Subharmonic Solutions and Minimal Periodic Solutions of First-order Hamiltonian Systems with Anisotropic Growth

Using a homologically link theorem in variational theory and iteration inequalities of Maslov-type index, we show the existence of a sequence of subharmonic solutions of non-autonomous Hamiltonian systems with the Hamiltonian functions satisfying some anisotropic growth conditions, i.e., the Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Moreover, we also consider the minimal period problem of some autonomous Hamiltonian systems with anisotropic growth.


Introduction
In this paper, we first consider subharmonic solutions of the following Hamiltonian system where H ′ z is the gradient of H with respect to the variables z = (p 1 , · · · , p n , q 1 , · · · , q n ) ∈ R 2n and J = 0 −I n I n 0 with I n being the n × n identity matrix.
Note that the Hamiltonian function is an example satisfying (H1)-(H5) with anisotropic growth when σ i = τ i for some i. When σ i = τ i for all i, it is an almost quadratic growth function which is slower growing than any super-quadratic function at infinity in the sense of Ambrosetti-Rabinowitz.
Now we list our main results of subharmonic solutions as following. (H3) ′ there exist constants ξ i , η i > 0 with ξ i + η i = 1 (i = 1, 2, · · · , n) such that Then for each integer k ≥ 1, the system (1.1) possesses a kτ -periodic nonconstant solution z k such that z k and z pk are geometrically distinct provided p > 2n + 1. If all z k are non-degenerate, then z k and z pk (p > 1) are geometrically distinct.
Remark 1.1. In the case where H(t, z) = 1 2 (B(t)z, z) +Ĥ(t, z) withB(t) being a τ -periodic, continuous symmetric matrix function andĤ satisfying the conditions as stated in Theorems 1.1, 1.2 and Corollary 1.1, we also obtain the similar results with some restrictive conditions on B in Section 4 below. But compared with the results in [18], we note that the condition B(t) is semi-positive-definite required in [18] is not necessary here (see Remark 4.1).
In Section 5, we consider the minimal periodic problem of some autonomous Hamiltonian systems with the Hamiltonian functions H(z) satisfying the anisotropic growth conditions as stated in Theorems 1.1, 1.2 and Corollary 1.1. With the same tricks, we show that the critical points (z, τ ) obtained from the homological link method in fact is the minimal periodic solution of the Hamiltonian systems provided the Hessian H ′′ zz (z) is positively definite for z ∈ R 2n \ {0}.
In the pioneer work [29], Rabinowitz obtained a sequence of subharmonic solutions of the system (1.1). Since then, many papers were devoted to the study of subharmonic solutions (see [4,6,7,17,18,25,26,31,35]). For the brake subharmonic solutions of Hamiltonian systems we refer to [14,16]. For the P -symmetric subharmonic solutions of Hamiltonian systems we refer to [23]. We note that all the results obtained in the references mentioned here are related with the Hamiltonian functions with superquadratic growth or subquadratic growth.
This paper is organized as follows, in Section 2, as preliminary we recall some notions about the Maslov-type index theory and the iteration inequalities developed by Y.Long and the first author of this paper in [21]. In this section we also recall the homologically link theorem in [1] . We consider minimal periodic problem for the autonomous Hamiltonian systems in Section 5.

Preliminaries
We first recall the notion of Maslov-type index and some iteration estimates. We refer to [1], [21] and [24] for details.
In the case of linear Hamiltonian systems where B(t) is a τ -periodic, symmetric and continuous matrix function. Its fundamental solution is also called the Maslov-type index of the matrix function B(t).
If z is a τ -periodic solution of the system (1.1), we denote by (i τ (z), ν τ (z)) : and ν kτ (j * z) = ν kτ (z) hold for 0 ≤ j ≤ k.    Now we introduce some concepts and results of Sobolev space theory.
where · denotes the norm on E. Given B(t) a τ -periodic, symmetric and continuous matrix function with Maslov-type index

There exists a linear bounded self-adjoint operators
then we have the following theorem. .
Finally, we recall the homologically link theorem in [1].
. Let Q ⊂ M be a topologically embedded closed q-dimensional ball and let S ⊂ M be a closed subset such that ∂Q S = ∅. Assume that ∂Q and S homologically link. Moreover, assume Then, if Γ denotes the set of all q-chains in M whose boundary has support ∂Q, the number Remark 2.1. If M is a finite dimensional Hilbert space, and f satisfies (C) condition instead of (PS) condition, the above theorem still holds, the proof is the same as that of Theorem 4.1.7 in [1] (see [27] for results obtained under (C) condition).
Recall that the functional f satisfies the so called Cerami condition ((C) condition for short) m → +∞ has a convergent subsequence.

Proofs of the Main Results
For simplicity, we first give a proof of Corollary 1.1.
, we have f ∈ C 2 (E, R). As usual, finding periodic solutions of the system (1.1) converts to looking for critical points of f .
We can suppose that there are only finitely many z ± m = 0. Dividing the two sides by z m · z ± m , it implies z ± m zm → 0 as m → +∞. By (3.1), we see |z 0 We note that if f satisfies (C) * condition on E, then f m satisfies (C) condition on E m .
We note that B ρ is a linear bounded and invertible operator and B ρ ≤ 1, if ρ ≤ 1.
Let B be the operator for B(t) = H ′′ zz (t, z(t)) defined in Section 2, then we have by (3.15), there exists a constant κ > 0 such that Then for m large enough, we have x ∈B(z, κ) E m .
Similarly, we have By (3.14), (3.17) and (3.18), for m large enough, Lemma 2.4 shows that and The above two estimates show that (3.13) holds. Proof of Corollary 1.1. The proof is the same as that in [18]. For readers convenience' we give the details here.
Since H is kτ -periodic, by Theorem 3.1, the system 1.1 possesses a nonconstant kτ -periodic solution z k satisfying If z k and z pk are not geometrically distinct, by definition, there exist integers l and m such that l * z k = m * z pk . By Proposition 2.1, we have i kτ (l * z k ) = i kτ (z k ), ν kτ (l * z k ) = ν kτ (z k ) and i pkT (m * z pk ) = i pkT (z pk ), ν pkT (m * z pk ) = ν pkT (z pk ).
If all z k are non-degenerate, then ν kτ (z k ) = 0 and i kτ (z k ) = n + 1 for k ∈ N. Proposition 2.3 shows that p + n ≤ n + 1, so we get p = 1. Hence z k and z pk are geometrically distinct when p > 1. We complete the proof of Theorem 1.1. Indeed, for any K > 0, we take a cut-off function defined by Choosing λ 0 ∈ (γ, 1 + β) and If K > 0 is large enough, it is easy to show that H K satisfies (H2) and (H3) ′ with the constants independent of K (see [34]). The modified function H K also satisfies (H1), (H3)-(H5).
. By the choice of λ 0 , there exists a constant A 2 > 0 such that In all the arguments before, we replace H, λ and f by H K , λ 0 and f K respectively, we see By (H3) ′ , it is easy to prove that z = z K is independent of K and a τ -periodic nonconstant solution of the system (1.1) for K large enough (see [34]). In fact, we also take the cut-off function χ ∈ C ∞ ([0, +∞), R) as before.
Then H K satisfies (C2) and (C3) with the constants independent of K (see [2]) if R(K) and R are large enough.
. It is easy to show that f K satisfies (PS) * condition (see [2]). By the definition of H K , we can choose From [2], we know that there exist constants d 1 > 0 and d 2 > 0 such that which indicates an inequality similar to (3.8).

The case: H contains a quadratic term
Now we consider the case where H(t, z) = 1 2 (B(t)z, z) +Ĥ(t, z). The proof of the following results are similar to that of Theorems 1.1, 1.2 and Corollary 1.1, we only state the results.
Then for each integer k ≥ 1 and k < 2π ωτ , the system (1.1) possesses a kT -periodic nonconstant solution z k such that z k and z pk are geometrically distinct provided p > 2n + 1 and pk < 2π ωτ . If all z k are non-degenerate, then z k and z pk (p > 1) are geometrically distinct.
Similarly we have the following results.
Remark 4.2. The key point of the proof of Corollary 4.1 is that if (H6) holds, then we have z))dt and BB ρ z, B ρ z = ρ η−2 Bz, z , where z ∈ E, and f , η and B ρ (ρ > 0) are defined in Section 3. Note that H satisfies (H3) ifĤ does. Then the proof of (C) * condition is the same as that of Lemma 3.1. We can define B µ for small µ ∈ {̺ m } and B ν for large ν ∈ {̺ m } as in Section 3. So the arguments can be applied to the current case.
Then we have the same results as in Theorem 4.1.

Minimal periodic solutions for the autonomous Hamiltonian systems
In this section, we consider the minimal periodic problem of the following autonomous Hamiltonian systems ż = JH ′ (z), z ∈ R 2n , z(τ ) = z(0). (5.1) We say that (z, τ ) is a minimal periodic solution of (5.1) if z solves the problem (5.1) with τ being the minimal period of z.
As shown in [21], we can also obtain minimal periodic solutions for the autonomous Hamil- Note that the Hamiltonian function H of (1.2) satisfies (H7).
Proof of Theorem 5.1. The proof is almost the same as that in [21]. For readers' convenience, we estimate the iteration number of the solution (z, τ ) now.
Assume (z, τ ) has minimal period τ k , i.e., its iteration number is k ∈ Z. Since the nonlinear Hamiltonian system in (5.1) is autonomous and (H7) holds, we have ν τ k (z) ≥ 1 and i τ k (z) ≥ n by Lemma 2.1. From Lemma 2.2, we see k = 1, that is, the solution (z, τ ) has minimal period τ .
In his pioneer work [28], P. Rabinowitz proposed a conjecture on whether a superquadratic Hamiltonian system possesses a periodic solution with a prescribed minimal period. This conjecture has been deeply studied by many mathematicians. We refer to [5,6,8,11,12,21,24] for the original Rabinowitz's conjecture under some further conditions (for example the convex case). For the minimal periodic problem of brake solution of Hamiltonian systems, we refer to [15,19,33]. For the minimal periodic problem of P -symmetric solution of Hamiltonian systems, we refer to [20,23]. Up to our knowledge, Theorem 5.1 is the first result on the minimal periodic problem of nonlinear Hamiltonian systems with anisotropic growth.