Periodic solutions for nonautonomous second order Hamiltonian systems with sublinear nonlinearity

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.

Consider the second order systems

where T > 0 and F : [0, T] × ℝ^ℕ → ℝ satisfies the following assumption:

1. (A)

   F (t, x) is measurable in t for every x ∈ ℝ^ℕ and continuously differentiable in x for a.e. t ∈ [0, T], and there exist a ∈ C(ℝ⁺, ℝ⁺), b ∈ L¹(0, T ; ℝ⁺) such that

   \left|F\left(t,x\right)\right|\le a\left(\left|x\right|\right)b\left(t\right),|\nabla F\left(t,x\right)|\le a\left(\left|x\right|\right)b\left(t\right)

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 [1–13] and the reference therein. In particular, under the assumptions that the nonlinearity ∇F (t, x) is bounded, that is, there exists p(t) ∈ L¹(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) ∈ L¹(0, T ; ℝ⁺) and α ∈ [0, 1) such that

for all x ∈ ℝ^ℕ and a.e. t ∈ [0, T], Tang in [7] have generalized the above results under the hypotheses

Subsequently, Meng and Tang in [13] further improved condition (1.5) with α ∈ (0, 1) by using the following assumptions

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

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) ∈ L¹(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 α = 0, h(t) only need to satisfy conditions (a)-(c), here C*, K₁ and K₂ are positive constants. Moreover, α ∈ [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 \frac{\left|x\right|}{ln\left(100+|x{|}^2\right)}, f(t) satisfies some certain restrictions and α is required in a more wider range, say, α ∈ [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) ∈ L¹(0, T ; ℝ⁺) and α ∈ [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:

(S₁) There exist constants C ≥ 0, C* > 0 and a positive function h ∈ C(ℝ⁺, ℝ⁺) with the properties:

where H\left(t\right):={\int }_0^{t}h\left(s\right)ds. Moreover, there exist f ∈ L¹(0, T; ℝ⁺) and g ∈ L¹(0, T; ℝ⁺) such that

for all x ∈ ℝ^ℕ and a.e. t ∈ [0, T];

(S₂) 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 φ given by

on the Hilbert space H_T^1 defined by

with the norm

Theorem 1.2. Suppose that (S₁) and assumption (A) hold. Assume that

Then, problem (1.1) has at least one solution in H_T^1.

Theorem 1.3. Suppose that (S₁), (S₃) 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 \omega =\frac{2\pi }{T}. Then, problem(1.1) has at least two distinct solutions in H_T^1.

Theorem 1.4. Suppose that (S₁), (S₂) 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 solutions in H_T^1.

Remark 1.1.

1. (i)

   Let α ∈ [0, 1), in Theorems 1.1-1.4, ∇F(t, x) does not need to be controlled by |x|^2αat infinity; in particular, we can not only deal with the case in which ∇F(t, x) grows slightly faster than |x|^2αat infinity, such as the example (1.8), but also we can treat the cases like (1.4), (1.5).

2. (ii)

   Compared with [12], we remove the restriction on the function f(t) as well as the restriction on the range of α ∈ [0, 1] when we are concerned with the cases like (1.8).

3. (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).

Remark 1.2. From (i) of (S₁), we see that, nonincreasing control functions h(t) can be permitted. With respect to the detailed example on this assertion, one can see Example 4.3 of Section 4.

Remark 1.3. There are functions F(t, x) satisfying our theorems and not satisfying the results in [1–14]. For example, consider function

where f(t) ∈ L¹(0, T; ℝ⁺) and f(t) > 0 for a.e. t ∈ [0, T]. It is apparent that

for all x ∈ ℝ^ℕ and t ∈ [0, T]. (1.12) shows that (1.4) does not hold for any α ∈ [0, 1), moreover, note f(t) only belongs to L¹(0, T; ℝ⁺) and no further requirements on the upper bound of {\int }_0^{T}f\left(t\right)dt are posed, then the approach of [12] cannot be repeated. This example cannot be solved by earlier results, such as [1–13].

On the other hand, take h\left(t\right)=\frac{t}{ln\left(100+t^2\right)}, H\left(t\right)={\int }_0^t\frac{s}{ln\left(100+s^2\right)}ds, C = 0, C* = 1, then by simple computation, one has

and

Hence, (S₁) and (S₂) are hold, by Theorem 1.1, problem (1.1) has at least one solution which minimizes the functional φ in H_T^1.

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) ∈ L¹(0, T; ℝ^ℕ). It is not difficult to see that

for all x ∈ ℝ^ℕ and a.e. t ∈ [0, T]. Choose h\left(t\right)=t^{\frac{1}{2}}, H\left(t\right)=\frac{2}{3}{t}^{\frac{3}{2}}, C = 0, C* = 1, f\left(t\right)=\frac{3}{2}|0.6T-t| and g(t) = |q(t)|, then (S₁) and (S₂) hold, by Theorem 1.1, problem (1.1) has at least one solution which minimizes the functional φ in H_T^1. 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.

For u\in H_T^1, let ū:=\frac{1}{T}{\int }_0^{T}u\left(t\right)dt and ũ\left(t\right):=u\left(t\right)-ū, then one has

and

where \left|\right|ũ|{|}_{\infty }:=\underset{0\le t\le T}{max}|ũ\left(t\right)|.

It follows from assumption (A) that the corresponding function φ on H_T^1 given by

is continuously differentiable and weakly lower semi-continuous on H_T^1 (see[2]). Moreover, one has

for all u,v\in H_T^1. It is well known that the solutions of problem (1.1) correspond to the critical point of φ.

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 (S₁), then we have the following estimates:

Proof. It follows from (iv) of (S₁) that, for any ε > 0, there exists M₁ > 0 such that

By (iii) of (S₁), there exists M₂ > 0 such that

which implies that

where M := max{M₁, M₂}. Hence, we obtain

for all t > 0 by (i) of (S₁). Obviously, h(t) satisfies (1) due to the definition of h(t) and (2.4).

Next, we come to check condition (2). Recalling the property (iv) of (S₁) and (2.2), we get

Therefore, condition (2) holds.

Finally, we show that (3) is also true. By (iii) of (S₁), one arrives at, for every β > 0, there exists M₃ > 0 such that

Let θ ≥ 1, using (2.5) and integrating the relation

over an interval [1, S] ⊂ [1, +∞), we obtain

Thus, since \underset{S\to +\infty }{lim}\frac{H\left(St\right)}{S^2}=0 by (iv) of (S₁), one has

for all t ≥ M₃. That is,

which completes the proof. □

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.

Proof of Theorem 1.1. For u\in H_T^1, it follows from (S₁), Lemma 2.1 and Sobolev's inequality that

which implies that

Taking into account Lemma 2.1 and (S₂), one has

As ||u|| → +∞ if and only if {\left(|ū{|}^2+{\int }_0^T|\stackrel{}{u}\left(t\right){|}^{2}dt\right)}^{1∕2}\to +\infty, for ε small enough, (3.2) and (3.3) deduce that

Hence, by the least action principle, problem (1.1) has at least one solution which minimizes the function φ in H_T^1. □

Proof of Theorem 1.2. First, we prove that φ satisfies the (PS) condition. Suppose that \left\{u_n\right\}\subset H_T^1 is a (PS) sequence of φ, that is, φ'(u _n ) → 0 as n → +∞ and {φ(u _n )} is bounded. In a way similar to the proof of Theorem 1.1, we have

for all n. Hence, we get

for large n. On the other hand, it follows from Wirtinger's inequality that

for all n. Combining (3.4) with (3.5), we obtain

for all large n. By (3.1), (3.6), Lemma 2.1 and (S₃), one has

This contradicts the boundedness of {φ(u _n )}. So, \left\{{ū}_n\right\} 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:

(φ 1) φ(u) → +∞ as ||u|| → +∞in {\tilde{H}}_T^1, where {\tilde{H}}_T^1:=\left\{u\in H_T^1|ū=0\right\},

(φ 2) φ(u) → -∞ as |u(t)| → +∞.

In fact, for all u\in {\tilde{H}}_T^1, by (S₁), Sobolev's inequality and Lemma 2.1, we have

which implies that

By Wirtinger's inequality, one has

Hence, for ε small enough, (φ₁) follows from (3.7).

On the other hand, by (S₃) and Lemma 2.1, we get

which implies that

Thus, (φ₂) is verified. The proof of Theorem 1.2 is completed. □

Proof of Theorem 1.3. Let E=H_T^1,

and ψ = -φ. Then, ψ ∈ C¹(E, ℝ) satisfies the (PS) condition by the proof of Theorem 1.2. In view of Theorem 5.29 and Example 5.26 in [2], we only need to prove that

We see that

for all x ∈ ℝ^ℕ and a.e. t ∈ [0, T]. By (S₁) and Lemma 2.1, one has