Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems

This paper considers the Hamiltonian systems 
with new generalized super-quadratic conditions. Using the variational principle and critical point theory, 
we obtain infinitely many distinct subharmonic solutions and an unbounded sequence of periodic solutions.

1. Introduction. We consider the non-autonomous Hamiltonian system which also can be written asż = JH z (t, z), where H ∈ C 1 (R × R 2n , R), z(t) = (p(t), q(t)) (t ∈ R, p(t), q(t) ∈ R n ), H z = ∂H ∂z , H p = ∂H ∂p , H q = ∂H ∂q and J = 0 −I n I n 0 with I n being the n × n identity matrix.
In the pioneer work [11], the author deals with the existence of periodic solutions of the Hamiltonian systems under a classical super-quadratic condition, that is, (S) there exist constantsθ ∈ (0, 1 2 ) and R > 0 such that θH z (t, z) · z ≥ H(t, z) > 0, (t, z) ∈ R × R 2n with |z| ≥ R.
Paper [12] considers subharmonic solutions of the system (1) under classical super-quadratic and sub-quadratic assumptions. Paper [7] uses the Maslov-type index to solve the subharmonic solutions of the system (1). Paper [2] considers periodic and subharmonic solutions of the system (1) under the generalized superquadratic condition in paper [4]. Paper [6] considers brake subharmonic solutions by using the Galerkin approximation method and the iteration inequalities of the L-Maslov type index theory. To the authors' knowledge, the subharmonic solution problem originates from paper [12] and focuses on when two solutions are geometrically distinct. A subharmonic solution of the system (1) is a kT -periodic solution for some k ≥ 2, if the Hamiltonian function H is T -periodic with respect to t. A k 1 T -periodic solution z 1 and a k 2 T -periodic z 2 are geometrically distinct, if m 1 * z 1 = m 2 * z 2 holds for any m 1 , m 2 ∈ Z, where (m * z)(t) = z(t+mT ). Books [10] and [14] contain some results of the existence and multiplicity of periodic solutions and subharmonic solutions of the system (1).
Our aim is to use the the variational principle and critical point theory to verify the multiplicity of subharmonic solutions and periodic solutions of the Hamiltonian system (1) with the super-quadratic condition (NS).
We shall prepare some preliminaries in Section 2, give the multiplicity results of subharmonic solutions of the system (1) in Section 3 and give the multiplicity result of periodic solutions of the system (1) in Section 4.
Our three results are divided into two parts. Firstly, two results about subharmonic solutions are listed as following.
Next, we shall use a generalized critical point theorem of even functionals to search for an unbounded sequence of periodic solutions of the system (1).
During the proof of the above three theorems, we need two inequalities stated below.

Preliminaries.
We introduce some notations and conclusions which are used later.
We will use the following two lemmas in our proof.
Lemma 2.1 (See [14]). E can be compactly embedded into L s (S 1 , R 2n ) (s ≥ 1), in particular, there exists a constant C s > 0 such that z L s ≤ C s z holds for z ∈ E.
Suppose that I ∈ C 1 (E, R), I satisfies the (PS) condition means that if a sequence {z m } satisfies that {I(z m )} is bounded and I (z m ) → 0 as m → +∞, then {z m } has a convergent subsequence.
satisfies the (PS) condition, and (I1) L is a linear, bounded and self-adjoint operator, transforms bounded sets into bounded sets, and Ψ is compact.
For e = (p + , q + ) ∈ E 1 with e = 1, setẼ = span{e} ⊕ E 2 and W = {z ∈Ẽ 1 ≤ z ≤ 2 and z − ≤ z + + z 0 }, then we have Lemma 3.1 (See [18]). There exists a constant ε 1 > 0 such that Furthermore, we set and ∂Q refers to the boundary of Q relative tô Proof. The proof is similar to that of Lemma 3.3 in our previous paper [18].
Step 1. We will prove that the system (2) possesses a nontrivial classical 2πperiodic solution for every k ∈ N * .
Since H satisfies (H1)-(H5), for every k ∈ N * , H k also satisfies (H1)-(H5), which leads to the following results: book [14] implies that I k ∈ C 1 (E, R) satisfies (I1) and (I2) in Lemma 2.2, where A(z) = 1 2 Lz, z and b k (z) = − 2π 0 H k (t, z(t))dt, z ∈ E; Lemma 3.1 in our paper [18] indicates that I k satisfies the (PS) condition; Lemma 3.4 in our paper [18] shows that I k satisfies (I3) in Lemma 2.2 for r and k defined as in Lemma 3.2; Lemma 3.2 and the arguments before it show that I k satisfies (I4) in Lemma 2.2. Thus Lemma 2.2 holds for I k , if H satisfies (H1)-(H5). Then there exists a critical point z k of I k with l k = I k (z k ) ≥ γ k > 0. Pages 40 and 41 in book [14] indicate that z k (t) is a nontrivial classical 2π-periodic solution of the system (2) for every fixed k ∈ N * .
Step 2. We will prove that the system (1) possesses a distinct sequence of subharmonic solutions.
Repeating the above arguments, it follows that we have a distinct sequence of subharmonic solutions of (1), i.e., z 1 (t), z k1  (H5), then H k also satisfies those conditions. As our paper [18] demonstrates, due to the lack of growth control for (H k ) z , to obtain a polynomial growth at infinity, for any K > R ≥ √ 2, we take a cut-off function χ ∈ C ∞ ([0, +∞), [0, 1]) defined as where A 0 is as in Lemma 3.2, define Lemma 3.5 in our paper [18] indicates that if H satisfies (H1), (H2), (H3) and (H4), then H k,K also satisfies those conditions. Clearly, H k,K satisfies (H5). So H k,K satisfies the conditions of Theorem 1.2. Let I k,K (z) = A(z)− 2π 0 H k,K (t, z)dt, z ∈ E, then the proof of Lemma 3.2 in our paper [18] shows that for every k ∈ N * , there exists a number k,K ∈ (0, 1) depending on k and K such that Similarly, we set B k,K (p, q) = ( τ −1 k,K p, σ−1 k,K q), (p, q) ∈ E, then B k,K is linear, bounded and invertible. Proof. The proof is similar to Lemma 3.2.
Step 1. We will prove that the system (2) possesses a nontrivial classical 2πperiodic solution for every fixed k ∈ N * .
The proof is similar to that of Theorem 1.3 in our paper [18].
Since H satisfies (H1), (H2), (H3) , (H4) and (H5), for every k ∈ N * , H k,K also satisfies those conditions, which leads to the following results: book [14] implies that I k,K ∈ C 1 (E, R) satisfies (I1) and (I2) in Lemma 2.2, where A(z) = 1 2 Lz, z and b k,K (z) = − 2π 0 H k,K (t, z(t))dt, z ∈ E; Lemma 3.1 in our paper [18] indicates that I k,K satisfies the (PS) condition; Lemma 3.4 in our paper [18] shows that I k,K satisfies (I3) in Lemma 2.2 for r and k,K defined as in Lemma 3.3; and finally Then there exists a critical point z k,K of I k,K with l k,K = I k,K (z k,K ) ≥ γ k,K > 0. Pages 40 and 41 in book [14] indicate that for every fixed k ∈ N * , z k,K is a nontrivial classical 2π-periodic solution of the system We claim that for every fixed k ∈ N * , there exists a constant In fact, following the proof of Theorem 1.3 in our paper [18], I k,K (z k,K ) = inf h∈Γ sup z∈Q I k,K (h (1, z)), where Γ is as in Lemma 2.2. Since Id ∈ Γ, for z ∈ Q, from (5) and (H1), we have Since H k,K satisfies (H2) with coefficients independent of K, then whereĈ β is independent of K. By (H3) and the factż k,K = J(H k,K ) z (t, z k,K ), we see The above two estimates imply that for every k ∈ N * , there exists a constant C(k) > 0 such that z k,K L 1 ≤ C(k) and ż k,K L 1 ≤ C(k).
For t ∈ [0, 2π], we have Then Inequalities (13) and (14) imply that hence we verify that z K L ∞ ≤ K 0 holds for some K 0 independent of K.
The fact z k,K L ∞ ≤ K 0 and the definition of H k,K imply H k,K (t, z k,K ) = H k (t, z k,K ) holds for any K > K 0 , then (H k,K ) z (t, z k,K ) = (H k ) z (t, z k,K ), which implies that z k,K is a 2π-periodic solution of the system (2).
For K ≥ K 0 (K 0 depends on k), set z k = z k,K and l k = I k,K (z k ) > 0.
Step 2. We will prove that the system (1) possesses a distinct sequence of subharmonic solutions.
The proof is the same as Step 2 of the proof for Theorem 1.1.

4.
Multiplicity of periodic solutions. Now we turn to the multiplicity of 2πperiodic solutions of the system (1). For this aim, we introduce a theorem coming from paper [5]. Let E be a Hilbert space with E = X ⊕Y , where X and Y are infinite dimensional subspaces of E. Assume we have sequences of finite dimensional subspaces X m ⊂ X and Y m ⊂ Y such that E m = X m ⊕ Y m and E = ∪ +∞ m=1 E m . For m large enough, T i : E → E (i = 1, 2) is linear, bounded and invertible, and satisfies T i (E m ) = E m . For constants ρ, r, M > 0 with r T −1 1 T 2 y 1 > ρ and for some fixed y 1 ∈ Y independent of m with y 1 = 1, set Suppose that I ∈ C 1 (E, R), I satisfies the (PS) * condition with respect to {E m } means that if a sequence {z m |z m ∈ E Nm } satisfies that I| E Nm (z m ) is bounded and (I| E Nm ) (z m ) → 0 as m → +∞, then {z m } has a convergent subsequence, where {N m } is a strictly increasing subsequence of N.  Proof. The proof is similar to the proof of Lemma 3.1 in our paper [18]. Let {z m |z m ∈ E Nm } be a (PS) * sequence, that is, {I(z m )} is bounded and (I E Nm ) (z m ) → 0 as m → +∞. In the process, we regard I (z) as an element in E and still write I (z), ζ as I (z)ζ, where z, ζ ∈ E. Let P m : E → E Nm denote the projective operator, then for z, ζ ∈ E Nm , we have (I E Nm ) (z) = P m I (z) and (I E Nm ) (z)ζ = P m I (z), ζ = I (z)ζ. Firstly, we show that {z m } is bounded. If not, we may assume that z m → +∞ as m → +∞.