\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{mathrsfs}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 292, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2015/292\hfil Multiplicity of positive solutions]
{Multiplicity of positive solutions for second-order differential 
 inclusion systems depending on two parameters}

\author[Z. Yuan, L. Huang, C. Zeng \hfil EJDE-2015/292\hfilneg]
{Ziqing Yuan, Lihong Huang, Chunyi Zeng}

\address{Ziqing Yuan \newline
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan
410082, China}
\email{junjyuan@sina.com}

\address{Lihong Huang (corresponding author)\newline
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan
410082, China}
\email{lhhuang@hnu.edu.cn}

\address{Chunyi Zeng \newline
 Department of Foundational Education, Southwest University for Nationalities,
\newline Chengdu, Sichuan 610000, China}
\email{ykbzcy@163.com}


\thanks{Submitted February 20, 2015. Published November 30, 2015.}
\subjclass[2010]{49J40, 35R70, 35L85}
\keywords{Neumann problem; differential inclusion system; locally Lipschitz;  
\hfill\break\indent nonsmooth critical point}

\begin{abstract}
 We consider the two-point boundary-value system
 \begin{gather*}
 -u''_i+u_i\in\lambda\partial_{u_i}F(u_1,\ldots,u_n)
 +\mu\partial_{u_i}G(u_1,\ldots,u_n),\\
 u'_i(a)=u'_i(b)=0\quad u_i\geq 0,\quad 1\leq i\leq n.
 \end{gather*}
 Applying a version of nonsmooth  three critical points theorem, 
 we show the existence of at least three positive solutions.
 \end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}

  In the previous decades, there has been a lot of interest in scalar periodic
 problems driven by the one dimension $p$-laplacian. Some results can be
found in \cite{b2,m2,w1,s1}  and references therein.  We mention the
works by Guo  \cite{g2}, Pino et al \cite{p2}, Fabry and Fayad  \cite{f1} and
Dang and Oppenheimer \cite{d1}. The authors used degree theory and assumed
that the right-hand side nonlinearity $f(t,\zeta)$ is jointly  continuous
in $t\in T=[a,b]$ and $\zeta\in\mathbb{R}$. Their conditions on $f$ are also
 asymptotic and there is no interaction between the nonlinearity  and
the Fu\v{c}ik spectrum of the one-dimensional  $p$-laplacian. Especially,
 Heidarkhani and Yu  \cite{h1} considered the existence of at least
three solutions for a class of two-point boundary-value systems of the form
\begin{equation}\label{1.20}
 \begin{gathered}
-u''_i+u_i=\lambda\partial_{u_i}F(u_1,\ldots,u_n)
 +\mu\partial_{u_i}G(u_1,\ldots,u_n),\\
u'_i(a)=u'_i(b)=0
\end{gathered}
\end{equation}
for $1\leq i\leq n$, where $F,G:[a,b]\times\mathbb{R}^n\to\mathbb{R}$ are
$\rm C^1$-functionals with respect to $(u_1,\ldots,u_n)\in\mathbb{R}^n$
for a.e. $x\in [a,b]$.

From the above results, a natural question arises:
what will happen when the potential functions $F$ and $G$ are not differentiable
in \eqref{1.20}?
This is the main point of interest in our paper. Here,  we extend the main
results in \cite{h1} to a class of perturbed Motreanu-Panagiotopoulos
functionals  \cite{m3}, which raises some essential difficulties.
The presence of non-differentiable function  probably leads to no
solution of \eqref{1.20} in general.  Therefore  to overcome this difficulty,
setting  $f_i=\partial_{u_i}F(u_1,\ldots,u_n)$ and
$g_i=\partial_{u_i}G(u_1,\ldots,u_n)$,  we consider such functions $f_i$ and
$g_i$, which are locally essentially bounded measurable and we fill
 the discontinuity gaps of $f_i$ and $g_i$, replacing $f_i$ and $g_i$ by
intervals $[f^-_i(u_1,\ldots,u_n),f^+_i(u_1,\ldots,u_n)]$ and
$[g^-_i(u_1,\ldots,u_n),g^+_i(u_1,\ldots,u_n)]$, where
\begin{gather*}
f^-_i(u_1,\ldots,u_n)
=\lim_{\delta\to 0^+}\operatorname{ess\,inf}_{|u'_i-u_i|<\delta} 
 \partial_{u_i}F(u_1,\ldots,u'_i,\ldots,u_n),
\\
f^+_i(u_1,\ldots,u_n)=\lim_{\delta\to 0^+}\operatorname{ess\,sup}_{|u'_i-u_i|
<\delta}\partial_{u_i}F(u_1,\ldots,u'_i,\ldots,u_n),
\\
g^-_i(u_1,\ldots,u_n)=\lim_{\delta\to 0^+}\operatorname{ess\,inf}_{|u'_i-u_i|<\delta}
\partial_{u_i}G(u_1,\ldots,u'_i,\ldots,u_n),
\\
g^+_i(u_1,\ldots,u_n)=\lim_{\delta\to 0^+}\operatorname{ess\,sup}_{|u'_i-u_i|
<\delta}\partial_{u_i}G(u_1,\ldots,u'_i,\ldots,u_n).
\end{gather*}
Then  $f^-_i(u_1,\ldots,u_n)$, $g^-_i(u_1,\ldots,u_n)$ are lower semi-continuous,
and  $f^+_i(u_1,\ldots,u_n)$, $g^+_i(u_1,\ldots,u_n)$ are upper semi-continuous.

So instead of \eqref{1.20} we consider the
following second-order Neumann inclusion systems on a bounded interval
 $[a,b]$ in $\mathbb{R}$ $(a<b)$ with  nonsmooth potentials
(hemivariational inequality):
\begin{equation}\label{1.1}
 \begin{gathered}
-u''_i+u_i\in\lambda\partial_{u_i}F(u_1,\ldots,u_n)
 +\mu\partial_{u_i}G(u_1,\ldots,u_n),\\
u'_i(a)=u'_i(b)=0\,\,,u_i\geq 0\quad\text{for }1\leq i\leq n,
   \end{gathered}
\end{equation}
where $\lambda,\mu$ are two positive parameters,
$F,G:\mathbb{R}^n\to\mathbb{R}$ are measurable and  locally Lipschitz functions.
We denote by $\partial_{u_i}F(u_1,\dots ,u_n)$ $(1\leq i\leq n)$ the partial
generalized gradient of $F(u_1,\dots ,u_n)$ with respect to $u_i$ $(1\leq i\leq n)$,
and by $\partial_{u_i}G(u_1,\dots ,u_n)$ $(1\leq i\leq n)$ the partial generalized
gradient  of $G(u_1,\dots ,u_n)$ to $u_i$ $(1\leq i\leq n)$. Hemivariational
inequality is a new type of variational expressions which arises in problems
of engineering and mechanics, when one deals with nonsmooth and nonconvex
energy functionals. For several concrete applications, we refer the reader
to the monographs of  \cite{g1,m3,m4,n1,p1}
and references  \cite{d2,i2,f2} 
and  therein. More precisely, Iannizzoto  \cite{i1} established a nonsmooth
three critical points theorem and give two applications for
variational-hemivariational inequalities depending on two parameters.
Marano and Motreanu  \cite{m1} obtained a nonsmooth version of Ricceri's
theorem and used the theorem to discuss  the existence of solutions to the
following discontinuous variational-hemivariational inequality problem
\begin{align*}
&-\int_\Omega\nabla u(x)\cdot\nabla(v(x)-u(x))\,{\rm d}x\\
&\leq \lambda\int_\Omega[J^\circ(x,u(x);v(x)-u(x))
+(\mu K)^\circ (x,u(x);v(x)-u(x))]\,{\rm d}x.
\end{align*}
Krist\'aly et al  \cite{k1} generalized a result of Ricceri concerning
the existence of three critical points of certain nonsmooth functional,
also gave two applications, both in the theory of differential inclusions.
Our approach is based on the nonsmooth critical point for non-differential
functions due to Chang \cite{c2} and the nonsmooth  three critical points
which was proved by Iannizzotto  \cite{i1}. Compared with the results
in  \cite{i1,k1,m1}, our framework presents new nontrivial difficulties.
 In particular, the presence of set-valued reaction terms $\partial G$ and
$\partial F$ require completely different devices in order to verify the
appropriate conditions.


We say that  $u=(u_1,\dots ,u_n)\in (W^{1,2}([a,b]))^n$  is a weak solution
of \eqref{1.1} if the following conditions are satisfied
\begin{equation}\label{1.2}
\begin{aligned}
&\int_a^b u'_i(x)v_i'(x)\,{\rm d}x +\int_a^b u_i(x)v_i(x)\,{\rm d}x\\
&+\lambda \int_a^b F^0_{u_i}(u_1(x),\ldots,u_n(x);-v_i(x))\,{\rm d}x \\
& +\mu \int_a^b G^0_{u_i}(u_1(x),\ldots,u_n(x);-v_i(x))\,{\rm d}x\geq 0
\end{aligned}
\end{equation}
for $1\leq i\leq n$ and all $v=(v_1,\ldots,v_n)\in (W^{1,2}([a,b]))^n$.
Moreover we assume that the nonsmooth potential functions $F$ and $G$
satisfy the following assumptions:
\begin{itemize}
\item[(A1)] $F$ and $G$ are regular on $\mathbb{R}^n$ (in the sense of
 Clarke  \cite{c3});

\item[(A2)] There exists $k_1>0$ and $a_1>0$ such that
$|\omega_1|+\ldots+|\omega_n|\leq k_1(|u_1|+\ldots+|u_n|)+a_1$
for all $(u_1,\ldots,u_n)\in\mathbb{R}^n$ and all
$\omega_i\in \partial_{u_i}F(u_1,\ldots,u_n)$ $(1\leq i\leq n)$;

\item[(A3)] There exists $k_2>0$ and $a_2>0$ such that
$|\xi_1|+\ldots+|\xi_n|\leq k_2(|u_1|+\ldots+|u_n|)+a_2$ for all
$(u_1,\ldots,u_n)\in\mathbb{R}^n$ and all
$\xi_i\in \partial_{u_i}G(u_1,\ldots,u_n)~(1\leq i\leq n)$.
\end{itemize}
Our main results are the following:

\begin{theorem} \label{thm1.1}
 Assume that {\rm (A1)--(A3)} are satisfied and there exist
$2n+3$ positive constants $d,e,r\eta_i,\gamma_i$, for $1\leq i\leq n$,
such that  $d+e<b-a$, $0<k_1<\frac{M_1}{2nr}$ and
\[
\sum^n_{i=1}\eta_i^2\leq c\sum^n_{i=1}\gamma_i^2,
\]
 where
$c=de(b-a-\frac{4}{5}(d+e))+\frac{4}{3}(d+e)$, and
\begin{itemize}
\item[(A4)] $F(\zeta_1,\ldots,\zeta_n)\geq 0$ for each
$\zeta_i\in [0,de\gamma_i]$ $(1\leq i\leq n)$ and $F(0,\ldots,0)=0$;

\item[(A5)]
\begin{align*}
M_1&=\frac{\sum_{i=1}^{n}\eta_i^2}{c \sum_{i=1}^{n}\gamma_i^2}
(b-a-(d+e))F(de\gamma_1,\ldots,de\gamma_n)\\
&\quad -(b-a)\max_{(\zeta_1,\ldots,\zeta_n)
\in A_1}F(\zeta_1,\ldots,\zeta_n)>0,
\end{align*}
 where
\[
A_1=\{(\zeta_1,\ldots,\zeta_n)|\sum_{i=1}^{n}\zeta_i^2
\leq \frac{2de}{b-a}\sum^n_{i=1}\eta_i^2\};
\]
\end{itemize}
then, there exist $\lambda',\lambda''\in(0,\nu]$,
$0<\nu<\frac{1}{2nk_1}$, $\lambda'<\lambda''$, $\mu_1>0$  and
 $\sigma_1>0$ such that
for every $\lambda\in[\lambda',\lambda'']$ and $\mu\in(0,\mu_1)$,
 system  \eqref{1.1} has at least three positive solutions
in $(W^{1,2}([a,b]))^n$ whose norms are less than $\sigma_1$.
\end{theorem}

If $n=1$, then system   \eqref{1.1} turns into
\begin{equation}\label{1.3}
 \begin{gathered}
-u''+u\in\lambda\partial_{u}F(u)+\mu\partial_{u}G(u),\\
u'(a)=u'(b)=0, \quad u\geq 0.
   \end{gathered}
\end{equation}
From  Theorem \ref{thm1.1}, we have the following result.

\begin{corollary} \label{coro1.1}
 Assume that the following conditions are satisfied:
\begin{itemize}
\item[(A1')] $F$ and $G$ are regular on $\mathbb{R}$;

\item[(A2')] There exists $k_3>0$ and $a_3>0$ such that
$|\omega|\leq k_3|u|+a_3$ for all $u\in\mathbb{R}$ and all
$\omega\in \partial_{u}F(u)$;

\item[(A3')] There exists $k_4>0$ and $a_4>0$ such that
$|\xi|\leq k_4|u|+a_4$ for all $u\in\mathbb{R}$ and all $\xi\in \partial_{u}G(u)$;
\end{itemize}
and there exist five positive constants $d,e,r,\eta,\gamma$ such that
$d+e<b-a$, $k_3<\frac{M_2}{2r}$ and $\eta^2\leq c\gamma^2$,
where $c=de(b-a-\frac{4}{5}(d+e))+\frac{4}{3}(d+e)$, and
\begin{itemize}
\item[(A4')] $F(u)\geq 0$ for all $u\in [0,de\gamma]$  and $F(0)=0$;

\item[(A5')] $M_2=\frac{\eta^2}{c \gamma^2}(b-a-(d+e))F(de\gamma)
    -(b-a)\max_{u\in A_2}F(u)>0$, where
     $A_2=\{u|-\eta(\frac{2de}{b-a})^{1/2}\leq u
\leq \eta(\frac{2de}{b-a})^{1/2}\}$;
\end{itemize}
then, there exist $\lambda',\lambda''\in(0,\nu]$, $0<\nu<\frac{1}{2nk_3}$,
$\lambda'<\lambda''$, $\mu_1>0$  and  $\sigma_1>0$ such that
for every $\lambda\in[\lambda',\lambda'']$ and $\mu\in(0,\mu_1)$,
 system  \eqref{1.3} has at least three positive solutions
in $W^{1,2}([a,b])$ whose norms are less than $\sigma_1$.
\end{corollary}

Next, we give an example that illustrate Theorem \ref{thm1.1} 
(and Corollary \ref{coro1.1}).
Set
\[
F(u)=\begin{cases}
10^{199}e^{-9900}ue^{-u^3}& u\geq 10,\\
u^{200} e^{-u^4} &0<u<10,\\
0 & u\leq 0,
\end{cases}
\quad
G(u)=\begin{cases}
u & u\geq 1,\\
u^2 &0<u<1,\\
0 &u\leq 0,
\end{cases}
\]
and choose $a=0$, $b=1$, $d=0.25$, $e=0.5$, $\eta=0.5$, $\gamma=16$.
Then it is easy to check that $F(u)$ and $G(u)$ satisfy
the assumptions in Theorem \ref{thm1.1}.

\section{Preliminaries}

In this section we state some definitions and lemmas, which will be
used in this article.  First of all, we give some definitions:
$(X,\|\cdot\|)$ denotes a (real)
 Banach space and $(X^{*},\|\cdot\|_{*})$ its topological dual. 
While $x_{n}\to x$ (respectively, $x_{n}\rightharpoonup x$) in $X$ means 
the sequence $\{x_{n}\}$ converges strongly (respectively, weakly) in $X$.


\begin{definition} \label{def2.1} \rm
 A function $\varphi$: $X\to  \mathbb{R}$ is locally Lipschitz if for
 every $u\in X$ there exist  a neighborhood $U$ of $u$ and $L>0$ such that 
for every $\nu,\omega\in U$,
$$
|\varphi (\nu)-\varphi(\omega)|\leq L\|\nu-\omega\|.
$$
If $\varphi$ is locally Lipschitz on bounded sets, then clearly it 
is locally Lipschitz.
\end{definition}

\begin{definition} \label{def2.2} \rm
 Let $\varphi: X\to \mathbb{R}$ be a locally Lipschitz functional and
$u, \nu\in X$, the 
generalized derivative of $\varphi$ in $u$ along the direction $\nu$, is
$$
\varphi^{0}(u; \nu)=\limsup_{\omega\to u, \tau\to
0^{+}} \frac{\varphi(\omega+\tau\nu)-\varphi(\omega)}{\tau}.
$$
It is easy to see that the function $\nu\to\varphi^{0}(u;\nu)$ is sublinear, 
continuous and so is the support function of a
nonempty, convex and $w^{*}-$ compact set 
$\partial \varphi (u)\subset X^{*}$,
$$
\partial \varphi (u)=\{u^{*}\in X^{*}:\langle u^{*},\nu\rangle_{X}
\leq\varphi^{0}(u; \nu)\}.
$$
If $\varphi\in C^{1}(X)$, then
$\partial\varphi(u)=\{\varphi'(u)\}$.
\end{definition}

Clearly, these definitions extend those of the G\^{a}teaux
directional derivative and gradient.

\begin{definition} \label{def2.3} \rm
A mapping $A:X\to X^*$ is of type $(S)_+$, for every sequence $\{u_n\}$ 
such that $u_n\rightharpoonup u \in X $ and
$$
\limsup_n\langle A(u_n), u_n-u\rangle\leq 0,
$$
one has $u_n\to u$.
\end{definition}

\begin{definition} \label{def2.4} \rm
Let $\varphi:X\to\mathbb{R}$ be a locally Lipschitz functional and 
$\mathcal{X}:X\to\mathbb{R}\cup\{+\infty\}$ be a proper, convex, 
lower semicontinuous (l.s.c.) functional whose restriction to the set
$$
\operatorname{dom}(\mathcal{X})=\{u\in X:\mathcal{X}(u)<+\infty\}
$$
is continuous, then $\varphi+\mathcal X$ is a Motreanu-Panagiotopouls functional.
\end{definition}

In most applications,  $C$ is a nonempty, closed, convex subset of $X$; 
the indicator of $C$ is the function $\mathcal{X}_C:X\to \mathbb{R}\cup\{+\infty\}$ 
defined by 
$$
\mathcal{X}_C=\begin{cases}
0&\text{if }u\in C,\\
+\infty&\text{if }u\not\in C.
\end{cases}
$$
It is easy to see that $\mathcal{X}_C$ is proper, convex and l.s.c., 
while its restriction to ${\rm dom}(\mathcal{X})=C$ is the constant 0.

\begin{definition} \label{def2.5} \rm
Let $\varphi+\mathcal{X}$ be a Motreanu-Panagiotopouls functional, 
$u\in X$. Then, $u$ is a critical point of $\varphi+\mathcal{X}$ if 
for every $v\in X$
$$
\varphi^\circ(u;v-u)+\mathcal{X}(v)-\mathcal{X}(u)\geq 0.
$$
\end{definition}

The next propositions will be used later. 


\begin{proposition}[\cite{c3}] \label{prop2.1}
 Let $h:X\to\mathbb{R}$ be locally Lipschitz on $X$. Then
\begin{itemize}
\item[(i)] $h^{\circ}(u;v)=\max\{\langle\omega,v\rangle_X:\omega\in\partial h(u)\}$ 
for all $u,v\in X$,

\item[(ii)] (Lebourg's mean value theorem) 
Let $u$ and $v$ be two points in $X$, then there exists a point $\zeta$ in 
the open segment between $u$ and $v$ and 
$\omega_{\zeta}\in\partial h(\zeta)$ such that
$$
h(u)-h(v)=\langle\omega_{\zeta},u-v\rangle_X.
$$
\end{itemize}
\end{proposition}

We say that $h$ is regular at $u\in X$ (in the sense of Clarke  \cite{c3}) 
if for all $z\in X$ the usual one-sided directional derivative
$$
h'(u;z)=\lim_{t\to 0^+}\frac{h(u+tz)-h(u)}{t}
$$
exists and $h'(u;z)=h^\circ(u;z)$. Moreover, we say that  $h$ is 
regular on $X$, if it is regular in every point $u\in X$.

\begin{proposition} \label{prop2.2}
 Let $h:X\to\mathbb{R}$ be a locally Lipschitz function which is regular 
at $(u_1,\ldots,u_n)\in X$, then
\begin{itemize}
\item[(i)]  
\[
\partial h(u_1,\ldots,u_n)\subset\partial_{u_1}h(u_1,
\ldots,u_n)\times\ldots\times\partial_{u_n}h(u_1,\ldots,u_n),
\]
 where 
$\partial_{u_i}h(u_1,\ldots,u_n)$  denotes the partial generalized 
gradient of \\
$h(u_1,\ldots,u_i,\ldots,u_n)$ to $u_i$ for $1\leq i\leq n$.

\item[(ii)] $h^\circ (u_1,\ldots,u_n; v_1,\ldots,v_n)\leq 
h^\circ_1(u_1,\ldots,u_n;v_1)+\ldots+
 h^\circ_n(u_1,\ldots,u_n;v_n)$ for all $(v_1,\ldots,v_n)\in X$.
\end{itemize}
\end{proposition}

\begin{proof}
 For the proof of (i), see  \cite[Proposition 2.3.15]{c3}. 
From  Proposition \ref{prop2.1} (i), it follows that there exists a 
$\omega\in\partial h(u,v)$ such that $h^\circ(u;v)=\langle\omega,v\rangle_X$. 
From  (i) we have $\omega=(\omega_1,\ldots,\omega_n)$, where 
$\omega_i\in \partial_{u_i}h(u_1,\ldots,u_n)$ $(1\leq i\leq n)$, 
and using the definition of the generalized gradient, we derive 
$h^\circ (u;v)=\langle\omega_1,v_1\rangle_{W^{1,2}([a,b])}+\ldots
+\langle\omega_n,v_n\rangle_{W^{1,2}([a,b])}
\leq h^\circ_1(u_1,\ldots,u_n;v_1)+\ldots+
 h^\circ_n(u_1,\ldots,u_n;v_n)$.
\end{proof}

The following theorems are the main tools for proving our main results.

\begin{theorem}[see  \cite{i1}] \label{thm2.1} 
 Let $(X,\|\cdot\|)$ be a reflexive Banach space, $\Lambda\subset\mathbb{R}$ 
an interval, $C$ a nonempty, closed, convex subset of $X$, 
$\mathcal{N}\in C^1(X,\mathbb{R})$ a sequentially weakly l.s.c. functional, 
bounded on any bounded subset of $X$, such that $\mathcal{N}'$ is of type
 $(S)_+$, $\Gamma:X\to\mathbb{R}$ is a locally Lipschitz functional with 
compact gradient, and $\rho_1\in\mathbb{R}$. Assume also that the following 
conditions hold:
\begin{itemize}
\item[(i)] 
 $\sup_{\lambda\in\Lambda}\inf_{u\in C}[\mathcal{N}(u)
 +\lambda (\rho_1-\Gamma(u))]<\inf_{u\in C}
\sup_{\lambda\in\Lambda}[\mathcal{N}(u)+\lambda (\rho_1-\Gamma(u))]$;

\item[(ii)] $\lim_{\|u\|\to+\infty}[\mathcal{N}(u)-\lambda \Gamma(u)]
=+\infty$ for every $\lambda\in\Lambda$.
\end{itemize}
Then there exist $\lambda',~\lambda''\in\Lambda~(\lambda'<\lambda'')$ and 
$\sigma_1>0$ such that for every $\lambda\in[\lambda',\lambda'']$ and every 
locally Lipschitz functional $\mathcal{G}:X\to\mathbb{R}$ with compact gradient, 
there exists $\mu_1>0$ such that for every $\mu\in (0,\mu_1)$, the functional
$\mathscr{N}-\lambda\Gamma-\mu\mathcal{G}+\mathcal{X}_C$ has at least 
three critical points whose norms are less than $\sigma_1$.
\end{theorem}

\begin{theorem}[\cite{b1}] \label{thm2.2} 
 Let $X$ be a nonempty set and $\Phi$, $\Psi$ two real functions on $X$. 
Assume that $\Phi(u)\geq 0$ for every  $u\in X$ and there exists $u_0\in X$ 
such that $\Phi(u_0)=\Psi(u_0)=0$. Further, assume that there exist 
$u_1\in X$, $r>0$ such that $\Phi(u_1)>r$ and
$$
\sup_{\Phi(u)<r}\Psi(u)<r\frac{\Psi(u_1)}{\Phi(u_1)}.
$$
Then, for every $h>1$ and for every $\rho \in\mathbb{R}$ satisfying
$$
\sup_{\Phi(u)<r}\Psi(u)+\frac{r\frac{\Psi(u_1)}
{\Phi(u_1)}-\sup_{\Phi(u)<r}\Psi(u)}{h}<\rho<r\frac{\Psi(u_1)}{\Phi(u_1)},
$$
one has
$$
\sup_{\lambda\in\mathbb{R}}\inf_{u\in X}
[\Phi(u)+\lambda (-\Psi(u)+\rho)]<\inf_{u\in X}
\sup_{\lambda\in[0,\nu]}[\Phi(u)+\lambda (-\Psi(u)+\rho))],
$$
where 
\[
\nu=\frac{hr}{r\frac{\Psi(u_1)}{\Phi(u_1)}-\sup_{\Phi(u)<r}(\Phi(u))}.
\]
\end{theorem}

\section{Proof of main results}

Let $X=(W^{1,2}([a,b]))^n$ equipped with the norm
$$
\|(u_1,\dots ,u_n)\|=\Big(\sum^n_{i=1}\|u_i\|^2\Big)^{1/2},
$$
where $\|u_i\|=(\int^b_a(|u'_i(x)|^2+|u_i(x)|^2)\,{\rm d}x)^{1/2}$ for
$1\leq i\leq n$, and we introduce the functionals 
$\Phi,\Psi:X\to\mathbb{R}$ 
for each $u=(u_1,\ldots,u_n)\in X$, as follows
\[
\Phi(u)=\sum^n_{i=1}\frac{1}{2}\|u_i\|^2, \quad
\Psi(u)=\int^b_a F(u_1(x),\ldots,u_n(x))\,{\rm d}x
\]
and
\[
J(u)=\int^b_a G(u_1(x),\ldots,u_n(x))\,{\rm d}x.
\]
Let $C=\{u\in X:u(x)\geq 0\text{ for every }x\in[a,b]\}$, 
then for all $\lambda,\mu>0$ and $u\in X$,
$$
\varphi(u)=\Phi(u)-\lambda\Psi(u)-\mu J(u)+\mathcal{X}_C(u).
$$
The next lemma displays some properties of $\Phi$.

\begin{lemma} \label{lem3.1}
 $\Phi\in C^1(X,\mathbb{R})$ and its gradient, defined for $u,v\in X$ by
$$
\langle \Phi'(u),v\rangle=\int^b_a\nabla u(x)\nabla v(x)\,{\rm d}x
$$
is of type $(S)_+$,
where $u=(u_1,\ldots,u_n)$ and $v=(v_1,\ldots,v_n)$.
\end{lemma}

The proof of the above lemma is similar to the one in Chabrowski 
\cite[Section 2.2]{c1}.  We omit  it here.
Next we consider some  properties of $\Psi$.


\begin{lemma} \label{lem3.2}
 If {\rm (A1)--(A2)} are satisfied, then $\Psi(u):X\to\mathbb{R}$ is a 
locally Lipschitz function with compact gradient. Moreover,
\begin{equation}\label{3.1}
\Psi^\circ (u_1,\ldots,u_n;v_1,\ldots,v_n)
\leq\int^b_a F^\circ(u_1,\ldots,u_n;v_1,\ldots,v_n)\,{\rm d}x,
\end{equation}
for all $(u_1,\ldots,u_n),(v_1,\ldots,v_n)\in X$.
\end{lemma}

\begin{proof} 
First, let $u=(u_1,\ldots,u_n)$, $v=(v_1,\ldots,v_n)\in X$ be fixed elements. 
Using the regularity of $F$ and Lebourg's mean value theorem 
(see Proposition \ref{prop2.1}) we derive a $\omega\in\partial F(\zeta_1,\ldots,\zeta_n)$ 
such that
$$
F(u)-F(v)=\langle\omega,u-v\rangle,
$$
where $(\zeta_1,\ldots,\zeta_n)$ is in the open line segment between 
$(u_1,\ldots,u_n)$ and $(v_1,\ldots,v_n)$.
Using  Proposition \ref{prop2.2}, there exist 
$\omega_i\in \partial_{\zeta_i}F(\zeta_1,\ldots,\zeta_n)$ $(1\leq i\leq n)$, 
such that
\begin{equation}\label{3.2}
F(u_1,\ldots,u_n)-F(v_1,\ldots,v_n)=\omega_1(u_1-v_1)+\ldots+\omega_n(u_n-v_n).
\end{equation}
From (A1) and \eqref{3.2}, we obtain

\begin{equation}\label{3.3}
\begin{aligned}
&|F(u_1,\ldots,u_n)-F(v_1,\ldots,v_n)|\\
&\leq (|\omega_1|+\ldots+|\omega_n|)(|u_1-v_1|+\ldots+|u_n-v_n|)\\
&\leq [k_1(|u_1|+\ldots+|u_n|+|v_1|+\ldots+|v_n|)+a_1](|u_1-v_1|+\ldots+|u_n-v_n|) .
\end{aligned}
\end{equation}
 Using \eqref{3.3}, H\"older's
inequality and the fact the embedding  $W^{1,2}([a,b])\hookrightarrow L^2([a,b])$ 
is continuous, we derive
\begin{align*}
&|\Psi (u_1,\ldots, u_n)-\Psi (v_1,\ldots, v_n)|\\
&\leq nk_1\sum^n_{i=1}(\|u_i\|_2+\|v_i\|_2+m_1)(\|u_1-v_1\|_2+\ldots+\|u_n-v_n\|_2)\\
&\leq c_1 \sum^n_{i=1}(\|u_i\|+\|v_i\|+m_1)(\|u_1-v_1\|+\ldots+\|u_n-v_n\|)
\end{align*}
for some $c_1>0$, $m_1>0$ and $\|\cdot\|_2$ denotes the $L^2-$norm. 
From this relation it follows that $\Psi$ is locally Lipschitz on $X$.

 Now choose $u=(u_1,\ldots,u_n)$, $h=(h_1,\ldots,h_n)\in X$, since 
$F(u_1,\ldots,u_n)$ is continuous, $F^\circ (u_1,\ldots,u_n;h_1,\ldots,h_n)$ 
can be expressed as the upper limit of
$$
\frac{F(u^0_1+th_1,\ldots,u^0_n+th_n)-F(u^0_1,\ldots,u^0_n)}{t},
$$
where $t\to 0^+$ taking rational values and 
$(u^0_1,\ldots,u^0_n)\to (u_1,\ldots,u_n)$ taking values in a countable 
dense subset of $X$. Therefore, the map 
\[
x\mapsto F^\circ (u_1(x),\ldots,u_n(x);h_1(x),\ldots,h_n(x))
\] 
is also measurable. By  (A2), the map 
$x\mapsto F^\circ (u_1(x),\ldots,u_n(x);h_1(x),\ldots,h_n(x))$ 
belongs to $L^1([a,b])$. Since $X$ is separable, there exist functions 
$(u^k_1,\ldots,u^k_n)\in X$ and numbers $t_k\to 0^+$ such that 
$(u^k_1,\ldots,u^k_n)\to (u_1,\ldots,u_n)$ in $X$ and
$$
\Psi^\circ(u_1,\ldots,u_n;h_1,\ldots,h_n)
=\lim_{k\to+\infty}\frac{\Psi(u^k_1+t_k h_1,\ldots,u^k_n+t_k h_n)
-\Psi(u^k_1,\ldots,u^k_n)}{t_k}.
$$
We define $g_n:[a,b]\to\mathbb{R}\cup\{+\infty\}$ by
\begin{align*}
g_k(x)
&=-\frac{F(u^k_1+t_k h_1,\ldots,u^k_n+t_k h_n)-F(u^k_1,\ldots,u^k_n)}{t_k}
 +k_1(|h_1|+\ldots+|h_n|)\\
&\quad\times \Big(|u^k_1|+\ldots+|u^k_n|+|u^k_1+t_k h_1|
+\ldots+|u^k_n+t_k h_n|+\frac{a_1}{k_1}\Big).
\end{align*}
Then the function $g_k$ is measurable and nonnegative (see \eqref{3.3}). 
From Fatou's Lemma, we have
$$
I=\limsup_{k\to+\infty}\int^b_a[-g_k(x)]\,{\rm d}x
\leq\int^b_a\limsup_{k\to+\infty}[-g_k(x)]\,{\rm d}x=H.
$$
Let $L_k=B_k+g_k$, where
$$
B_k(x)=\frac{F(u^k_1+t_k h_1,\ldots,u^k_n+t_k h_n)-F(u^k_1,\ldots,u^k_n)}{t_k}.
$$
From the Lebesgue dominated convergence theorem, we obtain
\begin{align*}
&\limsup_{k\to+\infty}\int^b_a L_k(x)\,{\rm d}x\\
&=2k_1\int_{a}^b(|h_1(x)|+\ldots+|h_n(x)|)\Big(|u_1(x)|+\ldots
+|u_n(x)|+\frac{a_1}{2k_1}\Big)\,{\rm d}x.
\end{align*}
Hence, we derive
\begin{align*}
I&=\limsup_{k\to+\infty}\frac{\Psi(u^k_1+t_k h_1,\ldots,u^k_n+t_k h_n)
-\Psi(u^k_1,\ldots,u^k_n)}{t_k}-\lim_{k\to+\infty}\int^b_aL_k\,{\rm d}x\\
&=\Psi^\circ(u_1,\ldots,u_n;h_1,\ldots,h_n)
 -2k_1\int_{a}^b(|h_1(x)|+\ldots+|h_n(x)|)\\
&\quad\times \Big(|u_1(x)|+\ldots+|u_n(x)|+\frac{a_1}{2k_1}\Big)\,{\rm d}x.
\end{align*}
Now, we obtain the estimates  $H\leq H_B-H_L$, where 
$H_B=\int^b_a\limsup_{k\to+\infty}B_k(x)\,{\rm d}x$ and 
$H_L=\int^b_a\liminf_{k\to+\infty}L_k(x)\,{\rm d}x$. 
Since $(u^k_1(x),\ldots,u^k_n(x))\to(u_1(x),\ldots,u_n(x))$ a.e. in 
$[a,b]$ and $t_k\to 0^+$, we derive
$$
H_L=2k_1\int_{a}^b(|h_1(x)|+\ldots+|h_n(x)|)
\Big(|u_1(x)|+\ldots+|u_n(x)|+\frac{a_1}{2k_1}\Big)\,{\rm d}x.
$$
On the other hand,
\begin{align*}
H_B
&=\int^b_a\limsup_{k\to+\infty}\frac{F(u^k_1+t_k h_1,\ldots,u^k_n
 +t_k h_n)-F(u^k_1,\ldots,u^k_n)}{t_k}\,{\rm d}x\\
&\leq\int^b_a\limsup_{(u^0_1,\ldots,u^0_n)\to(u_1,\ldots,u_n),\, t\to 0^+}
 \frac{F(u^0_1+t h_1,\ldots,u_n^0+t h_n)-F(u^0_1,\ldots,u^0_n)}{t}\,{\rm d}x\\
&=\int^b_a F^\circ (u_1,\ldots,u_n;h_1,\ldots,h_n)\,{\rm d}x,
\end{align*}
which implies \eqref{3.1}.

At last, we prove that $\partial\Psi$ is compact. Let $\{u^k\}_{k\geq 1}$ 
be a sequence in $X$, where $u^k=(u_{1}^k,\ldots,u_{n}^k)$, such that 
$\|u^k\|\leq M$ and choose $\omega^k\in\partial\Psi(u^k)$, where 
$\omega^k=(\omega_{1}^k,\ldots,\omega_{n}^k)$, $k\geq 1$, $k\in\mathbb{N}$ and $M>0$. 
From (A1), for every $v=(v_1,\ldots,v_n)\in X$, we obtain
\begin{align*}
&\langle\omega^k,v\rangle\\
&\leq\int^b_a|\omega^k(x)||v(x)|\,{\rm d}x
 \leq\int^b_a k_1 \Big(|u_1^k|+\ldots+|u_n^k|+\frac{a_1}{k_1}\Big)(|v_1|+\ldots+|v_n|)\,{\rm d}x\\
&\leq k_1\Big[\Big(\int^b_a (|u_1^k|+\ldots+|u_n^k|)^2\,{\rm d}x\Big)^{1/2}
 +\frac{a_1}{k_1}(b-a)^{1/2}\Big]\\
&\quad\times \Big(\int^b_a (|v_1|+\ldots  +|v_n|)^2\,{\rm d}x\Big)^{1/2}\\
&=k_1\Big[\Big(\int^b_a (|u_1^k|^2+\ldots+|u_n^k|^2+2|u_1^k||u_2^k|+\ldots
 +2|u_{n-1}^k||u_n^k|)\,{\rm d}x\Big)^{1/2} \\
&\quad +\frac{a_1}{k_1}(b-a)^{1/2}\Big]
\Big(\int^b_a (|v_1|^2+\ldots+|h_n|^2+2|v_1||v_2|+\ldots
 +2|v_{n-1}||v_n|)\,{\rm d}x\Big)^{1/2}\\
&\leq k_1\Big[n\Big(\int^b_a(|u_1^k|^2+\ldots+|u_n^k|^2)\,{\rm d}x\Big)^{1/2}
 +\frac{a_1}{k_1}(b-a)^{1/2}\Big]\\
&\quad\times \Big(\int^b_a(|v_1|^2+\ldots+|v_n|^2)\,{\rm d}x\Big)^{1/2}\\
&\leq \Big(c^1\|u^k\|+\frac{a_1}{k_1}(b-a)^{1/2}\Big)\|v\|\\
&\leq \Big(c^1M+\frac{a_1}{k_1}(b-a)^{1/2}\Big)\|v\|,
\end{align*}
where $c^1$ is a positive constant.
Hence
$$
\|\omega^k\|_*\leq c^1M+\frac{a_1}{k_1}(b-a)^{1/2}=c^2\,.
$$
This means that $\{\omega^k\}$ is bounded. Passing to a subsequence 
$\omega^k\rightharpoonup\omega\in X^*$, where 
$\omega=(\omega_1,\ldots,\omega_n)$. We  need to prove that the convergence 
is strong.

We proceed by contradiction. Suppose that there exists $\varepsilon>0$, 
such that for every $k\in\mathbb N$
$$
\|\omega^k-\omega\|_*>\varepsilon,
$$
where $\omega^k=(\omega_{1}^k,\ldots,\omega_{n}^k)$.
That is  for all $k\in\mathbb N$, there exists a 
$v^k=(v_{1}^k,\ldots,v_{n}^k)\in B(0,1)\times\ldots\times B(0,1)$ such that
\begin{equation}\label{3.4}
\langle\omega^k-\omega,v^k\rangle>\varepsilon\,.
\end{equation}
Since $\{v^k\}_{k\geq 1}$ is  bounded, passing to a subsequence, 
$v_n\rightharpoonup v=(v_{1}^0,\ldots,v_{n}^0)\in X$ and $\|v^k-v\|\to 0$, 
so for $k$ big enough,
$$
|\langle\omega^k-\omega,v\rangle|<\frac{\varepsilon}{3},\quad
|\langle\omega,v^k-v\rangle|<\frac{\varepsilon}{3},\quad
\|v^k-v\|<\frac{\varepsilon}{3c^2},
$$
this implies
\begin{align*}
\langle\omega^k-\omega,v^k\rangle
&=\langle\omega^k-\omega,v\rangle+\langle\omega^k,v^k-v\rangle
-\langle\omega,v^k-v\rangle\\
&\leq \frac{\varepsilon}{3}+\frac{\varepsilon}{3}+c^2\|v^k-v\|\\
&<\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}
=\varepsilon,
\end{align*}
which contradicts \eqref{3.4}. The proof is complete.
\end{proof}

 Analogously, we deduce the properties of the function $J$.

\begin{lemma} \label{lem3.3}
If {\rm (A1)} and {\rm (A3)}  are satisfied, then $J:X\to \mathbb{R}$ 
is a locally Lipschitz function with compact gradient and
$$
J^\circ(u_1,\ldots,u_n;v_1,\ldots,v_n)
\leq \int^b_aG^\circ (u_1,\ldots,u_n;v_1,\ldots,v_n)\,{\rm d}x
$$
for all $(u_1,\ldots,u_n)$, $(v_1,\ldots,v_n)\in X$.
\end{lemma}

Now, we are in a position to establish the following proposition.

\begin{lemma} \label{lem3.4}
  If  {\rm (A1)--(A3)} are satisfied, then, for every $\lambda,\mu>0$, 
$\varphi:X\to\mathbb{R}\cup\{+\infty\}$ is a Motreanu-Panagiotopoulos 
function and  the critical points $(u_1,\ldots,u_n)$ belong to $X$ of $\varphi$ 
is a weak solution of \eqref{1.20}.
\end{lemma}

\begin{proof}
 From Lemmas \ref{lem3.1}--\ref{lem3.3} the function $I=\Phi-\lambda\Psi-\mu J$ 
is locally Lipschitz; furthermore, $C$ is a closed convex subset of $X$ and 
$C\neq\emptyset$; thus $\varphi$ is a Motreanu-Panagiotopoulos function.
Since $(u_1,\ldots,u_n)\in X$ is a critical point of $\varphi$, then $u\in C$ 
and for all $v=(v_1,\ldots,v_n)\in C$ we have
\begin{align*}
0
&\leq I^\circ(u_1,\ldots,u_n;v_1,\ldots,v_n)\\
&=\int^b_a\sum^n_{i=1}(u'_iv'_i+u_iv_i)\,{\rm d}x+\lambda(-\Psi)^\circ
(u_1,\ldots,u_n;v_1,\ldots,v_n)\\
&\quad+\mu(-J)^\circ
(u_1,\ldots,u_n;v_1,\ldots,v_n)\\
&\leq \int^b_a\sum^n_{i=1}(u'_iv'_i+u_iv_i)\,{\rm d}x+\lambda\int^b_aF^\circ
(u_1,\ldots,u_n;-v_1,\ldots,-v_n)\,{\rm d}x\\
&\quad +\mu\int^b_aG^\circ (u_1,\ldots,u_n;-v_1,\ldots,-v_n)\,{\rm d}x.
\end{align*}
From Proposition \ref{prop2.2} (ii), we have
\begin{align*}
0&\leq \int^b_a\sum^n_{i=1}(u'_iv'_i+u_iv_i)\,{\rm d}x
 +\lambda\int^b_aF_{u_1}^\circ (u_1,\ldots,u_n;-v_1)\,{\rm d}x+\ldots\\
&\quad +\lambda\int^b_aF_{u_n}^\circ (u_1,\ldots,u_n;-v_n)\,{\rm d}x\\
&\quad +\mu\int^b_aG_{u_1}^\circ (u_1,\ldots,u_n;-v_1)\,{\rm d}x+\ldots
 +\mu\int^b_aG_{u_n}^\circ (u_1,\ldots,u_n;-v_n)\,{\rm d}x.
\end{align*}
Taking $v_1=\ldots=v_{i-1}=v_{i+1}=\ldots=v_n=0$ in the above inequality 
for $1\leq i\leq n$, then we lead to \eqref{1.2}, i.e., $(u_1,\ldots,u_n)$ 
is a weak solution of \eqref{1.1}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
We apply Theorem \ref{thm2.1} to prove this theorem. For this purpose, it is  
easy to see that $X$ is a reflexive Banach space. 
We put $\Lambda=(0,\nu]$, where $0<\nu<\frac{1}{2nk_1}$.  
The functional $\Phi\in C^1 (X,\mathbb{R})$ is continuous and convex, 
hence weakly l.s.c. and obviously bounded on any bounded subset of  $X$. 
Moreover, $\Phi'$ is of type $(S_+)$ (Lemma \ref{lem3.1}) and  $\Psi$ is a 
locally Lipschitz function with compact gradient (Lemma \ref{lem3.2}). 
We only need to test conditions (i) and (ii) in Theorem \ref{thm2.1}.

We first check condition (i). Let $v(x)=(v_1(x),\ldots,v_n(x))$ 
such that for $1\leq i\leq n$,
\begin{equation}\label{3.5}
v_i(x)=\begin{cases}
\frac{e\gamma_i}{d}(x-a)^2 &\text{if } a\leq x<a+d,\\
de\gamma_i &\text{if } a+d\leq x\leq b-e, \\
\frac{d\gamma_i}{e}(b-x)^2 &\text{if } b-e\leq x\leq b.
\end{cases}
\end{equation}
It is obvious that $v\in X$. From a simple computation, we have
\begin{equation}\label{3.4a}
\int^b_a(|v'_i(x)|^2+|v_i(x)|^2)\,{\rm d}x
=\Big(d^2e^2\Big(b-a-\frac{4(d+e)}{5}\Big)+\frac{4}{3}de(d+e)\Big)\gamma_i^2.
\end{equation}
Set $r=\frac{de}{2}\sum^n_{i=1}\eta_i^2$, since 
$\sum^n_{i=1}\eta_i^2\leq c\sum^n_{i=1}\gamma_i^2$, from \eqref{3.4a},
 we obtain
\[
\Phi(v_1,\ldots,v_n)=\frac{dec}{2}\sum^n_{i=1}\gamma_i^2
>\frac{de}{2}\sum^n_{i=1}\eta_i^2=r.
\]
 Let $u_0=(0,\ldots,0)$, $u_1=(v_1,\ldots,v_n)$. From (A4), we have
 $\Psi(u_0)=0$,  and $\Phi(u_0)=0$. From  \cite{a1}, we obtain
 $$
\max_{x\in[a,b]}|u_i(x)|\leq \Big(\frac{2}{b-a}\Big)^{1/2}\|u_i\|
$$
for all $u_i\in W^{1,2}([a,b]),~1\leq i\leq n$.
Hence
\begin{equation}\label{3.5a}
\sup_{x\in[a,b]}\sum^n_{i=1}\frac{|u_i(x)|^2}{2}
\leq \frac{2}{b-a}\sum^n_{i=1}\frac{\|u_i(x)\|^2}{2}
\end{equation}
for all $u=(u_1,\ldots,u_n)\in X$. From \eqref{3.5a}, for each $r>0$,
\begin{equation}\label{3.5b}
\begin{aligned}
&\Phi^{-1}((-\infty,r))\\
&=\{u=(u_1,\ldots,u_n)\in X:\Phi(u)<r\} \\
&=\big\{u=(u_1,\ldots,u_n)\in X:\sum^n_{i=1}\frac{\|u_i\|^2}{2}<r\big\}\\
&\subset \big\{u=(u_1,\ldots,u_n)\in X:\sum^n_{i=1}|u_i|^2<\frac{4r}{b-a}
\text{ for all }x\in [a,b]\big\}.
\end{aligned}
\end{equation}
By  (A5), we have
\begin{equation}\label{3.6}
\frac{\sum^n_{i=1}\eta^2_i}{c\sum^n_{i=1}\gamma^2_i}
(b-a-(d+e))F(de\gamma_1,\ldots,de\gamma_n)>(b-a)
\max_{(\zeta_1,\ldots,\zeta_n)\in A_1}F(\zeta_1,\ldots,\zeta_n).
\end{equation}
From (A4), (A5), \eqref{3.5}, \eqref{3.5b} and \eqref{3.6}, for all 
$u=(u_1,\ldots,u_n)\in X$, we have
\begin{equation}\label{eqn:(3.7)}
\begin{aligned}
\sup_{u\in\Phi^{-1}((-\infty,r))}\Psi(u)
&\leq\sup_{\sum^n_{i=1}|u_i(x)|^2
\leq\frac{4r}{b-a}}\int^b_a F(u_1(x),\ldots,u_n(x))\,{\rm d}x\\
&\leq \int^b_a\sup_{(\zeta_1,\ldots,\zeta_n)\in A_1}F(\zeta_1,\ldots,\zeta_n)
 \,{\rm d}x\\
&<\frac{\sum^n_{i=1}\eta_i^2}{c\sum^n_{i=1}\gamma^2_i}(b-a-(d+e))
 F(de\gamma_1,\ldots,de\gamma_n)\\
&\leq \frac{r\int^{b-e}_{a+d}F(de\gamma_1,\ldots,de\gamma_n)
 \,{\rm d}x}{\Phi(v_1,\ldots,v_n)}\\
&\leq \frac{r\int^b_aF(v_1,\ldots,v_n)\,{\rm d}x}{\Phi(v_1,\ldots,v_n)}
=\frac{r\Psi(v)}{\Phi(v)}.
\end{aligned}
\end{equation}
Note that $0<k_1<\frac{M_1}{2nr}$.
Fix $1<h<\frac{M_1}{2nk_1r}$ and $\rho$ such that
$$
\sup_{\Phi(u)<r}\Phi(u)+\frac{r\frac{\Psi(v)}{\Phi(v)}-\sup_{\Phi(u)<r}\Psi(u)}{h}
<\rho<r\frac{\Psi(v)}{\Phi(v)}.
$$
From Theorem \ref{thm2.2}, we obtain
$$
\sup_{\lambda\in\mathbb{R}}\inf_{u\in X}[\Phi (u)
 +\lambda (\rho-\Psi(u))]<\inf_{u\in X}
\sup_{\lambda\in\Lambda}[\Phi (u)+\lambda (\rho-\Psi(u))].
$$
Next, we test condition (ii). From (A2),  Lebourg's mean value theorem 
and Proposition \ref{prop2.2}, there exist 
$\omega_i\in\partial_{\zeta_i}F(\zeta_1,\ldots,\zeta_n)$, where $\zeta_i$ 
is between the segment $u_i$ and 0 for $1\leq i\leq n$, such that
\begin{equation}\label{3.8}
\begin{aligned}
F(u_1,\ldots,u_n)
&=F(u_1,\ldots,u_n)-F(0,\ldots,0) 
 =\omega_1 u_1+\ldots+\omega_n u_n\\
&\leq (|\omega_1|+\ldots+|\omega_n|)(|u_1|+\ldots+|u_n|)\\
&\leq k_1(|u_1|+\ldots+|u_n|)^2+a_1(|u_1|+\ldots+|u_n|)\\
& =k_1(|u_1|^2+\ldots+|u_n|^2+2|u_1||u_2|+\ldots+2|u_{n-1}||u_n|)\\
&\quad +a_1(|u_1|+\ldots+|u_n|)\\
&\leq nk_1(|u_1|^2+\ldots+|u_n|^2)+a_1(|u_1|+\ldots+|u_n|).
\end{aligned}
\end{equation}
By  \eqref{3.8}, we can find two positive constants $\mathcal{K}$ 
and $\tau$ satisfying
$\mathcal{K}\leq nk_1$ and
 $$
 F(\zeta_1,\ldots,\zeta_n)\leq \mathcal{K}\sum^n_{i=1}\zeta^2_i+\tau
$$
for all $(\zeta_1,\ldots,\zeta_n)\in\mathbb{R}^n$. Let $u=(u_1,\ldots,u_n)\in X$, 
then it is easy to see that
\begin{equation}\label{3.9}
F(u_1,\ldots,u_n)\leq \mathcal{K}\sum^n_{i=1}|u_i(x)|^2+\tau\quad
\text{a.e. }x\in [a,b].
\end{equation}
Choosing $\lambda\in (0,\nu]$, from \eqref{3.9}, we obtain
\begin{align*}
\Phi(u)-\lambda\Psi(u)
&=\sum^n_{i=1}\frac{\|u_i\|^2}{2}-\lambda\int^b_aF(u_1(x),\ldots,u_n(x))\,{\rm d}x\\
&\geq \sum^n_{i=1}\frac{\|u_i\|^2}{2}-\lambda \mathcal{K}\sum^n_{i=1}
 \int^b_a|u_i(x)|^2\,{\rm d}x-\lambda(b-a)\tau\\
&\geq \sum^n_{i=1}\frac{\|u_i\|^2}{2}-\lambda \mathcal{K}
 \sum^n_{i=1}\int^b_a|u_i(x)|^2\,{\rm d}x-\frac{b-a}{2nk_1}\tau\\
&\geq \Big(\frac{1}{2}-\lambda \mathcal{K}\Big)
 \sum^n_{i=1}\|u_i\|^2-\frac{b-a}{2nk_1}\tau.
\end{align*}
Since $0<\mathcal{K}\leq nk_1$ and $0<\nu<\frac{1}{2nk_1}$, we have
$$
\lim_{\|u\|\to+\infty}(\Phi(u)-\lambda\Psi(u))=+\infty,
$$
then we have tested condition (ii). The proof is complete.
\end{proof}


\subsection*{Acknowledgements}
This research was partly supported by the National Natural Science
 Foundation of China (11371127). The authors would like to thank the 
 editor and the reviewer for his/her valuable comments and constructive 
suggestions, which help  to improve the presentation of this paper.

\begin{thebibliography}{99}
 

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