u ∈ H b m , let u u u, where |u| b m , u ∈ H 1 T .It follows from 1.7 and Sobolev's inequality that

2.19
For fixed n, by Lemma 2.3, we can choose m such that b m > a n and γ k B a n cannot intersect the hyperplane H b m .Let w k w k w k , where w k ∈ R N and w k ∈ H 1 T .Then we have for b m large enough.Besides, by Sobolev's inequality and 1.7 , it is obvious that

2.21
As |u| 2 u t L 2 1/2 is an equivalent norm in H 1 T , it follows that w k t is bounded.Hence, w k is bounded.Also {v k } is bounded in H 1 T .We assume that as k → ∞.Moreover, an easy computation shows that

2.27
The first result of Theorem 1.1 is obtained.
For fixed m, define the subset P m of H 1 T by

2.28
For u ∈ P m , we have

2.29
It ∈ C R N ; R N is a homotopy of γ 0 id with γ 1 π • γ.Moreover, γ t | ∂B an id for all t ∈ 0, 1 .By homotopy invariance and normalization of the degree, we have ∈ π γ B a n .Thus γ B a n intersects the hyperplane H 1 T .By Lemma 2.1, the functional ϕ is coercive on H1 if and only if u → ∞, then the lemma follows from 1.9 and the above inequality.Now we give the proof Theorem 1.1.T .There is a constant M such that max , u n is critical point and c n is critical value of functional ϕ.For anyγ ∈ S n , if a n > b m , γ B a n intersects the hyperplane H b m {x ∈ R N , |u| b m } ⊕ H 1 follows that ϕ is bounded below on P m .Define } be a minimizing sequence in P m , that is,ϕ u k −→ μ m as k −→ ∞. 2.31 From 2.29 , {u k } is bounded in H 1 T .Then, there is a subsequence, we also denoted by {u k }, The case that P m is a convex closed subset of H 1 T implies that u * m ∈ P m .As ϕ is weakly lower semicontinuous, we have * m is in the interior of P m , then u * m is a local minimum of functional ϕ.In fact, let u * m u * m u * m .For large m, from Lemmas 2.2 and 2.3, we have |u * m | / b m , which means that u * m is not on the boundary of P m .Finally, as u * m is a minimum of ϕ in P m ,