Periodic solutions for nonautonomous second order Hamiltonian systems with sublinear nonlinearity

* Correspondence: mathswzhy@126.com Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, Jiangsu, People’s Republic of China Full list of author information is available at the end of the article Abstract Some existence and multiplicity of periodic solutions are obtained for nonautonomous second order Hamiltonian systems with sublinear nonlinearity by using the least action principle and minimax methods in critical point theory. Mathematics Subject Classification (2000): 34C25, 37J45, 58E50.


Introduction and main results
for all x ℝ N and a.e. t [0, T]. The existence of periodic solutions for problem (1.1) has been studied extensively, a lot of existence and multiplicity results have been obtained, we refer the readers to [1][2][3][4][5][6][7][8][9][10][11][12][13] and the reference therein. In particular, under the assumptions that the nonlinearity ∇F (t, x) is bounded, that is, there exists p(t) L 1 (0, T ; ℝ + ) such that for all x ℝ N and a.e. t [0, T], and that Mawhin and Willem in [3] have proved that problem (1.1) admitted a periodic solution. After that, when the nonlinearity ∇F (t, x) is sublinear, that is, there exists f(t), g (t) L 1 (0, T ; ℝ + ) and a [0, 1) such that |∇F(t, x)| ≤ f (t)|x| α + g(t) (1:4) for all x ℝ N and a.e. t [0, T], Tang in [7] have generalized the above results under the hypotheses 1 |x| 2α T 0 F(t, x)dt → ±∞ as |x| → +∞. (1:5) Subsequently, Meng and Tang in [13] further improved condition (1.5) with a (0, 1) by using the following assumptions (1:7) Recently, authors in [14] investigated the existence of periodic solutions for the second order nonautonomous Hamiltonian systems with p-Laplacian, here p > 1, it is assumed that the nonlinearity ∇F (t, x) may grow slightly slower than |x| p-1 , a typical example with p = 2 is , (1:8) solutions are found as saddle points to the corresponding action functional. Furthermore, authors in [12] have extended the ideas of [14], replacing in assumptions (1.4) and (1.5) the term |x| with a more general function h(|x|), which generalized the results of [3,7,10,11]. Concretely speaking, it is assumed that there exist f(t), g(t) L 1 (0, T; ℝ + ) and a nonnegative function h C([0, +∞), [0, +∞)) such that for all x ℝ N and a.e. t [0, T], and that where h be a control function with the properties: if a = 0, h(t) only need to satisfy conditions (a)-(c), here C*, K 1 and K 2 are positive constants. Moreover, a [0, 1) is posed. Under these assumptions, periodic solutions of problem (1.1) are obtained. In addition, if the nonlinearity ∇F (t, x) grows more faster at infinity with the rate like |x| ln (100+|x| 2 ) , f(t) satisfies some certain restrictions and a is required in a more wider range, say, a [0,1], periodic solutions have also been established in [12] by minimax methods.
An interesting question naturally arises: Is it possible to handle both the case such as (1.8) and some cases like (1.4), (1.5), in which only f(t) L 1 (0, T ; ℝ + ) and a [0, 1)? In this paper, we will focus on this problem.
We now state our main results. Theorem 1.1. Suppose that F satisfies assumption (A) and the following conditions: (S 1 ) There exist constants C ≥ 0, C* >0 and a positive function h C(ℝ + , ℝ + ) with the properties: Moreover, there exist f L 1 (0, T; ℝ + ) and g L 1 (0, T; ℝ + ) such that for all x ℝ N and a.e. t [0, T]; (S 2 ) There exists a positive function h C(ℝ + , ℝ + ) which satisfies the conditions (i)-(iv) and Then, problem (1.1) has at least one solution which minimizes the functional given by with the norm Suppose that (S 1 ) and assumption (A) hold. Assume that Then, problem (1.1) has at least one solution in H 1 T . Theorem 1.3. Suppose that (S 1 ), (S 3 ) and assumption (A) hold. Assume that there exist δ >0, ε >0 and an integer k >0 such that for all x ℝ N and a.e. t [0, T], and for all |x| ≤ δ and a.e. t [0, T], where ω = 2π T . Then, problem (1.1) has at least two distinct solutions in H 1 T . Theorem 1.4. Suppose that (S 1 ), (S 2 ) and assumption (A) hold. Assume that there exist δ >0, ε >0 and an integer k ≥ 0 such that for all |x| ≤ δ and a.e. t [0, T]. Then, problem (1.1) has at least three distinct solu- does not need to be controlled by | x| 2a at infinity; in particular, we can not only deal with the case in which ∇F(t, x) grows slightly faster than |x| 2a at infinity, such as the example (1.8), but also we can treat the cases like (1.4), (1.5).
(ii) Compared with [12], we remove the restriction on the function f(t) as well as the restriction on the range of a [0, 1] when we are concerned with the cases like (1.8).
(iii) Here, we point out that introducing the control function h(t) has also been used in [12,14], however, these control functions are different from ours because of the distinct characters of h(t).
On the other hand, take h(t) = t ln (100+t 2 ) , H(t) = t 0 s ln (100+s 2 ) ds, C = 0, C* = 1, then by simple computation, one has Hence, (S 1 ) and (S 2 ) are hold, by Theorem 1.1, problem (1.1) has at least one solution which minimizes the functional in H 1 T . What's more, Theorem 1.1 can also deal with some cases which satisfy the conditions (1.4) and (1.5). For instance, consider function where q(t) L 1 (0, T; ℝ N ). It is not difficult to see that  T . However, we can find that the results of [14] cannot cover this case. More examples are drawn in Section 4.
Our paper is organized as follows. In Section 2, we collect some notations and give a result regrading properties of control function h(t). In Section 3, we are devote to the proofs of main theorems. Finally, we will give some examples to illustrate our results in Section 4.  It follows from assumption (A) that the corresponding function on H 1 T given by

Preliminaries
is continuously differentiable and weakly lower semi-continuous on H 1 T (see [2]). Moreover, one has T . It is well known that the solutions of problem (1.1) correspond to the critical point of .
In order to prove our main theorems, we prepare the following auxiliary result, which will be used frequently later on.
Lemma 2.1. Suppose that there exists a positive function h which satisfies the conditions (i), (iii), (iv) of (S 1 ), then we have the following estimates: Proof. It follows from (iv) of (S 1 ) that, for any ε >0, there exists M 1 >0 such that (2:1) By (iii) of (S 1 ), there exists M 2 >0 such that which implies that where M := max{M 1 , M 2 }. Hence, we obtain for all t >0 by (i) of (S 1 ). Obviously, h(t) satisfies (1) due to the definition of h(t) and (2.4).

Proof of main results
For the sake of convenience, we will denote various positive constants as C i , i = 1, 2, 3,.... Now, we are ready to proof our main results.
which implies that (3:2) Taking into account Lemma 2.1 and (S 2 ), one has  (∇F(t, u n (t)),ũ n (t))dt for all n. Hence, we get This contradicts the boundedness of {(u n )}. So, {ū n } is bounded. Notice (3.6) and (1) of Lemma 2.1, hence {u n } is bounded. Arguing then as in Proposition 4.1 in [3], we conclude that the (PS) condition is satisfied.
In order to apply the saddle point theorem in [2,3], we only need to verify the following conditions: -∞ as |u(t)| +∞.
In fact, for all u ∈H 1 T , by (S 1 ), Sobolev's inequality and Lemma 2.1, we have (∇F(t,ū + su(t)), u(t)) dsdt By Wirtinger's inequality, one has Hence, for ε small enough, ( 1 ) follows from (3.7). On the other hand, by (S 3 ) and Lemma 2.1, we get which implies that  [2], we only need to prove that We see that for all u H k . Then, (ψ 1 ) follows from the above inequality.
So, (ψ 2 ) is obtained. At last, (ψ 3 ) follows from ( 1 ) which are appeared in the proof of Theorem 1.2. Then the proof of Theorem 1.3 is completed. □ Proof of Theorem 1.4. From the proof of Theorem 1.1, we know that is coercive which implies that satisfies the (PS) condition. With the similar manner to [4,7], we can get the multiplicity results, here we omit the details. □ and