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\fancyhead[CE]{M.R.  Heidari Tavani} 
\fancyhead[CO]{Existence results  for a class of  $p$-Hamiltonian systems}



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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
\vspace*{9mm}

\begin{center}

{\Large \bf 
Existence results  for a class of  $p$-Hamiltonian systems\\}



\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

{\bf Mohammad Reza Heidari Tavani}\vspace*{-2mm}\\
\vspace{2mm} {\small % name of the department, where author works 
Ramhormoz Branch, % name of the university, where author works 
 Islamic Azad University} \vspace{2mm}


\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In the present paper, by using the variational methods in critical point theory,~the existence and multiplicity of periodic solutions for a class of $p$-Hamiltonian systems is established.~In fact, using two fundamental theorems that are attributed to Bonanno, we get some
important results.~Are presented the results were extention
of some existing results.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 35J40; 34K13; 34B15

\noindent{\bf Keywords and Phrases:}  $p$-Hamiltonian systems, periodic solutions, critical point theory, variational methods.
\end{quotation}}

\section{Introduction}
Consider the following $p$-Hamiltonian system
\begin{equation}\label{1}
\left\{\begin{array}{ll}-(|u'|^{p-2}u')'-q(t)|u'|^{p-2}u'
+A(t)|u|^{p-2}u=\lambda \nabla F(t,u) \;\;\;\;\;\;\; a.e.\;\;t\in
[0,T],\\\\
u(0)-u(T)={u'}(0)-e^{Q(T)}{u'}(T)=0,
\end{array}\right.
\end{equation}
where $T>0$, $p>1$, $q\in L^{1}(0,T;\mathbb{R})$, $Q(t)=\int_0^t q(s) ds$, $A:[0,T]\to \mathbb{R}^{N\times N}$ is a continuous map from the
interval $[0,T]$ to the set of $N$-order symmetric matrices, 
$\lambda>0$ and $F:[0,T]\times{\Bbb R}^N\to\Bbb R$ is function
measurable with respect to $t$, for all $x\in{\Bbb R}^N$,
continuously differentiable in $x$, for almost every $t\in[0,T]$,and
there exists $a\in C({\mathbb{R}^+},{\mathbb{R}^+})$, $b\in L^{1}(0,T;\mathbb{R}^+)$ such that
\begin{equation}\label{2}|F(t,x)|\leq\,a(|x|)b(t) ,\,\,\, |\nabla F(t,x)|\leq\,a(|x|)b(t)
\end{equation} for all $x\in{\Bbb R}^N$ and a.e.\;\;$t\in
[0,T]$.\\

 
 
Hamiltonian systems are a special case of dynamical systems.This type of equations play an important role in fluid mechanics and gas dynamics. For the study of Hamiltonian systems can be considered \cite{MW,Ra2}. When $p = 2$, problem \eqref{1} becomes the second order Hamiltonian systems. In recent years, the existence of periodic solutions for the second order Hamiltonian systems have been studied in many papers (see
\cite{BL2,C,CR,F,FL,HAH,T1,TW,WCT} and the
references therein). For example in \cite{HAH}, the authors proved the existence of periodic solutions by the variational methods in critical point theory for the following second-order Hamiltonian system



\begin{equation}\label{j1}
\left\{\begin{array}{ll}-\ddot{u}(t)-
q(t)\dot{u}(t)+A(t)u(t)=\lambda \nabla F(t,u(t))+\mu \nabla
G(t,u(t)) \;\;\;\;\;\;\; a.e.\;\;t\in
[0,T],\\
u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,
\end{array}\right.
\end{equation}
where $\mu\geq 0$ and $G:[0,T]\times{\Bbb R}^N\to\Bbb R$ is function
measurable with respect to $t$, for all $x\in{\Bbb R}^N$,
continuously differentiable in $x$, for almost every $t\in[0,T].$
 For the general case $p > 1$, some authors presented interesting results (see \cite{MZ,ME,XT,ZOT}).\\\\\
For example, Xu and Tang in \cite{XT}, by using minimax methods in critical point theory established the existence of periodic solutions for problem 
\begin{equation}\label{un}
\left\{\begin{array}{ll}-(|u'(t)|^{p-2}u'(t))'=\lambda \nabla F(t,u(t)) \;\;\;\;\;\;\; a.e.\;\;t\in
[0,T],\\\\
u(0)-u(T)={u'}(0)-{u'}(T)=0.
\end{array}\right.
\end{equation}

Also, some authors in \cite{ZOT} have considered the following problem 
\begin{equation}\label{unz}
\left\{\begin{array}{ll}-(|u'|^{p-2}u')'
+A(t)|u|^{p-2}u=\lambda \nabla F(t,u)+\mu \nabla G(t,u) \;\;\;\;\;\;\; a.e.\;\;t\in
[0,T],\\\\
u(0)-u(T)={u'}(0)-{u'}(T)=0.
\end{array}\right.
\end{equation}

They studied the existence of at least three  periodic solutions for problem \eqref{unz}  using two theorems due respectively to Ricceri (see reference  [17] in \cite{ZOT}  ) and Averna-Bonanno( see reference  [13] in \cite{ZOT}).\\ In this paper, using two kinds of  critical points theorem obtained in \cite{B} and \cite{B00} which we recall in the next section , we ensure the existence of periodic solutions   for  problem \eqref{1}.\\ The paper is organized as follows. In section $2$ we establish all the preliminary results that we 
need, and in section $3$ we present our main results.

  
\section{Preliminaries}
\hspace*{.5cm}\bigskip
 We assume that the matrix $A$ satisfies the following conditions:

 (i) $A(t)=(a_{ij}(t))$, $i=1,\ldots,N$, $j=1,\ldots,N$, is a symmetric matrix with $a_{ij}\in
L^\infty{[0,T]}$ for any $t\in[0,T]$,

(ii)  there exists a positive constant $\underline{\delta}$  such that $\langle A(t)|x|^{p-2}x,x\rangle\geq\underline{\delta}\, |x|^{p}$ for all $x\in\mathbb{R}^{N}$ and $t\in
[0,T]$, where $\langle\cdot,\cdot\rangle$ denotes the inner product
in $\mathbb{R}^N$ and in the other hand we know that $\langle A(t)|x|^{p-2}x,x\rangle\leq\bar{\delta}\, |x|^{p}$ for any $x\in\mathbb{R}^N$ and
for every $t\in[0,T]$ where  $\bar{\delta}\leq \sum\limits_{i,j=1}^{N}\|a_{ij}\|$ ( see \cite{ZOT}) .

 Let us recall some basic concepts. Denote $$E=
\{u : [0,T]\rightarrow {\Bbb R}^N ,\ u\
 \textrm{is absolutely continuous}, \;\; u(0)= u(T),\;\; u' \in L^p([0,T],{\Bbb
 R}^N)\}.$$ Assume that $E$ is equipped with the following norm$$\|
u\|_{E}=\Bigg{(}\int_{0}^{T} (|u'(t)|^p+ |u(t)|^p)dt\Bigg{)}^\frac{1}{p},\ \forall\ u\in
E.$$Also , we consider $E$ with the norm$$\| u\|=\Bigg{(}\int_{0}^{T}e^{Q(t)}
[|u'(t)|^p+ \langle A(t)|u(t)|^{p-2}u(t),u(t)\rangle]dt\Bigg{)}^\frac{1}{p}.$$
 The Banach space $E$ is a separable and reflexive.
 Obviously, $E$
is also a uniformly convex Banach space.

Due to the inequality   $$K_{1} \min\{1,\underline\delta\}\|u\|_{E}^p\leq\|u\|^p\leq K_{2}\max\{1,\bar\delta\}\|u\|_{E}^p,$$ where $K_{1}=\min_{t\in [0,T]}e^{Q(t)}$ , and $K_{2}=\max_{t\in [0,T]}e^{Q(t)}$ ,the
 norm $\|\cdot\|$ is equivalent to the norm $\|\cdot\|_{E}$ .

 Since $(E,\|\cdot\|)$ is compactly embedded in $C([0,T],\mathbb{R}^N)$ (see \cite{MW}), there
exists a positive constant \begin{equation}\label{z}c\leq c_0=\sqrt[q]{2}\max\{T^{\frac{1}{q}},T^{\frac{-1}{p}}\}(K_{1}\min\{1,\underline\delta\})^\frac{-1}{p}\end{equation} where $q=\displaystyle\frac{p}{p-1}$ ,such that
\begin{equation}\label{3}\|u\|_\infty \leq c\parallel u\parallel
,\end{equation} where $\|u\|_\infty =\max_{t\in[0,T]}\mid
u(t)\mid$.The proof is similar to the corresponding
parts in  \cite{ZOT}.\\


  Let $\Phi,\ \Psi:E\to \mathbb{R}$ be defined by
\begin{equation}\Phi(u)=\frac{1}{p}\parallel u \parallel ^p=\frac{1}{p}\int_{0}^{T}e^{Q(t)}
[|u'(t)|^p+ \langle A(t)|u(t)|^{p-2}u(t),u(t)\rangle]dt\end{equation}and
\begin{equation}\Psi(u)=\displaystyle\int ^{T}_{0}e^{Q(t)}F(t,u(t))dt\end{equation}for every $u\in E$ . It is well known that
$\Psi$ is a continuously G\^ateaux differentiable functional whose G\^ateaux  dervative is compact (see \cite{WCT}) and for each $u,v\in E$ 
$$\Psi'(u)(v)=\int_0^T e^{Q(t)}\langle\nabla F(t,u(t)),v(t)\rangle  dt,$$ and, $\Phi$ is continuously G\^ateaux differentiable and sequentially weakly lower semicontinuous functional. Moreover , the G\^ateaux  dervative of $\Phi$ admits a continuous inverse on $E^*$ (see \cite{ZOT}). For each $u,v\in E$ we have
$$\Phi'(u)(v)=\int_0^T e^{Q(t)}[\langle |u'(t)|^{p-2}u'(t),v'(t)\rangle + \langle A(t)|u(t)|^{p-2}u(t),v(t)\rangle] dt.$$ 



\begin{definition}\label{jord}Let $\Phi$ and $\Psi$ be defined as above.
Put $I_\lambda=\Phi-\lambda \Psi$, $\lambda>0$.\,\,\, We say that $u\in E$ is a critical point of $I_\lambda$ when $I'_\lambda(u)=0_{\{E^*\}}$, that is, $I'_\lambda(u)(v)=0$ for all $v\in E$.
\end{definition}


\begin{definition}
\rm A function $u\in
E$ is a weak solution to  problem \eqref{1}, if
 $$\int_0^T e^{Q(t)}[\langle |u'(t)|^{p-2}u'(t),v'(t)\rangle + \langle A(t)|u(t)|^{p-2}u(t),v(t)\rangle $$$$ -\lambda \langle\nabla F(t,u(t)),v(t)\rangle ] dt= 0$$ for every $v\in E$.\end{definition}

\begin{remark}\label{456}We clearly observe that the weak solutions of the problem \eqref{1} are exactly the solutions of the equation $I'_\lambda(u)(v)=\Phi'(u)(v)-\lambda \Psi'(u)(v)=0$.  \end{remark}
\begin{definition}\label{23} 
A G\^{a}tuax differentiable function $I$ satisfies the Palais-Smale condition  $($in short ${(PS)}$
-condition$)$ if any sequence $\{u_n\}$ such that \\$(a)$ $\{I(u_n)\}$ is bounded,\\$(b)$ 
$\mathop {\lim }\limits_{n \to + \infty } \|I'({u_n})\|_{{X^*}} = 0,\,\,\,\,\forall n \in \mathbb{N}$,\\
has a convergent subsequence.
\end{definition}
  Below , we will present a non-standard state of the Palais-Smale condition that is introduced in \cite{B}.
\begin{definition}\label{24} Fix $r \in ] - \infty , + \infty ]$ . 
A G\^{a}tuax differentiable function $I$ satisfies the Palais-Smale condition cut off upper at $r$ (in short ${(PS)^{[{r}]}}$
-condition) if any sequence $\{u_n\}$ such that \\$(a)$ $\{I(u_n)\}$ is bounded,\\$(b)$ 
$\mathop {\lim }\limits_{n \to + \infty } \|I'({u_n})\|_{{X^*}} = 0$,\\$(c)$ $ \Phi ({u_n}) < {r}\,\,\,\,\forall n \in \mathbb{N},$\\
has a convergent subsequence.
\end{definition}

Now we present two propositions that will be needed to prove   of the main theorems of this paper.
\begin{proposition}\label{tir}$($\cite{B}-Proposition 2.1$)$ Let $X$ be a reflexive real Banach space, $\Phi : X \to \mathbb{R}$ be a sequentially weakly lower semicontinuous, coercive and continuously G\^ateaux differentiable function whose G\^ateaux derivative admits a continuous inverse on $X^*$, $\Psi : X \to \mathbb{R}$ be a continuously G\^ateaux differentiable function whose G\^ateaux derivative is compact.Then, for all $r\in\mathbb{R}$,  the function $\Phi-\Psi$ satisfies the $(PS)^{[r]}$-condition.
\end{proposition}


\begin{remark}\label{qwe}Fix $\lambda >0$ . Then 

according to proposition \ref{tir}, the functional $I_\lambda=\Phi-\lambda\Psi$
  satisfies the ${(PS)^{[{r}]}}$-condition for any $r >0$.

\end{remark}
In the next proposition , using one of the types of  
Ambrosetti-Rabinowitz  conditions  obtained in \cite{AR} ,we will ensure that functional $I_\lambda$  is unbounded from below .
\begin{proposition}\label{sir}
Assume that there are $M > 0$ and $\theta > p$ such that \begin{equation*}\label{pir}0 <\theta F(t,x)\leq \langle \nabla F(t,x),x\rangle \end{equation*} for all $x \in \mathbb{R}^N$ with $|x|\geq M$ and a.e.\, $t\in [0,T]$.
 Then $I_\lambda=\Phi-\lambda\Psi$ satisfies the ${(PS)}$-condition and it is unbounded from below.

\end{proposition}
\begin{proof}
First we prove that $I_\lambda$ satisfies ${(PS)}$-condition for every $\lambda >0$. For this purpose we will prove that for arbitrary sequence $\{u_n\}\subset E$ satisfying 

\begin{equation}\label{mir}
|I_\lambda(u_n)|\leq D\,\,\,\, \textit{for some $D > 0$ and for all n}\in \mathbb{N},
\end{equation}



\begin{equation}\label{nir}
\mathop {\lim }\limits_{n \to + \infty } \|I'({u_n})\|_{{E^*}} = 0,\,\,\,\,\forall n \in \mathbb{N}
\end{equation}
contains a convergent subsequence. For $n$ large enough,  we have by \eqref{mir} 

$$D\geq I_{\lambda}(u_n)=\frac{1}{p}{\|u_n\|}^p-\lambda\int _{0}^{T}F(t,u_n)dt\,\geq$$
$$\frac{1}{p}{\|u_n\|}^p-\frac{\lambda}{\theta}\int _{0}^{T}\langle \nabla F(t,u_n),u_n\rangle dt\,=$$
\begin{equation}\label{hers}\Big{(}\displaystyle\frac{1}{p}-\frac{1}{\theta}\Big{)}{\|u_n\|}^p+\frac{1}{\theta}I'_\lambda(u_n)(u_n).\end{equation}

From $\mathop {\lim }\limits_{n \to + \infty } \|I_{\lambda}'({u_n})\|_{{E^*}} = 0$, there exists a sequence $\{\varepsilon_n\}$ ,with $\varepsilon_n \rightarrow 0^+ $ , such that 
\begin{equation}\label{her}
|I_{\lambda}'({u_n})({v_n})|\leq \varepsilon_n \end{equation} for all $n\in \mathbb{N}$ and for all $v\in E$ with $\parallel v \parallel \leq 1$. Taking into account $v(x)=\frac{u_n(x)}{\parallel u_n \parallel}$, from \eqref{her} one has 
\begin{equation} \label{hersss}  |I_{\lambda}'({u_n})({u_n})|\leq\varepsilon_n \parallel u_n \parallel \end{equation} for all $n\in \mathbb{N}$. Hence from \eqref{hers} and \eqref{hersss} we have 
\begin{equation}\label{herss}
D+ \frac{\varepsilon_n}{\theta} \parallel u_n \parallel\,\geq \Big{(}\displaystyle\frac{1}{p}-\frac{1}{\theta}\Big{)}{\|u_n\|}^p.
\end{equation}
Thus , \eqref{herss} ensures that $\{u_n\}$ is bounded in $E$
and hence , passing to a subsequence if necessary we can assume that there exists $u_0 \in E$ such that $u_{n}\rightharpoonup u_0 $ (see \cite{BOOO}-Theorem 3.18). Now since $\Psi'$ is compact then $\Psi'(u_n)\to \Psi'(u_0)$. But from \eqref{nir} we have $I_{\lambda}'(u_n)=\Phi'(u_n)-\lambda \Psi'(u_n)\to 0 .$ This implies that $u_{n}\to\Phi' {^{ - 1}}(\lambda \Psi '(u_0))$ (because $ \Phi$ admits a continuous inverse on $E^*$ ) and finally according to the uniqueness of the weak limit, $u_{n}\to u_0$  in $E$ and so $I_\lambda$ satisfies ${(PS)^{[{r}]}}$-condition.\\Take $h(t):=\min_{|\xi|=M}F(t,\xi).$
From \eqref{pir}, by standard computations , we have

 \begin{equation}\label{tez}F(t,x)\geq\, h(t)\frac{|x|^\theta}{M^\theta}-\big{(}\max_{s\in[0,M]}a(s)\big{)}b(t)\end{equation} 
  for all $x\in {\mathbb{R}}^N$ and a.e.\, $t\in[0,T]$. Fixed $u_0\in E-\{0\} $. For each $s>1$ , we have $$I_\lambda(s u_0)=\frac{1}{p}\parallel s u_0 \parallel^p-\lambda\int_{0}^TF(t,s u_0)dx.$$Taking into account \eqref{tez} , one has

$$I_\lambda(s u_0)\leq \frac{s^p}{p}\parallel  u_0 \parallel^p-\lambda \int_{0}^T\bigg{(} h(t)\frac{s^\theta\,{|u_0|}^\theta}{M^\theta}-\big{(}\max_{z\in[0,M]}a(z)\big{)}b(t)\bigg{)}dt$$
 and since $\theta > p$, this condition guarantees that $I_\lambda$ is unbounded from below.\end{proof}
Our main tools are the following critical
point theorems.


\begin{theorem}[\cite{B00}, Theorem 2.3]\label{t1} Let $X$ be a  real Banach
space, and let  $ \Phi,\Psi:X \longrightarrow \mathbb{R}$ be two
continuously G\^{a}teaux differentiable functionals such that $\inf_{X}\Phi =\Phi(0)=\Psi(0)=0$.\\
 Assume that there are $r\in\mathbb{R} $ and  $\tilde{u}\in X$, with $0< \Phi(\tilde{u})<r$, such
 that \begin{equation}\label{ghaz}
 \displaystyle\frac{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}{r}<
\displaystyle\frac{\Psi(\tilde{u})}{\Phi(\tilde{u})},\end{equation}
and , for each $\lambda\in
\displaystyle\left]\frac{\Phi(\tilde{u})}{\Psi(\tilde{u})},
\frac{r}{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}\right[$ the functional $
\Phi-\lambda
\Psi$ satisfies the ${(PS)^{[{r}]}}$-condition .Then, for each $\lambda\in
\displaystyle\left]\frac{\Phi(\tilde{u})}{\Psi(\tilde{u})},
\frac{r}{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}\right[$ there is $u_\lambda\in \Phi^{-1}(]0,r[) $ (hence $u_\lambda \neq 0 $) such that $I_{\lambda}(u_\lambda)<I_{\lambda}(u)$ for all $u\in \Phi^{-1}(]0,r[) $ and $I'_{\lambda}(u_\lambda)=0$.
\end{theorem}
\begin{theorem}[\cite{B}, Theorem 3.2]\label{t2}Let $X$ be a  real Banach
space, and let  $ \Phi,\Psi:X \longrightarrow \mathbb{R}$ be two
continuously G\^{a}teaux differentiable functionals such that $\Phi$ is bounded from below and

  $\Phi(0)=\Psi(0)=0$. Fix $r>0$ such that $\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)<+\infty$ and assume that for each $\lambda\in
\displaystyle\left]0,
\frac{r}{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}\right[$ the functional $
\Phi-\lambda
\Psi$ satisfies the ${(PS)}$-condition and it is unbounded from below. Then, for each 
$\lambda\in
\displaystyle\left]0,
\frac{r}{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}\right[$ the functional $I_\lambda$ admits two distinct critical points.
\end{theorem}
\section{Main results} \hspace{.5cm}
In this section ,we use the following notation:  
\begin{equation}\label{w10}
F^\theta:=\displaystyle\int ^{T}_{0}e^{Q(t)}\sup _{|x|\leq \theta}F(t,x)dt,\ \ t\in
[0,T], \ \ \forall \theta>0 .\end{equation}







Now, we formulate our main result.
  
 \begin{theorem}\label{t3}Let   $F:[0,T]\times \mathbb{R}^{N}\to \mathbb{R}$ satisfy assumption   \eqref{2} and $\int_0^T e^{Q(t)}F(t,0)=0$ for  a.e.\, $t\in [0,T]$.  Assume that the following condition hold:\\
$(A)$ there exists positive constant $\theta$ 
and a point $0\neq x_{0}\in \mathbb{R}^{N}$ with $|x_{0}|c(\bar{\delta}\,\,\int_0^T e^{Q(t)} dt)^{\frac{1}{p}}<\theta,$   such that 
\begin{equation}\label{zrth}\frac{F^\theta }{\theta^p}<\frac{\displaystyle\int_{0}^{T} e^{Q(t)} F(t,x_0)dt}{c^{p}\bar\delta\, {|x_0|^{p}\int_0^T e^{Q(t)} dt}}.\end{equation} Then for
every 
\begin{equation}\lambda\in\Lambda:=\displaystyle
\left]\frac{\bar\delta\, {|x_0|^{p}\int_0^T e^{Q(t)} dt}}{p\displaystyle\int_{0}^{T} e^{Q(t)} F(t,x_0)dt},\frac{\theta^p}{p\,c^{p}F^\theta }\right[, \end{equation} 
 
  the problem
\eqref{1} admits at least one non-trivial  weak 
 solution $u_\lambda\in E $ such that $\|u_\lambda\|_\infty<\theta$.
\end{theorem}
\begin{proof}Our aim is to apply Theorem \ref{t1} , to problem \eqref{1}.  Fix $\lambda\in\Lambda$. Take $X= E$ and $\Phi$
and $\Psi$ as in the previous section. We observe that
the regularity assumptions of Theorem \ref{t1} on $\Phi$ and
$\Psi$ are satisfied and also  according to Remark  \ref{qwe}, the functional $I_\lambda=\Phi-\lambda \Psi$ satisfies the $(PS)^{[r]}$-condition for all $r > 0$.




 Put $r=\dfrac{1}{p}(\dfrac{\theta}{c})^{p}$ and  $\tilde{u}(t)=x_0$ for all $t\in [0,T]$. We clearly observe that $\tilde{u}\in E $ and according to assumption $|x_{0}|c(\bar{\delta}\,\,\int_0^T e^{Q(t)} dt)^{\frac{1}{p}}<\theta,$ one has $0< \Phi(\tilde{u})<r$.
 
 

 
  For each $u\in E$
 bearing
in mind  \eqref{3}, we see that
\begin{eqnarray*}
 \Phi^{-1}(]-\infty,r[)&=&\{u\in E;\ \Phi(u)< r\}
 \\&=&\left\{u\in E;\ \frac{\|u\|^{p}}{p}< r\right\}\\&\subseteq&
  \left\{ u\in E;\
  |u(t)|\leq
 \theta \textrm{ for each} \ t\in [0,T]\right\}.\end{eqnarray*} Now we have
\begin{gather}
\displaystyle{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}=\sup _{\Phi(u)< r}\displaystyle\int ^{T}_{0}e^{Q(t)}F(t,u(t)dt\leq \nonumber\\
\displaystyle\int ^{T}_{0}e^{Q(t)} \sup _{|x|\leq
\theta}F(t,x)dt  =\nonumber F^{\theta}. \label{9}
\end{gather}

 Therefore, we have 
 
\begin{eqnarray}\label{30}\displaystyle\frac{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}{r}
&\leq &\frac{F^{\theta}}{\displaystyle\frac{1}{p}\Bigg{(}\frac{\theta}{c}\Bigg{)}^p}
=  \frac{p\,c^p\,F^\theta}{{\theta}^p}<\frac{1}{\lambda}.
 \end{eqnarray} 
 On the other hand
 
 

 \begin{equation}\label{31}
\frac{\Psi(\tilde{u})}
 {\Phi(\tilde{u})}= \dfrac{\displaystyle\int ^{T}_{0}e^{Q(t)}F(t,x_{0})dt}{\dfrac{1}{p}\|x_{0}\|^{p}}\geq  \dfrac{\displaystyle\int ^{T}_{0}e^{Q(t)}F(t,x_{0})dt}{\dfrac{1}{p}\bar{\delta}|x_{0}|^{p}\int ^{T}_{0}e^{Q(t)}dt}>\frac{1}{\lambda}.\end{equation}
 
Now from \eqref{30} and \eqref{31} we have,

$$\displaystyle\frac{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}{r}<
\displaystyle\frac{\Psi(\tilde{u})}{\Phi(\tilde{u})}$$
and \eqref{ghaz} is proved. Finally, for each 
   $\lambda\in\Lambda\subseteq \left]\dfrac{\Phi(\tilde{u})}
{\Psi(\tilde{u})},\dfrac{r}{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}\right[$,
    Theorem \ref{t1}   guarantees  the existence of at least one non-trivial critical point for the functional   $I_\lambda=\Phi-\lambda \Psi$, and the proof is complete.\end{proof}  
  
  

A corollary of Theorem \ref{t3},  is as follows.

 \begin{corollary}\label{2000}Let $F:\mathbb{R}^N\to \mathbb{R}$ be a non-negative function
such that $F(0)=0$ and $\nabla F$  be
continuous in $\Bbb R^N$ .Moreover, suppose that 
\begin{equation}\label{x}
  \limsup_{|\xi|\to +0}\frac{F(\xi)}{|\xi|^{p}}=+\infty.\end{equation}
  
 Then,  for each $\theta>0$ and $\lambda \in \big{]}0,\frac{\theta^p}{p\,c^p\, (\sup_{|\xi| \leq
\theta}F(\xi))\int ^{T}_{0}e^{Q(t)}dt}\big{[}$
  the
problem 
\begin{equation}\label{1000}
\left\{\begin{array}{ll}-(|u'|^{p-2}u')'-q(t)|u'|^{p-2}u'
+A(t)|u|^{p-2}u=\lambda \nabla F(u) \;\;\;\;\;\;\; a.e.\;\;t\in
[0,T],\\
u(0)-u(T)={u'}(0)-e^{Q(T)}{u'}(T)=0,
\end{array}\right.
\end{equation}
admits at least one non-trivial classical solution  $u_\lambda\in E $ such that $\|u_\lambda\|_\infty<\theta$.\end{corollary}
\begin{proof} Fix
$\theta>0$ , $\lambda \in \big{]}0,\frac{\theta^p}{p\,c^p\, (\sup_{|\xi| \leq
\theta}F(\xi))\int ^{T}_{0}e^{Q(t)}dt}\big{[}.$\\
By \eqref{x} , there exists $x_0\in {\mathbb{R}}^N$ with $|x_{0}|c(\bar{\delta}\,\,\int_0^T e^{Q(t)} dt)^{\frac{1}{p}}<\theta,$ such that $\displaystyle\frac{F(x_0)}{|x_0|^p}>\frac{\bar{\delta}}{p\,\lambda} $.\\Taking into account that $\lambda \in \big{]}0,\frac{\theta^p}{p\,c^p\, (\sup_{|\xi| \leq
\theta}F(\xi))\int ^{T}_{0}e^{Q(t)}dt}\big{[},$ one has 
\begin{equation*}\frac{(\sup_{|\xi| \leq
\theta}F(\xi))\int ^{T}_{0}e^{Q(t)}dt}{\theta^p}<\frac{1}{\lambda\,p\,c^p}<\frac{F(x_0)}{c^p\,\bar{\delta}|x_0|^p}\end{equation*}
and so condition \eqref{zrth} of Theorem \ref{t3} is verified.
Now the desired result, can be obtained from Theorem \ref{t3}. \end{proof}





Now, we will present an example for Corollary \ref{2000}.
  \begin{example}\label{t1.2}

\rm Let $ T=1,p=3 $ and $ A(t)=I, $ where $I$ is identity matrix of order $ N\times N $,\,$q(t)=1$ and therefore $Q(t)=t$ for all $ t\in [0,1] $ . Due to the \eqref{z},we can consider $c=\sqrt[3]{4}$.




 Also
let\, $F(x)=\sinh(|x|^2)$ for all $x\in \mathbb{R}^N$ and hence $\sup_{|\xi| \leq
\theta}F(\xi)=\sinh(\theta^2)$ for all $\theta>0$
  . 

Then for every $\lambda\in\big{]}0,\displaystyle\frac{\theta^3}{12(e-1)\sinh(\theta^2)}\big{[}$ all the hypotheses  of Corollary \ref{2000} are satisfied and therefore  the problem 
\begin{equation*}
\left\{\begin{array}{ll}-(|u'|u')'-|u'|u'
+|u|u=2\lambda u \cos(|u|^2) \;\;\;\;\;\;\; a.e.\;\;t\in
[0,1],\\
u(0)-u(1)={u'}(0)-e\,{u'}(1)=0,
\end{array}\right.
\end{equation*}admits at least one non-trivial classical solution  $u_\lambda\in E $ such that $\|u_\lambda\|_\infty<\theta$.
\end{example} 




Now, we point out the following existence results, as
 consequences of Theorem \ref{t2}. 
 
 \begin{theorem}\label{200m}Let   $F:[0,T]\times \mathbb{R}^{N}\to \mathbb{R}$ satisfy assumption   \eqref{2} and $\int_0^T e^{Q(t)}F(t,0)=0$ for  a.e.\, $t\in [0,T]$. Moreover
 Suppose that there are $M > 0$ and $\theta > p$ such that \begin{equation}\label{pirzd}0 <\theta F(t,x)\leq \langle \nabla F(t,x),x\rangle \end{equation} for all $x \in \mathbb{R}^N$ with $|x|\geq M$ and a.e.\, $t\in [0,T]$. Then for each $\lambda \in ]0,\lambda^*[$ where $\lambda^*=\displaystyle\frac{1}{F^{c\,{p^{\frac{1}{p}}}}}$ , problem \eqref{1} admits at least two distinct  weak solutions.
 \end{theorem}
 
\begin{proof} Our aim is to apply Theorem \ref{t2} , to problem \eqref{1}. Put $r=1$ and fixed $\lambda \in ]0,\lambda^*[$. Let $E$ , $\Phi$ and $\Psi$ be as given in the proof of Theorem \ref{t3}.
  We observe that
the regularity assumptions of Theorem \ref{t2} on $\Phi$ and
$\Psi$ are satisfied and also  according to proposition \ref{sir}, the functional $I_\lambda$ satisfies the $(PS)$-condition and it is
unbounded from below. If $u\in\Phi^{-1}(]-\infty,1[)$ then $\Phi(u)<1$ and so $\|u\|<p^{\frac{1}{p}}.$ Hence  according to \eqref{3}  we get 

 $$\displaystyle\frac{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}{r}=\displaystyle{\sup_{ u\in\Phi^{-1}(]-\infty,1[)}\Psi(u)}=\sup_{u\in\Phi^{-1}(]-\infty,1[)}\int_{0}^T e^{Q(t)}\,\,F(t,u) dt
\leq$$  \begin{equation}\label{zse}
\int_{0}^Te^{Q(t)}\,\,\sup_{|x|\leq c\,p^{\frac{1}{p}}} F(t,x) dt = F^{c\,{p^{\frac{1}{p}}}}=\frac{1}{\lambda^*} < \frac{1}{\lambda} .\end{equation}From \eqref{zse} we have 

  \begin{equation*}\lambda\in ]0,\lambda^*[\,\, \subseteq \displaystyle\left]0,\frac{r}{\sup_{ u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}\right[ .\end{equation*}So all hypotheses of Theorem \ref{t2} are verified. Therefore, for each $\lambda\in ]0,\lambda^*[$, the functional
$I_\lambda$ admits at least  two distinct critical points which are,  weak solutions of problem  \eqref{1} and the proof is complete.\end{proof}


Finally, we present the following example to illustrate Theorem  \ref{200m}.



\begin{example}

 Let $T=2\pi$, $p=3$ and $\theta=4$. Again we can consider $c=\sqrt[3]{4}$ $($see example \ref{t1.2}$)$ .\\\\ Now if we consider $F(t,x)=e^{-Q(t)}(\sin t+|x|^6)$ , one has 
 $$0<\theta\, e^{-Q(t)}(\sin t+|x|^6)\leq 6\,e^{-Q(t)}|x|^6$$ for all $x\in{\mathbb{R}}^N$ with $|x| \geq \sqrt[6]{2}$ and a.e. \,$t\in[0,2\pi]$  and so \eqref{pirzd} is verified . Therefore  according to Theorem \ref{200m} for each $\lambda \in \Big{]}0,\frac{1}{F^{c\,{p^{\frac{1}{p}}}}} \Big{[},$ 
 
where $F^{c\,{p^{\frac{1}{p}}}}=F^{\sqrt[3]{12}}=\displaystyle\int_0^{2\pi}e^{Q(t)}\sup_{|x|\leq \sqrt[3]{12}} F(t,x) dt=\\\displaystyle\int_0^{2\pi}(\sin t+144)dt=288\pi$ 
 
 
 
 the problem 


\begin{equation}\label{01k}
\left\{\begin{array}{ll}-(|u'|u')'-q(t)|u'|u'
+A(t)|u|u=\lambda  e^{-Q(t)}|u|^{4}\,u \;\;\;\;\;\;\; a.e.\;\;t\in
[0,2\pi],\\\\
u(0)-u(2\pi)={u'}(0)-e^{Q(2\pi)}{u'}(2\pi)=0,
\end{array}\right.
\end{equation}
 admits at least two classical solutions.
\end{example}

  
 % Acknowledgments 
\section*{\large Acknowledgments} 
  The author is very thankful for the many helpful suggestions
and corrections given by the refrees who reviewed this paper.

  
  
\def\bibname{\vspace*{-30mm}{\centerline{\normalsize References}}} 
  

\bibliographystyle{amsplain}
\begin{thebibliography}{XX}
  
\bibitem{B}{ G. Bonanno }, { A critical point theorem via the Ekeland variational principle}, \textit {Nonlinear Analysis}, 75 (2012), 2992-3007.
\bibitem{B00}G. Bonanno, { Relations between the mountain pass theorem and local minima,}\textit{ Adv. Nonlinear Analysis}, 1 (2012), 205-220.


\bibitem{BL2}{ G. Bonanno, R. Livrea},{ Periodic solutions for a
class of second-order Hamiltonian systems},\textit{ Electron. J.
Differential Equations}, 115 (2005), 1-13.

\bibitem{BOOO} { H. Brezis},{ Functional Analysis, Sobolev Spaces and Partial Differential Equations}, 585 DOI 10.1007/978-0-387-70914-7,  Springer Science+Business Media, LLC (2011)





\bibitem{C}{ G. Cordaro}, { Three periodic
solutions to an eigenvalue problem for a class of second order
Hamiltonian systems},\textit{ Abstr. Appl. Anal}, 18 (2003), 1037-1045.
\bibitem{CR}{ G. Cordaro, G. Rao},{ Three periodic solutions for perturbed second
order Hamiltonian systems}, \textit{ J. Math. Anal. Appl}, 359 (2009),
780-785.
\bibitem{F}{ F. Faraci}, {Multiple periodic
solutions for second order systems with changing sign potential},
\textit {J. Math. Anal. Appl}, 319 (2006), 567-578.
\bibitem{FL}{ F. Faraci, R. Livrea}, {Infinitely many periodic solutions
for a second-order nonautonomous system},\textit{ Nonlinear Anal}, 54 (2003), 417-429.
\bibitem{HAH}{ M.R. Heidari Tavani, G. Afrouzi, S. Heidarkhani},{ Multiple solutions for a class of perturbed damped vibration problems }, \textit {J. Math. Computer Sci}, 16 (2016), 351-363

\bibitem{MZ}{ S.Ma , Y.Zhang},{ Existence of infinitely many periodic solutions for ordinary $p$-Laplacian systems},\textit{ J. Math. Anal. Appl}, 351 (2009), 469-479


\bibitem{MW}{ J. Mawhin, M. Willem},{ Critical Point Theory and
Hamiltonian Systems}, Springer-Verlag, New York, Berlin,
Heidelberg, London, Paris, Tokyo, 1989.
\bibitem{ME}{ Q.Meng}, {Three periodic solutions for a class of ordinary
$p$-Hamiltonian systems},\textit{ Boundary Value Problems} 150 (2014), 1-6.
 \bibitem{Ra2}{ P. H. Rabinowitz}, { Variational methods for Hamiltonian systems, in: Handbook of
Dynamical Systems}, vol. 1, North-Holland, 2002, Part 1, Chapter
14, pp. 1091-1127.
\bibitem{AR}A. Ambrosetti and P. H. Rabinowitz,{ Dual variational methods in critical point theory and applications},\textit{ J. Funct. Analysis },14 (1973), 349-381.
\bibitem{T1}{ C.L. Tang},{ Periodic solutions of non-autonomous second order systems with $\gamma$-quasisubadditive potential},
 \textit{ J. Math. Anal. Appl} 189 (1995),
671-675.
\bibitem{TW}{ C.L. Tang, X.P. Wu},{  Periodic solutions for a class of nonautonomous
subquadratic second order Hamiltonian systems},\textit{ J. Math. Anal.
Appl}, 275 (2002), 870-882.


\bibitem{WCT}{ X. Wu, S. Chen, K. Teng}, {\em On variational methods for
a class of damped vibration problems},\textit{ Nonlinear Analysis}, 68 (2008), 1432-1441.


\bibitem{XT}{ B. Xu, C.L. Tang},{  Some existence results on  periodic solutions of ordinary $p$-Laplacian systems}
,\textit{ J. Math. Anal Appl}, 333 (2007), 1228-1236.
\bibitem{ZOT}{ C.L. Zeng, Q. Ou, C.L. Tang}, { Three periodic solutions for $p$-Hamiltonian systems}, \textit {Nonlinear Anal}, 74 (2011), 1596-1606.




  



\end{thebibliography}




{\small

\noindent{\bf Mohammad Reza Heidari Tavani}\\
\noindent Assistant Professor of Mathematics\\
\noindent Department of Mathematics



\noindent Ramhormoz Branch, Islamic Azad University

\noindent Ramhormoz, Iran

\noindent E-mail: m.reza.h56@gmail.com}

\end{document}