\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{afterpage} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \fancyhead[CE]{M.R. Heidari Tavani} \fancyhead[CO]{Existence results for a class of $p$-Hamiltonian systems} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\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 $0< \Phi(\tilde{u})0$ 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})\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\|