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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 163, pp. 1--9.\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-2012/163\hfil Infinitely many solutions]
{Infinitely many solutions for class of
Navier boundary $(p,q)$-biharmonic systems}

\author[M. Massar, E. M. Hssini, N. Tsouli \hfil EJDE-2012/163\hfilneg]
{Mohammed Massar, El Miloud Hssini, Najib Tsouli}  

\address{Mohammed Massar \newline
University Mohamed I, Faculty of Sciences, Department of
Mathematics, Oujda, Morocco} 
\email{massarmed@hotmail.com}

\address{El Miloud Hssini \newline
University Mohamed I, Faculty of Sciences, Department of
Mathematics, Oujda, Morocco} 
\email{hssini1975@yahoo.fr}

\address{Najib Tsouli \newline
University Mohamed I, Faculty of Sciences, Department of
Mathematics, Oujda, Morocco} 
\email{tsouli@hotmail.com}

\thanks{Submitted June 4, 2012. Published September 21, 2012.}
\subjclass[2000]{35J40, 35J60}
\keywords{Navier value problem; infinitely many solutions; \hfill\break\indent
  Ricceri's variational principle}

\begin{abstract}
 This article shows the existence and multiplicity of weak solutions
 for the  $(p,q)$-biharmonic type system
 \begin{gather*}
 \Delta(|\Delta u|^{p-2}\Delta u)=\lambda F_u(x,u,v)\quad\text{in }\Omega,\\
 \Delta(|\Delta v|^{q-2}\Delta v)=\lambda F_v(x,u,v)\quad\text{in }\Omega,\\
 u=v=\Delta u=\Delta v=0\quad  \text{on }\partial\Omega.
 \end{gather*}
 Under certain conditions on $F$, we show the existence of infinitely many 
 weak  solutions. Our technical approach is based on Bonanno and 
 Molica Bisci's general  critical point theorem.
\end{abstract}

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

\section{Introduction}

In this paper we are concerned with the existence and multiplicity
of weak solutions for the  $(p,q)$-biharmonic type system
 \begin{equation} \label{E11}
 \begin{gathered} 
 \Delta(|\Delta u|^{p-2}\Delta u)=\lambda F_u(x,u,v) \quad \text{in }\Omega, \\
 \Delta(|\Delta v|^{q-2}\Delta v)=\lambda F_v(x,u,v) \quad \text{in }\Omega, \\
 u=v=\Delta u=\Delta v=0 \quad \text{on }\partial\Omega, 
 \end{gathered}
\end{equation}
where $\Omega$ is an open bounded subset of
$\mathbb{R}^{N}$ $(N\geq1)$, with smooth boundary,
$\lambda\in(0,\infty)$, $p>\max\{1,\frac{N}2\}$,
$q>\max\{1,\frac{N}2\}$.
$F: \overline{\Omega}\times \mathbb{R}^2\to\mathbb{R}$ is a
function such that $F(.,s,t)$ is continuous in $\overline{\Omega }$, 
for all $(s,t)\in \mathbb{R}^2$ and $F(x,.,.)$ is $C^1$ in
$\mathbb{R}^2$ for every $x\in\Omega$, and $F_u,F_v$ denote the
partial derivatives of $F$, with respect to $u,v$ respectively.

The investigation of existence and multiplicity of solutions for
problems involving biharmonic and p-biharmonic operators has drawn
the attention of many authors, see \cite{CL,LT,SZ,WS} and references
therein.  Candito and Livrea \cite{CL} considered the
 nonlinear elliptic Navier boundary-value problem
 \begin{equation} \label{E12}
\begin{gathered}
 \Delta(|\Delta u|^{p-2}\Delta u)=\lambda f(x,u) \quad \text{i n}\Omega, \\
 u=\Delta u=0\quad\text{on }\partial\Omega. 
 \end{gathered}
\end{equation}
There the authors established the existence of infinitely many solutions.

In the present paper, we look for the existence of infinitely many
solutions of system \eqref{E11}. More precisely, we will prove the
existence of well precise intervals of parameters such that problem
\eqref{E11} admits either an unbounded sequence of solutions
provided that $F(x,u,v)$ has a suitable behaviour at infinity or a
sequence of nontrivial solutions converging to zero if a similar
behaviour occurs at zero. Our main tool is a general critical points
theorem due to Bonanno and Molica Bisci \cite{BM} that is a
generalization of a previous result of Ricceri \cite{R}.

In the sequel, $X$ will denote the space $\big(W^{2,p}(\Omega)\cap
W_0^{1,p}(\Omega)\big)\times\big(W^{2,q}(\Omega)\cap
W_0^{1,q}(\Omega)\big)$, which is a reflexive Banach space endowed
with the norm
$$
\|(u,v)\|=\|u\|_p+\|v\|_q,
$$
where
$$
\|u\|_p=\Big(\int_\Omega|\Delta u|^pdx\Big)^{1/p}\quad\text{and}
\quad \|v\|_q=\Big(\int_\Omega|\Delta v|^qdx\Big)^{1/q}.
$$
Let
\begin{equation}\label{E13}
K:=\max\Big\{\underset{u\in W^{2,p}(\Omega)\cap
W_0^{1,p}(\Omega)\setminus\{0\}}
\sup\frac{\underset{x\in\Omega}\max|u(x)|^p}{\|u\|_p^p},
\underset{v\in W^{2,q}(\Omega)\cap
W_0^{1,q}(\Omega)\setminus\{0\}}\sup
\frac{\underset{x\in\Omega}\max|v(x)|^q}{\|v\|_q^q}\Big\}.
\end{equation}
Since $p>\max\{1,\frac{N}2\}$ and $q>\max\{1,\frac{N}2\}$, the
Rellich Kondrachov theorem assures that
 $W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)\hookrightarrow C(\overline{\Omega})$ and
$W^{2,q}(\Omega)\cap W_0^{1,q}(\Omega)\hookrightarrow
C(\overline{\Omega})$ are compact, and hence $K<\infty$. 

\begin{definition} \label{def1.1} \rm
We say that $(u,v)\in X$ is a weak solution of problem \eqref{E11}
if
\begin{align*}
&\int_\Omega|\Delta u|^{p-2}\Delta u\Delta\varphi\, dx
+\int_\Omega|\Delta v|^{q-2}\Delta v \Delta\psi\, dx\\
&-\lambda\int_\Omega F_u(x,u,v)\varphi\, dx
 -\lambda\int_\Omega F_v(x,u,v)\psi \,dx=0,
\end{align*}
for all $(\varphi,\psi)\in X$.
\end{definition}

Define the functional $I_\lambda: X\to\mathbb{R}$, given by
$$
I_\lambda(u,v)=\Phi(u,v)-\lambda\Psi(u,v),
$$
for all $(u,v)\in X$, where
$$
\Phi(u,v)=\frac1p\|u\|_p^p+\frac1q\|v\|_q^q\quad\text{and}\quad
\Psi(u,v)=\int_\Omega F(x,u,v)dx.
$$
Since $X$ is compactly embedded in $C^0(\overline{\Omega})\times
C^0(\overline{\Omega})$, it is well known that $\Phi$ and $\Psi$ are
well defined G\^{a}teaux differentiable functionals whose
G\^{a}teaux derivatives at $(u,v)\in X$ are given by
\begin{gather*}
\langle\Phi'(u,v),(\varphi,\psi)\rangle=\int_\Omega|\Delta
u|^{p-2}\Delta u \Delta\varphi dx+\int_\Omega|\Delta v|^{q-2}\Delta
v \Delta\psi dx,
\\
\langle\Psi'(u,v),(\varphi,\psi)\rangle=\int_\Omega
F_u(x,u,v)\varphi dx +\int_\Omega F_v(x,u,v)\psi dx,
\end{gather*}
for all $(\varphi,\psi)\in X$. Moreover, by the weakly lower
semicontinuity of norm, we see that $\Phi$ is sequentially weakly
lower semi continuous. Since $\Psi$ has compact derivative, it
follows that $\Psi$ is sequentially weakly continuous.

In view of \eqref{E13}, for every $(u,v)\in X$, we have
$$
\sup_{x\in\Omega}|u(x)|^p\leq K\|u\|_p^p\quad\text{and}\quad
\sup_{x\in\Omega}|v(x)|^q\leq K\|v\|_q^q,
$$
thus
\begin{equation}\label{E14}
\sup_{x\in\Omega}\Big(\frac1p|u(x)|^p+\frac1q|v(x)|^q\Big)\leq
K\Big(\frac1p\|u\|_p^p+\frac1q\|v\|_q^q\Big).
\end{equation}
Hence, for every $r>0$
\begin{equation} \label{E15}
\begin{split}
\Phi^{-1}(]-\infty,r[):&= \big\{(u,v)\in X: \Phi(u,v)<r\big\} \\
&= \big\{(u,v)\in X:
\frac1p\|u\|_p^p+\frac1q\|v\|_q^q<r\big\} \\
&\subseteq \big\{(u,v)\in X:
\frac1p|u(x)|^p+\frac1q|v(x)|^q<Kr,\,\forall \,x\in \Omega\big\}.
\end{split}
\end{equation}

Let us  recall for the reader's convenience a smooth version of
a previous result of Ricceri \cite{R}.

\begin{theorem}\label{theo11}
Let $X$ be a reflexive real Banach space, let $\Phi, \Psi :
X\to\mathbb{R}$ be two G\^{a}teaux differentiable
functionals such that $\Phi$ is sequentially weakly lower
semicontinuous and coercive and $\Psi$ is sequentially weakly upper
semicontinuous. For every $r>\inf_X\Phi$, let us put
$$
\varphi(r):=\inf_{u\in\Phi^{-1}(]-\infty,r[)}\frac
{\big(\sup_{v\in\Phi^{-1}(]-\infty,r[)}\Psi(v)\big)-\Psi(u)}{r-\Phi(u)}
$$
and
$$
\gamma:=\liminf_{r\to+\infty}\varphi(r),\quad
\delta:=\liminf_{r\to(\inf_X\Phi)^+}\varphi(r).
$$
Then, one has

(a) for every $r>\inf_X\Phi$ and every
$\lambda\in]0,\frac1{\varphi(r)}[$, the restriction of
the functional $I_\lambda=\Phi-\lambda\Psi$ to
$\Phi^{-1}(]-\infty,r[)$ admits a global minimum, which is a
critical point (local minimum) of $I_\lambda$ in $X$.

(b) If $\gamma<+\infty$ then, for each
$\lambda\in]0,\frac1\gamma[$, the following alternative
holds: either
\begin{itemize}
\item[(b1)] $I_\lambda$ possesses a global minimum,or

\item[(b2)] there is a sequence $(u_n)$ of critical points (local
minima) of $I_\lambda$ such that
$\lim_{n\to+\infty}\Phi(u_n)=+\infty$.
\end{itemize}

(c) If $\delta<+\infty$ then, for each
$\lambda\in]0,\frac1\delta[$, the following alternative
holds: either
\begin{itemize}
\item[(c1)] there is a global minimum of $\Phi$ which is a local
minimum of $I_\lambda$,or
\item[(c2)] there is a sequence of pairwise distinct critical points
(local minima) of $I_\lambda$ which weakly converges to global
minimum of $\Phi$.
\end{itemize}
\end{theorem}

\section{Main results}

Fix $x^0\in\Omega$ and pick $R_2>R_1>0$ such that
$B(x^0,R_2)\subseteq\Omega$. Set
\begin{equation}\label{E21}
\begin{gathered}
L_p:=\frac{\Gamma(1+N/2)}
{\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)} \pi^{N/2}}
\Big(\frac{R_2^2-R_1^2}{2N}\Big)^p\frac1{R_2^N-R_1^N},\\
L_q:=\frac{\Gamma(1+N/2)}{\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}\pi^{N/2}}
\Big(\frac{R_2^2-R_1^2}{2N} \Big)^q\frac1{R_2^N-R_1^N}
\end{gathered}
\end{equation}
where $\Gamma$ denotes the Gamma function and $K$ is given by
\eqref{E13}.
Now we are ready to state our main results.

\begin{theorem}\label{theo21}
Assume that
\begin{itemize}
\item[(i1)] $F(x,s,t)\geq0$ for every
$(x,s,t)\in\Omega\times[0,+\infty)^2;$

\item[(i2)] There exist $x^0\in\Omega$, $0<R_1<R_2$ as considered
in \eqref{E21} such that, if we put
$$
\alpha:=\liminf_{\sigma\to+\infty}
\frac{\int_\Omega\sup_{|s|+|t|\leq\sigma}F(x,s,t)dx}{\sigma^{\min(p,q)}},\quad
\beta:=\limsup_{s,t\to+\infty}
 \frac{\int_{B(x^0,R_1)}F(x,s,t)dx}{\frac{s^p}p+\frac{t^q}q},
$$
one has
\begin{equation}\label{E22}
\alpha<L\beta,
\end{equation}
where $L:=\min\{L_p, L_q\}$.
\end{itemize}
Then, for every
$$
\lambda\in\Lambda:=\frac1{\left((Kp)^{1/p}+(Kq)^{1/q}\right)
^{\min(p,q)}}\big]\frac1{L\beta},\frac1\alpha\big[
$$
problem \eqref{E11} admits an unbounded sequence of weak solutions.
\end{theorem}

\begin{theorem}\label{theo22}
Assume that {\rm (i1)} holds and
\begin{itemize}
\item[(i3)] $F(x,0,0)=0$ for every $x\in\Omega$.
\item[(i4)] There exist $x^0\in\Omega$, $0<R_1<R_2$ as considered
in \eqref{E21} such that, if we put
$$
\alpha^0:=\liminf_{\sigma\to0^+}\frac{\int_\Omega\sup_{|s|+|t|\leq\sigma}F(x,s,t)dx}{\sigma^{\min(p,q)}}
,\quad
\beta^0:=\limsup_{s,t\to0^+}\frac{\int_{B(x^0,R_1)}F(x,s,t)dx}{\frac{s^p}p+\frac{t^q}q},
$$
one has
\begin{equation}\label{E23}
\alpha^0<L\beta^0.
\end{equation}
where
$L:=\min\{L_p, L_q\}$.
\end{itemize}
Then, for every
\[
\lambda\in\Lambda:=\frac1{\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}}
\big]\frac1{L\beta^0},\frac1{\alpha^0}\big[\,,
\]
problem \eqref{E11} admits a sequence $(u_n)$ of weak solutions such
that $u_n\rightharpoonup0$.
\end{theorem}

\section{Proofs of main results}

\begin{proof}[Proof of Theorem \ref{theo21}]
To apply Theorem \ref{theo11}, we set
$$
\varphi(r):=\inf_{(u,v)\in\Phi^{-1}(]-\infty,r[)}\frac
{\left(\sup_{(w,z)\in\Phi^{-1}(]-\infty,r[)}\Psi(w,z)\right)
-\Psi(u,v)}{r-\Phi(u,v)}
$$
Note that $\Phi(0,0)=0$, and by (i1), $\Psi(0,0)\geq 0$.
Therefore, for every $r>0$,
\begin{equation} \label{E31}
\begin{split}
\varphi(r)&= \inf_{(u,v)\in\Phi^{-1}(]-\infty,r[)}\frac
{\left(\sup_{(w,z)\in\Phi^{-1}(]-\infty,r[)}\Psi(w,z)\right)
-\Psi(u,v)}{r-\Phi(u,v)} \\
&\leq \frac {\sup_{\Phi^{-1}(]-\infty,r[)}\Psi}r \\
&= \frac{\sup_{\Phi(u,v)<r}\int_\Omega F(x,u,v)dx}r.
\end{split}
\end{equation}
Hence, from \eqref{E15}, we have
$$
\varphi(r)\leq
\frac{1}{r}
\sup_{\{(u,v)\in X: \frac{|u(x)|^p}p+\frac{|v(x)|^q}q<Kr,\,\forall \,x\in \Omega\}}
\int_\Omega F(x,u,v)dx
$$
Let $(\sigma_n)$ a sequence of positive numbers such that
$\sigma_n\to+\infty$ and
\begin{equation}\label{E32}
\lim_{n\to+\infty}\frac{\int_\Omega\sup_{|s|+|t|\leq\sigma_n}F(x,s,t)dx}
{\sigma_n^{\min(p,q)}}
=\alpha<+\infty.
\end{equation}
Put
$$
r_n:=\Big(\frac{\sigma_n}{(Kp)^{1/p}+(Kq)^{1/q}}\Big)^{\min(p,q)}
$$
Let $(u,v)\in\Phi^{-1}(]-\infty,r_n[)$, from \eqref{E15} we have
$$
\frac{|u(x)|^p}p+\frac{|v(x)|^q}q<Kr_n,\quad \text{for all }\;x\in\Omega.
$$
Thus
$$
|u(x)|\leq(Kpr_n)^{1/p}\quad\text{and}\quad |v(x)|\leq(Kqr_n)^{1/q},
$$
hence, for $n$ large enough $(r_n>1)$,
\begin{align*}
|u(x)|+|v(x)|&\leq (Kpr_n)^{1/p}+(Kqr_n)^{1/q}\\
&\leq \left((Kp)^{1/p}+(Kq)^{1/q}\right)r_n^{\frac1{\min(p,q)}}
= \sigma_n.
\end{align*}
Therefore,
\begin{equation}\label{E33}
\begin{split}
\varphi(r_n)&\leq \frac{\sup_{\{(u,v)\in X:
|u(x)|+|v(x)|<\sigma_n,\,\forall \,x\in \Omega\}}\int_\Omega
F(x,u,v)dx}{\big(\frac{\sigma_n}
{(Kp)^{1/p}+(Kq)^{1/q}}\big)^{\min(p,q)}} \\
&\leq \left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}
\frac{\int_\Omega\sup_{|s|+|t|<\sigma_n}
F(x,s,t)dx}{\sigma_n^{\min(p,q)}}.
\end{split}
\end{equation}
Let
$$
\gamma:=\liminf_{r\to+\infty}\varphi(r).
$$
It follows from \eqref{E32} and \eqref{E33} that
\begin{equation} \label{E34}
\begin{split}
\gamma
&\leq \liminf_{n\to+\infty}\varphi(r_n) \\
&\leq \left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}\lim_{n\to+\infty}
\frac{\int_\Omega \sup_{|s|+|t|<\sigma_n}F(x,s,t)}{\sigma_n^{\min(p,q)}} \\
&= \alpha \left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}<+\infty.
\end{split}
\end{equation}
From \eqref{E34}, it is clear that
$\Lambda\subseteq ]0,\frac1\gamma[$.

For $\lambda\in\Lambda$, we claim that the functional $I_\lambda$ is
unbounded from below. Indeed, since
$\frac1\lambda<\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}L\beta$,
we can consider a sequence $(\tau_n)$ of positive numbers and
$\eta>0$ such that $\tau_n\to+\infty$ and
\begin{equation}\label{E35}
\frac1\lambda<\eta<L\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}\frac{\int_{B(x^0,R_1)}F(x,\tau_n,\tau_n)dx}
{\frac{\tau_n^p}p+\frac{\tau_n^q}q},
\end{equation}
for $n$ large enough. Define a sequence $(u_n)$ as follows
\begin{equation}\label{E36}
u_n(x)=\begin{cases}
0,&x\in\overline{\Omega}\backslash B(x_0,R_2)\\
\frac{\tau_n}{R_2^2-R_1^2}[R_2^2-\big(\sum_{i=1}^N(x^i-x_0^i)^2\big)],
 &x\in B(x_0,R_2)\backslash B(x_0,R_1)\\
\tau_n, &x\in B(x_0,R_1)
\end{cases}
\end{equation}
Then $(u_n,u_n)\in X$ and
\begin{gather*}
\|u_n\|_p^p=\frac{\pi^{N/2}}{\Gamma(1+N/2)}
\Big(\frac{2N\tau_n}{R_2^2-R_1^2}\Big)^p(R_2^N-R_1^N),
\\
\|u_n\|_q^q=\frac{\pi^{N/2}}{\Gamma(1+N/2)}
\Big(\frac{2N\tau_n}{R_2^2-R_1^2}\Big)^q(R_2^N-R_1^N).
\end{gather*}
This and \eqref{E21} imply that
\begin{equation}\label{E37}
\Phi(u_n,u_n)=\frac1{\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}}
\Big(\frac{\tau_n^p}{pL_p}+\frac{\tau_n^q}{qL_q}\Big).
\end{equation}
By (i1), we have
\begin{equation} \label{E38}
\Psi(u_n,u_n)= \int_\Omega F(x,u_n,u_n)dx
\geq \int_{B(x^0,R_1)} F(x,\tau_n,\tau_n)dx.
\end{equation}
Combining \eqref{E35}, \eqref{E37} and \eqref{E38}, we obtain
\begin{equation} \label{E39}
\begin{split}
&I_\lambda(u_n,u_n)\\
&= \Phi(u_n,u_n)-\lambda\Psi(u_n,u_n) \\
&\leq \frac1{\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}}
\Big(\frac{\tau_n^p}{pL_p}+\frac{\tau_n^q}{qL_q}\Big)
 -\lambda\int_{B(x^0,R_1)} F(x,\tau_n,\tau_n)dx \\
&\leq \frac1{L\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}}
\Big(\frac{\tau_n^p}p+\frac{\tau_n^q}q\Big)
-\lambda\int_{B(x^0,R_1)} F(x,\tau_n,\tau_n)dx \\
&< \frac{1-\lambda\eta}{L\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}}
\Big(\frac{\tau_n^p}p+\frac{\tau_n^q}q\Big),
\end{split}
\end{equation}
for $n$ large enough, so
$$
\lim_{n\to+\infty}I_\lambda(u_n,u_n)=-\infty,
$$
and hence the claim follows.

The alternative of Theorem \ref{theo11} case $(b)$ assures the
existence of unbounded sequence $(u_n)$ of critical points of the
functional $I_\lambda$ and the proof of Theorem \ref{theo21} is
complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{theo22}] First, note that
\begin{equation}\label{E310}
\min_X\Phi=\Phi(0,0)=0.
\end{equation}
Let $(\sigma_n)$ be a sequence of positive numbers such that
$\sigma_n\to0^+$ and
\begin{equation}\label{E311}
\lim_{n\to+\infty}\frac{\int_\Omega\sup_{|s|+|t|\leq\sigma_n}F(x,s,t)dx}
{\sigma_n^{\min(p,q)}} =\alpha^0<+\infty.
\end{equation}
Put
$$
r_n=\Big(\frac{\sigma_n}{(Kp)^{1/p}+(Kq)^{1/q}}\Big)^{\min(p,q)},\quad
\delta:=\liminf_{r\to0^+}\varphi(r).
$$
It follows from \eqref{E31} and \eqref{E311} that
\begin{equation}\label{E312}
\begin{split}
\delta&\leq \liminf_{n\to+\infty}\varphi(r_n) \\
&\leq \left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}\lim_{n\to+\infty}\frac{\int_\Omega
\sup_{|s|+|t|<\sigma_n}F(x,s,t)}{\sigma_n^{\min(p,q)}} \\
&= \alpha^0\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}<+\infty.
\end{split}
\end{equation}
By \eqref{E312}, we see that
$\Lambda\subseteq ]0,\frac1\delta[$.

Now, for $\lambda\in\Lambda$, we claim that $I_\lambda$ has not a
local minimum at zero. Indeed, since
$\frac1\lambda<\big((Kp)^{1/p}+(Kq)^{1/q}\big)^{\min(p,q)}L\beta^0$,
we can consider a sequence $(\tau_n)$ of positive numbers and
$\eta>0$ such that $\tau_n\to0^+$ and
\begin{equation}\label{E313}
\frac1\lambda<\eta<L\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}
\frac{\int_{B(x^0,R_1)}F(x,\tau_n,\tau_n)dx}
{\frac{\tau_n^p}p+\frac{\tau_n^q}q},
\end{equation}
for $n$ large enough. Let $(u_n)$ be the sequence defined in
\eqref{E36}. By combining \eqref{E37}, \eqref{E38} and \eqref{E313},
and taking into account $(i_3)$, we obtain
\begin{equation}\label{E314}
\begin{split}
&I_\lambda(u_n,u_n)\\
&= \Phi(u_n,u_n)-\lambda\Psi(u_n,u_n) \\
&\leq \frac1{\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}}
\Big(\frac{\tau_n^p}{pL_p}+\frac{\tau_n^q}{qL_q}\Big)
-\lambda\int_{B(x^0,R_1)}
F(x,\tau_n,\tau_n)dx \\
&\leq \frac1{L\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}}
\Big(\frac{\tau_n^p}p+\frac{\tau_n^q}q\Big) -\lambda\int_{B(x^0,R_1)}
F(x,\tau_n,\tau_n)dx \\
&< \frac{1-\lambda\eta}{L\left((Kp)^{1/p}+(Kq)^{1/q}\right)^{\min(p,q)}}
\Big(\frac{\tau_n^p}p +\frac{\tau_n^q}q\Big) \\
&< 0=I_\lambda(0,0)
\end{split}
\end{equation}
for $n$ large enough. This together with the fact that
$\|(u_n,u_n)\|\to0$ shows that $I_\lambda$ has not a local
minimum at zero, and the claim follows.

The alternative of Theorem \ref{theo11} case $(c)$ ensures the
existence of sequence $(u_n)$ of pairwise distinct critical points
(local minima) of $I_\lambda$ which weakly converges to $0$. This
completes the proof of Theorem \ref{theo22}.
\end{proof}

\noindent\textbf{Example.}
 It could be possible to consider the same
example given in \cite{SA} for the $p$-Laplacian system. Let
$\Omega\subset\mathbb{R}^2, p=3, q=4$ and
$F:\mathbb{R}^2\to\mathbb{R}$ be a function defined by
\begin{equation}\label{E25}
F(s,t)=\begin{cases}
(a_{n+1})^5e^{-\frac{1}{1-[(s-a_{n+1})^2+(t-a_{n+1})^2]}}
&(s,t)\in\cup_{n\geq1}B((a_{n+1},a_{n+1}),1) \\
0 &\text{otherwise},
\end{cases}
\end{equation}
for all $x\in\Omega$, where
$$
a_1:=2,\quad a_{n+1}:=n!(a_n)^{5/4}+2
$$
and $B((a_{n+1},a_{n+1}),1)$ is an open unit ball of center
$(a_{n+1},a_{n+1})$.

We see that $F$ is non-negative and $F\in C^1(\mathbb{R}^2)$. For
every $n\in\mathbb{N}$, the restriction of $F$ on
$B((a_{n+1},a_{n+1}),1)$ attains its maximum in $(a_{n+1},a_{n+1})$
and
$$
F(a_{n+1},a_{n+1})=(a_{n+1})^5e^{-1},
$$
then
$$
\limsup_{n\to+\infty}\frac{F(a_{n+1},a_{n+1})}
 {\frac{a_{n+1}^3}3+\frac{a_{n+1}^4}4}=+\infty.
$$
So,
\begin{align*}
\beta:
&= \limsup_{s,t\to+\infty}\frac{\int_{B(x^0,R_1)}F(s,t)dx}{\frac{s^3}3+\frac{t^4}4}\\
&= |B(x^0,R_1)|\limsup_{s,t\to+\infty}\frac{F(s,t)}{\frac{s^3}3+\frac{t^4}4}
= +\infty.
\end{align*}
On the other hand, for every $n\in\mathbb{N}$, we have
$$
\sup_{|s|+|t|\leq a_{n+1}-1}F(s,t)=a_n^5e^{-1}\quad \text{for all }n\in\mathbb{N}.
$$
Then
$$
\lim_{n\to+\infty}\frac{\sup_{|s|+|t|\leq
a_{n+1}-1}F(s,t)}{(a_{n+1}-1)^3}=0,
$$
and hence
$$
\lim_{\sigma\to+\infty}\frac{\sup_{|s|+|t|\leq
\sigma}F(s,t)}{\sigma^3}=0.
$$
Finally
\begin{align*}
\alpha:&= \liminf_{\sigma\to+\infty}\frac{\int_\Omega
\sup_{|s|+|t|\leq \sigma}F(s,t)dx}{\sigma^3}\\
&= |\Omega|\liminf_{\sigma\to+\infty}\frac{\sup_{|s|+|t|\leq
\sigma}F(s,t)}{\sigma^3}\\
&= 0 <L\beta
\end{align*}
So, applying Theorem \ref{theo21}, we have that for every
$\lambda\in]0,+\infty[$ the  system
 \begin{equation} \label{E319}
\begin{gathered} 
\Delta(|\Delta u|\Delta u)=\lambda F_u(u,v)  \quad\text{in }\Omega, \\
\Delta(|\Delta v|^2\Delta v)=\lambda F_v(u,v) \quad\text{in }\Omega, \\
 u=v=\Delta u=\Delta v=0\quad\text{on }\partial\Omega, 
 \end{gathered}
\end{equation}
admits an unbounded sequence of weak solutions.

\begin{thebibliography}{00}

\bibitem{AH}{G. A. Afrouzi, S. Heidarkhani};
{Existence of three solutions for a class of Dirichlet
quasilinear elliptic systems involving the
$(p_1,\dots,p_n)$-Laplacian},
\emph{Nonlinear Anal.} 70, 135-143 (2009).

\bibitem{BM}{G. Bonanno, G. Molica Bisci};
 {Infinitely many solutions for a boundary value problem with
discontinuous nonlinearities},
\emph{Bound. Value Probl.}, 2009 (2009), 1-20.

\bibitem{BMB}{G. Bonanno, G. Molica Bisci};
{Infinitely many solutions for a Dirichlet problem involving
the p-Laplacian},
\emph{Proc. Royal Soc. Edinburgh}, 140A, 737-752 (2010).

\bibitem{BMO}{G. Bonanno, G. Molica Bisci, D. O'Regan};
{Infinitely many weak solutions for a class of
quasilinear elliptic systems},
 \emph{Math. Comput. Model.} 52,152-160 (2010).

\bibitem{CL}{P. Candito, R. Livrea};
{Infintely many solution for a nonlinear
Navier boundary value problem involving $p$-biharmonic},
\emph{Studia Univ."Babe\c{s}-Bolyai", Mathematica}, 4 (2010).

\bibitem{CLL} P. Candito, L. Li, R. Livrea;
{Infinitely many solutions for a perturbed
nonlinear Navier boundary value problem involving the p-biharmonic},
\emph{Nonlinear Anal.}, 75 (17) (2012) 6360-6369.

\bibitem{DS} G. D'Agui, A. Sciammetta;
{Infinitely many solutions to elliptic problems
with variable exponent and nonhomogeneous Neumann conditions},
\emph{Nonlinear Anal.}, 75, (14) (2012) 5612-5618.

\bibitem{HGG}{H. M. Guo  D. Geng};
{Infinitely many solutions for the Dirichlet problem involving the
p-biharmonic-like equation}, \emph{J. South China Normal Univ.
Natur. Sci. Ed.}, 28 (2009), 18-21.

\bibitem{LT}{C. Li, C. L. Tang};
{Three solutions for a Navier boundary
value problem involving the p-biharmonic},
\emph{Nonlinear Anal.}, 72 (2010), 1339-1347.

\bibitem{LT1}{C. Li, C. L. Tang};
{Existence of three solutions for (p,q)-biharmonic systems},
\emph{Nonlinear Anal.}, 73 (2010), 796-805.

\bibitem{R}{B. Ricceri},
{A general variational principle and some of its applications},
\emph{J. Comput. Appl.Math.}, 133 (2000), 401-410.

\bibitem{SZ}{Y. Shen, J. Zhang},
{Existence of two solutions for a Navier boundary
value Problem involving the p-biharmonic}, \emph{Differential
Equations and Applications}, 3 (2011), 399-414.

\bibitem{SZ1}{Y. Shen, J. Zhang};
{Multiplicity of positive solutions for a
Navier boundary-value problem involving the p-biharmonic with
critical exponent},
\emph{Electronic J. of Differential Equations}, 2011, no. 47 (2011) 1-14.

\bibitem{SA} {D. M. Shoorabi, G. A. Afrouzi};
{Infinitely many solutions for class of Neumann quasilinear
elliptic systems}, \emph{Bound. Value Probl.} 2012, 2012:54
doi:10.1186/1687-2770-2012-54

\bibitem{WS} {Y. Wang, Y. Shen};
{Infinitely many sign-changing solutions for a class of biarmonic
equations without symmetry}, \emph{Nonlinear Anal.}, 71 (2009),
967-977.
\bibitem{WZ}{W. Wang, P. Zhao};
{Nonuniformly nonlinear elliptic equations of p-biharmonic
type}, \emph{J. Math. Anal. Appl.}, 348 (2008) 730-738.

\bibitem{YGY}{ Z. Yang, D. Geng,  H. Yan};
{Existence of multiple solutions for a semilinear biharmonic
equations with critical exponent},
\emph{ActaMath. Sci. Ser. A}, 27 (1)(2006), 129-142.

\end{thebibliography}
\end{document}