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.
Keywords:
Control function; Periodic solutions; The least action principle; Minimax methods1 Introduction and main results
Consider the second order systems
where T > 0 and F : [0, T] × ℝ^{ℕ }→ ℝ satisfies the following assumption:
(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 ∈ L^{1}(0, T ; ℝ^{+}) such that
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 [113] 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 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 pLaplacian, here p > 1, it is assumed that the nonlinearity ∇F (t, x) may grow slightly slower than x^{p1}, 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) ∈ 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 α = 0, h(t) only need to satisfy conditions (a)(c), here C*, K_{1 }and K_{2 }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 , 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) ∈ L^{1}(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:
(S_{1}) There exist constants C ≥ 0, C* > 0 and a positive function h ∈ C(ℝ^{+}, ℝ^{+}) with the properties:
where . 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 φ given by
on the Hilbert space defined by
with the norm
Theorem 1.2. Suppose that (S_{1}) and assumption (A) hold. Assume that
Then, problem (1.1) has at least one solution in .
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 . Then, problem (1.1) has at least two distinct solutions in .
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 solutions in .
Remark 1.1.
(i) Let α ∈ [0, 1), in Theorems 1.11.4, ∇F(t, x) does not need to be controlled by x^{2α }at infinity; in particular, we can not only deal with the case in which ∇F(t, x) grows slightly faster than x^{2α }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 α ∈ [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).
Remark 1.2. From (i) of (S_{1}), 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 [114]. For example, consider function
where f(t) ∈ L^{1}(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 L^{1}(0, T; ℝ^{+}) and no further requirements on the upper bound of are posed, then the approach of [12] cannot be repeated. This example cannot be solved by earlier results, such as [113].
On the other hand, take , , C = 0, C* = 1, then by simple computation, one has
and
Hence, (S_{1}) and (S_{2}) are hold, by Theorem 1.1, problem (1.1) has at least one solution which minimizes the functional φ in .
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
for all x ∈ ℝ^{ℕ }and a.e. t ∈ [0, T]. Choose , , C = 0, C* = 1, and g(t) = q(t), then (S_{1}) and (S_{2}) hold, by Theorem 1.1, problem (1.1) has at least one solution which minimizes the functional φ in . 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.
2 Preliminaries
and
It follows from assumption (A) that the corresponding function φ on given by
is continuously differentiable and weakly lower semicontinuous on (see[2]). Moreover, one has
for all . 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 (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
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).
Next, we come to check condition (2). Recalling the property (iv) of (S_{1}) and (2.2), we get
Therefore, condition (2) holds.
Finally, we show that (3) is also true. By (iii) of (S_{1}), one arrives at, for every β > 0, there exists M_{3 }> 0 such that
Let θ ≥ 1, using (2.5) and integrating the relation
over an interval [1, S] ⊂ [1, +∞), we obtain
Thus, since by (iv) of (S_{1}), one has
for all t ≥ M_{3}. That is,
which completes the proof. □
3 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.
Proof of Theorem 1.1. For , it follows from (S_{1}), Lemma 2.1 and Sobolev's inequality that
which implies that
Taking into account Lemma 2.1 and (S_{2}), one has
As u → +∞ if and only if , 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 . □
Proof of Theorem 1.2. First, we prove that φ satisfies the (PS) condition. Suppose that 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 (S_{3}), one has
This contradicts the boundedness of {φ(u_{n})}. So, 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:
(φ1) φ(u) → +∞ as u → +∞in , where ,
(φ2) φ(u) → ∞ as u(t) → +∞.
In fact, for all , by (S_{1}), Sobolev's inequality and Lemma 2.1, we have
which implies that
By Wirtinger's inequality, one has
Hence, for ε small enough, (φ_{1}) follows from (3.7).
On the other hand, by (S_{3}) and Lemma 2.1, we get
which implies that
Thus, (φ_{2}) is verified. The proof of Theorem 1.2 is completed. □
and ψ = φ. Then, ψ ∈ C^{1}(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 (S_{1}) and Lemma 2.1, one has
for all x ≥ δ, a.e. t ∈ [0, T] and some Q(t) ∈ L^{1}(0, T; ℝ^{+}) given by
Now, it follows from (1.10) that
for all x ∈ ℝ^{ℕ }and a.e. t ∈ [0, T]. Hence, we obtain
for all u ∈ H_{k}. Then, (ψ_{1}) follows from the above inequality.
So, (ψ_{2}) is obtained. At last, (ψ_{3}) follows from (φ_{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 φ is coercive which implies that φ satisfies the (PS) condition. With the similar manner to [4,7], we can get the multiplicity results, here we omit the details. □
4 Examples
In this section, we give some examples to illustrate our results.
Example 4.1. Consider the function
where d(t) ∈ L^{1}(0, T; ℝ^{ℕ}). Let , then , by a direct computation, (S_{1}) and (S_{3}) hold. Then, by Theorem 1.2, we conclude that problem (1.1) has one solution in . However, as the reason of Remark 1.3, the results in [113] cannot be applied.
Example 4.2. Consider the function
where A(t), B(t) are suitable functions which insure assumption (A) hold. Also, put , , we see that (S_{1}), (S_{2}) and (1.11) hold. By virtue of Theorem 1.4, problem (1.1) has at least three distinct solutions in .
Example 4.3. Consider the function
We observe that
which means ∇F(t, x) is bounded, moreover, one has
Then, by the results in [3,7,12], problem (1.1) has one solution which minimizes the functional φ in .
In fact, our Theorem 1.1 can also handle this case. In this situation, let , , and choose C = 2, C* = 1 , g(t) ≡ 0, we infer
and
So, by Theorem 1.1, problem (1.1) has one solution which minimizes the functional φ in .
Remark 4.1. Unlike the control functions in [12], where h(t) is nondecreasing, here control function is bounded but not increasing.
Example 4.4. Consider the function
where k(t) ∈ L^{1}(0, T; ℝ^{ℕ}). It is easy to check that
The above inequality leads to (1.4) hold with
So, by the theorems in [3,7,12,13], problem (1.1) has at least one solution in .
Indeed, our Theorem 1.2 can also deal with this case. Let , , and choose C = 0, C* = 1, , g(t) = k(t), we know
Furthermore, one has
Hence, (S_{1}) and (S_{3}) are true, by Theorem 1.2, problem (1.1) has at least one solution in . However, we can find that the results in [14] cannot deal with this case.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
The authors would like to thank Professor Huicheng Yin for his help and many valuable discussions, and the first author takes the opportunity to thank Professor Xiangsheng Xu and the members at Department of Mathematics and Statistics at Mississippi State University for their warm hospitality and kindness. This Project is Supported by National Natural Science Foundation of China (Grant No. 11026213, 10871096), Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 10KJB110006) and Foundation of Nanjing University of Information Science and Technology (Grant No. 20080280).
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