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\title{Three weak solutions for a class of Neumann boundary value
systems involving the $(p_1,\ldots,p_n)$-Laplacian}%
\author{Armin Hadjian \\\\
Department of Mathematics, Faculty of Basic Sciences,\\
University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran\\E-mail:
hadjian83@gmail.com, a.hadjian@ub.ac.ir}
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\date{}

\begin{document}
\maketitle
\begin{abstract}
In this paper, we establish the existence of two intervals of
positive real parameters $\lambda$ for which a class of Neumann
boundary value equations involving the $(p_1,\ldots,p_n)$-Laplacian
admits three weak solutions, whose norms are uniformly bounded with
respect to $\lambda$ belonging to one of the two intervals. The
approach is based on variational methods.
\end{abstract}
{\bf Keywords}: $p$-Laplacian; Neumann problem;
Multiplicity results; Variational methods.\\
2010 Mathematics Subject Classification: 35D30; 35B38; 35J50.

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\section{Introduction}

\quad Throughout the paper, $\Omega\subset \mathbb{R}^{N}~(N\geq1)$
is a bounded domain with a smooth boundary $\partial\Omega$, $p_i>N$
(and $p_i\geq2$) for $1\leq i\leq n$ are natural numbers
 and $\lambda$ is
a positive parameter.

The aim of this paper is to investigate the following quasilinear
elliptic system
\begin{equation}\label{1}
\left\{\begin{array}{ll} \Delta_{p_i}u_i+\lambda
F_{u_i}(x,u_1,\ldots,u_n)
=a_i(x)|u_i|^{p_i-2}u_i & \textrm{ in } \Omega,\\
\partial u_i/\partial\nu=0 & \textrm{ on } \partial\Omega,
\end{array}
\right.
\end{equation}
for $1\leq i\leq n,$ where $\Delta_{p_i}u_i:=\textrm{div}(|\nabla
u_i|^{p_i-2} \nabla u_i)$ is the so called $p_i$-Laplacian operator,
$a_1,\ldots,a_n\in L^{\infty}(\Omega)$ be $n$ functions such that
$\min_{1\leq i\leq n}\{\textrm{ess}\inf_{\Omega}a_i\}>0$,
$F:\Omega\times \mathbb{R}^n\to\mathbb{R}$ is a function such that
the map $x\mapsto F(x,t_1,t_2,\ldots,t_n)$ is measurable in $\Omega$
for all $(t_1,\ldots,t_n)\in \mathbb{R}^n$ and the map
$(t_1,t_2,\ldots,t_n)\mapsto F(x,t_1,t_2,\ldots,t_n)$ is $C^1$ in
$\mathbb{R}^n$ for a.e. $x\in\Omega,$ $F_{u_i}$ denotes the partial
derivative of $F$ with respect to $u_i$, and $\nu$ is the outer
unite normal to $\partial\Omega$.

Moreover, $F$ satisfy the following additional assumptions:

\begin{enumerate}
\item[$\rm(F_1)$] for every $M>0$,
$$
\sup\limits_{|(t_1,\ldots,t_n)|\leq M}|F_{u_i}(x,t_1,\ldots,t_n)|\in
L^1(\Omega).
$$
\item[$\rm(F_2)$] $F(x,0,\ldots,0)=0$ for a.e. $x\in \Omega$.
\end{enumerate}

In recent years, many publications (see, e.g.,
\cite{BF,BHO,BMO,BM,DM,DT1,DT2,K,LT} and references therein) have
appeared concerning quasilinear elliptic systems which have been
used in a great variety of applications. Multiplicity results for
this kind of systems have been broadly investigated in which the
technical approach is based on the three critical points theorems.

Bonanno in \cite{B} established the existence of two intervals of
positive real parameters $\lambda$ for which the functional
$\Phi-\lambda J$ has three critical points, whose norms are
uniformly bounded with respect to $\lambda$ belonging to one of the
two intervals. He illustrated the result for a two point boundary
value problem. In this paper, by assuming that $F(x,\cdot)$ has a
$(p-1)$-sublinear growth at $\infty$ and satisfies a certain local
condition near to $0$, we prove the existence of two intervals
$\Lambda_1^{'}$ and $\Lambda_2^{'}$ such that, for each
$\lambda\in\Lambda_1^{'}\cup\Lambda_2^{'}$, the system (\ref{1})
admits at least three weak solutions whose norms are uniformly
bounded with respect to $\lambda\in\Lambda_2^{'}$.

This paper is arranged as follows. In Section 2, we recall some
basic notations and definitions and our main tool (Theorem
\ref{t1}), while Section 3 is devoted to our main result, some
consequences and one example that illustrates the result. For a
thorough account on the subject, we refer the reader to the very
recent monographs \cite{GR,KRV}.

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\section{Preliminaries}
\quad First we recall for the reader's convenience Theorem 2.1 of
\cite{B} to transfer the existence of three solutions of the system
(\ref{1}) into the existence of critical points of the Euler
functional. Here, $X^*$ denotes the dual space of $X$.

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\begin{theorem}[Theorem 2.1 of \cite{B}]\label{t1}
Let $X$ be a separable and reflexive real Banach space;
$\Phi:X\to\mathbb{R}$ a non-negative continuously G\^{a}teaux
differentiable and sequentially weakly lower semicontinuous
functional whose G\^{a}teaux derivative admits a continuous inverse
on $X^*$; $J:X\to\mathbb{R}$ a continuously G\^{a}teaux
differentiable functional whose G\^{a}teaux derivative is compact.
Assume that there exists $x_0\in X$ such that $\Phi(x_0)=J(x_0)=0$
and that
\begin{description}
\item {\rm(i)} $\lim_{\|x\|\to+\infty}(\Phi(x)-\lambda
J(x))=+\infty$\hspace{.5cm} for all $\lambda\in[0,+\infty[$.
\end{description}
Further, assume that there are $r>0$, $x_1\in X$ such that:
\begin{description}
\item{\rm(ii)} $r<\Phi(x_1);$
\item {\rm(iii)}
$\sup\limits_{x\in\overline{\Phi^{-1}(]-\infty,r[)}^{w}}J(x)<\frac{r}{r+\Phi(x_1)}J(x_1).$
\end{description}
Here $\overline{\Phi^{-1}(]-\infty,r[)}^w$ denotes the closure of
$\Phi^{-1}(]-\infty,r[)$ in the weak topology. Then, for each
$$
\lambda\in
\Lambda_1:=\left]\frac{\Phi(x_1)}{J(x_1)-\sup\limits_{x\in\overline{\Phi^{-1}(]-\infty,r[)}^w}J(x)},
\frac{r}{\sup\limits_{x\in\overline{\Phi^{-1}(]-\infty,r[)}^w}J(x)}\right[,
$$
the equation
\begin{equation}\label{2}
\Phi^{\prime}(x)-\lambda J^{\prime}(x)=0
\end{equation}
has at least three solutions in $X$ and, moreover, for each $h>1$,
there exists an open interval
$$
\Lambda_2\subseteq\left[0,\frac{hr}{r\frac{J(x_1)}{\Phi(x_1)}
-\sup\limits_{x\in\overline{\Phi^{-1}(]-\infty,r[)}^w}J(x)}\right]
$$
and a positive real number $\sigma$ such that, for each
$\lambda\in\Lambda_2$, the equation (\ref{2}) has at least three
solutions in $X$ whose norms are less than $\sigma$.
\end{theorem}

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In the sequel, $X$ will denote the Cartesian product of the $n$
Sobolev spaces $W^{1,p_i}(\Omega)$ for $1\leq i\leq n$, i.e.,
$X=W^{1,p_1}(\Omega)\times W^{1,p_2}(\Omega)\times\cdots\times
W^{1,p_n}(\Omega)$ equipped with the norm
$$
\|(u_1,u_2,\ldots,u_n)\|:=\sum_{i=1}^n\|u_i\|_{p_i},
$$
where
$$
\|u_i\|_{p_i}:=\left(\int_{\Omega}|\nabla u_i(x)|^{p_i}dx
+\int_{\Omega}a_i(x)|u_i(x)|^{p_i}dx\right)^{1/p_i}
$$
for $1\leq i\leq n$, which is equivalent to the usual one.

Put
\begin{equation}\label{3}
k:=\max\left\{\sup\limits_{u_i\in
W^{1,p_i}(\Omega)\setminus\{0\}}\frac{\max_{x\in\overline{\Omega}}|u_i(x)|^{p_i}}
{\|u_i\|_{p_i}^{p_i}}: \textrm{ for }1\leq i\leq n\right\}.
\end{equation}
Since $p_i>N$ for $1\leq i\leq n$, the embedding $X\hookrightarrow
(C^{0}(\overline{\Omega}))^n$ is compact, and so $k<+\infty$. It
follows from Proposition 4.1 of \cite{AC} that
$$
\sup\limits_{u_i\in W^{1,p_i}(\Omega)\setminus\{0\}}\frac{\max_{x\in\overline{\Omega}}|u_i(x)|^{p_i}}
{\|u_i\|_{p_i}^{p_i}}>\frac{1}{\|a_i\|_1}\ \ \textrm{for} \ \  1\leq
i\leq n,
$$
where $\|a_i\|_1:=\int_{\Omega}|a_{i}(x)|dx$ for $1\leq
i\leq n$, and so $\frac{1}{\|a_i\|_1}\leq k$ for $1\leq i\leq n$. In
addition, if $\Omega$ is convex, it is known \cite{AC} that
$$
\sup\limits_{u_i\in
W^{1,p_i}(\Omega)\setminus\{0\}}\frac{\max_{x\in
\overline{\Omega}}|u_i(x)|}{\|u_i\|_{p_i}}\leq
2^{\frac{p_i-1}{p_i}}\max\Bigg{\{}\Big{(}\frac{1}{\|a_i\|_1}\Big{)}^{\frac{1}{p_i}},
\frac{\textrm{diam}(\Omega)}{N^{\frac{1}{p_i}}}\Big{(}\frac{p_i-1}{p_i-N}m(\Omega)\Big{)}^{\frac{p_i-1}{p_i}}
\frac{\|a_i\|_\infty}{\|a_i\|_1}\Bigg{\}}
$$
for $1\leq i\leq n,$ where $m(\Omega)$ is the Lebesgue measure of
the set $\Omega,$ and equality occurs when $\Omega$ is a ball.

We recall that a function $u=(u_1,\ldots,u_n)\in X$ is said to be a
(weak) solution of the system (\ref{1}) if
\begin{eqnarray*}
\int_{\Omega}\sum_{i=1}^n|\nabla u_{i}(x)|^{p_{i}-2}\nabla
u_{i}(x)\nabla v_{i}(x)dx&-&\lambda\,\int_{\Omega}\sum_{i=1}^n F_{u_i}(x,u_1(x),\ldots,u_n(x))v_i(x)dx\\
&+&\int_{\Omega}\sum_{i=1}^n
a_i(x)|u_i(x)|^{p_i-2}u_{i}(x)v_i(x)dx=0
\end{eqnarray*}
for all $v=(v_1,\ldots,v_n)\in X$.

For all $c>0$ we denote by $K(c)$ the set
$$
\left\{(t_{1},\ldots,t_{n})\in \mathbb{R}^n~:~~
\sum_{i=1}^{n}\frac{|t_{i}|^{p_{i}}}{p_{i}}\leq c\right\}.
$$
This set will be used in some of our hypotheses with appropriate
choices of $c$.

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\section{Main results}

\quad Our main result is the following theorem.

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\begin{theorem}\label{t2}
Assume that there exist $n+1$ positive constants $r$ and $s_i$ for
$1\leq i\leq n,$ with $s_i<p_i$ for $1\leq i\leq n,$ and two
functions $\alpha\in L^1(\Omega)$ and $w=(w_{1},\ldots,w_{n})\in X$
such that
\begin{description}
\item {\rm(j)} $\sum_{i=1}^n\frac{\|w_i\|_{p_i}^{p_i}}{p_i}>r;$
\item {\rm(jj)} $\displaystyle\int_{\Omega}\sup\limits_{(t_1,\ldots,t_n)\in
K(kr)}F(x,t_1,\ldots,t_n)dx< \frac{\displaystyle
r\int_{\Omega}F(x,w_1(x),\ldots,w_{n}(x))dx} {\displaystyle
2\sum_{i=1}^n\frac{\|w_i\|_{p_i}^{p_i}}{p_i}};$
\item {\rm(jjj)} $F(x,t_1,\ldots,t_n)\leq\alpha(x)\left(1+\sum_{i=1}^n
|t_i|^{s_i}\right)$ for a.e. $x\in\Omega$ and all
$(t_1,\ldots,t_n)\in\mathbb{R}^n.$
\end{description}
Then, for each
\begin{eqnarray*}
&&\lambda\in\Lambda_1^{'}:=\Bigg{]}\frac{\displaystyle\sum_{i=1}^n\frac{\|w_i\|_{p_i}^{p_i}}{p_i}}
{\displaystyle\int_{\Omega}F(x,w_1(x),\ldots,w_n(x))dx
-\int_{\Omega}\sup\limits_{(t_1,\ldots,t_n)\in
K(kr)}F(x,t_1,\ldots,t_n)dx},\\
&&\hspace{2cm}\frac{r}{\displaystyle\int_{\Omega}\sup\limits_{(t_1,\ldots,t_n)\in
K(kr)}F(x,t_1,\ldots,t_n)dx}\Bigg{[},
\end{eqnarray*}
the system (\ref{1}) admits at least three weak solutions in $X$
and, moreover, for each $h>1$, there exist an open interval
$$
\Lambda_2^{'}\subseteq\Bigg{[}0,\frac{hr}{\displaystyle
r\frac{\int_{\Omega}F(x,w_{1}(x),\ldots,w_{n}(x))dx}
{\sum_{i=1}^{n}\frac{\|w_{i}\|_{p_i}^{p_{i}}}{p_{i}}}
-\int_{\Omega}\sup\limits_{(t_{1},\ldots,t_{n})\in
K(kr)}F(x,t_{1},\ldots,t_{n})dx}\Bigg{]}
$$
and a positive real number $\sigma$ such that, for each
$\lambda\in\Lambda_2^{'}$, the system (\ref{1}) admits at least
three weak solutions in $X$ whose norms are less than $\sigma$.
\end{theorem}
\begin{proof}
In order to apply Theorem \ref{t1}, we begin by setting
\begin{equation}\label{4}
\Phi(u):=\sum_{i=1}^{n}\frac{\|u_{i}\|_{p_i}^{p_i}}{p_i}
\end{equation}
and
\begin{equation}\label{5}
J(u):=\int_{\Omega}F(x,u_1(x),\ldots,u_n(x))dx
\end{equation}
for all $u=(u_1,\ldots,u_n)\in X.$ It is known that $\Phi$ and $J$
are well defined and continuously G\^{a}teaux differentiable
functionals with
$$
\Phi'(u)(v)=\int_\Omega\sum_{i=1}^{n}|\nabla u_i(x)|^{p_i-2}\nabla u_i(x)\nabla v_i(x)dx
+\int_{\Omega}\sum_{i=1}^{n}a_{i}(x)|u_{i}(x)|^{p_{i}-2}u_{i}(x)v_{i}(x)dx
$$
and
$$
J'(u)(v)=\int_\Omega\sum_{i=1}^{n}F_{u_i}(x,u_1(x),\ldots,u_n(x))v_i(x)dx
$$
for every $u=(u_1,\ldots,u_n), v=(v_1,\ldots,v_n)\in X,$ as well as
$\Phi$ is sequentially weakly lower semicontinuous (see Proposition
25.20 of \cite{Z}). Also, $\Phi':X\rightarrow X^*$ is a uniformly
monotone operator in $X$ (for more details, see (2.2) of \cite{S}),
and since $\Phi'$ is coercive and hemicontinuous in $X$, by applying
Theorem 26.A of \cite{Z}, $\Phi'$ admits a continuous inverse on
$X^*.$

We claim that $J':X \to X^*$ is a compact operator. To this end, it
is enough to show that $J'$ is strongly continuous on $X$. For this,
for fixed $(u_1,\ldots,u_n)\in X$, let
$(u_{1m},\ldots,u_{nm})\to(u_1,\ldots,u_n)$ weakly in $X$ as $m\to
+\infty$. Then we have $(u_{1m},\ldots,u_{nm})$ converges uniformly
to $(u_1,\ldots,u_n)$ on $\Omega$ as $m\to +\infty$ (see \cite{Z}).
Since $F(x,\cdot,\ldots,\cdot)$ is $C^1$ in $\mathbb{R}^n$ for every
$x\in\Omega$, the derivatives of $F$ are continuous in
$\mathbb{R}^n$ for every $x\in\Omega$, so for $1\leq i\leq n,$
$F_{u_i}(x,u_{1m},\ldots,u_{nm})\to F_{u_i}(x,u_1,\ldots,u_n)$
strongly as $m\to +\infty$. By the Lebesgue control convergence
theorem, $J'(u_{1m},\ldots,u_{nm})\to J'(u_1,\ldots,u_n)$ strongly
as $m\to +\infty$. Thus we proved that $J'$ is strongly continuous
on $X$, which implies that $J'$ is a compact operator by
\cite[Proposition 26.2]{Z}. Hence the claim is true.

Thanks to the assumption (jjj), for each $\lambda>0$ one has
$$
\lim_{\|u\|\to+\infty}(\Phi(u)-\lambda J(u))=+\infty.
$$

Also, from (j) and (\ref{4}) we get $\Phi(w)>r$. Due to (\ref{3}),
for each $u_i\in W^{1,p_i}(\Omega)$
$$
\sup\limits_{x\in\Omega}|u_{i}(x)|^{p_i}\leq k\|u_i\|_{p_i}^{p_i}
$$
for $1\leq i\leq n$, so we have
\begin{equation}\label{6}
\sup\limits_{x\in\Omega}\sum_{i=1}^n\frac{|u_i(x)|^{p_i}}{p_i} \leq
k\sum_{i=1}^n\frac{\|u_i\|_{p_i}^{p_i}}{p_i}=k\Phi(u)
\end{equation}
for every $u=(u_{1},\ldots,u_{n})\in X$. From (\ref{6}), for each
$r>0$ we obtain
\begin{eqnarray*}
\Phi^{-1}(]-\infty,r])&=&\bigg{\{}u=(u_1,\ldots,u_n)\in X:\Phi(u)\leq r\bigg{\}}\\
&=&\bigg{\{}u=(u_1,\ldots,u_n)\in
X:\sum_{i=1}^n\frac{\|u_i\|_{p_i}^{p_i}}{p_i}\leq r\bigg{\}}\\
&\subseteq&\bigg{\{}u=(u_1,\ldots,u_n)\in
X:\sum_{i=1}^n\frac{|u_i(x)|^{p_i}}{p_i}\leq k r \ \ \textrm{for
all}\; x\in\Omega\bigg{\}},
\end{eqnarray*}
and, since
$\overline{\Phi^{-1}(]-\infty,r[)}^w=\Phi^{-1}(]-\infty,r])$, owing
to our assumptions, we have
\begin{eqnarray*}
\sup\limits_{u\in\overline{\Phi^{-1}(]-\infty,r[)}^w}J(u)&\leq&\int_{\Omega}\sup\limits_{(t_1,\ldots,t_n)\in
K(kr)}F(x,t_1,\ldots,t_n)dx\\
&<&\frac{\displaystyle r\int_{\Omega}F(x,w_1(x),\ldots,w_{n}(x))dx}
{\displaystyle 2\sum_{i=1}^n\frac{\|w_i\|_{p_i}^{p_i}}{p_i}}\\
&<&r\frac{\displaystyle\int_{\Omega}F(x,w_1(x),\ldots,w_n(x))dx}
{\displaystyle r+\sum_{i=1}^n\frac{\|w_i\|_{p_i}^{p_i}}{p_i}}\\
&=&\frac{r}{r+\Phi(w)}J(w).
\end{eqnarray*}

We can apply Theorem \ref{t1} at this point and obtain two intervals
$\Lambda_1$ and $\Lambda_2$ such that if
$\lambda\in\Lambda_1\cup\Lambda_2$, then system \eqref{1} has at
least three weak solutions. Next, we derive the upper and lower
bounds of $\Lambda_1$ and $\Lambda_2$. For each $x\in\Omega$ we have
$$
\frac{r}{\sup\limits_{u\in\overline{\Phi^{-1}(]-\infty,r[)}^{w}}J(u)}
\geq\frac{r}{\displaystyle\int_{\Omega}\sup\limits_{(t_{1},\ldots,t_{n})\in
K(kr)}F(x,t_{1},\ldots,t_{n})dx}
$$
and
\begin{eqnarray*}
&&\frac{\Phi(w)}{J(w)-\sup\limits_{u\in\overline{\Phi^{-1}(]-\infty,r[)}^w}J(u)}\\
&&\leq
\frac{\displaystyle\sum_{i=1}^n\frac{\|w_{i}\|_{p_i}^{p_i}}{p_i}}
{\displaystyle\int_{\Omega}F(x,w_1(x),\ldots,w_n(x))dx-\int_{\Omega}\sup\limits_{(t_1,\ldots,t_n)\in
K(kr)}F(x,t_1,\ldots,t_n)dx}.
\end{eqnarray*}
Note also that (jj) immediately implies
\begin{eqnarray*}
&&\frac{\displaystyle\sum_{i=1}^n\frac{\|w_i\|_{p_i}^{p_i}}{p_i}}
{\displaystyle\int_{\Omega}F(x,w_{1}(x),\ldots,w_{n}(x))dx-\int_{\Omega}\sup\limits_{(t_{1},\ldots,t_{n})\in
K(kr)}F(x,t_1,\ldots,t_n)dx}\\
&&<\frac{\displaystyle\sum_{i=1}^n\frac{\|w_{i}\|_{p_i}^{p_i}}{p_i}}
{\displaystyle\left(\frac{2\sum_{i=1}^n\frac{\|w_{i}\|_{p_i}^{p_i}}{p_i}}{r}-1\right)
\int_{\Omega}\sup\limits_{(t_1,\ldots,t_n)\in
K(kr)}F(x,t_1,\ldots,t_n)dx}\\
&&<\frac{\displaystyle\sum_{i=1}^n\frac{\|w_{i}\|_{p_i}^{p_i}}{p_i}}
{\displaystyle\left(\frac{r+\sum_{i=1}^n\frac{\|w_{i}\|_{p_i}^{p_i}}{p_i}}{r}-1\right)
\int_{\Omega}\sup\limits_{(t_1,\ldots,t_n)\in
K(kr)}F(x,t_1,\ldots,t_n)dx}\\
&&=\frac{r}{\displaystyle\int_{\Omega}\sup\limits_{(t_{1},\ldots,t_{n})\in
K(kr)}F(x,t_1,\ldots,t_n)dx}.
\end{eqnarray*}
Also
\begin{eqnarray*}
&&\frac{hr}{r\frac{J(w)}{\Phi(w)}-\sup\limits_{u\in\overline{\Phi^{-1}(]-\infty,r[)}^w}J(u)}\\
&&~~~\leq\frac{hr}{\displaystyle
r\frac{\displaystyle\int_{\Omega}F(x,w_1(x),\ldots,w_n(x))dx}
{\displaystyle\sum_{i=1}^n\frac{\|w_i\|_{p_i}^{p_i}}{p_i}}
-\int_{\Omega}\sup\limits_{(t_1,\ldots,t_n)\in
K(kr)}F(x,t_1,\ldots,t_n)dx}=\rho.
\end{eqnarray*}
So
$$
\int_{\Omega}\sup\limits_{(t_1,\ldots,t_n)\in
K(kr)}F(x,t_1,\ldots,t_n)dx<
r\frac{\displaystyle\int_{\Omega}F(x,w_1(x),\ldots,w_n(x))dx}{\sum_{i=1}^n\frac{\|w_i\|_{p_i}^{p_i}}{p_i}},
$$
and now apply (jj). Thus, by choosing $x_0=0,~x_1=w$, from Theorem
\ref{t1} it follows that, for each $\lambda\in\Lambda_1^{'}$ the
system (\ref{1}) admits at least three weak solutions and there
exist an open interval $\Lambda_2^{'}\subseteq[0,\rho]$ and a real
positive number $\sigma$ such that, for each
$\lambda\in\Lambda_2^{'}$, the system (\ref{1}) admits at least
three weak solutions whose norms in $X$ are less than $\sigma$.
\end{proof}

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Now, we give a particular consequence of Theorem \ref{t2} for a
fixed test function $w$. Moreover, $F$ dose not depend on
$x\in\Omega.$

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\begin{corollary}\label{c1}
Let $F:\mathbb{R}^n\to \mathbb{R}$ be a $C^1$-function and assume
that there exist $n+3$ positive constants $\gamma,\,\delta,\,\alpha$
and $s_i$ for $1\leq i\leq n,$ with $s_i<p_i$ for $1\leq i\leq n$,
such that
\begin{description}
\item {\rm(k)} $\sum_{i=1}^n\frac{\delta^{p_i}}{p_i}>\frac{\gamma}{\prod_{i=1}^n p_i};$
\item {\rm(kk)} $\max\limits_{(t_1,\ldots,t_n)\in
K(\frac{\gamma}{\prod_{i=1}^n
p_i})}F(t_1,\ldots,t_n)<\frac{\gamma}{2k\prod_{i=1}^n p_i}
\frac{F(\delta,\ldots,\delta)}
{\sum_{i=1}^n\frac{\delta^{p_i}}{p_i}\|a_i\|_1};$
\item {\rm(kkk)}
$F(t_1,\ldots,t_n)\leq\alpha(1+\sum_{i=1}^n|t_i|^{s_i})$ for all
$(t_1,\ldots,t_n)\in\mathbb{R}^n.$
\end{description}
Then, for each
\begin{eqnarray*}
&&\lambda\in\Lambda_1^{'}:=\Bigg{]}\frac{\sum_{i=1}^n\frac{\delta^{p_i}}{p_i}\|a_i\|_1}
{m(\Omega)\Big{(}F(\delta,\ldots,\delta)-\max\limits_{(t_1,\ldots,t_n)\in
K(\frac{\gamma}{\prod_{i=1}^n p_i})}F(t_1,\ldots,t_n)\Big{)}},\\
&&\hspace{2cm}\frac{\frac{\gamma}{k\prod_{i=1}^{n}p_i}}{m(\Omega)\max\limits_{(t_1,\ldots,t_n)\in
K(\frac{\gamma}{\prod_{i=1}^n p_i})}F(t_1,\ldots,t_n)}\Bigg{[},
\end{eqnarray*}
the system
\begin{equation}\label{7}
\left\{\begin{array}{ll} \Delta_{p_i}u_i+\lambda
F_{u_i}(u_1,\ldots,u_n)=a_i(x)|u_i|^{p_i-2}u_i & \textrm{ in } \Omega,\\
\partial u_i/\partial\nu=0 & \textrm{ on } \partial\Omega,
\end{array}\right.
\end{equation}
for $1\leq i\leq n,$ admits at least three weak solutions in $X$
and, moreover, for each $h>1$, there exist an open interval
$$
\Lambda_2^{'}\subseteq\Bigg{[}0,\frac{\frac{h\gamma}{k\prod_{i=1}^n p_i}}{m(\Omega)\Big{(}\frac{\gamma}
{k\prod_{i=1}^n p_i}\frac{F(\delta,\ldots,\delta)}{\sum_{i=1}^n
\frac{\delta^{p_i}}{p_i}\|a_i\|_1} -\max\limits_{(t_1,\ldots,t_n)\in
K(\frac{\gamma}{\prod_{i=1}^n
p_i})}F(t_1,\ldots,t_n)\Big{)}}\Bigg{]}
$$
and a positive real number $\sigma$ such that, for each
$\lambda\in\Lambda_2^{'}$, the system (\ref{7}) admits at least
three weak solutions in $X$ whose norms are less than $\sigma$.
\end{corollary}
\begin{proof}
We prove that all assumptions of Theorem \ref{t2} are fulfilled with
$w(x):=(\delta,\ldots,\delta)$ and $r:=\frac{\gamma}{k\prod_{i=1}^n
p_i}$. If we put $w(x):=(\delta,\ldots,\delta)$ for each
$x\in\Omega$, then we have
$\|w_i\|_{p_i}=\|a_i\|_1^{\frac{1}{p_i}}\delta$ for $1\leq i\leq n$.
By (k) and the fact that $\frac{1}{\|a_i\|_1}\leq k$ for $1\leq
i\leq n$, we get
$\Phi(w)=\sum_{i=1}^n\frac{\delta^{p_i}}{p_i}\|a_i\|_1>r$. The other
assumptions of Theorem \ref{t2} are clearly satisfied.
\end{proof}

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\begin{corollary}\label{c2}
Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. Put
$F(t)=\int_0^t f(\xi)d\xi$ for each $t\in\mathbb{R}$. Assume that
there exist four positive constants $\gamma,\,\delta,\,\alpha$ and
$s$ with $\delta^p>\gamma$ and $s<p,$ such that
\begin{description}
\item {\rm(l)} $\max\limits_{t\in[-\sqrt[p]{\gamma},\sqrt[p]{\gamma}]}F(t)
<\frac{F(\delta)}{2k\|a\|_1};$
\item {\rm(ll)} $F(t)\leq\alpha(1+|t|^s)$ for all $t\in\mathbb{R}.$
\end{description}
Then, for each
\begin{eqnarray*}
\lambda\in\Lambda_1^{'}:=\Bigg{]}\frac{\delta^p\|a\|_1}
{p\,m(\Omega)\Big{(}F(\delta)-\max\limits_{t\in[-\sqrt[p]{\gamma},\sqrt[p]{\gamma}]}F(t)\Big{)}},
\frac{\gamma}{(kp)m(\Omega)\max\limits_{t\in[-\sqrt[p]{\gamma},\sqrt[p]{\gamma}]}F(t)}\Bigg{[},
\end{eqnarray*}
the problem
\begin{equation}\label{8}
\left\{\begin{array}{ll}
\Delta_p u+\lambda f(u)=a(x)|u|^{p-2}u & \textrm{ in } \Omega,\\
\partial u/\partial\nu=0 & \textrm{ on } \partial\Omega
\end{array}\right.
\end{equation}
admits at least three weak solutions in $W^{1,p}(\Omega)$ and,
moreover, for each $h>1$, there exist an open interval
$$
\Lambda_2^{'}\subseteq\Bigg{[}0,\frac{h\gamma}{(kp)m(\Omega)\Big{(}\frac{\gamma F(\delta)}{k
\delta^p\|a\|_1}-\max\limits_{t\in[-\sqrt[p]{\gamma},\sqrt[p]{\gamma}]}F(t)\Big{)}}\Bigg{]}
$$
and a positive real number $\sigma$ such that, for each
$\lambda\in\Lambda_2^{'}$, the problem (\ref{8}) admits at least
three weak solutions in $W^{1,p}(\Omega)$ whose norms are less than
$\sigma$.
\end{corollary}

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Finally, we present the application of Theorem \ref{t2} in the
ordinary case with $p=2,$ that Example \ref{e1} illustrates the
result. For simplicity, we put $\Omega=(0,1)$. Note that in this
situation we have
$$
k=2\max\{\|a\|_1^{-1},\|a\|_\infty^2\|a\|_1^{-2}\}.
$$

%**************************************************************************************************

\begin{corollary}\label{c3}
Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. Put
$F(t)=\int_0^t f(\xi)d\xi$ for each $t\in\mathbb{R}$. Assume that
there exist four positive constants $\gamma,\,\delta,\,\alpha$ and
$s$ with $\delta^2>\gamma$ and $s<2,$ such that assumption (ll) in
Corollary \ref{c2} holds, and
\begin{description}
\item {\rm(m)} $\max\limits_{t\in[-\sqrt{\gamma},\sqrt{\gamma}]}F(t)
<\frac{F(\delta)}{2k\|a\|_1}.$
\end{description}
Then, for each
\begin{eqnarray*}
\lambda\in\Lambda_1^{'}:=\Bigg{]}\frac{\delta^2\|a\|_1}
{2\Big{(}F(\delta)-\max\limits_{t\in[-\sqrt{\gamma},\sqrt{\gamma}]}F(t)\Big{)}},
\frac{\gamma}{2k\max\limits_{t\in[-\sqrt{\gamma},\sqrt{\gamma}]}F(t)}\Bigg{[},
\end{eqnarray*}
the problem
\begin{equation}\label{9}
\left\{\begin{array}{ll}
u''+\lambda f(u)=a(x)u & \textrm{ in } (0,1),\\
u'(0)=u'(1)=0
\end{array}\right.
\end{equation}
admits at least three classical solutions in $C^2([0,1])$ and,
moreover, for each $h>1$, there exist an open interval
$$
\Lambda_2^{'}\subseteq\Bigg{[}0,\frac{h\gamma}{2k\Big{(}\frac{\gamma F(\delta)}{k
\delta^2\|a\|_1}-\max\limits_{t\in[-\sqrt{\gamma},\sqrt{\gamma}]}F(t)\Big{)}}\Bigg{]}
$$
and a positive real number $\sigma$ such that, for each
$\lambda\in\Lambda_2^{'}$, the problem (\ref{9}) admits at least
three classical solutions in $C^2([0,1])$ whose norms are less than
$\sigma$.
\end{corollary}

%**************************************************************************************************

\begin{example}\label{e1}
Consider the problem
\begin{equation}\label{10}
\left\{\begin{array}{ll}
u''+\lambda e^{-u}u^{11}(12-u)=xu & \textrm{ in } (0,1),\\
u'(0)=u'(1)=0.
\end{array}\right.
\end{equation}
Set $f(t)=e^{-t}t^{11}(12-t)$ for all $t\in\mathbb{R}.$ A direct
calculation yields $F(t)=e^{-t}t^{12}$ for all $t\in\mathbb{R}.$
Note that by choosing $\delta=2,~\gamma=1$ and $a(x)=x,$ we have
$k=8.$ A simple computation shows
$$
\frac{F(\delta)}{2k\|a\|_1}-\max\limits_{t\in[-\sqrt{\gamma},\sqrt{\gamma}]}F(t)
=\frac{2^9}{e^2}-e>0.
$$
Moreover, with $s=1$ and $\alpha$ sufficiently large, the assumption
(ll) is satisfied. So, by Corollary \ref{c3}, for each
$\lambda\in\Lambda_1^{'}:=]\frac{1}{2^{12}e^{-2}-e},\frac{1}{16\,e}[,$
the problem (\ref{10}) admits at least three classical solutions in
$C^2([0,1])$ and, moreover, for each $h>1$, there exist an open
interval
$\Lambda_2^{'}\subseteq\big{[}0,\frac{h}{16(2^8e^{-2}-e)}\big{]}$
and a positive real number $\sigma$ such that, for each
$\lambda\in\Lambda_2^{'}$, the problem (\ref{10}) admits at least
three classical solutions in $C^2([0,1])$ whose norms are less than
$\sigma$.
\end{example}

\subsection*{Acknowledgments}
This research work has been supported by a research grant from {\it
National Elites Foundation}.


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