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

Chungen Liu and Xiaofei Zhang
School of Mathematics and LPMC, Nankai University,
Tianjin 300071, P. R. China
Partially supported by the NSF of China (11071127, 10621101), 973 Program of
MOST (2011CB808002) and SRFDP. E-mail<EMAIL_ADDRESS>
Abstract: 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.

Key Words: Maslov-type index; Morse index; homologically link; Subharmonic
solution; minimal period.

##  1 Introduction

In this paper, we first consider subharmonic solutions of the following
Hamiltonian system

$\left\\{\begin{array}[]{ll}-J\dot{z}=H^{\prime}_{z}(t,z),\\\
z(k\tau)=z(0),\;\;k\in\mathbf{Z},\end{array}\right.$ (1.1)

where $H^{\prime}_{z}$ is the gradient of $H$ with respect to the variables
$z=(p_{1},\cdots,p_{n},q_{1},\cdots,q_{n})\in\mathbf{R}^{2n}$ and
$J=\left(\begin{array}[]{cc}0&-I_{n}\\\ I_{n}&0\end{array}\right)$ with
$I_{n}$ being the $n\times n$ identity matrix.

Denote any principal diagonal matrix
diag$\\{a_{1},\cdots,a_{n},b_{1},\cdots,b_{n}\\}\in\mathbf{R}^{2n}$ by
$V(a,b)$ with $a=(a_{1},\cdots,a_{n})$ and $b=(b_{1},\cdots,b_{n})$, then
$V(a,b)(z)=(a_{1}p_{1},\cdots,a_{n}p_{n},b_{1}q_{1},\cdots,b_{n}q_{n})$. Now
we suppose the Hamiltonian function $H$ satisfying the following conditions as
in [32] with a bit difference.

(H1) $H\in C^{2}(\mathbf{R}\times\mathbf{R}^{2n},\mathbf{R})$ is nonnegative
and $\tau$-periodic with respect to $t$.

(H2) There exist $\beta>1$ and $c_{1},c_{2},\alpha_{i},\beta_{i}>0$ with
$\alpha_{i}+\beta_{i}=1$ $(i=1,2,\cdots,n)$ such that

$H^{\prime}_{z}(t,z)\cdot V_{1}(z)-H(t,z)\geq c_{1}|z|^{\beta}-c_{2},\ \ \
(t,z)\in\mathbf{R}\times\mathbf{R}^{2n},$

where $V_{1}=V(\alpha,\beta)$.

(H3) There exist ${\sigma}_{i},{\tau}_{i}>0$ and $\lambda\in\Lambda$ with
$\Lambda=\\{\lambda\in\mathbf{R}:\max_{1\leq i\leq
n}\\{\frac{{\sigma}_{i}}{{\tau}_{i}},\frac{{\tau}_{i}}{{\sigma}_{i}}\\}<\lambda<1+\beta\\}$
such that

$|H^{\prime\prime}_{zz}(t,z)|\leq c_{2}(|z|^{\lambda-1}+1),\ \ \
(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}.$

(H4) $\displaystyle\frac{H(t,z)}{\omega(z)}\rightarrow 0$ as $|z|\rightarrow
0$ uniformly in $t$, where
$w(z)=\sum_{i=1}^{n}\left(|p_{i}|^{1+\frac{{\sigma}_{i}}{{\tau}_{i}}}+|q_{i}|^{1+\frac{{\tau}_{i}}{{\sigma}_{i}}}\right)$.

(H5) $\displaystyle\frac{H(t,z)}{w(z)}\rightarrow+\infty$ as
$|z|\rightarrow+\infty$ uniformly in $t$.

Note that the Hamiltonian function

$H(t,z)=\displaystyle\sum_{i=1}^{n}\left(|p_{i}|^{1+\frac{{\sigma}_{i}}{{\tau}_{i}}}\ln(1+p_{i}^{2})+|q_{i}|^{1+\frac{{\tau}_{i}}{{\sigma}_{i}}}\ln(1+q_{i}^{2})\right),\
\ \ \ (t,z)\in\mathbf{R}\times\mathbf{R}^{2n}$ (1.2)

is an example satisfying (H1)-(H5) with anisotropic growth when
${\sigma}_{i}\neq{\tau}_{i}$ for some $i$. When ${\sigma}_{i}={\tau}_{i}$ 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.

Given $j\in\mathbf{Z}$ and a $k\tau$-periodic solution $(z,k\tau)$ of the
system (1.1), the phase shift $j\ast z$ of $z$ is defined by $j\ast
z(t)=z(t+j\tau)$. Recall that two solutions $(z_{1},k_{1}\tau)$ and
$(z_{2},k_{2}\tau)$ are geometrically distinct if $j\ast z_{1}\neq l\ast
z_{2}$, $j,l\in\mathbf{Z}$.

Now we list our main results of subharmonic solutions as following.

######  Theorem 1.1.

Suppose $H$ satisfies (H1), (H2), (H4), (H5) and

(H3)′ there exist constants $\xi_{i},\eta_{i}>0$ with $\xi_{i}+\eta_{i}=1$
$(i=1,2,\cdots,n)$ such that

$H^{\prime}_{z}(t,z)\cdot V_{2}(z)-H(t,z)\geq
c_{1}|H_{z}^{\prime}(t,z)|-c_{2},\ \ \
(t,z)\in\mathbf{R}\times\mathbf{R}^{2n},$

where $V_{2}=V(\xi,\eta)$ and $\max_{1\leq i\leq
n}\\{\frac{\alpha_{i}}{\beta_{i}},\frac{\beta_{i}}{\alpha_{i}},\frac{\xi_{i}}{\eta_{i}},\frac{\eta_{i}}{\xi_{i}},\frac{{\sigma}_{i}}{{\tau}_{i}},\frac{{\tau}_{i}}{{\sigma}_{i}}\\}<1+\beta$.

Then for each integer $k\geq 1$, the system (1.1) possesses a $k\tau$-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.

Note that (H3)′ is weaker than (H3), so we have a similar result stated as a
corollary of Theorem 1.1.

######  Corollary 1.1.

Suppose $H$ satisfies (H1)-(H5), then we have the same results as in Theorem
1.1.

######  Theorem 1.2.

The conclusions of Theorem 1.1 still hold if $H$ satisfies the conditions

(C1) $H\in C^{2}(\mathbf{R}\times\mathbf{R}^{2n},\mathbf{R})$ is nonnegative
and $\tau$-periodic with respect to $t$,

(C2) there exist constants $0<\theta<1$, $R,\varphi_{i},\psi_{i}>0$ with
$\frac{1}{\varphi_{i}}+\frac{1}{\psi_{i}}=1$ $(i=1,2,\cdots,n)$ such that, by
setting $V_{3}=V(\varphi^{-1},\psi^{-1})$, we have

$\theta H^{\prime}_{z}(t,z)\cdot V_{3}(z)\geq H(t,z)>0,\ \ \
(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}\ \text{with}\ |z|\geq R,$

(C3) there exist constants $b_{1}>0$ and $b_{2}>0$ such that

$|H^{\prime}_{z}(t,z)|\leq b_{1}H^{\prime}_{z}(t,z)\cdot V_{3}(z)+b_{2},\ \ \
(t,z)\in\mathbf{R}\times\mathbf{R}^{2n},$

(C4)
$\displaystyle\frac{H(t,z)}{\sum_{i=1}^{n}\left(|p_{i}|^{\varphi_{i}}+|q_{i}|^{\psi_{i}}\right)}\rightarrow
0$ as $|z|\rightarrow 0$ uniformly in $t$.

The above conditions (C1)-(C4) are similar to that of [2] with minor
difference.

######  Remark 1.1.

In the case where $H(t,z)=\frac{1}{2}(\hat{B}(t)z,z)+\hat{H}(t,z)$ with
$\hat{B}(t)$ being a $\tau$-periodic, continuous symmetric matrix function and
$\hat{H}$ 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 $\hat{B}$ 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 $(z,\tau)$
obtained from the homological link method in fact is the minimal periodic
solution of the Hamiltonian systems provided the Hessian
$H_{zz}^{\prime\prime}(z)$ is positively definite for
$z\in\mathbf{R}^{2n}\setminus\\{0\\}$.

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 [6, 7, 4, 25, 26, 17, 18, 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] from which we can
find a critical point of the corresponding functional together with index
information. Under the conditions as in Theorems 1.1, 1.2 and Corollary 1.1,
we show that there is a homologically link structure for the functional. In
Section 3, we give a proof of Theorems 1.1, 1.2 and Corollary 1.1. In Section
4, we consider the existence of subharmonic solutions of the Hamiltonian
systems in the case where $H$ may contain a quadratic term as
$H(t,z)=\frac{1}{2}(\hat{B}(t)z,z)+\hat{H}(t,z)$. We consider minimal periodic
problem for the autonomous Hamiltonian systems in Section 5.

##  2 Preliminaries

We first recall the notion of Maslov-type index and some iteration estimates.
We refer to [1], [21] and [24] for details.

Denote by $Sp(2n)=\\{M\in\mathcal{L}(\mathbf{R}^{2n})|M^{t}JM=J\\}$ the
$2n\times 2n$ symplectic group, where $M^{t}$ denotes the transpose of $M$.
Define $\mathcal{P}(2n)=\\{\gamma|\gamma\in
C([0,\tau],Sp(2n)),\;\;\gamma(0)=I\\}$.

For $\gamma\in\mathcal{P}(2n)$, according to [1] and [24], there is a Maslov-
type index theory which assigns to $\gamma$ a pair of integers

$(i_{\tau},\nu_{\tau}):=(i_{\tau}(\gamma),\nu_{\tau}(\gamma))\in\mathbf{Z}\times\\{0,1,\cdots,2n\\},$

where $i_{\tau}$ is the index part of $\gamma$ and $\nu_{\tau}$ is the
nullity.

For $\gamma\in\mathcal{P}(2n)$, define $\gamma^{k}:[0,k\tau]\rightarrow
Sp(2n)$ by

$\gamma^{k}(t)=\gamma(t-j\tau)\gamma(\tau)^{j},\ \ \ \ j\tau\leq
t\leq(j+1)\tau,\ 0\leq j\leq k-1.$

We denote the Maslov index of $\gamma^{k}$ on the interval $[0,k\tau]$ by
$(i_{k\tau},\nu_{k\tau}):=(i_{k\tau}(\gamma^{k}),\nu_{k\tau}(\gamma^{k}))$.

In the case of linear Hamiltonian systems

$\left\\{\begin{array}[]{l}-J\dot{z}=B(t)z\\\ z(0)=z(\tau),\\\
\end{array}\right.$

where $B(t)$ is a $\tau$-periodic, symmetric and continuous matrix function.
Its fundamental solution is denoted by $\gamma_{B}\in C([0,\tau],Sp(2n))$ with
$\gamma(0)=I_{2n}$. The Maslov-type index
$(i_{\tau}(B),\nu_{\tau}(B)):=(i_{\tau}(\gamma_{B}),\nu_{\tau}(\gamma_{B}))$
is also called the Maslov-type index of the matrix function $B(t)$.

If $z$ is a $\tau$-periodic solution of the system (1.1), we denote by
$(i_{\tau}(z),\nu_{\tau}(z)):=(i_{\tau}(B),\nu_{\tau}(B))$ with
$B(t)=H_{z}^{\prime\prime}(t,z(t))$. The solution $z$ is non-degenerate if
$\nu_{\tau}(z)=0$.

######  Proposition 2.1.

([18]) If $z$ is a $k\tau$-periodic solution of the system (1.1), then
$i_{k\tau}(j\ast z)=i_{k\tau}(z)$ and $\nu_{k\tau}(j\ast z)=\nu_{k\tau}(z)$
hold for $0\leq j\leq k$.

######  Proposition 2.2.

([18]) For $k\in\mathbf{N}$, there holds

$k(i_{\tau}+\nu_{\tau}-n)-n\leq i_{k\tau}\leq k(i_{\tau}+n)+n-\nu_{k\tau}.$

######  Proposition 2.3.

([21]) For $m\in\mathbf{N}$, there holds

$m(i_{\tau}+\nu_{\tau}-n)+n-\nu_{\tau}\leq i_{m\tau}\leq
m(i_{\tau}+n)-n-(\nu_{m\tau}-\nu_{\tau}).$

######  Lemma 2.1.

([1]) Let $B(t)$ be a $\tau$-periodic, symmetric and continuous matrix
function. Assume $B(t)$ are positive for $t\in[0,\tau]$ and $B(t_{0})$ is
strictly positive for some $t_{0}\in[0,\tau]$. Then $i_{\tau}(B)\geq n$.

######  Lemma 2.2.

([5]) Let $B(t)$ be a $\tau$-periodic, symmetric and continuous matrix
function. Assume for some $k\in\mathbf{N}$, there hold $i_{k\tau}(B)\leq n+1$,
$i_{\tau}(B)\geq n$ and $\nu_{\tau}(B)\geq 1$. Then $k=1.$

Now we introduce some concepts and results of Sobolev space theory.

Let $E=W^{\frac{1}{2},2}(S_{\tau},\mathbf{R}^{2n})=\bigg{\\{}z\in
L^{2}(S_{\tau},\mathbf{R}^{2n})\big{|}\displaystyle\sum_{j\in\textbf{Z}}|j||a_{j}|^{2}<+\infty\bigg{\\}}$,
where $S_{\tau}:=\mathbf{R}/\tau\textbf{Z}$,
$z(t)=\sum_{k\in\textbf{Z}}\exp{(\frac{2k\pi
t}{\tau}J)}a_{k},a_{k}\in\mathbf{R}^{2n}$.

For $\zeta\in E$, $\zeta(t)=\sum_{k\in\textbf{Z}}\exp{(\frac{2k\pi
t}{\tau}J)}b_{k},b_{k}\in\mathbf{R}^{2n}$, the inner product on $E$ is

$\langle
z,\zeta\rangle=\tau(a_{0},b_{0})+\tau\sum_{k\in\textbf{Z}}|k|a_{k}\cdot
b_{k}.$

######  Lemma 2.3.

([30]) (Embedding Theorem) The space $E$ compactly embeds into
$L^{s}(S_{\tau},\mathbf{R}^{2n})$ $(s\geq 1)$, in particular, there exists a
constant $C_{s}>0$ such that $\|z\|_{L^{s}}\leq C_{s}\|z\|$ holds for $z\in
E$, where $\|\cdot\|$ denotes the norm on $E$.

There exists a linear bounded self-adjoint operators $A$ such that $\langle
Az,\zeta\rangle=2\pi\sum_{k\in\textbf{Z}}ka_{k}\cdot b_{k}$, obtained by
extending the bilinear form $\langle
Az,\zeta\rangle=\int^{2\pi}_{0}(-J\dot{z},\zeta)\text{d}t$, $z,\zeta\in
W^{1,2}(S_{\tau},\mathbf{R}^{2n})$.

Set $E^{\pm}=\\{z\in E|z(t)=\sum_{\pm k>0}\exp{(\frac{2k\pi
t}{\tau}J)}a_{k},a_{k}\in\mathbf{R}^{2n}\\}$ and $E^{0}=\mathbf{R}^{2n}$, then
$Az^{\pm}=\pm\frac{2\pi}{\tau}z^{\pm}$, $z^{\pm}\in E^{\pm}$ (see [18]).
Moreover, we set $E_{m}=\\{z\in E|z(t)=\sum_{k=-m}^{m}\exp{(\frac{2k\pi
t}{\tau}J)}a_{k},a_{k}\in\mathbf{R}^{2n}\\}$ and $E_{m}^{\pm}=E^{\pm}\bigcap
E_{m}$, and let $P_{m}$ be the corresponding orthogonal projection.

For $d>0$, we denote by $M_{d}^{+}(C)$, $M_{d}^{-}(C)$ and $M_{d}^{0}(C)$ the
eigenspaces of any linear bounded self-adjoint Fredholm operator $C$
corresponding to the eigenvalue $\lambda$ belonging to $(d,+\infty)$,
$(-\infty,d)$ and $[-d,d]$ respectively.

Given $B(t)$ a $\tau$-periodic, symmetric and continuous matrix function with
Maslov-type index $(i_{\tau}(B),\nu_{\tau}(B))$, define $\langle
Bz,\zeta\rangle=\int^{\tau}_{0}(B(t)z,\zeta)\text{d}t$, $z,\zeta\in E$. Set
$(A-B)^{\sharp}:=(A-B|_{R(A-B)})^{-1}$, then we have the following theorem.

######  Lemma 2.4.

([10]) Suppose $0<d<\frac{1}{4}\|(A-B)^{\sharp}\|^{-1}$, then for $m$ large
enough, there hold

$\text{dim}M_{d}^{+}(P_{m}(A-B)P_{m})=\frac{1}{2}\text{dim}(P_{m}E)-i_{\tau}(B)-\nu_{\tau}(B),$
$\text{dim}M_{d}^{-}(P_{m}(A-B)P_{m})=\frac{1}{2}\text{dim}(P_{m}E)+i_{\tau}(B),$
$\text{dim}M_{d}^{0}(P_{m}(A-B)P_{m})=\nu_{\tau}(B).$

For $e\in E_{1}\bigcap E^{+}$ with $\|e\|=1$, set
$W=\big{\\{}z\in\text{span}\\{e\\}\bigoplus E^{-}\bigoplus
E^{0}\big{|}1\leq\|z\|\leq 2,\|z^{-}\|\leq\|z^{+}+z^{0}\|\big{\\}}$, then we
have

######  Lemma 2.5.

([9]) There exists a constant $\varepsilon_{1}>0$ such that

$\emph{measure}\left\\{t\in[0,T]\big{|}|z(t)|\geq\varepsilon_{1}\right\\}\geq\varepsilon_{1},\
\ \ \ z\in W.$

Finally, we recall the homologically link theorem in [1].

######  Definition 2.1.

([3]) Let $Q$ be a topologically embedded closed $q$-dimensional ball on a
Hilbert manifold $M$ and let $S\subset M$ be a closed subset such that
$\partial Q\bigcap S=\emptyset$. We say that $\partial Q$ and $S$
homotopically link if $\varphi(Q)\bigcap S\neq\emptyset$ for $\varphi\in
C(Q,M)$ with $\varphi|_{\partial Q}=\emph{id}|_{\partial Q}$.

######  Definition 2.2.

([1]) Let $Q$ be a topologically embedded closed $q$-dimensional ball on a
Hilbert manifold $M$ and let $S\subset M$ be a closed subset such that
$\partial Q\bigcap S=\emptyset$. We say that $\partial Q$ and $S$
homologically link if $\partial Q$ is the support of a non-vanishing homology
class in $H_{q-1}(M\backslash S)$.

######  Lemma 2.6.

Let $M=M_{1}\bigoplus M_{2}$ be a Hilbert space with dim$M_{2}=q-1$,
$S=\partial B_{\mu}\bigcap M_{1}$ and $Q=(\bar{B}_{\nu}\bigcap
M_{2})\bigoplus[0,\nu]e$, where $e\in M_{1}$ with $\|e\|=1$ and $\nu>\mu>0$.
Let $B_{\mu}$, $B_{\nu}$ be two bounded linear invertible operators on $M$
such that $\nu>\mu\|B_{\nu}^{-1}B_{\mu}\|$ and
$PB_{\mu}^{-1}B_{\nu}:M_{2}\rightarrow M_{2}$ is invertible, where
$P:M\rightarrow M_{2}$ is the orthogonal projection. Then $B_{\nu}(\partial
Q)$ and $B_{\mu}(S)$ homologically link.

###### Proof.

It is easy to prove $B_{\nu}(\partial Q)$ and $B_{\mu}(S)$ homotopically link
(see [13]).

Indeed, to show $B_{\nu}(Q)\bigcap B_{\mu}(S)\neq\emptyset$, it is equivalent
to proving $\psi_{0}(t,v)=(\mu,0)$ has a solution in
$[0,\nu]\times(\bar{B}_{\nu}\bigcap M_{2})$, where

$\psi_{0}(t,v)=(\|B_{\mu}^{-1}B_{\nu}(te+v)\|,P_{2}B_{\mu}^{-1}B_{\nu}(te+v)),\
\ \ \ (t,v)\in[0,\nu]\times(\bar{B}_{\nu}\bigcap M_{2}).$

Note that
$t=\mu\|B_{\mu}^{-1}B_{\nu}e-B_{\mu}^{-1}B_{\nu}B_{0}^{-1}B_{0}e\|^{-1}$
$(<\nu)$ and $v=-tB_{0}^{-1}B_{0}e$ $(\in B_{\nu}\bigcap M_{2})$ is the unique
solution of $\psi_{0}$ in $[0,\nu]\times(\bar{B}_{\nu}\bigcap M_{2})$ , where
$B_{0}=P_{2}B_{\mu}^{-1}B_{\nu}$ and $B_{0}^{-1}$ denotes the inverse of
$B_{0}|_{M_{2}}$. Thus
$(\mu,0)\notin\psi_{0}(\partial([0,\nu]\times(\bar{B}_{\nu}\bigcap M_{2})))$,
deg$(\psi_{0},(0,\nu)\times(B_{\nu}\bigcap M_{2}),(\mu,0))=\pm 1$ and
$B_{\nu}(\partial Q)\bigcap B_{\mu}(S)=\emptyset$.

For $\varphi\in C(B_{\nu}(Q),M)$ with $\varphi|_{B_{\nu}(\partial
Q)}=\emph{id}|_{B_{\nu}(\partial Q)}$, define
$\psi:[0,\nu]\times(\bar{B}_{\nu}\bigcap M_{2})\rightarrow\mathbf{R}\times
M_{2}$ as

$\psi(t,v)=(\|B_{\mu}^{-1}\varphi B_{\nu}(te+v)\|,P_{2}B_{\mu}^{-1}\varphi
B_{\nu}(te+v)).$

In order to show $\varphi(B_{\nu}(Q))\bigcap B_{\mu}(S)\neq\emptyset$, it is
equivalent to proving $\psi(t,v)=(\mu,0)$ has a solution in
$[0,\nu]\times(\bar{B}_{\nu}\bigcap M_{2})$.

Since $\psi=\psi_{0}$ on $\partial([0,\nu]\times(\bar{B}_{\nu}\bigcap
M_{2}))$, by the Brouwer degree theory, deg$(\psi,(0,\nu)\times(B_{\nu}\bigcap
M_{2}),(\mu,0))$= deg$(\psi_{0},(0,\nu)\times(B_{\nu}\bigcap
M_{2}),(\mu,0))=\pm 1$, then $\psi(t,v)=(\mu,0)$ has a solution in
$[0,\nu]\times(\bar{B}_{\nu}\bigcap M_{2})$. Hence, $B_{\nu}(\partial Q)$ and
$B_{\mu}(S)$ homotopically link.

From Theorem II.1.2 in [3], we see $B_{\nu}(\partial Q)$ and $B_{\mu}(S)$
homologically link. ∎

Let $f$ be a $C^{2}$ functional on a Hilbert manifold $M$. Recall that the
Morse index $m(x)$ of $f$ at a critical point $x$ is the dimension of a
maximal subspace on which $D^{2}f(x)$ is strictly negative definite, while the
large Morse index $m^{*}(x)$ is $m(x)$+dim Ker$D^{2}f(x)$.

######  Lemma 2.7.

([1]) Let $f$ be a $C^{2}$ functional on a Hilbert manifold $M$ with Fredholm
gradient. Let $Q\subset M$ be a topologically embedded closed $q$-dimensional
ball and let $S\subset M$ be a closed subset such that $\partial Q\bigcap
S=\emptyset$. Assume that $\partial Q$ and $S$ homologically link. Moreover,
assume

(i) $\sup_{\partial Q}f<\inf_{S}f$,

(ii) $f$ satisfies (PS) condition on some open interval containing
$[\inf_{S}f,\sup_{Q}f]$.

Then, if $\Gamma$ denotes the set of all $q$-chains in $M$ whose boundary has
support $\partial Q$, the number
$c=\inf_{\xi\in\Gamma}\sup_{|\xi|}f\in[\inf_{S}f,\sup_{Q}f]$ is a critical
value of $f$, where $|\xi|$ denotes the support of the chain $\xi$. Moreover,
$f$ has a critical point $\bar{x}$ such that $f(\bar{x})=c$ and
$m(\bar{x})\leq q\leq m^{*}(\bar{x})$.

######  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) on $J\subset\mathbf{R}\bigcup\\{\pm\infty\\}$ if
$\\{z_{m}\\}\subset M$ such that $f(z_{m})\rightarrow c\in J$ and
$(1+\|z_{m}\|)\|\nabla f(z_{m})\|\rightarrow 0$ as $m\rightarrow+\infty$ has a
convergent subsequence.

##  3 Proofs of the Main Results

For simplicity, we first give a proof of Corollary 1.1.

Define $f(z)=\frac{1}{2}\langle Az,z\rangle-\int_{0}^{\tau}H(t,z)\text{d}t$,
$z\in E$, by (H3), we have $f\in C^{2}(E,\mathbf{R})$. As usual, finding
periodic solutions of the system (1.1) converts to looking for critical points
of $f$.

Let $f_{m}=f|_{E_{m}}$, $X_{m}=E_{m}^{-}\bigoplus E^{0}$ and
$Y_{m}=E_{m}^{+}$. Now we check the conditions in Lemma 2.7 for $f_{m}$ when
$H$ satisfies (H1)-(H5). The proofs are similar to those in [32, 34].

######  Lemma 3.1.

The functional $f$ satisfies (C)∗ condition with respect to
$\\{E_{m}|m=1,2,\cdots\\}$, that is, any sequence $\\{z_{m}\\}$ such that
$z_{m}\in E_{m}$, $\\{f_{m}(z_{m})\\}$ is bounded and $(1+\|z_{m}\|)\|\nabla
f_{m}(z_{m})\|\rightarrow 0$ as $m\rightarrow+\infty$ has a convergent
subsequence.

###### Proof.

Let $\\{z_{m}\\}$ be such a sequence, we only need to prove $\\{z_{m}\\}$ is
bounded. Otherwise, we may suppose $\|z_{m}\|\rightarrow+\infty$ as
$m\rightarrow+\infty$.

Note that for $z\in E_{m}$, we have $\nabla f_{m}(z)=P_{m}\nabla f(z)$. By
using $\alpha_{i}+\beta_{i}=1$ and the integration by parts, there hold

$f_{m}(z)-\langle\nabla f_{m}(z),V_{1}(z)\rangle=f(z)-\langle\nabla
f(z),V_{1}(z)\rangle=\int_{0}^{\tau}(H^{\prime}_{z}(t,z)\cdot
V_{1}(z)-H(t,z))\text{d}t,$

where $V_{1}(z)$ is defined in (H2). By (H2), we have

$\displaystyle f_{m}(z_{m})-\langle\nabla f_{m}(z_{m}),V_{1}(z_{m})\rangle$
(3.1) $\displaystyle=$
$\displaystyle\int_{0}^{\tau}(H^{\prime}_{z}(t,z_{m})\cdot
V_{1}(z_{m})-H(t,z_{m}))\text{d}t$ $\displaystyle\geq$ $\displaystyle
c_{1}\|z_{m}\|_{L^{\beta}}^{\beta}-\tau c_{2}.$

Hence $\\{\|z_{m}\|_{L^{\beta}}\\}$ is bounded.

By using the constants $\beta,\;\lambda$ defined in (H2), (H3), we set
$p=\frac{2\beta+1}{2\lambda-1}$. It is obvious that $p>1$. Take $q$ such that
$\frac{1}{p}+\frac{1}{q}=1$. It is easy to see
$\lambda-\frac{\beta}{p}=\frac{\lambda+\beta}{2\beta+1}<1$ and
$2q(\lambda-\frac{\beta}{p})=\frac{\lambda+\beta}{\beta+1-\lambda}>1$. By
(H3), Lemma 2.3 and (3.1), we have

$\displaystyle\int_{0}^{\tau}|H^{\prime}_{z}(t,z_{m})\cdot
z^{\pm}_{m}|\text{d}t$ (3.2) $\displaystyle\leq$ $\displaystyle
c_{3}\int_{0}^{\tau}|z_{m}|^{\lambda}|z^{\pm}_{m}|\text{d}t+c_{4}\|z^{\pm}_{m}\|$
$\displaystyle=$ $\displaystyle
c_{3}\int_{0}^{\tau}|z_{m}|^{\frac{\beta}{p}}|z|^{\lambda-\frac{\beta}{p}}|z^{\pm}_{m}|\text{d}t+c_{4}\|z^{\pm}_{m}\|$
$\displaystyle\leq$ $\displaystyle
c_{3}\left(\int^{\tau}_{0}|z_{m}|^{\beta}\text{d}t\right)^{\frac{1}{p}}\left(\int^{\tau}_{0}|z_{m}|^{\left(\lambda-\frac{\beta}{p}\right)q}|z^{\pm}_{m}|^{q}\text{d}t\right)^{\frac{1}{q}}+c_{4}\|z^{\pm}_{m}\|$
$\displaystyle\leq$ $\displaystyle
c_{5}\left(\int^{\tau}_{0}|z_{m}|^{\left(\lambda-\frac{\beta}{p}\right)2q}\text{d}t\right)^{\frac{1}{2q}}\left(\int^{\tau}_{0}|z^{\pm}_{m}|^{2q}\text{d}t\right)^{\frac{1}{2q}}+c_{4}\|z^{\pm}_{m}\|$
$\displaystyle\leq$ $\displaystyle
c_{6}\|z_{m}\|^{\lambda-\frac{\beta}{p}}\|z^{\pm}_{m}\|+c_{6}\|z^{\pm}_{m}\|,$

where $c_{i}>0$ are suitable constants.

By (3.2), we have

$\displaystyle\|\nabla f_{m}(z_{m})\|\cdot\|z_{m}^{\pm}\|$ $\displaystyle\geq$
$\displaystyle\pm\langle\nabla f_{m}(z_{m}),z_{m}^{\pm}\rangle$ (3.3)
$\displaystyle=$ $\displaystyle\pm\langle
Az_{m},z_{m}^{\pm}\rangle\mp\int_{0}^{\tau}H^{\prime}_{z}(t,z_{m})\cdot
z^{\pm}_{m}\text{d}t$ $\displaystyle\geq$
$\displaystyle\frac{2\pi}{\tau}\|z^{\pm}_{m}\|^{2}-c_{6}\|z_{m}\|^{\lambda-\frac{\beta}{p}}\|z^{\pm}_{m}\|-c_{6}\|z^{\pm}_{m}\|.$

We can suppose that there are only finitely many $z_{m}^{\pm}=0$. Dividing the
two sides by $\|z_{m}\|\cdot\|z_{m}^{\pm}\|$, it implies
$\frac{\|z^{\pm}_{m}\|}{\|z_{m}\|}\rightarrow 0$ as $m\rightarrow+\infty$. By
(3.1), we see $\frac{|z_{m}^{0}|}{\|z_{m}\|}\rightarrow 0$,
$m\rightarrow+\infty$. From
$\frac{\|z^{+}_{m}\|^{2}+\|z^{-}_{m}\|^{2}+\tau|z_{m}^{0}|^{2}}{\|z_{m}\|^{2}}=1$,
we obtain a contradiction. ∎

We note that if $f$ satisfies (C)∗ condition on $E$, then $f_{m}$ satisfies
(C) condition on $E_{m}$.

There exists a constant $\eta>0$ such that
$\tilde{\sigma}_{i}=\frac{\eta{\sigma}_{i}}{{\sigma}_{i}+{\tau}_{i}}\geq 1$
and $\tilde{\tau}_{i}=\frac{\eta{\tau}_{i}}{{\sigma}_{i}+{\tau}_{i}}\geq 1$.
For $\rho>0$ and $z=(p_{1},\cdots,p_{n},q_{1},\cdots,q_{n})\in E$, we set
$B_{\rho}(z)=(\rho^{\tilde{\tau}_{1}-1}p_{1},\cdots,\rho^{\tilde{\tau}_{n}-1}p_{n},\rho^{\tilde{\sigma}_{1}-1}q_{1}\cdots,\rho^{\tilde{\sigma}_{n}-1}q_{n}).$
We note that $B_{\rho}$ is a linear bounded and invertible operator and
$\|B_{\rho}\|\leq 1$, if $\rho\leq 1$.

For $z=z^{+}+z^{0}+z^{-}\in E$, we have

$\langle AB_{\rho}z,B_{\rho}z\rangle=\rho^{\eta-2}\langle
Az,z\rangle=\frac{2\pi}{\tau}\rho^{\eta-2}(\|z^{+}\|^{2}-\|z^{-}\|^{2}).$
(3.4)

######  Lemma 3.2.

There exist $\mu\in(0,1)$ and $\delta>0$ independent of $m$ such that
$\inf_{B_{\mu}(S_{m})}f_{m}\geq\delta$, where $S_{m}=S\cap Y_{m}$ and
$S=\\{z\in E^{+}|\|z\|=\mu\\}$.

###### Proof.

It suffices to show $\inf_{B_{\mu}(S)}f\geq\delta$.

By (H3) and (H4), for any $\varepsilon>0$, there exists $M_{\varepsilon}>0$
such that

$H(t,z)\leq\varepsilon\sum_{i=1}^{n}\left(|p_{i}|^{1+\frac{\sigma_{i}}{\tau_{i}}}+|q_{i}|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right)+M_{\varepsilon}\sum_{i=1}^{n}\left(|p_{i}|^{1+\lambda}+|q_{i}|^{1+\lambda}\right),\
\ \ \ (t,z)\in\mathbf{R}\times\mathbf{R}^{2n}.$ (3.5)

By (3.5), for $z=(p_{1},\cdots,p_{n},q_{1},\cdots,q_{n})\in E$, $\|z\|=\mu$,
we have

$\displaystyle\int_{0}^{\tau}H(t,B_{\mu}z)\text{d}t$ (3.6) $\displaystyle\leq$
$\displaystyle\varepsilon\sum_{i=1}^{n}\int_{0}^{\tau}\left(|\mu^{\tilde{\tau}_{i}-1}p_{i}|^{1+\frac{\sigma_{i}}{\tau_{i}}}+|\mu^{\tilde{\sigma}_{i}-1}q_{i}|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right)\text{d}t+M_{\varepsilon}\sum_{i=1}^{n}\int_{0}^{\tau}\left(|\mu^{\tilde{\tau}_{i}-1}p_{i}|^{1+\lambda}+|\mu^{\tilde{\sigma}_{i}-1}q_{i}|^{1+\lambda}\right)\text{d}t$
$\displaystyle\leq$
$\displaystyle\varepsilon\sum_{i=1}^{n}\int_{0}^{\tau}\left(\mu^{(\tilde{\tau}_{i}-1)(1+\frac{\sigma_{i}}{\tau_{i}})}|z|^{1+\frac{\sigma_{i}}{\tau_{i}}}+\mu^{(\tilde{\sigma}_{i}-1)(1+\frac{\tau_{i}}{\sigma_{i}})}|z|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right)\text{d}t$
$\displaystyle+M_{\varepsilon}\sum_{i=1}^{n}\int_{0}^{\tau}\left(\mu^{(\tilde{\tau}_{i}-1)(1+\lambda)}|z|^{1+\lambda}+\mu^{(\tilde{\sigma}_{i}-1)(1+\lambda)}|z|^{1+\lambda}\right)\text{d}t$
$\displaystyle\leq$ $\displaystyle
2\varepsilon\mu^{\eta}\sum_{i=1}^{n}C(\sigma_{i},\tau_{i})+M_{\varepsilon}\mu^{\eta}\sum_{i=1}^{n}C(\lambda)\left(\mu^{\tilde{\tau}_{i}(\lambda-\frac{\sigma_{i}}{\tau_{i}})}+\mu^{\tilde{\sigma}_{i}(\lambda-\frac{\tau_{i}}{\sigma_{i}})}\right)$
$\displaystyle\leq$ $\displaystyle
c_{7}\mu^{\eta}\left[2\varepsilon+M_{\varepsilon}\sum_{i=1}^{n}\left(\mu^{\tilde{\tau}_{i}(\lambda-\frac{\sigma_{i}}{\tau_{i}})}+\mu^{\tilde{\sigma}_{i}(\lambda-\frac{\tau_{i}}{\sigma_{i}})}\right)\right],$

where $C(\sigma_{i},\tau_{i}),\;C(\lambda)>0$ are the embedding constants.

By (3.4) and (3.6), for $z\in E^{+}$, $\|z\|=\mu$, we have

$\displaystyle f(B_{\mu}z)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\langle
AB_{\mu}z,B_{\mu}z\rangle-\int_{0}^{\tau}H(t,B_{\mu}z)\text{d}t$ (3.7)
$\displaystyle\geq$
$\displaystyle\frac{\pi}{\tau}\mu^{\eta}-c_{7}\mu^{\eta}\left[2\varepsilon+M_{\varepsilon}\sum_{i=1}^{n}\left(\mu^{\tilde{\tau}_{i}(\lambda-\frac{\sigma_{i}}{\tau_{i}})}+\mu^{\tilde{\sigma}_{i}(\lambda-\frac{\tau_{i}}{\sigma_{i}})}\right)\right].$

Choose $\varepsilon>0$ and $0<\mu<1$ so small that
$f(B_{\mu}z)\geq\delta:=\frac{\pi}{3\tau}\mu^{\eta}$ for $z\in E^{+}$ and
$\|z\|=\mu$. Thus $\inf_{B_{\mu}(S)}f\geq\delta>0$. ∎

######  Lemma 3.3.

Set $Q=[\bar{B}_{\nu}\bigcap(E^{-}\bigoplus E^{0})]\bigoplus[0,\nu]e$ and
$Q_{m}=Q\bigcap(X_{m}\bigoplus\text{span}\\{e\\})$, where
$\bar{B}_{\nu}=\\{z\in E|\|z\|\leq\nu\\}$. For any $\nu$ with $\nu>\mu>0$, we
have $B_{\nu}(\partial Q_{m})$ and $B_{\mu}(S_{m})$ homologically link.

###### Proof.

Since $\nu>\mu>0$, then
$\nu>\mu\|B_{\nu}^{-1}B_{\mu}\|=\mu\|B_{\frac{\mu}{\nu}}\|$ and
$PB^{-1}_{\mu}B_{\nu}:E^{-}\bigoplus E^{0}\rightarrow E^{-}\bigoplus E^{0}$ is
linear bounded and invertible (see [2, 32]), where $P:E\rightarrow
E^{-}\bigoplus E^{0}$ denotes the orthogonal projection. Furthermore, by
noting that $B_{i}(E_{m})\subset E_{m}$ $(i=\mu,\nu)$, and
$B_{i}|_{E_{m}}:E_{m}\rightarrow E_{m}$ is linear bounded and invertible, then
$\tilde{P}_{m}(B_{\mu}|_{E_{m}})^{-1}B_{\nu}|_{E_{m}}:X_{m}\rightarrow X_{m}$
is linear bounded and invertible, where $\tilde{P}_{m}:E_{m}\rightarrow X_{m}$
is the orthogonal projection. From Lemma 2.6, we complete the proof. ∎

For $\varepsilon_{1}>0$ as in Lemma 2.5, we set
$\frac{2\pi}{\tau}\cdot\sqrt{2n}A^{-1}_{1}=\varepsilon_{1}\min_{1\leq i\leq
n}\left\\{\left(\frac{\varepsilon_{1}}{\sqrt{2n}}\right)^{1+\frac{\sigma_{i}}{\tau_{i}}},\left(\frac{\varepsilon_{1}}{\sqrt{2n}}\right)^{1+\frac{\tau_{i}}{\sigma_{i}}}\right\\}$,
by (H5), there exists a constant $A_{2}>0$ such that

$H(t,z)\geq
A_{1}\sum_{i=1}^{n}\left(|p_{i}|^{1+\frac{\sigma_{i}}{\tau_{i}}}+|q_{i}|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right),\
\ \ \ (t,z)\in\mathbf{R}\times\mathbf{R}^{2n}\ \text{with}\ |z|\geq A_{2}.$
(3.8)

######  Lemma 3.4.

Choose $\nu>\frac{A_{2}}{\varepsilon_{1}}+1$, then $f_{m}|_{B_{\nu}(\partial
Q_{m})}\leq 0$.

###### Proof.

Since $\partial Q_{m}\subset\partial Q$, we show $f|_{B_{\nu}(\partial Q)}\leq
0$.

For $z\in\partial Q$, $z=se+z^{-}+z^{0}$, then $f(B_{\nu}z)\leq 0$. We show
this in two cases.

Case 1. If $s=0$, by (H1) and (3.4), we have $f(B_{\nu}z)\leq 0$.

Case 2. If $s\neq 0$, then $s=\nu$ and $\|z^{-}+z^{0}\|\leq\nu$ or $0\leq
s\leq\nu$ and $\|z^{-}+z^{0}\|=\nu$. In the two situations, we always have
$\nu\leq\|z\|\leq 2\nu$. We now consider two subcases.

Subcase 1. If $\|se+z^{0}\|<\|z^{-}\|$, so $\|se\|<\|z^{-}\|$, by (H1) and
(3.4), then $f(B_{\nu}z)\leq 0$.

Subcase 2. If $\|se+z^{0}\|\geq\|z^{-}\|$, set
$\Omega_{z}=\\{t\in[0,\tau]||z(t)|\geq\nu\varepsilon_{1}\\}$, Lemma 2.5 shows
that measure $\Omega_{z}\geq\varepsilon_{1}$. From definition, we have

$\frac{\sqrt{2n}}{\nu\varepsilon_{1}}|z(t)|\geq\sqrt{2n},\quad t\in\Omega_{z}$
(3.9)

and

$|B_{\nu}z(t)|\geq|z(t)|\geq\nu\varepsilon_{1}>A_{2},\quad t\in\Omega_{z}.$
(3.10)

From (3.9) and Remark 1.4 of [32], there hold

$\displaystyle\sum_{i=1}^{n}\left(\left|\frac{\sqrt{2n}}{\nu\varepsilon_{1}}p_{i}(t)\right|^{1+\frac{\sigma_{i}}{\tau_{i}}}+\left|\frac{\sqrt{2n}}{\nu\varepsilon_{1}}q_{i}(t)\right|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right)$
(3.11) $\displaystyle\geq$
$\displaystyle\frac{1}{2n}\sum_{i=1}^{n}\left(\frac{\sqrt{2n}}{\nu\varepsilon_{1}}|p_{i}(t)|+\frac{\sqrt{2n}}{\nu\varepsilon_{1}}|q_{i}(t)|\right)\geq\frac{1}{2n}\frac{\sqrt{2n}}{\nu\varepsilon_{1}}|z(t)|,\quad
t\in\Omega_{z}.$

By (3.8), (3.10) and (3.11), we have

$\displaystyle H(t,B_{\nu}z(t))$ $\displaystyle\geq$ $\displaystyle
A_{1}\sum_{i=1}^{n}\left(|\nu^{\tilde{\tau}_{i}-1}p_{i}(t)|^{1+\frac{\sigma_{i}}{\tau_{i}}}+|\nu^{\tilde{\sigma}_{i}-1}q_{i}(t)|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right)$
(3.12) $\displaystyle\geq$ $\displaystyle
A_{1}\nu^{\eta}\sum_{i=1}^{n}\left(|\nu^{-1}p_{i}(t)|^{1+\frac{\sigma_{i}}{\tau_{i}}}+|\nu^{-1}q_{i}(t)|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right)$
$\displaystyle\geq$ $\displaystyle A_{1}\nu^{\eta}\min_{1\leq i\leq
n}\left\\{\left(\frac{\varepsilon_{1}}{\sqrt{2n}}\right)^{1+\frac{\sigma_{i}}{\tau_{i}}},\left(\frac{\varepsilon_{1}}{\sqrt{2n}}\right)^{1+\frac{\tau_{i}}{\sigma_{i}}}\right\\}\cdot$
$\displaystyle\sum_{i=1}^{n}\left(\left|\frac{\sqrt{2n}}{\nu\varepsilon_{1}}p_{i}(t)\right|^{1+\frac{\sigma_{i}}{\tau_{i}}}+\left|\frac{\sqrt{2n}}{\nu\varepsilon_{1}}q_{i}(t)\right|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right)$
$\displaystyle\geq$ $\displaystyle A_{1}\nu^{\eta}\min_{1\leq i\leq
n}\left\\{\left(\frac{\varepsilon_{1}}{\sqrt{2n}}\right)^{1+\frac{\sigma_{i}}{\tau_{i}}},\left(\frac{\varepsilon_{1}}{\sqrt{2n}}\right)^{1+\frac{\tau_{i}}{\sigma_{i}}}\right\\}\frac{1}{2n}\frac{\sqrt{2n}}{\nu\varepsilon_{1}}|z(t)|$
$\displaystyle\geq$
$\displaystyle\frac{2\pi}{\tau\varepsilon_{1}}\nu^{\eta},\quad
t\in\Omega_{z}.$

By (H1), (3.4) and (3.12), we have
$f(B_{\nu}z)\leq\frac{2\pi}{\tau}\nu^{\eta}-\int_{\Omega_{z}}H(t,B_{\nu}z(t))\text{d}t\leq
0.$ ∎

######  Theorem 3.1.

If $H$ satisfies (H1)-(H5), then there exists a nonconstant solution $z$ of
the system (1.1) satisfying

$i_{\tau}(z)\leq n+1\leq i_{\tau}(z)+\nu_{\tau}(z).$ (3.13)

###### Proof.

We follow the ideas of [18]. Now Lemmas 3.1-3.4 show that all conditions of
Lemma 2.7 are satisfied for $f_{m}$ (see Remark 2.1). By (3.4) and (H1), we
have $f|_{B_{\nu}(Q)}\leq\frac{2\pi}{\tau}\nu^{\eta}$. So $f_{m}$ has a
critical point $z_{m}$ satisfying

$\delta\leq f_{m}(z_{m})\leq\frac{2\pi}{\tau}\nu^{\eta}\ \ \text{and}\ \
m(z_{m})\leq\ \text{dim}X_{m}+1\leq m^{*}(z_{m}).$ (3.14)

By Lemma 3.1, we may assume $z_{m}\rightarrow z\in E$ with $\delta\leq
f(z)\leq\frac{2\pi}{\tau}\nu^{\eta}$ and $\nabla f(z)=0$. By (H1), we see $z$
is a nonconstant solution of the system (1.1). Now we show that the critical
point $z$ satisfies (3.13).

Let $B$ be the operator for $B(t)=H_{zz}^{\prime\prime}(t,z(t))$ defined in
Section 2, then we have

$\|f^{\prime\prime}(x)-f^{\prime\prime}(z)\|=\|f^{\prime\prime}(x)-(A-B)\|\rightarrow
0,\ \ \ \ \|x-z\|\rightarrow 0.$ (3.15)

Let $0<d<\|(A-B)^{\sharp}\|^{-1}$, by (3.15), there exists a constant
$\kappa>0$ such that

$\|f^{\prime\prime}(x)-(A-B)\|<\frac{d}{3},\ \ \ \
x\in\bar{B}(z,\kappa):=\\{z\in E|\|x-z\|\leq\kappa\\}.$

Then for $m$ large enough, we have

$\|f_{m}^{\prime\prime}(x)-P_{m}(A-B)P_{m}\|<\frac{d}{2},\ \ \ \
x\in\bar{B}(z,\kappa)\bigcap E_{m}.$ (3.16)

For $x\in\bar{B}(z,\kappa)\bigcap E_{m}$, Eq. (3.16) implies that

$\displaystyle\langle f_{m}^{\prime\prime}(x)u,u\rangle$ $\displaystyle\leq$
$\displaystyle\langle
P_{m}(A-B)P_{m}u,u\rangle+\|f_{m}^{\prime\prime}(x)-P_{m}(A-B)P_{m}\|\|u\|^{2}$
$\displaystyle\leq$ $\displaystyle-\frac{d}{2}\|u\|^{2}<0,\ \ \ \ u\in
M_{d}^{-}(P_{m}(A-B)P_{m})\backslash\\{0\\}.$

Thus,

$\text{dim}M^{-}(f_{m}^{\prime\prime}(x))\geq\text{dim}M_{d}^{-}(P_{m}(A-B)P_{m}),\
\ \ \ x\in\bar{B}(z,\kappa)\bigcap E_{m}.$ (3.17)

Similarly, we have

$\text{dim}M^{+}(f_{m}^{\prime\prime}(x))\geq\text{dim}M_{d}^{+}(P_{m}(A-B)P_{m}),\
\ \ \ x\in\bar{B}(z,\kappa)\bigcap E_{m}.$ (3.18)

By (3.14), (3.17) and (3.18), for $m$ large enough, Lemma 2.4 shows that

$\displaystyle\frac{1}{2}\text{dim}E_{m}+n+1$ $\displaystyle=$
$\displaystyle\text{dim}X_{m}+1\geq m(z_{m})$ $\displaystyle\geq$
$\displaystyle\text{dim}M_{d}^{-}(P_{m}(A-B)P_{m})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\text{dim}E_{m}+i_{\tau}(z)$

and

$\displaystyle\frac{1}{2}\text{dim}E_{m}+n+1$ $\displaystyle=$
$\displaystyle\text{dim}X_{m}+1\leq m^{*}(z_{m})$ $\displaystyle\leq$
$\displaystyle\text{dim}M_{d}^{-}(P_{m}(A-B)P_{m})+M_{d}^{0}(P_{m}(A-B)P_{m})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\text{dim}E_{m}+i_{\tau}(z)+\upsilon_{T}(z).$

The above two estimates show that (3.13) holds. ∎

######  Remark 3.1.

Under either the conditions of Theorem 1.1 or Theorem 1.2, the conclusion of
Theorem 3.1 still holds (see Remarks below).

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 $k\tau$-periodic, by Theorem 3.1, the system 1.1 possesses a
nonconstant $k\tau$-periodic solution $z_{k}$ satisfying

$i_{k\tau}(z_{k})\leq n+1\leq i_{k\tau}(z_{k})+\nu_{k\tau}(z_{k}).$ (3.19)

If $z_{k}$ and $z_{pk}$ are not geometrically distinct, by definition, there
exist integers $l$ and $m$ such that $l\ast z_{k}=m\ast z_{pk}$. By
Proposition 2.1, we have $i_{k\tau}(l\ast z_{k})=i_{k\tau}(z_{k})$,
$\nu_{k\tau}(l\ast z_{k})=\nu_{k\tau}(z_{k})$ and $i_{pkT}(m\ast
z_{pk})=i_{pkT}(z_{pk})$, $\nu_{pkT}(m\ast z_{pk})=\nu_{pkT}(z_{pk})$.

Eq. (3.19) shows that $i_{pkT}(z_{pk})\leq n+1$ and
$i_{k\tau}(z_{k})+\nu_{k\tau}(z_{k})\geq n+1$. Proposition 2.2 shows that
$p-n\leq n+1$ contradicting with the assumption $p>2n+1$. Hence if $p>2n+1$,
then $z_{k}$ and $z_{pk}$ are geometrically distinct.

If all $z_{k}$ are non-degenerate, then $\nu_{k\tau}(z_{k})=0$ and
$i_{k\tau}(z_{k})=n+1$ for $k\in\mathbf{N}$. Proposition 2.3 shows that
$p+n\leq n+1$, 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. $\Box$

From Remark 3.1, we see the proofs of Theorem 1.1 and Theorem 1.2 are similar
to the proof of Corollary 1.1.

######  Remark 3.2.

Under the conditions of Theorem 1.1, the conclusion of Theorem 3.1 still
holds.

Indeed, for any $K>0$, we take a cut-off function defined by

$\chi(s)=\left\\{\begin{array}[]{cc}\begin{aligned} 1,\ \ \ &\mbox{$0\leq
s\leq K$,}\\\ 0,\ \ \ &\mbox{$s\geq K+1$,}\end{aligned}\ \ \ \ \mbox{and}\ \
\chi^{\prime}|_{(K,K+1)}<0.\end{array}\right.$

We set $\gamma=\max_{1\leq i\leq
n}\\{\frac{\alpha_{i}}{\beta_{i}},\frac{\beta_{i}}{\alpha_{i}},\frac{\xi_{i}}{\eta_{i}},\frac{\eta_{i}}{\xi_{i}},\frac{{\sigma}_{i}}{{\tau}_{i}},\frac{{\tau}_{i}}{{\sigma}_{i}},\beta-1\\}$.
Choosing $\lambda_{0}\in(\gamma,1+\beta)$ and

$C_{K}\geq\max\bigg{\\{}\max\limits_{{}^{\ \ \ \ t\in\textbf{R}}_{K\leq|z|\leq
K+1}}{\frac{H(t,z)}{|z|^{\lambda_{0}+1}}},\ \ \frac{c_{1}}{\min_{1\leq i\leq
n}\\{\alpha_{i}\lambda_{0}-\beta_{i},\beta_{i}\lambda_{0}-\alpha_{i}\\}},\ \
A_{1}\bigg{\\}},$

where $A_{1}$ is defined in (3.8), $c_{1},\;\alpha_{i},\;\beta_{i}$ are
defined in (H2). For $(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}$, we set

$H_{K}(t,z)=\chi(|z|)H(t,z)+(1-\chi(|z|))C_{K}|z|^{\lambda_{0}+1}.$

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).

Let $f_{K}(z)=\frac{1}{2}\langle
Az,z\rangle-\int^{\tau}_{0}H_{K}(t,z)\text{d}t$, then $f_{K}\in
C^{2}(E,\textbf{R})$.

By the choice of $\lambda_{0}$, there exists a constant $A_{2}>0$ such that

$|z|^{\lambda_{0}+1}\geq\sum_{i=1}^{n}\left(|p_{i}|^{1+\frac{\sigma_{i}}{\tau_{i}}}+|q_{i}|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right),\
\ \ |z|\geq A_{2},$

then we have

$H_{K}(t,z)\geq
A_{1}\sum_{i=1}^{n}\left(|p_{i}|^{1+\frac{\sigma_{i}}{\tau_{i}}}+|q_{i}|^{1+\frac{\tau_{i}}{\sigma_{i}}}\right),\
\ \ \ (t,z)\in\mathbf{R}\times\mathbf{R}^{2n}\ \text{with}\ |z|\geq A_{2}.$

In all the arguments before, we replace $H$, $\lambda$ and $f$ by $H_{K}$,
$\lambda_{0}$ and $f_{K}$ respectively, we see $f_{K}$ possesses a critical
point $z_{K}$ satisfying
$0<\delta_{K}<f_{K}(z_{K})\leq\frac{2\pi}{\tau}\nu^{\eta}$ and
$i_{\tau}(z_{K})\leq n+1\leq i_{\tau}(z_{K})+\nu_{\tau}(z_{K}).$

By (H3)′, it is easy to prove that $z=z_{K}$ is independent of $K$ and a
$\tau$-periodic nonconstant solution of the system (1.1) for $K$ large enough
(see [34]).

######  Remark 3.3.

Under the conditions of Theorem 1.2, the conclusion of Theorem 3.1 still
holds.

In fact, we also take the cut-off function $\chi\in
C^{\infty}([0,+\infty),\mathbf{R})$ as before.

For $R$ defined in (C3), we set $R(K)\geq\max\limits_{{}^{\ \ \ \
t\in\textbf{R}}_{K\leq|z|\leq
K+1}}\frac{H(t,z)}{\sum_{i=1}^{n}(|p_{i}|^{\varphi_{i}}+|q_{i}|^{\psi_{i}})}$
$(K>R)$ and define

$H_{K}(t,z)=\chi(|z|)H(t,z)+(1-\chi(|z|))R(K)\sum_{i=1}^{n}\left(|p_{i}|^{\theta^{-1}\varphi_{i}}+|q_{i}|^{\theta^{-1}\psi_{i}}\right),\;\;(t,z)\in{\mathbf{R}}\times{\mathbf{R}}^{2n}.$

Then $H_{K}$ satisfies (C2) and (C3) with the constants independent of $K$
(see [2]) if $R(K)$ and $R$ are large enough.

Let $f_{K}(z)=\frac{1}{2}\langle
Az,z\rangle-\int^{\tau}_{0}H_{K}(t,z)\text{d}t$, then $f_{K}\in
C^{2}(E,\textbf{R})$. It is easy to show that $f_{K}$ satisfies (PS)∗
condition (see [2]). By the definition of $H_{K}$, we can choose
$\lambda_{1}>\max_{1\leq i\leq
n}\\{\frac{\varphi_{i}}{\psi_{i}},\frac{\psi_{i}}{\varphi_{i}}\\}$ such that
$H_{K}$ satisfies (H4), then $f_{K}|_{B_{\mu(K)}(S)}\geq\delta_{K}>0$ (see
Lemma 3.2).

From [2], we know that there exist constants $d_{1}>0$ and $d_{2}>0$ such that

$\displaystyle H_{K}(t,z)$ $\displaystyle\geq$ $\displaystyle
d_{1}\sum_{i=1}^{n}\left(|p_{i}|^{\theta^{-1}\varphi_{i}}+|q_{i}|^{\theta^{-1}\psi_{i}}\right)-d_{2}$
$\displaystyle=$ $\displaystyle
d_{1}\sum_{i=1}^{n}\left(|p_{i}|^{\theta^{-1}(1+\frac{\varphi_{i}}{\psi_{i}})}+|q_{i}|^{\theta^{-1}(1+\frac{\psi_{i}}{\varphi_{i}})}\right)-d_{2},\
\ \ \ (t,z)\in\mathbf{R}\times\mathbf{R}^{2n},$

which indicates an inequality similar to (3.8).

##  4 The case: $H$ contains a quadratic term

Now we consider the case where
$H(t,z)=\frac{1}{2}(\hat{B}(t)z,z)+\hat{H}(t,z)$. 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.

We set $\omega=\max_{t\in\mathbf{R}}|\hat{B}(t)|$ and suppose $H(t,z)\geq
0,\;(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}$.

######  Theorem 4.1.

Suppose $\hat{H}$ satisfies (H1), (H2), (H3)′, (H4), (H5) and $\hat{B}(t)$
satisfies
$(\hat{B}(t)z,z)=2(\hat{B}(t)z,V_{1}(z)),(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}$
and

(H6) $\hat{B}(t)$ is a $\tau$-periodic, symmetric and continuous matrix
function and satisfies

$(\hat{B}(t)z,z)=2(\hat{B}(t)z,V_{2}(z)),\ \ \ \
(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}.$

We also require there exists an unbounded sequence
$\\{\varrho_{m}\\}\subset(0,+\infty)$ with $\inf_{m}\varrho_{m}=0$ such that

$(\hat{B}(t)B_{\varrho}z,B_{\varrho}z)=\varrho^{\eta-2}(\hat{B}(t)z,z),\ \
(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}$

holds for $\varrho\in\\{\varrho_{m}\\}$, where
$B_{\varrho}z=(\varrho^{\tilde{\tau}_{1}-1}p_{1},\cdots,\varrho^{\tilde{\tau}_{n}-1}p_{n},\varrho^{\tilde{\sigma}_{1}-1}q_{1},\cdots,\varrho^{\tilde{\sigma}_{n}-1}q_{n})$,
$\varrho>0$ with $\eta,\tilde{\sigma}_{i},\tilde{\tau}_{i}$ defined as in
Section 3.

Then for each integer $k\geq 1$ and $k<\frac{2\pi}{\omega\tau}$, 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
$pk<\frac{2\pi}{\omega\tau}$. If all $z_{k}$ are non-degenerate, then $z_{k}$
and $z_{pk}$ $(p>1)$ are geometrically distinct.

Note that (H6) is satisfied if $b_{ij}(t)=0$ whenever $|i-j|\neq n$. If
$\alpha_{i}=\beta_{i}=2$, $\sigma_{i}=\tau_{i}=1$ $(i=1,2,\cdots,n)$, then
$\hat{B}(t)$ is just a $\tau$-periodic, symmetric and continuous matrix
function.

Similarly we have the following results.

######  Corollary 4.1.

Replace (H3)′ with (H3), then we have the same results as in Theorem 4.1.

######  Theorem 4.2.

Suppose $\hat{H}$ satisfies (C1)-(C4) and $\hat{B}(t)$ satisfies

(C5) $\;\hat{B}(t)$ is a $\tau$-periodic, symmetric and continuous matrix
function and satisfies

$(\hat{B}(t)z,z)=2(\hat{B}(t)z,V_{3}(z)),\ \ \ \
(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}.$

Moreover, set $\hat{\eta}=\max_{1\leq i\leq n}\\{\varphi_{i}+\psi_{i}\\}$,
$\hat{\sigma}_{i}=\frac{\hat{\eta}}{\varphi_{i}+\psi_{i}}\varphi_{i}$ and
$\hat{\tau}_{i}=\frac{\hat{\eta}}{\varphi_{i}+\psi_{i}}\psi_{i}$, we require
that there exists an unbounded sequence $\\{\varrho_{m}\\}\subset(0,+\infty)$
with $\inf_{m}\varrho_{m}=0$ such that

$(\hat{B}(t)B_{\varrho}z,B_{\varrho}z)=\varrho^{\hat{\eta}-2}(\hat{B}(t)z,z),\
\ (t,z)\in\mathbf{R}\times\mathbf{R}^{2n},$

holds for $\varrho\in\\{\varrho_{m}\\}$, where
$B_{\varrho}z=(\varrho^{\hat{\tau}_{1}-1}p_{1},\cdots,\varrho^{\hat{\tau}_{n}-1}p_{n},\varrho^{\hat{\sigma}_{1}-1}q_{1},\cdots,\varrho^{\hat{\sigma}_{n}-1}q_{n})$,
$\varrho>0$. Then we have the same results as in Theorem 4.1.

######  Remark 4.1.

If $\varphi_{i}=\psi_{i}$ $(i=1,2,\cdots,n)$, then (C1)-(C5) are the
conditions in [18] with the difference that we require
$H(t,z)=\frac{1}{2}(\hat{B}(t)z,z)+\hat{H}(t,z)\geq 0$ instead of $\hat{B}(t)$
being semi-positive-definite. Thus Theorem 4.2 generalizes the theorems in
[18] in the semi-positive-definite case.

For $z,\zeta\in E$, define $\langle
Bz,\zeta\rangle=\int^{\tau}_{0}(\hat{B}(t)z,\zeta)\text{d}t$, then $B$ is a
linear bounded and self-adjoint operator on $E$ and $|\langle
Bz,z\rangle|\leq\omega\|z\|^{2}.$

######  Remark 4.2.

The key point of the proof of Corollary 4.1 is that if (H6) holds, then we
have
$f(z)-f^{\prime}(z)V_{1}(z)=\int_{0}^{\tau}(\hat{H}^{\prime}_{z}(t,z)\cdot
V_{1}(z)-\hat{H}(t,z))\text{d}t$ and $\langle
BB_{\rho}z,B_{\rho}z\rangle=\rho^{\eta-2}\langle Bz,z\rangle$, where $z\in E$,
and $f$, $\eta$ and $B_{\rho}$ $(\rho>0)$ are defined in Section 3. Note that
$H$ satisfies (H3) if $\hat{H}$ does. Then the proof of (C)∗ condition is the
same as that of Lemma 3.1. We can define $B_{\mu}$ for small
$\mu\in\\{\varrho_{m}\\}$ and $B_{\nu}$ for large $\nu\in\\{\varrho_{m}\\}$ as
in Section 3. So the arguments can be applied to the current case.

The first equation in (H6) implies that
$\hat{B}(t)=\hat{B}(t)V_{1}+V_{1}\hat{B}(t)$, $t\in\mathbf{R}$. For $\beta$ in
(H2), we require that $\beta\geq 2$, so there exists $c_{\beta}>0$ such that
$\|z\|_{L^{\beta}}\geq c_{\beta}\|z\|_{L^{2}}$, $z\in
L^{\beta}(S_{\tau},\mathbf{R}^{2n})$. So similarly we have the following
result.

######  Theorem 4.3.

Suppose $\hat{H}$ satisfies (H1)-(H5) and $\hat{B}(t)$ satisfies

(H6)′ $\hat{B}(t)$ is a $\tau$-periodic, symmetric and continuous matrix
function with
$\max_{t\in\mathbf{R}}|\hat{B}(t)-\hat{B}(t)V_{1}-V_{1}\hat{B}(t)|<c_{1}c_{\beta}$
and satisfies

$\displaystyle\limsup_{\varrho\rightarrow
0^{+}}(\hat{B}(t)B_{\varrho}z,B_{\varrho}z)\leq\omega_{1}\varrho^{\eta-2}\ \
and\ \
\displaystyle\liminf_{\varrho\rightarrow+\infty}(\hat{B}(t)B_{\varrho}z,B_{\varrho}z)\geq\omega_{2}\varrho^{\eta-2}$

hold uniformly for $(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}$ and $|z|=1$,
where $c_{1}$ is as in (H2) and $\omega_{1},\omega_{2}\geq 0$.

Then we have the same results as in Theorem 4.1.

For $\theta,\varphi_{i},\psi_{i}$ $(i=1,2,\cdots,n)$ in (C2), we require that
$\gamma\geq 2\theta$, so there exists $c_{\gamma}>0$ such that
$\|z\|_{L^{\gamma}}\geq c_{\gamma}\|z\|_{L^{2}}$, $z\in
L^{\gamma}(S_{\tau},\mathbf{R}^{2n})$, where $\gamma=\varphi_{i},\psi_{i}$
$(i=1,2,\cdots,n)$. So similarly we also have the following result.

######  Theorem 4.4.

Suppose $H$ satisfies a condition similar to (C3), $\hat{H}$ satisfies
(C1)-(C5) and $\hat{B}(t)$ satisfies

(C5)′ $\hat{B}(t)$ is a $\tau$-periodic, symmetric and continuous matrix
function with
$\max_{t\in\mathbf{R}}|\hat{B}(t)-\hat{B}(t)V_{3}-V_{3}\hat{B}(t)|<c_{1}\min_{1\leq
i\leq n}\\{c_{\varphi_{i}},c_{\psi_{i}}\\}$ and satisfies

$\displaystyle\limsup_{\varrho\rightarrow
0^{+}}(\hat{B}(t)B_{\varrho}z,B_{\varrho}z)\leq\omega_{3}\varrho^{\hat{\eta}-2}\
\ and\ \
\displaystyle\liminf_{\varrho\rightarrow+\infty}(\hat{B}(t)B_{\varrho}z,B_{\varrho}z)\geq\omega_{4}\varrho^{\hat{\eta}-2}$

hold uniformly for $(t,z)\in\mathbf{R}\times\mathbf{R}^{2n}$ and $|z|=1$,
where $\hat{\eta}$ and $B_{\varrho}$ are as in Theorem 4.2 and
$\omega_{3},\omega_{4}\geq 0$.

Then we have the same results as in Theorem 4.1.

##  5 Minimal periodic solutions for the autonomous Hamiltonian systems

In this section, we consider the minimal periodic problem of the following
autonomous Hamiltonian systems

$\left\\{\begin{array}[]{ll}\dot{z}=JH^{\prime}(z),\ \ \ \
z\in\mathbf{R}^{2n},\\\ z(\tau)=z(0).\end{array}\right.$ (5.1)

We say that $(z,\tau)$ is a minimal periodic solution of (5.1) if $z$ solves
the problem (5.1) with $\tau$ being the minimal period of $z$.

As shown in [21], we can also obtain minimal periodic solutions for the
autonomous Hamiltonian systems (5.1).

From Theorem 3.1 and Remark 3.1, we see that for any $\tau>0$ the Hamiltonian
system (5.1) possesses a nontrivial $\tau$-periodic solution $(z,\tau)$
satisfying $i_{\tau}(z)\leq n+1$ provided the function $H$ satisfying the
conditions in one of the Theorems 1.1, 1.2 and Corollary 1.1 with some
necessary modifications in an obvious way (since the function $H$ does not
depend on time $t$). In the following result we understand the statements of
the conditions in this sense.

######  Theorem 5.1.

Suppose the autonomous Hamiltonian function $H(z)$ satisfies the conditions in
one of the Theorems 1.1, 1.2 and Corollary 1.1 and

(H7) $H^{\prime\prime}_{zz}(z)$ is strictly positive for every
$z\in\mathbf{R}^{2n}\setminus\\{0\\}$.

Then $(z,\tau)$ is a minimal periodic solution of the nonlinear Hamiltonian
system (5.1).

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
$(z,\tau)$ now.

Assume $(z,\tau)$ has minimal period $\frac{\tau}{k}$, i.e., its iteration
number is $k\in\mathbf{Z}$. Since the nonlinear Hamiltonian system in (5.1) is
autonomous and (H7) holds, we have $\nu_{\frac{\tau}{k}}(z)\geq 1$ and
$i_{\frac{\tau}{k}}(z)\geq n$ by Lemma 2.1. From Lemma 2.2, we see $k=1$, that
is, the solution $(z,\tau)$ has minimal period $\tau$. $\Box$

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 [19, 15, 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.

## References

  * [1] A. Abbondandolo, Morse theory for Hamiltonian systems, _Chapman, Hall, London_ , 2001.
  * [2] T. An, Z. Wang, _Periodic solutions of Hamiltonian systems with anisotropic growth_ , Comm. Pure Appl. Ana. 9 (2010), 1069-1082.
  * [3] K. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, _Birkh-äuser, Boston_ , 1993.
  * [4] S. Chen, C. Tang, _Periodic and subharmonic solutions of a class of superquadratic Hamiltonian systems_ , J. Math. Anal. Appl. 297 (2004), 267-284.
  * [5] D. Dong, Y. Long, _The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems_ , Trans. Amer. Math. Soc. 349 (1997), 2619-2661.
  * [6] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, _Springer, Berlin_ , 1990.
  * [7] I. Ekeland, H. Hofer, _Subharmonics of convex nonautonomous Hamiltonian systems_ , Comm. Pure Appl. Math. 40 (1987), 1-37.
  * [8] G. Fei, Relative morse index and its application to Hamiltonian systems in the presence of symmetries, J. Differential Equations, 122 (1995), 302-315.
  * [9] G. Fei, _On periodic solutions of superquadratic Hamiltonian systems_ , Electro. J. Differential Equations 8 (2002), 1-12.
  * [10] G. Fei, Q. Qiu, _Periodic solutions of asymptotically linear Hamiltonian systems_ , Chinese Ann. Math. Ser. B 18 (1997), 359-372.
  * [11] G. Fei, Q. Qiu, _Minimal period solutions of nonlinear Hamiltonian systems_ , Nonlinear Analysis, T.M.A. 27 (1996), 821-839.
  * [12] G. Fei, S. Kim, T. Wang, _Minimal period estimates of period solutions for superquadratic Hamiltonian systmes_ , J. Math. Anal. Appl. 238 (1999), 216-233.
  * [13] P.L. Felmer, _Periodic solutions of “superquadratic” Hamiltonian systems_, J. Differential Equations 102 (1993), 188-207.
  * [14] C. Li, _Brake subharmonic solutions of subquadratic Hamiltonian systems_ , Chin. Ann. Math. Ser. B 37 (2016), 405-418.
  * [15] C. Li, _The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems_ , Acta Math. Sin. (Engl. Ser.) 31 (2015), 1645-1658.
  * [16] C. Li, C.Liu, _Brake subharmonic solutions of first order Hamiltonian systems_ SCIENCE CHINA (Mathematics), 53 (2010), 2719-2732.
  * [17] C. Li, Z. Ou, C. Tang, _Periodic and subharmonic solutions for a class of non-autonomous Hamiltonian systems_ , Nonlinear Analysis, T.M.A. 75 (2012), 2262- 2272\.
  * [18] C. Liu, _Subharmonic solutions of Hamiltonian systems_ , Nonlinear Anal. 42 (2000), 185-198.
  * [19] C. Liu, _Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems_ , Discrete and continuous dynamical systems 27 (2010), 337-355.
  * [20] C. Liu, _Relative index theories and applications_ , preprint.
  * [21] C. Liu, Y. Long, _Iteration inequalities of the Maslov-type index theory with applications_ , J. Differential Equations 165 (2000), 355-376.
  * [22] C. Liu, S. Tang, _Subharmonic $P$-solutions of first order Hamiltonian systems_, Preprint.
  * [23] C. Liu, S. Tang, _Iteration inequalities of the Maslov P-index theory with applications_ , Nonlinear Anal. T.M.A. 127 (2015), 215-234.
  * [24] Y. Long, Index Theory for Symplectic Paths with Applications, _Birkhauser Verlag Basel $\cdot$ Boston $\cdot$ Berlin_, 2002.
  * [25] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, _Springer_ , 1989.
  * [26] R. Michalek, G. Tarantello, _Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems_ , J. Differential Equations 72 (1988), 28-55.
  * [27] K. Perera, M. Schechter, Topics in critical point theory, _Cambridge University Press_ , 2013.
  * [28] P. H. Rabinowitz, _Periodic solutions of Hamiltonian systmes_ , Comm. Pure Appl. Math. 31 (1978), 157-184.
  * [29] P.H. Rabinowitz, _On subhamonic solutions of Hamiltonian systems_ , Comm. Pure Appl. Math. 33 (1980), 609-633.
  * [30] P.H. Rabinowitz, _Mini-max methods in critical point theory with applications to differential equations_ , CBMS Reg. Conf. Ser. Math. 65, 1986.
  * [31] E. Silva, _Subharmonic solutions for subquadratic Hamiltonian systems_ , J. Differential Equations 115 (1995), 120-145.
  * [32] Q. Xing, F. Guo, X. Zhang, _One generalized critical point theorem and its applications on super-quadratic Hamiltonian systems_ , preprint.
  * [33] D. Zhang, _Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems_ , Science China, Mathematics, 57 (2014), 81-96.
  * [34] X. Zhang, F. Guo, _Existence of Periodic Solutions of a Particular Type of Super-quadratic Hamiltonian Systems_ , J. Math. Anal. Appl. 421 (2015), 1587-1602.
  * [35] X. Zhang, F. Guo, _Multiplicity of Subharmonic Solutions and Periodic Solutions of a Particular Type of Super-quadratic Hamiltonian Systems_ , Comm. Pure Appl. Ana. 15 (2016), 1625-1642.