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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 183, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/183\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions for nonlinear elliptic 
 Dirichlet systems}

\author[G. Bonanno, E. Tornatore \hfil EJDE-2012/183\hfilneg]
{Gabriele Bonanno, Elisabetta Tornatore}  % in alphabetical order

\address{Gabriele Bonanno \newline
Department of Civil, Information Technology,
Construction, Environmental Engineering and Applied Mathematics,
University of Messina, 98166 - Messina, Italy}
\email{bonanno@unime.it}

\address{Elisabetta Tornatore \newline
Dipartimento di Ingegneria Elettrica,
Elettronica e delle Telecomunicazioni  di Tecnologie Chimiche,
Automatica e Modelli Matematici (DIEETCAM)\\
Universit\`a degli studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy}
\email{elisa.tornatore@unipa.it}

\thanks{Submitted August 28, 2012. Published October 28, 2012.}
\subjclass[2000]{35J65, 35J20}
\keywords{Elliptic systems; variational problems; multiple solutions}

\begin{abstract}
 The existence and multiplicity of solutions for systems of nonlinear
 elliptic equations with Dirichlet boundary conditions is investigated.
 Under suitable assumptions on the potential of the nonlinearity,
 the existence of one, two, or three solutions is established.
 Our approach is based on variational methods.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

The aim of this article is to establish the existence of solutions
to the  system
\begin{equation}
\begin{gathered}
   -\Delta u=\lambda \nabla_uF(x,u)\quad \text{in } \Omega,\\
    u=0 \quad \text{on }\partial \Omega,
  \label{problema}
  \end{gathered}
\end{equation}
where $\Omega\subset \mathbb{R}^N$ (with $N\geq 3$) is a non-empty
bounded open set with smooth boundary $\partial \Omega$,
 $\lambda$ is a positive parameter. In the statement of problem
 \eqref{problema}, $u:\Omega\to \mathbb{R}^m$ (with $m\geq 1$) and
 $F:\Omega\times \mathbb{R}^m\to \mathbb{R}$ is a $C^1$-function,
$F(x,0)=0$ for every
$x\in \Omega$ and $\nabla_u F=(F_{u_i})_{i=1,\dots,m}$ where $F_{u_i}$
denotes the partial derivative of $F$ respect on $u_i$ ($i=1,\dots,m$).

Existence results for nonlinear elliptic systems of type \eqref{problema}
have received a great deal of interest in recent years. We refer the reader
to \cite{Defi} for a complete overview on this subject, and to \cite{Tang}
and the references therein for more recent developments.

In this article, at first, we prove the existence of a non-zero solution
of problem \eqref{problema}, without assuming any asymptotic condition
neither at zero nor at infinity (see Theorem \ref{1soluzione}) and,
as a consequence, we obtain the existence of one solution, by assuming only that
the potential $F$ has a suitable behavior at zero (see Corollary \ref{Cor1}).
Next, we obtain the existence of two solutions, possibly both non-zero, assuming
only the classical Ambrosetti-Rabinowitz condition; that is, without requiring
that the potential $F$ satisfies the usual condition at zero (see Theorem \ref{Th2}).
Finally, we present a three solutions existence result under appropriate condition
on the potential $F$ (see Theorem \ref{3soluzioni}).

It is worth noticing that in \cite{Tang}
the nonlinear elliptic Dirichlet system involves the $(p,q)$-Laplacian with $p,q>N$,
since in a such result the compact embedding of the Sobolev space in
 $C^0(\bar\Omega)$ is a crucial point in the proof; while in our results,
the case $p=q=2<N$ is investigated.   Some examples illustrate the obtained
results (see Examples \ref{examp3.1}, \ref{examp3.2} and \ref{examp3.3}). 
Our approach is based on critical
point theorems contained in \cite{bo2} and \cite{bo-ma}.
The paper is arranged as follows. In Section 2 we recall our main tools,
while Section 3 is devoted to our main results.

\section{Preliminaries}

In this section, we recall definitions and theorems to be used in this article.
Let $(X,\|\cdot\|)$ be a real Banach space and   $\Phi$, $\Psi:X\to \mathbb{R}$
be  two continuously G\^ateaux differentiable functionals; put
$$
I=\Phi-\Psi
$$
and fix $r_1$, $r_2\in[-\infty,+\infty]$, with $r_1<r_2$. We say that
functional $I$ satisfies the \emph{Palais-Smale condition cut off
lower at $r_1$ and upper at $r_2$} ($^{[r_1]}(PS)^{[r_2]}$-condition)
if any sequence $\{u_n\}\in X$ such that
\begin{itemize}
\item  $\{I(u_n)\}$ is bounded,
\item  $\lim_{n\to +\infty}\|I'(u_n)\|_{X^*}=0$,
\item  $r_1<\Phi(u_n)<r_2\quad \forall n\in \mathbb{N}$,
\end{itemize}
has a convergent subsequence.

If $r_1=-\infty$ and $r_2=+\infty$ it coincides with the classical $(PS)$-condition,
while if  $r_1=-\infty$ and $r_2\in \mathbb{R}$ it is denoted by
$(PS)^{[r_2]}$-condition.

Now we recall a result of local minimum obtained
in \cite{bo2}, which is based on \cite[Theorem 5.1]{bo1}.

\begin{theorem}[{\cite[Theorem 2.2]{bo2}}]  \label{critical0}
Let $X$ be a real Banach space, and let $\Phi$, $\Psi:X\to
\mathbb{R}$ be two continuously G\^ateaux differentiable
 functionals such that $\inf_X\Phi=\Phi(0)=\Psi(0)=0 $.
Assume that there exist $r\in \mathbb{R}$ and
$ \bar{u}\in X$, with $0<\Phi(\bar{u})<r $, such that
\begin{equation}
 \frac{\sup_{u\in \Phi^{-1}(]-\infty,r[)
}\Psi(u)}{r}<\frac{\Psi(\bar{u})}{\Phi(\bar{u})}\label{condizionealgebrica}
\end{equation}
and, for each $\lambda\in \Lambda:=\big]\frac{\Phi(\bar{u})}{\Psi(\bar{u})},
\frac{r}{\sup_{u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}\big[$ the functional
$I_\lambda=\Phi-\lambda \Psi$ satisfies the $(PS)^{[r]}$-condition.
Then, for each $\lambda\in
\Lambda:=\big]\frac{\Phi(\bar{u})}{\Psi(\bar{u})},
\frac{r}{\sup_{u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}\big[$,
there is $u_\lambda\in \Phi^{-1}(]0,r[)$ such that
$I_\lambda(u_\lambda)\leq I_\lambda(u)$ for all $u\in
\Phi^{-1}(]0,r[)$ and $I'_\lambda(u_\lambda)=0$.
\end{theorem}

Now, we also recall a recent result obtained in \cite{bo2} that
ensures the existence of two critical points and which is based
 on \cite[Theorem 3.1]{bo1} and on
the classical Ambrosetti-Rabinowitz Theorem (see \cite{A-R}).


\begin{theorem}[{\cite[Theorem 3.2]{bo2}}]  \label{critical2}
Let $X$ be a real Banach space and let $\Phi, \Psi:X\to \mathbb{R}$ be two
continuously G\^ateaux differentiable functionals such that
$\Phi$ is bounded from below and $ \Phi(0)=\Psi(0)=0$.

Fix $r>0$ and assume that, for each $\lambda\in
\big]0,\frac{r}{\sup_{u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}\big[$,
the functional $I_\lambda=\Phi-\lambda \Psi$ satisfies
(PS)-condition and it is unbounded from below.
Then, for each $\lambda\in
]0,\frac{r}{\sup_{u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}[$,
the functional $I_{\lambda}$ admits two distinct critical points in
$X$.
\end{theorem}

Finally we point out an other result, which insures the existence of
at least three critical points, that has been obtained in \cite{bo-ma}
and it is a more precise version of \cite[Theorem 3.2]{bo-ca}.

\begin{theorem}[{\cite[Theorem 3.6]{bo-ma}}]  \label{critical1}
Let $X$ be a reflexive real Banach space, $\Phi:X\to \mathbb{R}$ be a continuously
G\^ateaux differentiable, coercive and sequentially weakly
lower semicontinuous functional whose G\^ateaux derivative
admits a continuous inverse on $X^*$, $\Psi: X\to \mathbb{R}$ be a
continuously G\^ateaux differentiable functional whose
G\^ateaux derivative is compact, moreover
$$
\Phi(0)=\Psi(0)=0.
$$
Assume that there exist $r\in \mathbb{R}$ and $ \bar{u}\in X$, with
$0<r<\Phi(\bar{u}) $, such that
\begin{itemize}
\item[(i)] $\frac{\sup_{u\in\Phi^{-1}(]-\infty,r])}\Psi(u)}{r}
 <\frac{\Psi(\bar{u})}{\Phi(\bar{u})}$
\item[(ii)] for each $\lambda\in \Lambda
:=\big]\frac{\Phi(\bar{u})}{\Psi(\bar{u})},\frac{r}{\sup_{u\in\Phi^{-1}(]-\infty,r])}
\Psi(u)}\big[$ the functional $\Phi-\lambda \Psi$ is coercive.
\end{itemize}
Then, for each $\lambda\in \Lambda$, the functional
$I_{\lambda}=\Phi-\lambda \Psi$ has at least three distinct critical
points in $X$.
\end{theorem}

Throughout in the article we assume the following conditions:
\begin{itemize}
\item[(H0)] there exist two non negative constants $a_1$, $a_2$
and a constant $q\in ]1,\frac{2N}{N-2}[$ such that
$$
|F_{t_i}(x,t_1,\dots,t_m)|\leq a_1+a_2|t_i|^{q-1}\quad i=1,\dots,m
$$
for every $(x,t_1,\dots,t_m)\in \Omega\times \mathbb{R}^m$.
\end{itemize}

We consider the Sobolev space  $H_0^{1}(\Omega)$  endowed with the
norm
\begin{equation}
\|u\|_{H_0^{1}(\Omega)}:=\Big(\int_{\Omega} |\nabla
u(x)|^2dx\Big)^{1/2}\label{norma},
\end{equation}
 for all $u\in H_0^{1}(\Omega)$.

Now, let $X$ be the Cartesian product of $m$ Sobolev space
$H_0^{1}(\Omega)$; i.e., $X=\prod_{i=1}^m H_0^{1}(\Omega)$ endowed
with the norm
$$
\|u\|:=\sum_{i=1}^m\|u_i\|_{H_0^{1}(\Omega)}
$$
for all $u=(u_1,\dots,u_m)\in X$.

A function $u=(u_1,\dots,u_m)\in X$ is said a weak solution to
system \eqref{problema} if
$$
\int_{\Omega} \sum_{i=1}^m\nabla u_i(x)\cdot \nabla v_i(x)dx-\lambda
\int_{\Omega}\sum_{i=1}^m F_{u_i}(x,u_1(x),\dots,u_m(x))v_i(x)dx=0
$$
for every $ v=(v_1,v_2,\dots, v_m)\in X$.
Moreover, a weak solution $u\in X$ is called non negative if
$u_i(x)\geq 0$ for every $i=1,\dots,m$ and for each $x\in \Omega$.

Now, put $2^*=\frac{2N}{N-2}$ and denote by $\Gamma$ the Gamma
function defined by
$$
\Gamma(s)=\int_0^{+\infty}z^{s-1}e^{z}dz, \, \forall s>0.
$$
From the Sobolev embedding theorem, for every $u\in H_0^{1}(\Omega)$
there exists a constant $c\in \mathbb{R}_+$ such that

\begin{equation}
\|u\|_{{L^{2^{*}}(\Omega)}}\leq c \|u\|_{H_0^{1}(\Omega)}
\label{costante di Sobolev}
\end{equation}
the best (smallest) constant that appears in \eqref{costante di Sobolev}  is
\begin{equation}
c=\frac{1}{\sqrt{N(N-2)\pi}}\Big(\frac{N!}{2\Gamma(1+\frac{N}{2})}\Big)^{1/N}
\label{costante di Talenti}
\end{equation}
(see \cite{Talenti}).

Fixing $q\in [1,2^*[$ in virtue of Sobolev embedding
theorem, for every $u\in H_0^{1}(\Omega)$, there exists a positive
constant $c_q$ such that
\begin{equation}
\|u\|_{{L^{q}(\Omega)}}\leq c _q\|u\|_{H_0^{1}(\Omega)}
\label{costante di Sobolev-q}
\end{equation}
and, by the Rellich theorem the embedding is compact.

By using \eqref{costante di Talenti}, we have
\begin{equation}
c_q\leq\frac{\mu(\Omega)^{\frac{2^*-q}{2^*q}}}{\sqrt{N(N-2)\pi}}
\Big(\frac{N!}{2\Gamma(1+\frac{N}{2})}\Big)^{1/N}
\label{costante q}
\end{equation}
where $\mu(\Omega)$ denotes the Lebesgue measure of the set
$\Omega$.
Moreover, let
\begin{equation}
D:=\sup_{x\in \Omega}\operatorname{dist}(x,\partial \Omega).\label{D}
\end{equation}
Simple calculations show that there is $x_0\in \Omega$ such that
$B(x_0,D)\subseteq\Omega$.

Finally, we set
\begin{equation}
\kappa=\frac{D}{\sqrt{2}\pi^{\frac{N}{4}}}
\Big(\frac{\Gamma(1+\frac{N}{2})}{D^N-(D/2)^N}\Big)^{1/2},\label{k}
\end{equation}
and
\begin{equation}
K_1=\frac{2\sqrt{2}mc_1(2^N-1)}{D^2}\quad
K_2=\frac{2^{\frac{q+2}{2}}m^qc_q^q(2^N-1)}{qD^2}.\label{k1k2}
\end{equation}



To study system \eqref{problema}, we will
use the functionals $\Phi,\, \Psi\,:X\to \mathbb{R}$ defined by
putting
\begin{equation}
\Phi(u):=\frac{1}{2}\sum_{i=1}^m\|u_i\|_{H_0^{1}(\Omega)}^2,\quad
\Psi(u):=\int_{\Omega} F(x,u_1(x),\dots,u_m(x))dx\label{operatori}
\end{equation}
for every $ u=(u_1,u_2,\dots, u_m)\in X$.

Clearly,  $\Phi$ is a coercive, continuously
G\^ateaux differentiable and weakly sequentially lower
semicontinuous,
 whose G\^ateaux derivative
admits a continuous inverse on $X^*$. On the other hand
%taking intoaccount (H0),
$\Psi$ is well defined, continuously G\^ateaux
differentiable with compact derivative.
One has
\begin{gather*}
\Phi'(u)(v)=\int_{\Omega}\sum_{i=1}^m\nabla u_i(x)\cdot \nabla v_i(x)dx\\ \\
\Psi'(u)(v)=\int_{\Omega}\sum_{i=1}^m
F_{u_i}(x,u_1(x),\dots,u_m(x))v_i(x)dx,
\end{gather*}
for every $v=(v_1,v_2,\dots, v_m)$, $ u=(u_1,u_2,\dots, u_m)\in X$.

 A critical point for the functional
$I_\lambda:=\Phi-\lambda\Psi$ is any $u\in X$ such that
$$
\Phi'(u)(v)-\lambda\Psi'(u)(v)=0\quad \forall v\in X,
$$
Hence, the critical points for functional $I_\lambda:=\Phi-\lambda
\Psi$ are exactly the weak solutions to system \eqref{problema}.

\section{Main results}

In this Section, we present our main results. First, we
 establish the existence of one non-trivial solution.

\begin{theorem} \label{1soluzione}
We suppose that {\rm (H0)} holds and assume that
\begin{itemize}
\item [(J1)] $F(x,t)\geq 0$ for every $(x,t)\in \Omega\times \mathbb{R}^m_+$
       where $\mathbb{R}^m_+=\{t=(t_1,\dots,t_m)\in\mathbb{R}^m:
  t_i\geq 0\quad i=1,\dots,m\}$;


\item[(J2)] there exist a positive constant $\gamma$ and a vector
$\delta\in\mathbb{R}^m_+$ with $ |\delta|<\gamma\kappa, $ such that
$$
\frac{\inf_{x\in
\Omega}F(x,\delta)}{|\delta|^2}>a_1\frac{K_1}{\gamma}+a_2K_2\gamma^{q-2},
$$
where $a_1$, $a_2$, $q$ are given by {\rm (H0)} and $\kappa$, $K_1$, $K_2$
are given by \eqref{k} and \eqref{k1k2}.
\end{itemize}
Then,  for each $\lambda \in
\big]\frac{2(2^N-1)}{D^2}\frac{|\delta|^2}{\inf_{x\in
\Omega}F(x,\delta)},\frac{2(2^N-1)}{D^2}
\frac{1}{a_1\frac{K_1}{\gamma}+a_2K_2\gamma^{q-2}}\big[$,
the system \eqref{problema} has at least one non-zero weak solution.
\end{theorem}

\begin{proof} Our goal is to apply Theorem \ref{critical0}. Consider the
Sobolev space $X$ and the operators defined in \eqref{operatori}. By
using (H0) one has
\begin{equation}
|F(x,t_1,\dots,t_m)|\leq
a_1\sum_{i=1}^m|t_i|+\frac{a_2}{q}\sum_{i=1}^m|t_i|^q,\label{RelazioneF}
\end{equation}
for every $(x,t)\in \Omega\times \mathbb{R}^m$.
Taking into account \eqref{RelazioneF} it follows that
\begin{equation}
\Psi(u)=\int_\Omega F(x,u)dx\leq
a_1\sum_{i=1}^m\|u_i\|_{L^1(\Omega)}
+\frac{a_2}{q}\sum_{i=1}^m\|u_i\|^q_{L^q(\Omega)}.\label{PSI}
\end{equation}
Let $r\in ]0,+\infty[$, then for every $u=(u_1,\dots,u_m)\in X$
such that $\Phi(u)< r$, by using \eqref{costante di Sobolev-q} from
\eqref{PSI} we obtain
\begin{equation}
\Psi(u)\leq
a_1c_1m\sqrt{2r}+\frac{a_2}{q}m^qc_q^q2^{q/2}r^{q/2}\label{Psi}.
\end{equation}
Hence, from \eqref{Psi}, the following relation holds
\begin{equation}
\frac{\sup_{u\in \Phi^{-1}(]-\infty,r[)}\Psi(u)}{r}\leq
\sqrt{\frac{2}{r}}mc_1a_1+\frac{2^{q/2}m^qc_q^q
a_2}{q}r^{\frac{q}{2}-1},\label{supPsi}
\end{equation}
for every $r>0$.
Now, we choose the function
$\bar{u}=(\bar{u}_1,\dots,\bar{u}_m)\in X$ defined by
\begin{equation}
\bar{u}_i(x)=  \begin{cases}
   0 &\text{if }  x\in\Omega\setminus B(x_0,D) \\
    \frac{2\delta_i}{D}(D-\sqrt{\sum_{j=1}^N(x_j-x_{j0})^2}
&\text{if } x\in B(x_0,D)\setminus B(x_0,\frac{D}{2})\\
    \delta_i  &\text{ if}  x\in  B(x_0,\frac{D}{2})
  \end{cases}
\label{funzione_u}\end{equation}
for $i=1,\dots,m$.
Clearly $\bar{u}\in X$ and we have
\begin{equation}
\begin{split}
\Phi(\bar{u})&=\frac{1}{2}\sum_{i=1}^m\int_\Omega|\nabla u_i(x)|^2dx\\
&=  \frac{1}{2}\sum_{i=1}^m\int_{B(x_0,D)\setminus B(x_0,
 \frac{D}{2})}\frac{4\delta_i^2}{D^2}dx
  \\
&=\frac{2|\delta|^2}{D^2}(\mu(B(x_0,D))-\mu( B(x_0,\frac{D}{2})))
  \\
&=\frac{2|\delta|^2}{D^2}\frac{\pi^{\frac{N}{2}}}{\Gamma(1+\frac{N}{2})}(D^N-(D/2)^N).
\end{split}\label{Phi(u)}
\end{equation}
Put $r=\gamma^2$, bearing in mind that $|\delta|<\gamma\kappa$, we
obtain
$$
0<\Phi(\bar{u})<r
$$
 and by using (J1) we have
\begin{equation}
\Psi(\bar{u})=\int_\Omega F(x,\bar{u}(x))dx\geq
\int_{B(x_0,\frac{D}{2})} F(x,\delta)dx\geq \inf_{x\in
\Omega}F(x,\delta)\frac{\pi^{\frac{N}{2}}}{\Gamma(1+\frac{N}{2})}
\frac{D^N}{2^N}.\label{Psi(u)}
\end{equation}
 Hence, by \eqref{Phi(u)} and \eqref{Psi(u)}, one has
\begin{equation}
\frac{\Psi(\bar{u})}{\Phi(\bar{u})}\geq \frac{D^2\inf_{x\in
\Omega}F(x,\delta)}{2(2^N-1)|\delta|^2}.\label{formula1}
\end{equation}
By using \eqref{k1k2}, \eqref{supPsi}, \eqref{formula1} and taking
into account (J2), we obtain
\begin{align*}
\frac{\sup_{u\in\Phi^{-1}(]-\infty,r[)}\Psi(u)}{r}&\leq
\frac{\sqrt{2}}{\gamma}mc_1a_1+\frac{2^{q/2}m^qc_q^q
a_2}{q}\gamma^{q-2}
\\
&=\frac{D^2}{2(2^N-1)}\Big(a_1\frac{K_1}{\gamma}+a_2K_2\gamma^{q-2}\Big)
\\
&<\frac{D^2\inf_{x\in \Omega}F(x,\delta)}{2(2^N-1)|\delta|^2}\leq
\frac{\Psi(\bar{u})}{\Phi(\bar{u})}.
\end{align*}
Moreover, by using \cite[Proposition 2.1]{bo1},
it is easy to prove that the functional $I_\lambda=\Phi-\lambda\Psi$ satisfies
 $(PS)^{[r]}$-condition.

Therefore, all the assumptions of Theorem \ref{critical0} are satisfied.
So, for each $\lambda \in$
$ \big]\frac{2(2^N-1)}{D^2}\frac{|\delta|^2}{\inf_{x\in
\Omega}F(x,\delta)},\frac{2(2^N-1)}{D^2}\frac{1}{a_1\frac{K_1}{\gamma}
+a_2K_2\gamma^{q-2}}\big[\subseteq
\big]\frac{\Phi(\bar{u})}{\Psi(\bar{u})},\frac{\gamma^2}{\sup_{u\in\Phi^{-1}(]
-\infty,\gamma^2[)
}\Psi(u)}\big[\;$, the functional $I_\lambda$ has at least one
non-zero critical point that is weak solution of system
\eqref{problema}.
\end{proof}

We now point out the case when $F$ does not depend on $x\in \Omega$,
 we consider problem
\begin{equation}
\begin{gathered}
   -\Delta u=\lambda \nabla_uF(u)\quad \text{in } \Omega,\\
    u=0 \quad \text{on }\partial \Omega
  \label{problema1}
  \end{gathered}
\end{equation}
we have the following result.

\begin{corollary} \label{Cor1}
Let $F:\mathbb{R}^m\to \mathbb{R}$ be a non-negative and
$C^1$-function satisfying (H0) and assume that
$$
\limsup_{|t|\to 0^+}\frac{F(t)}{|t|^2}=+\infty.
$$
Then, there is $\lambda^*>0$ such that, for each $\lambda\in]0,\lambda^*[$,
problem \eqref{problema1} admits at least one non-zero weak
solution.
\end{corollary}


\begin{proof} Taking into account condition (H0), fix
$$
\lambda^*=\frac{1}{\sqrt{2} a_1c_1m + 2^{q/2}\frac{a_2}{q} c_q^q m^q}.
$$
From
$$
\limsup_{|t|\to 0^+}\frac{F(t)}{|t|^2}=+\infty
$$
for all $\lambda\in ]0,\lambda^*[$, there is a vector $\delta^*\in \mathbb{R}^m_+$
with $|\delta^*|<k$ such that
$$
\frac{D^2}{2(2^N-1)}\frac{F(\delta^*)}{|\delta^*|^2}>\frac{1}{\lambda}
$$
Put $\bar {u}\in X$ as in \eqref{funzione_u}, and by choosing
$\gamma=1$ we obtain
$$
\frac{F(\delta^*)}{|\delta^*|^2}>\frac{2(2^N-1)}{\lambda
D^2}>\frac{2(2^N-1)}{\lambda^* D^2}=a_1K_1+a_2K_2
$$
All the assumptions of Theorem \ref{1soluzione} are satisfied and the
 proof is complete.
\end{proof}

The following result, in which the global Ambrosetti-Rabinowitz
condition is also used,  ensures the existence at least two weak
solutions.

\begin{theorem} \label{Th2}
We suppose that {\rm (H0)} holds and  $\nabla_u F(x,0)\not=0$ for every
$x\in \Omega$. Assume that there are two positive constants $\mu>2$
and $R$ such that
\begin{equation}
0<\mu F(x,t)\leq t\cdot \nabla_t F(x,t)\label{AR}
\end{equation}
for all $x\in \Omega$ and $|t|\geq R$.
Then, there exists $\lambda^*>0$ such that for each $\lambda\in
]0,\lambda^*[$,  problem \eqref{problema} has at least two non trivial weak
solutions.
\end{theorem}


\begin{proof} Put
$$
\lambda^*=\frac{1}{\sqrt{2}a_1c_1m+2^{q/2}\frac{a_2}{q}c_q^qm^q},
$$
and fix $\lambda<\lambda^*$. From \eqref{AR}, by standard
computations, there is a positive constant $C$ such that
\begin{equation}
F(x,t)\geq C|t|^\mu\label{stimaF}
\end{equation}
for all $x\in \Omega$, $|t|>R$.
In fact, setting $a(x)=\min_{|\xi|=R}F(x,\xi) $
and
\begin{equation}
\varphi_t(s)=F(x,st)\quad \forall s>0,\label{varphi}
 \end{equation}
by \eqref{AR}, for every $ x\in \Omega$ and  $|t|>R$ one has
$$
0<\mu\varphi_t(s)=\mu F(x,st)\leq st\cdot \nabla
F(x,st)=s\varphi'_t(s)\quad \forall s>0.
$$
Therefore,
$$
\int_{R/|t|}^1\frac{\varphi'_t(s)}{\varphi_t(s)}ds\geq
\int_{R/|t|}^1\frac{\mu}{s}ds.
$$
Then
$$
\varphi_t(1)\geq \varphi_t\Big(\frac{R}{|t|}\Big)|t|^\mu.
$$
Taking into account of \eqref{varphi}, we obtain
$$
F(x,t)\geq F\Big(x,\frac{R}{|t|}t\Big)|t|^\mu\geq a(x)|t|^\mu\geq
C|t|^\mu
$$
and \eqref{stimaF} is proved. From \eqref{stimaF} it follows that
$I_\lambda$ is unbounded from below.

 Now, to verify the (PS)-condition it is sufficient to prove that
any sequence of Palais-Smale is bounded. To this end, taking into
account \eqref{AR} one has
\begin{equation}
 \begin{split}
&\mu I_\lambda(u_n)-\|I_\lambda'(u_n)\|_{X'}\|u_n\|\geq\mu
I_\lambda(u_n)- I_\lambda'(u_n)(u_n)
\\
&=\mu \Phi(u_n)-\lambda\mu
\Psi(u_n)-\Phi'(u_n)(u_n)+\lambda\mu\Psi'(u_n)(u_n)
\\
&=(\frac{\mu}{2} -1)\sum_{i=1}^m \|u_{in}\|^2-\lambda\int_\Omega(\mu
F(x,u_n(x))-\sum_{i=1}^m F_{u_i}(x,u_1(x),\dots,u_m(x))u_i(x))
\\
&\geq (\frac{\mu}{2} -1)\sum_{i=1}^m \|u_{in}\|^2\geq
\frac{1}{m}(\frac{\mu}{2} -1)\|u_n\|^2.
\end{split}\label{PS}
\end{equation}
If $\{u_n\}$ is not bounded from \eqref{PS} we have a contradiction.
Moreover, from \eqref{supPsi} by choosing $r=1$ one has
$$
\sup_{u\in \Phi^{-1}(]-\infty,1[)}\Psi(u)\leq
\sqrt{2}a_1c_1m+2^{q/2}\frac{a_2}{q}c_q^qm^q=\frac{1}{\lambda^*}.
$$
Hence, Theorem \ref{critical2} ensures that problem \eqref{problema}, for each
$\lambda\in ]0,\lambda^*[$, admits at least two weak solutions.
\end{proof}

Now, we point out the following result of three weak solutions.

\begin{theorem}  \label{3soluzioni}
We suppose that {\rm (H0)} holds and assume that
\begin{itemize}
\item [(H1)] $F(x,t)\geq 0$ for every $(x,t)\in \Omega\times \mathbb{R}^m_+$
 where $\mathbb{R}^m_+=\{t=(t_1,\dots,t_m)\in
\mathbb{R}^m: t_i\geq 0\quad i=1,\dots,m\}$;

\item[(H2)] there exist two positive constants $b$ and $s<2$ such
that
$$
F(x,t)\leq b(1+\sum_{i=1}^m|t_i|^s)
$$
for almost every $x\in\Omega$ and for every $t\in \mathbb{R}^m$;

\item[(H3)] there exist a positive constant $\gamma$ and a vector
$\delta\in\mathbb{R}^m_+$ such that $ |\delta|>\gamma\kappa$, such
that
$$
\frac{\inf_{x\in \Omega}F(x,\delta)}{|\delta|^2}>a_1\frac{K_1}{\gamma}
+a_2K_2\gamma^{q-2},
$$
where $a_1$, $a_2$, $q$ are given by (H0) and $\kappa$, $K_1$, $K_2$
are given by \eqref{k} and \eqref{k1k2}.
\end{itemize}
Then,  for each $\lambda \in
\big]\frac{2(2^N-1)}{D^2}\frac{\delta^2}{\inf_{x\in
\Omega}F(x,\delta)},\frac{2(2^N-1)}{D^2}\frac{1}{a_1\frac{K_1}{\gamma}
+a_2K_2\gamma^{q-2}}\big[$,
system \eqref{problema} has at least three weak solutions.
\end{theorem}

\begin{proof}
  Our goal is to apply Theorem \ref{critical1}. Consider the
Sobolev space $X$ and the operators defined in \eqref{operatori}
taking into account that the regularity assumptions on $\Phi$ \and
$\Psi$ are satisfied, our aim is to verify $(i)$ and $(ii)$. Arguing
as in the proof of Theorem \ref{1soluzione}, put $\bar{u}$ as in
\eqref{funzione_u} and  $r=\gamma^2$, bearing in mind that
$|\delta|>\gamma\kappa$, we obtain
$$
\Phi(\bar{u})>r>0.
$$
Therefore, the assumption (i) of Theorem \ref{critical1} is
satisfied.

We prove that the functional $I_\lambda=\Phi-\lambda\Psi$ is
coercive for all positive parameter, in fact by using condition
(H2) we have
\begin{align*}
I_\lambda(u)
&=\Phi(u) - \lambda \Psi(u)\geq \frac{1}{2m}\|u\|^2 -
\lambda \int_\Omega {F(x,u(x))dx}
\\
&\geq \frac{1}{2m}\|u\|^2 - \lambda
\int_\Omega b(1+\sum_{i=1}^{m}|u_i(x)|^s)dx
 \\
&\geq \frac{1}{2m}\|u\|^2 - \lambda b\mu(\Omega)-\lambda
bc_2^s\mu(\Omega)^{\frac{2-s}{2}}\|u\|^s.
\end{align*}
Then also condition (ii) holds, hence all the assumptions of
Theorem \ref{critical1} are satisfied.
 So, for each $\lambda$ in 
$\big]\frac{2(2^N-1)}{D^2}\frac{|\delta|^2}{\inf_{x\in
\Omega}F(x,\delta)},\frac{2(2^N-1)}{D^2}\frac{1}{a_1\frac{K_1}{\gamma}
+a_2K_2\gamma^{q-2}}\big[$, which is a subsect of 
$\big]\frac{\Phi(\bar{u})}{\Psi(\bar{u})},\frac{\gamma^2}{\sup_{
u\in\Phi^{-1}(]-\infty,\gamma^2[)}\Psi(u)}\big[$, the functional
$I_\lambda$ has at least three distinct critical points that are
weak solutions of system \eqref{problema}.
\end{proof}

An immediate consequence of Theorem \ref{3soluzioni} is the following result.

\begin{corollary} \label{coro3.2}
We suppose that {\rm (H0)} holds and assume that
\begin{itemize}
\item [(H1')] $F(t)\geq 0$ for every $t\in  \mathbb{R}^m_+$
 where $\mathbb{R}^m_+=\{t=(t_1,\dots,t_m)\in
\mathbb{R}^m: t_i\geq 0\quad i=1,\dots,m\}$;

\item[(H2')] there exist two positive constants $b$ and $s<2$ such
that $$F(t)\leq b(1+\sum_{i=1}^m|t_i|^s)$$  for every $t\in
\mathbb{R}^m$;

\item[(H3')] there exist a positive constant $\gamma$ and a vector
$\delta\in\mathbb{R}^m_+$ with $ |\delta|>\gamma\kappa, $ such that
$$
\frac{F(\delta)}{|\delta|^2}>a_1\frac{K_1}{\gamma}+a_2K_2\gamma^{q-2},
$$
where $a_1$, $a_2$ are given by (H1) and $\kappa$, $K_1$, $K_2$
are given by \eqref{k} and \eqref{k1k2}.
\end{itemize}
Then,  for each $\lambda \in
\big]\frac{2(2^N-1)}{D^2}\frac{\delta^2}{F(\delta)},
\frac{2(2^N-1)}{D^2}\frac{1}{a_1\frac{K_1}{\gamma}+a_2K_2\gamma^{q-2}}\big[$,
system \eqref{problema1} has at least three weak solutions.
\end{corollary}

\begin{remark} \label{rmk3.1} \rm
If we assume that $F_{u_i}:\Omega\times \mathbb{R}^m\to \mathbb{R}$
($i=1,\dots,m$) are non negative, continuous functions then the
previous theorems guarantee the existence of non negative weak
solutions. In fact, let $\bar{u}=(\bar{u_1},\dots, \bar{u_m})$ be a
weak solution of system \eqref{problema}. Fixed $i$, we consider the
problem
\begin{equation}
\begin{gathered}
   -\Delta u_i=\lambda F_{u_i}(x,\bar{u_1},\dots,  u_i, \dots, \bar{u_m})\quad
\text{in } \Omega,\\
    u_i\big|_{\partial \Omega}=0\quad i=1,\dots,m.
    \label{remark1}
\end{gathered}
\end{equation}
Clearly, one has $\bar{u_i}\in H_0^{1}(\Omega)$ and it is a weak
solution of \eqref{remark1}. Hence, the Strong Maximum Principle
ensures that either $\bar{u_i}(x)=0$ or $\bar{u_i}(x)>0$ on
$\Omega$.
\end{remark}

Now, we present some examples that illustrate our results.

\begin{example} \label{examp3.1} \rm
Let $\Omega$ be an open ball of radius one in $\mathbb{R}^3$.
Consider the function $F:\mathbb{R}^2\to \mathbb{R}$ defined by
$$
F(t_1,t_2)=|t_1|^{3/2}+|t_2|^{3/2}
$$
for every $(t_1,t_2)\in \mathbb{R}^2$.
We observe that
\begin{gather*}
F_{t_1}(t_1,t_2)=\frac{3}{2}|t_1|^{\frac{3}{2}-2}t_1,\\
F_{t_2}(t_1,t_2)=\frac{3}{2}|t_2|^{\frac{3}{2}-2}t_2\,.
\end{gather*}
Then, choosing  $q=3/2$, $a_1=0$ and $a_2=3/2$  the
condition (H0) holds.
Then by using  Corollary \ref{Cor1}, put
$$
\lambda^*=\frac{3^{3/2}\pi^{1/4}}{2^{19/4}}
$$
for all $\lambda\in ]0,\lambda^*[$, the  system
\begin{equation}
\begin{gathered}
   -\Delta u=\lambda F_u(u, v) \quad \text{in } \Omega,\\
  -\Delta v=\lambda F_v(u, v) \quad \text{in } \Omega,\\
    u=v=0 \quad \text{on }\partial \Omega
  \label{problema2}
  \end{gathered}
\end{equation}
admits at least one non-zero weak solution in $X=H^1_0(\Omega)\times
H^1_0(\Omega)$.
\end{example}


\begin{example} \label{examp3.2} \rm
 Let $\Omega$ be an open ball of radius one in $\mathbb{R}^3$.
Consider the function $F:\Omega\times \mathbb{R}^2\to \mathbb{R}$
defined by
$$
F(x,t_1,t_2)=\frac{1}{6}t_1+\frac{1}{6}t_2+\frac{1}{4}\big(|t_1|^4+|t_2|^4\big)
$$
for every $x\in\Omega$ and for every $(t_1,t_2)\in \mathbb{R}^2$.
We observe that
\begin{gather*}
F_{t_1}(x,t_1,t_2)=\frac{1}{6}+|t_1|^2t_1, \\
F_{t_2}(x,t_1,t_2)=\frac{1}{6}+|t_2|^2t_2,
\end{gather*}
therefore, $\nabla_u F(x,0)\not=0$ for every $x\in \Omega$, choosing
$q=4$, $a_1=1/6$ and $a_2=1$  the condition (H0) holds.
Moreover, choose $\mu=3$ we have
$$
0<3F(x,t_1,t_2)\leq t_1 F_{t_1}(x,t_1,t_2)+ t_2 F_{t_2}(x,t_1,t_2)
$$
for every $x\in\Omega$ and for every $t\in \mathbb{R}^2$. Then, by
using  Theorem \ref{Th2}, put
$$
\lambda^*=\frac{\pi^{7/12}3^{7/3}}{2^{17/6}(\pi^{3/4}+2^2
3^{7/4})}
$$
 for all $\lambda\in ]0,\lambda^*[$ the  system
\begin{equation}
\begin{gathered}
   -\Delta u=\lambda F_u(x,u, v)\quad \text{in } \Omega,\\
  -\Delta v=\lambda F_v(x,u, v)\quad \text{in } \Omega,\\
    u=v=0 \quad \text{on }\partial \Omega
  \label{problema2b}
  \end{gathered}
\end{equation}
admits at least two non-zero weak solutions in
$X=H^1_0(\Omega)\times H^1_0(\Omega)$.
\end{example}

 \begin{example} \label{examp3.3} \rm
Let $\Omega$ be an open ball of radius one in
$\mathbb{R}^3$. Set $q=5\in ]2,6[$, $s=3/2<2$, choose
$a_1=1$, $a_2=10/3$ and
$$
r=9>\Big(\frac{K_1+a_2K_2}{5}\Big)^{1/3}
$$
where $K_1$ and $K_2$ are given by \eqref{k1k2}.
Consider the function $F:\mathbb{R}^2\to \mathbb{R}$ defined by
$$
F(t_1,t_2)=\begin{cases}
 t_1+t_2+\frac{1}{5}(t_1^5+t^5_2)&
\text{if } t_1\leq 9,\; t_2\leq 9
\\
t_1+t_2-\frac{7}{5}3^9+\frac{1}{5}t_1^5+2\cdot 3^6
t_2^{3/2}&  \text{if } t_1\leq 9,\; t_2>9
\\
t_1+t_2-\frac{7}{5}3^9+\frac{1}{5}t_2^5+2\cdot 3^6
t_1^{3/2}&  \text{if } t_1> 9,\; t_2\leq 9
\\
t_1+t_2-\frac{14}{5}3^9+2\cdot 3^6
(t_1^{3/2}+t_2^{3/2})&  \text{if } t_1> 9,\;
t_2>9.
\end{cases}
$$
Clearly (H0) holds. Moreover, for each $(t_1,t_2)\in \mathbb{R}^2$,
one has
$$
F(t_1,t_2)\leq 2(9+2\cdot 3^6)(1+|t_1|^{3/2}+|t_2|^{3/2}),
$$
therefore, if we choose $\gamma=1$, $b= 2(9+2\cdot 3^6)$ and
$\delta=(9,9)$ the hypotheses of Corollary \ref{coro3.2} are satisfied. Then,
for each $\lambda\in]\frac{630}{6566},\;
\frac{\pi^{\frac{19}{6}}\cdot 3^{\frac{10}{3}}}{(3^{2}\cdot
\pi^{\frac{10}{3}}+2^{\frac{25}{6}})2^{\frac{23}{6}}}[$, the system
\begin{equation}
\begin{gathered}
   -\Delta u=\lambda F_u(u, v) \quad\text{in } \Omega,\\
  -\Delta v=\lambda F_v(u, v) \quad \text{in } \Omega,\\
    u=v=0 \quad \text{on }\partial \Omega
  \label{problema2c}
  \end{gathered}
\end{equation}
admits at least three non negative weak solutions in
$X=H^1_0(\Omega)\times H^1_0(\Omega)$.
\end{example}


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\end{document}