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\markboth{\hfil A note on a Liouville-type result \hfil EJDE--2002/99}
{EJDE--2002/99\hfil Ana Rute Domingos \& Yuxia Guo \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 99, pp. 1--20. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  A note on a Liouville-type result for a system of fourth-order
  equations in $\mathbb{R}^N$
 %
\thanks{ {\em Mathematics Subject Classifications:} 35J60.
\hfil\break\indent
{\em Key words:} Elliptic system of fourth order equations, moving-planes.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted November 27, 2001. Published November 27, 2002.} }
\date{}
%
\author{Ana Rute Domingos \& Yuxia Guo}
\maketitle

\begin{abstract}
 We consider the fourth order system
 $\Delta^2 u =v^\alpha,\Delta^2 v =u^\beta$
 in  $\mathbb{R}^N$, for $N\geq 5$, with $\alpha\geq 1$,
 $\beta\geq 1$, where $\Delta^2$ is the bilaplacian operator.
 For $1/(\alpha +1) +1/(\beta +1)>(N-4)/N$ we prove the
 non-existence of non-negative, radial, smooth solutions.
 For  $\alpha,\beta\leq (N+4)/(N-4)$ we show the non-existence
 of non-negative smooth solutions.
\end{abstract}

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{lem}[theorem]{Lemma}

\section{Introduction}

In this  work we consider  the fourth order nonlinear system
\begin{equation} \label{sist1}
\begin{gathered}
 \Delta^2 u  =v^\alpha\\
 \Delta^2 v  =u^\beta
\end{gathered}
\end{equation}
in the whole space $\mathbb{R}^N$.  We are interested in Liouville type results,
i.e.  we  want to  determine for which positive real values of the exponents
$\alpha$ and $\beta$ is $(u,v)=(0,0)$  the only non-negative solution $(u,v)$
of the system. In here, the solution is taken in the classical sense,
i.e.  $u,v\in C^4(\mathbb{R}^N)$.


This type of problems were studied for the Laplacian operator.
Mitidieri \cite{mi} proved that if $\alpha,\beta>1$ and
\begin{equation}
\label{conj}
\frac{1}{\alpha+1}+\frac{1}{\beta+1}>\frac{N-2}{N} ,
\end{equation}
then the system
\begin{equation} \label{sistlap}
\begin{gathered}
 \Delta u  +v^\alpha=0\\
\Delta v  +u^\beta=0
\end{gathered}
\end{equation}
has no  non-negative, radial, $C^2$ solutions in $\mathbb{R}^N$.
Souto in \cite{s} showed that if
$$
\frac{1}{\alpha+1}+\frac{1}{\beta+1}>\frac{N-2}{N-1},\quad \alpha,\beta>0,
$$
then (\ref{sistlap}) has no positive solutions.
 A further result  was given in a paper of Figueiredo and Felmer \cite{fife}.
The authors proved that if
$$
0<\alpha, \beta\leq \frac{N+2}{N-2}, \mbox{ but not both equal to }\frac{N+2}{N-2}
$$
then system (\ref{sistlap}) has no positive  $C^2$ solutions.

We point out that it is still an open problem to know whether (\ref{sistlap})
has no non-negative solutions under assumption
$$
\frac{N-2}{N-1}\geq\frac{1}{\alpha+1}+\frac{1}{\beta+1}>\frac{N-2}{N}
\quad\mbox{and}\quad (\alpha>\frac{N+2}{N-2}\mbox{ or } \beta>\frac{N+2}{N-2}) .
$$
Similar questions remain unsolved in the case where the Laplacian operator
in (\ref{sistlap}) is replaced by other self--adjoint operators. For example,
concerning the single  equation for the bilaplacian operator,
Mitidieri \cite{mi} also proved that the problem
$$
\Delta^2 u  =u^\alpha, \quad \Delta u\leq 0, \quad\mbox{in } \mathbb{R}^N
$$
has no radial $C^4$ positive solution, if $1<\alpha<(N+4)/(N-4)$.
 Using the method of the moving planes, for the same range of $\alpha$,
Lin \cite{lin} showed that
\begin{equation}
\label{ebilap}
\Delta^2 u  =u^\alpha \quad\mbox{in } \mathbb{R}^N,
\end{equation}
has no  $C^4$ positive solutions. Later on \cite{xu}, Xu proved the same
result using instead the method of the moving spheres.

We recall  that if $N\geq 5$ and $\alpha=(N+4)/(N-4)$ then problem
(\ref{ebilap}) has a whole family of positive solutions explicitely given by
$$
u(x)=\frac{c_1}{(1+c_2|x|^2)^\frac{N-4}{2}},
$$
where $c_1$ and $c_2$ are some appropriate positive constants.

A natural question that rises is to analyse the behaviour of system
(\ref{sist1}).  Here we show that the quoted results in \cite{fife,mi}
for system (\ref{sistlap}) can be extended to system (\ref{sist1}).
Precisely, we prove the following.

\begin{theorem} \label{thm1}
If $N\geq 5$, $\alpha,\beta \geq 1$, but not both equal to $1$  are
such that
\begin{equation}
\label{hipalbe}
\frac{1}{\alpha+1}+\frac{1}{\beta+1}>\frac{N-4}{N},
\end{equation}
then system {\em (\ref{sist1})} has no radial non-negative solutions in
$C^4(\mathbb{R}^N)$.
\end{theorem}

\begin{theorem} \label{thm2}
\begin{itemize}
\item[{\bf I)}]
 If
$  1\leq \alpha ,\beta \leq \frac{N+4}{N-4}$ but not both equal to  $1$
neither to $\frac{N+4}{N-4}$, then the only non-negative $C^4$ solution of
system {\em (\ref{sist1})} in the whole of $\mathbb{R}^N$ is the trivial one:
$(u,v)=(0,0)$.

\item[{\bf II)}] If $\alpha=\beta=\frac{N+4}{N-4}$, then $u$ and $v$ are
radially symmetric with respect to some point of $\mathbb{R}^N$.
\end{itemize}
\end{theorem}

Our proofs are strongly motivated by the works of Figueiredo and Felmer
\cite{fife}, Lin \cite{lin}, Xu \cite{xu} and Mitidieri \cite{mi}.
Indeed, the proof of  Theorem 1  presented in section 2 uses an idea of
Mitidieri, which  relies on the application of a Rellich type identity.
Section 3 is devoted to the proof of the fact that if $(u,v)$ is a
non-negative, $C^4$  solution of (\ref{sist1}) then $u$ and $v$ are
super-harmonic; here we extend  the result  in \cite{lin} for the single
equation. This result plays a key role in our work.
In section 4  we  apply the method of the moving planes in order to
prove Theorem 2. In our case, the main difficulty in applying the method
stems from the fact that the maximum principle cannot be applied directly
to $(u,v)$; to overcome this, we follow an idea of Xu \cite{xu} for
the single equation and  apply the moving planes method for both
$(-\Delta u,-\Delta v)$ and $(u,v)$. Contrarily to \cite{fife} we use no
additional change of variables except for Kelvin transforms.

 As far as we know this is the first work concerning Liouville type results
for a system involving the bilaplacian operator.  However, some related
questions remain unsolved. For example, it is not clear for us whether
Souto's result for system (\ref{sistlap}) can be extended to system
(\ref{sist1}).

\section{General auxiliary facts}

In this section  we state some general results that will be useful in
the sequel.

\begin{lem} \label{lemfife}
Let $u\in C^2(\mathbb{R}^N\setminus\{0\})$ be such that $u<0 $ and
$\Delta u\geq 0$. Then, for each $\varepsilon>0$, one has
$$
u(x)\leq M(\varepsilon):= \max\{u(y): |y|=\varepsilon \},
\quad 0<|x|\leq\varepsilon.
$$
\end{lem}

 The proof of this lemma is included in the proof of Lemma 1.1 of \cite{fife}
and is a consequence of the so called Hadamard Three Spheres
Theorem (\cite{pw}).

\begin{lem} \label{ode}
Suppose $y=y(r)\geq 0$ satisfies
$$
y''+\frac{N-1}{r}y'+\phi (r)\leq 0,\quad r>0,
$$
with $\phi $ non-negative and non-increasing and $y'$ bounded for $r$ near $0$. Then, for $r>0$,
\begin{gather}
\label{dyneg}  y'(r)\leq 0, \\
\label{miti}  ry'(r)+(N-2)y(r)\geq 0 ,\\
\label{ode2}  y(r)\geq cr^2\phi(r),
\end{gather}
where $c=c(N)$.
\end{lem}
\paragraph{Proof.}
The proofs of (\ref{miti}) and (\ref{ode2}) can be found in 
\cite[Lemma 3.1]{mi} and in \cite[Lemma 2.7]{sz}
(see also \cite[Lemma 3.1]{xu}) respectively.
It remains to prove (\ref{dyneg}).
 Multiplying the inequality in the Lemma by $r^{N-1}$  we get
$$
(r^{N-1}y'(r))'\leq- r^{N-1}\phi (r).
$$
Integrating from 0 to $r$ we obtain
$$
r^{N-1}y'(r)\leq-\int_o^r s^{N-1}\phi (s)\,ds\leq 0.
$$
\hfill$\square$

\section{Non existence of radial solutions. The  super-harmonic
property of the solutions}

In this section we prove that system (\ref{sist1}) does not admit
non-negative radial solutions. For that matter we use the following result.

\begin{theorem}
\label{txu}
If $(u,v)$ is a $C^4(\mathbb{R}^N)$, non-negative solution of
system {\rm (\ref{sist1})}, with $\alpha,\beta \geq 1$, but not both equal
to $1$, we have:
$$
\Delta u\leq 0 \quad\mbox{ and }\quad\Delta v\leq 0.
$$
\end{theorem}

This  theorem  is  the most powerful tool of the present work. We postpone its
proof to section 3.

The proof of Theorem \ref{thm1} stated in the Introduction makes use of the
following Rellich type identity in a smooth bounded domain $\Omega$,
obtained by Mitidieri in \cite{mi}:
\begin{equation}
\label{reid}
R_2(u,v)= R_1(\Delta u,v)+ R_1(u,\Delta v) -\int_{\partial \Omega}
\Delta u \Delta v (x,n)d\sigma+N\int_\Omega\Delta u \Delta v\,dx
\end{equation}
where
\begin{align*}
R_2(u,v)=&\int_\Omega(\Delta^2 u (x,\nabla v) +\Delta^2 v (x,\nabla u))\,dx,\\
R_1( u,v)=&\int_{\partial \Omega} (\frac{\partial u}{\partial n}(x,\nabla v)
+ \frac{\partial v}{\partial n}(x,\nabla u)-
  (\nabla u,\nabla v)(x,n))d\sigma \\
  &+(N-2)\int_\Omega (\nabla u, \nabla v)dx,
\end{align*}
 $n$ is the outward unit normal to $\partial \Omega$ and $(\cdot,\cdot)$
is the inner product in $\mathbb{R}^N$.

In what follows, for a rotationally symmetric function,
$$
 \Delta =\frac{d^2}{dr^2}+\frac{N-1}{r}\frac{d}{dr}
$$
and $'$ will denote differentiation with respect to $r$.

\begin{lem} \label{rellich}
Let $(u,v)$ be a non-negative, radial, $C^4$ solution of  system
{\rm (\ref{sist1})}. Then for all $r>0$ we have
\begin{align*}
( &\frac{N-4}{2}-\frac{N}{\alpha+1})\int_0^{r} v^{\alpha +1}(s) s^{N-1}\,ds
+(\frac{N-4}{2}-\frac{N}{\beta+1})\int_0^{r} u^{\beta +1}(s) s^{N-1}\,ds \\
=& -\frac{r^N}{\alpha+1} v^{\alpha +1}(r)-\frac{r^N}{\beta+1} u^{\beta +1}(r)
+ r^N(\Delta u)' (r)v'(r) + r^N(\Delta v)' (r)u'(r) \\
&-\frac{N-4}{2} r^{N-1}(\Delta u)' (r)v(r)+\frac{N-4}{2}r^{N-1}(\Delta v)'
(r)u(r)\\
&-r^N (\Delta u)(r)(\Delta v)(r) +\frac{N}{2}r^{N-1}(\Delta v) (r)u'(r)
+\frac{N}{2}r^{N-1}(\Delta u) (r)v'(r).
\end{align*}
\end{lem}
\paragraph{Proof.}
We obtain the desired identity by taking into account that $u$ and $v$ are
radial and after an integration by parts in (\ref{reid}), where we
take $\Omega =B_r(0)$.\hfill$\square$

\begin{lem} \label{estimatives}
Let $(u,v)$ be a non-negative, radial, $C^4$ solution of system {\rm (\ref{sist1})}. Then
there exists a positive constant $C$ such that for all $r>0$,
\begin{gather} \label{euv}
u(r)\leq C r^{-\frac{4(\alpha+1)}{\alpha\beta -1}}, \quad
 v(r) \leq C r^{-\frac{4(\beta+1)}{\alpha\beta -1}} ,\\
\label{elulv}
|\Delta u(r)|\leq C r^{-2-\frac{4(\alpha+1)}{\alpha\beta -1}},\quad
 |\Delta v(r)|\leq C r^{-2-\frac{4(\beta+1)}{\alpha\beta -1}}
\\
| r^{N-1}(\Delta v)'(r)u(r)|,\quad
| r^{N-1} (\Delta v)(r)u'(r)|,\quad
| r^{N-1}(\Delta u)'(r)v(r)|,\nonumber \\
|r^{N-1} (\Delta u)(r)v'(r)|,\quad
\label{eduv}
| r^N (\Delta v)'(r)u'(r)|,\quad |r^N (\Delta u)'(r)v'(r)|\\
\leq  C r^{N-4-\frac{4(\alpha+\beta+2)}{\alpha\beta -1}}. \nonumber
\end{gather}
 \end{lem}

\paragraph{Proof.} Since $u$ and $v$ are radial we can write
\begin{gather*}
 u''(r)+\frac{N-1}{r}u'(r)+w(r)=0,\quad
 w''(r)+\frac{N-1}{r}w'(r)+v^\alpha(r)=0\\
 v''(r)+\frac{N-1}{r}v'(r)+z(r)=0,\quad
  z''(r)+\frac{N-1}{r}z'(r)+u^\beta(r)=0 \\
u'(0)=w'(0)=v'(0)=z'(0)=0,
\end{gather*}
where $w:=-\Delta u$ and $z:=-\Delta v$. From now on we denote by $c$ some
positive constant possibly different from place to place. By
Theorem \ref{txu} $w$ and $z$ are non--negative functions.
Then by  (\ref{ode2}),
$$
u(r)\geq  c r^2 w(r),\quad w(r)\geq c r^2 v^\alpha(r),\quad
 v(r)\geq c r^2 z(r),\quad z(r)\geq c r^2 u^\beta(r).
$$
 So
$$
u^\beta(r)\leq c r^{-2}z(r)\leq c r^{-4}v(r)
\leq c r^{-4-\frac{2}{\alpha}}w^\frac{1}{\alpha}(r)
\leq c r^{-4-\frac{4}{\alpha}}u^\frac{1}{\alpha}(r),
$$
which implies that $u (r)\leq  c  r^{-4\frac{\alpha +1}{\alpha\beta -1}}$.
Then  $w(r)\leq c r^{-2-\frac{4(\alpha+1)}{\alpha\beta -1}}$,
$v(r)\leq c r^{-4\frac{\beta +1}{\alpha\beta -1}}$
and finally we obtain  $z(r)\leq cr^{-2-\frac{4(\beta+1)}{\alpha\beta -1}}$.

From (\ref{dyneg}), we know that $z'\leq 0$. Also, $rz'(r) +(N-2)z(r)\geq 0$
(cf. (\ref{miti})). Multiplying this by $r^{N-2} u$ and using the estimates
that we obtained before, yields
$$
|r^{N-1} z'(r)u(r)|= -r^{N-1} z'(r)u(r)\leq (N-2)r^{N-2} z(r)u(r)\leq
c r^{N-4-\frac{4(\alpha+\beta +2)}{\alpha\beta -1}}.
$$
Now, multiplying by $r^{N-1} u'$ the stated inequality and using the previous
estimate we get
$$
r^{N} z' (r)u'(r)\leq c r^{N-4-\frac{4(\alpha+\beta +2)}{\alpha\beta -1}}.
$$
Again from (\ref{dyneg}), $u'\leq 0$, so that
$$
| r^{N} z' (r)u'(r)|\leq c r^{N-4-\frac{4(\alpha+\beta +2)}{\alpha\beta -1}}.
$$
Using similar arguments we obtain  the remaining estimates in the statement
of the lemma.\hfill$\square$

\paragraph{Proof of Theorem \ref{thm1}.}
We multiply the first equation of system (\ref{sist1}) by $v$ and integrate
in $B_r(0)$, for $r>0$. We obtain
\begin{eqnarray*}
\int_{B_r(0)} v^{\alpha+1}\,dx & = &   \int_{B_r(0)} \Delta^2 u\, v\,dx \\
 &=&  -\int_{B_r(0)} (\nabla (\Delta u), \nabla v)\,dx+\int_{\partial B_r(0)}
\frac{\partial (\Delta u)}{\partial n} v \,d\sigma\\
& = & \int_{B_r(0)} \Delta u \Delta v\,dx-\int_{\partial B_r(0)} \frac{\partial v}{\partial n} \Delta u \,d\sigma +\int_{\partial B_r(0)} \frac{\partial(\Delta u)}{\partial n} v \,d\sigma.
 \end{eqnarray*}
Then
\begin{multline*}
\int_0^r
 v^{\alpha +1}(s)s^{N-1}\,ds= \\
\int_0^r (\Delta u)(s)(\Delta v)(s)s^{N-1}\,ds - \Delta u(r)v'(r)r^{N-1}
+(\Delta u)'(r)v(r)r^{N-1}.
\end{multline*}
Similarly,
\begin{multline*}
 \int_0^r  u^{\beta +1}(s)s^{N-1}\,ds \\
=\int_0^r (\Delta v)(s)(\Delta u)(s)s^{N-1}\,ds - (\Delta v)(r)u'(r)r^{N-1}+
(\Delta v)'(r)u(r)r^{N-1} .
\end{multline*}
So
\begin{eqnarray*} \label{ruv}
\lefteqn{\int_0^r  v^{\alpha +1}(s)s^{N-1}\,ds }\\
& = &\int_0^r u^{\beta +1}(s)s^{N-1}\,ds + (\Delta v)(r)u'(r)r^{N-1}-
(\Delta v)'(r)u(r)r^{N-1}\\
& & - \Delta u(r)v'(r)r^{N-1}+(\Delta u)'(r)v(r)r^{N-1}.
\end{eqnarray*}
Now, we observe that the assumption (\ref{hipalbe}) can be written as
$$
\frac{(\alpha+1)(\beta+1)}{\alpha\beta -1}>\frac{N}{4}.
$$
Thus from the estimates (\ref{euv})-(\ref{eduv}) we see that the boundary
terms in (\ref{ruv}) and also in the identity of Lemma \ref{rellich} all
have zero limits as $r\to +\infty$. Using (\ref{ruv}) in
Lemma \ref{rellich} and those facts we conclude that
$$
( N-4-\frac{N}{\alpha+1}-\frac{N}{\beta+1})
\int_0^{r} u^{\beta +1}(s) s^{N-1}\,ds =o(1)
\quad  (r\to +\infty).
$$
Passing to the limit we obtain $u=0$ and, as a consequence, also  $v=0$.\hfill$\square$



\section{Proof of Theorem \ref{txu}}

Let $(u,v)$ be  a non-negative solution of system (\ref{sist1}).
We define $w$ and $z$ as follows:
\begin{equation}
\label{defwz}
w(x):=-\Delta u(x)\quad\mbox{and}\quad z(x):=-\Delta v(x),
\end{equation}
for $x \in \mathbb{R}^N$.
We can write system (\ref{sist1}) as a system of four second order equations
\begin{equation}\label{sist4}
\begin{gathered}
\Delta u+w  = 0\\
\Delta w + v^\alpha  = 0\\
\Delta v + z  =0 \\
\Delta z +u^\beta  =0.
\end{gathered}
\end{equation}
Let $\bar u$ be the spherical average of $u$,  i.e.
$$
\bar u(r) :=\frac{1}{\omega_N Nr^{N-1}}\int_{\partial B_r(0)} u\, d\sigma,
$$
where $\omega_N$ denotes the measure of the unit sphere in $\mathbb{R}^N$.
Similarly we define $\bar v, \bar w, \bar z $ the spherical averages of $v,\,
 w,\, z$ respectively.
 From (\ref{sist4}) and the H\"older inequality, it is easy to see that
\begin{gather}
\label{w} \Delta \bar u+\bar w  =0 \\
\label{v}\Delta\bar w + \bar v^\alpha\leq 0\\
 \label{z}\Delta\bar v +\bar z=0\\
\label{u}\Delta \bar z+ \bar u^\beta\leq 0,
\end{gather}
where $\Delta f = f''+\frac{N-1}{r}f'$, $f'=\frac{d f}{d r}$,
for $f=\bar u,\,\bar w,\, \bar v,\,  \bar z$.
\vskip11pt
In order to prove Theorem \ref{txu} we begin with some auxiliary lemmas.

\begin{lem} \label{lk}
Let $N\geq 2$. Consider $p,q >0$ such that $pq\neq 1$ and let $(l_k)$
be the sequence defined by
$l_0=2$ and
$$ l_{k+1}= p(q l_k+4)+4, \quad k\geq 0.
$$
Then
\begin{enumerate}
\item[i)] $l_k=2(pq)^k+4(p+1)((pq)^k-1)/(pq-1)$,
\item[ii)] $(pq l_k+4p+N)(pq l_k+4p+2)(pq l_k+4 p+N+2)(pq l_k+4p+4)
\leq (N+2+2pq+4p)^{4(k+1)}$, for $k=0,1,2,\ldots$
\item[iii)] If $pq>1$ then $l_k\to +\infty$.
\end{enumerate}
\end{lem}

\begin{lem} \label{bk}
Let $p,q >0$ be such that $pq\neq 1$.
Suppose $b_0=0$ and define the sequence $(b_k)$ inductively by
$$
b_{k+1}= pq b_k +4p(k+1) +4(k+1),\quad \mbox{for } k\geq 0.
$$
Then
$$
b_k=4(p+1)\Big[ \frac{(pq)^{k+1}-(k+1)pq+k}{(pq-1)^2}\Big]
$$
for all non-negative integers $k$.
\end{lem}

\begin{lem} \label{nk}
Let  $p,q >0$ be such that $pq\neq 1$.
Suppose $n_0=0$ and define the sequence $(n_k)$ inductively by
$n_{k+1}= pq n_k +p$.
Then
$$
n_k=p\frac{(pq)^{k}-1}{pq-1}
$$
for all non-negative integers $k$.
\end{lem}
The three lemmas above can be proved by induction.

\begin{lem} \label{rk}
Let $N\geq 2$. Consider  $p,q >0$  such that $pq>1$, let $(b_k)$,  $(l_k)$
and $(n_k)$ be the sequences  defined in the previous lemmas,
$c_0$ a positive constant and $z_0$ be  a non-negative  constant.
Define the sequence as follows:
$r_0= 0$ and
$$r_k =\Big( \frac{z_0 2^{q n_{k-1}+1}(N+2+2pq+4p )^{q b_{k-1}}(N +q l_{k-1})
(2+q l_{k-1})}{c_0^{p^{k-1}q^{k}}}\Big)^\frac{1}{q l_{k-1}+2}
$$
for $k\geq 1$.
 Then there exists a positive number $a$ such that
$\lim_{k\to\infty} r_k= a$.
\end{lem}

\paragraph{Proof.} We can write $
r_k =z_0^\frac{1}{q l_{k-1}+2} \cdot ^1\! r_k\cdot ^2\!r_k\cdot^3\!r_k\cdot ^4\!r_k $
where
$$
^1\! r_k:=2^\frac{q n_{k-1}+1}{q l_{k-1}+2},\,\,
^2\! r_k:= (N+2+2pq+4p )^\frac{q b_{k-1}}{q l_{k-1}+2},
$$
$$
^3\! r_k:=[(N +q l_{k-1})(2+q l_{k-1}) ]^\frac{1}{q l_{k-1}+2},\,
^4\! r_k:=c_0^{-\frac{p^{k-1}q^{k}}{q l_{k-1}+2}}.
$$
We have
$$
\frac{q n_{k-1}+1}{q l_{k-1}+2}\to \frac{p}{2pq+4p+2},
$$
 and
$$
\frac{q b_{k-1}}{q l_{k-1}+2}\to \frac{2pq(p+1)}{(pq-1)^2+2(p+1)(pq-1)}.
$$
From this we deduce that both $^1\! r_k $ and $^2\! r_k$ converge.
Concerning the third sequence,
$$
\lim \mbox{ }^3\! r_k  =  \lim [(N +q l_{k-1})(2+q l_{k-1}) ]^\frac{1}{q l_{k-1}+2}=1.
$$
Finally, $
\frac{p^{k-1}q^{k}}{q l_{k-1}+2}\to \frac{(pq-1)}{2(pq-1)+4(p+1)}
$ yields
$$
\lim\mbox{ }^4\! r_k:=c_0^{-\frac{(pq-1)}{2(pq-1)+4(p+1)}}.
$$
Since $l_k\to +\infty$, $z_0^\frac{1}{q l_{k-1}+2}\to 1$ and we are done.
\hfill$\square$

\paragraph{Proof of Theorem \ref{txu}}
According to (\ref{defwz}), we want to prove that
$w \geq 0$ and $z\geq 0$. Suppose by contradiction that there exists
$x_0$ such that $w(x_0)<0$. Without loss of generality suppose that $x_0=0$.
Multiplying (\ref{v}) by $r^{N-1}$, we get
$$
r^{N-1}\bar w_{rr} +(N-1)r^{N-2}\bar w_r \leq -r^{N-1}\bar v^\alpha\leq 0,
$$
hence
$$
(r^{N-1}\bar w_r )_r\leq 0.
$$
By integrating the last inequality, we obtain $\bar  w_r \leq 0.$
Then $\bar w$ is non-increasing and we have
$$
\bar w(r)\leq \bar w(0)<0,\,\,\,\mbox{for all}\,\,\, r>0,
$$
since, from the definition, $\bar w(0)=w(0)$. So, from (\ref{w}),
$$
\Delta \bar u =-\bar w(r)\geq -\bar w(0).
$$
If we multiply the last inequality by $r^{N-1}$ and integrate twice, we get
\begin{equation}
\label{ineu2}
\bar u(r)\geq c r^2,\,\,\,\mbox{for all }\,\,\,r>0,
\end{equation}
where $c:=-\bar w(0)/(2N)>0$. Without loss of generality, we can assume
that $c<1$.
Next we prove by induction that
\begin{equation}
\label{ineuk}
\bar u(r)\geq \frac {c^{(\alpha\beta)^k}}{2^{n_k}(N+2+2\alpha\beta
+4\alpha)^{b_k}}r^{l_k},\,\,\,\mbox{for all }\,\,\,r> r_k,
\end{equation}
where $l_k,\, b_k,\,n_k$ and $r_k$ are defined in
Lemmas \ref{lk}, \ref{bk}, \ref{nk} and \ref{rk} with $p=\alpha$, $q=\beta$,
$z_0:=\bar z(0)$ if $\bar z(0)> 0$ and $z_0:=0$ otherwise and $c_0:= c$.

  We have (\ref{ineuk})  for $k=0$ by (\ref{ineu2}), since $l_0=2$, $b_0=0$,
$n_0=0$ and $r_0=0$. Assume that (\ref{ineuk}) is true for some $k$.
From (\ref{u}),
$$
\Delta \bar z \leq- \frac {c^{\alpha^k\beta^{k+1}}}{2^{\beta n_k}
(N+2+2\alpha\beta +4\alpha)^{\beta b_k}}r^{\beta l_k}.
$$
If we multiply both sides by $r^{N-1}$ and integrate twice, we obtain
$$
 \bar z (r)\leq -\frac {c^{\alpha^k\beta^{k+1}}}{2^{\beta n_k}(N+2
 +2\alpha\beta +4\alpha)^{\beta b_k}
(\beta l_k+N)(\beta l_k +2)}r^{\beta l_k+2} +\bar z(0).
$$
 Then, for all
$$
r\geq \left( \frac{ z_0 2^{\beta n_{k}+1}(N+2+2\alpha\beta+4\alpha )
^{\beta b_{k}}(\beta l_{k}+N)(\beta l_{k}+2)}{c^{\alpha^{k}\beta^{k+1}}}
\right)^\frac{1}{\beta l_{k}+2}
$$
we have
$$
\bar z(r) \leq -\frac {c^{\alpha^k\beta^{k+1}}}{2^{\beta n_k +1}(N+2
+2\alpha\beta +4\alpha)^{\beta b_k}
(\beta l_k+N)(\beta l_k +2)}r^{\beta l_k+2}.
$$
 From (\ref{z}) we get
$$
\Delta \bar v \geq \frac{c^{\alpha^k\beta^{k+1}}}{2^{\beta n_k +1}(N+2
+2\alpha\beta +4\alpha)^{\beta b_k}
(\beta l_k+N)(\beta l_k +2)}r^{\beta l_k+2}.
$$
After multiplying the last inequality by $r^{N-1}$ and integrating twice
we obtain
$$
\bar v(r)\geq \frac{c^{\alpha^k\beta^{k+1}}}{2^{\beta n_k +1}D_k}
r^{\beta l_k+4}+\bar v(0)\geq\frac{c^{\alpha^k\beta^{k+1}}}
{2^{\beta n_k +1}D_k}r^{\beta l_k+4} ,
$$
where
$$
D_k:=(N+2+2\alpha\beta +4\alpha)^{\beta b_k}(\beta l_k+N)(\beta l_k +2)(\beta l_k +N+2)
(\beta l_k +4).
$$
From (\ref{v}) we have
$$
\Delta \bar w \leq -\frac{c^{(\alpha\beta)^{k+1}}}{2^{\alpha\beta n_k
+\alpha}D_k^\alpha}r^{\alpha\beta l_k+4\alpha}.
$$
Once more, multiplying both sides of the above inequality and integrating
twice, we obtain, since $\bar w (0)<0$,
$$
\bar w(r) \leq -\frac{c^{(\alpha\beta)^{k+1}}}{2^{\alpha\beta n_k
+\alpha}D_k^\alpha (\alpha\beta l_k+4\alpha+N)(\alpha\beta l_k +4\alpha + 2)}r^{\alpha\beta l_k+4\alpha+2}.
$$
 At last, from (\ref{w}) we have
$$
\Delta \bar u \geq \frac{c^{(\alpha\beta)^{k+1}}}{2^{\alpha\beta n_k
+\alpha}D_k^\alpha (\alpha\beta l_k+4\alpha+N)(\alpha\beta l_k +4\alpha + 2)}r^{\alpha\beta l_k+4\alpha+2}.
$$
Using the same procedure that we used before, we get
$$
\bar u(r)\geq \frac{c^{(\alpha\beta)^{k+1}}}{2^{\alpha\beta n_k
+\alpha}D_k^\alpha E_k }r^{\alpha\beta l_k+4\alpha+4}+\bar u(0)
\geq \frac{c^{(\alpha\beta)^{k+1}}}{2^{\alpha\beta n_k
+\alpha}D_k^\alpha E_k }r^{\alpha\beta l_k+4\alpha+4}
$$
where
$$
E_k := (\alpha\beta l_k+4\alpha+N)(\alpha\beta l_k +4\alpha
+ 2)(\alpha\beta l_k+4\alpha+N+2)(\alpha\beta l_k +4\alpha +4).
$$
Taking into account that $\alpha\geq 1$,
\begin{align*}
D_k\leq &(N+2+2\alpha\beta +4\alpha)^{\beta b_k}(\alpha\beta l_k
+4\alpha+N)(\alpha\beta l_k+ 4\alpha+2) \\
&\times (\alpha\beta l_k+4\alpha+N+2)(\alpha\beta l_k+4\alpha +4).
\end{align*}
Thus, from  Lemma \ref{lk} ii),
$$
\bar u(r)\geq \frac{c^{(\alpha\beta)^{k+1}}}{2^{\alpha\beta n_k
+\alpha}(N+2+2\alpha\beta +4\alpha)^ {\alpha\beta b_k+4\alpha(k+1)
+4(k+1)} }r^{\alpha\beta l_k+4\alpha+4}.
$$
According to the definition of $(l_k),\,(b_k)$ and $(n_k)$ we
have (\ref{ineuk}) for $k+1$.

Now, fix  $k_0$ such that  $r_k < 2\lim r_k$, for all $k\geq k_0$.
Take $A>1$ such  that
$$
2A (N+2+2\alpha\beta+4 \alpha)^\frac{4(\alpha+1)}{\alpha\beta-1}>2\lim r_k.
$$
Taking $r=2c^{-1}A(N+2+2\alpha\beta+4 \alpha)^\frac{4(\alpha+1)}{\alpha\beta-1}$
 in (\ref{ineuk}), for all $k\geq k_0$ we get
$$
\bar u(r)\geq c^{(\alpha\beta)^{k}-l_k}2^{l_k-n_k}A^{l_k}(N+2+2\alpha\beta+4
\alpha)^{\frac{4(\alpha+1)}{\alpha\beta-1}l_k-b_k}.
$$
The right hand  member of the inequality goes to infinity, when $k\to +\infty$,
 since
\begin{gather*}
l_k-n_k= \frac{2(\alpha\beta)^{k+1}+(3\alpha+2)(\alpha\beta)^k
-(3\alpha+4)}{\alpha\beta-1}\to +\infty, \\
(\alpha\beta)^{k}-l_k\to -\infty
\end{gather*}
and also since  $\frac{4(\alpha+1)}{\alpha\beta-1}l_k-b_k$ equals
$$
\frac{4(\alpha+1)(\alpha\beta)^{k+1}}{(\alpha\beta-1)^2}
[1+\frac{4\alpha +2}{\alpha\beta} +\frac{k+1}{(\alpha\beta)^{k}}
-\frac{4\alpha +4+k}{(\alpha\beta)^{k+1}}]\to +\infty.
$$
This is a contradiction, since $\bar u(r)$ is a constant. Thus $w\geq 0$,
as claimed.

The case when there exists $y_0$ such that $z(y_0)<0$ can be
treated in a similar way, and this concludes the proof of Theorem
\ref{txu}. \hfill$\square$ \smallskip

For later purposes we prove the following results.

\begin{lem} \label{seqri}
Let $1\leq\alpha,\beta \leq \frac{N+4}{N-4}$ be  such that $\alpha\beta\neq 1$.
\begin{itemize}
\item[1)] There exists a sequence $(R_i)$ such that $R_i^3\bar w' (R_i) \to 0$
as $R_i\to +\infty$.
\item[2)] There exists a sequence $(\widetilde R_i)$ such that
$ \widetilde R_i^3\bar z'(\widetilde R_i) \to 0 $ as
$\widetilde  R_i\to +\infty$.
\end{itemize}
\end{lem}

\paragraph{Proof.} From Theorem \ref{txu}, $w$ and $z$ are non-negative
functions. Similarly to Lemma \ref{estimatives}, (\ref{ode2}) allows to
deduce the existence of a positive constant $c$ such that
\begin{equation}
\label{inewz}
\bar w(r)\leq c r^{-2-4\frac{\alpha+1}{\beta\alpha-1}},\quad
\bar z(r)\leq c r^{-2-4\frac{\beta+1}{\beta\alpha-1}}.
\end{equation}
We prove 1). The proof of 2) is similar thanks to the second inequality in
(\ref{inewz}).

Suppose by contradiction that there exist  $\delta_0>0$ and $r_0>0$ such that
for all $R>r_0$ we have   $-R^3\bar w'(R)\geq \delta_0>0$. Then
\begin{eqnarray*}
\delta_0 (R-r_0) &\leq & -\int_{r_0}^{R} s^3 \bar w'(s)ds\\
           &  =  & -R^3\bar w(R) +r_0^3\bar w(r_0) + 3\int_{r_0}^{R} s^2
       \bar w(s)ds\\
    & <   &3\int_{r_0}^{R} s^2 \bar w(s)ds + C
    \leq 3c\int_{r_0}^{R} s^{-4\frac{\alpha+1}{\beta\alpha-1}}ds+ C.
\end{eqnarray*}
In case $4\frac{\alpha+1}{\beta\alpha-1}=1$ (which is only possible if
$N=5$ or $N=6$), we get
$$
\delta_0 (R-r_0)<3c (\log R -\log r_0) +C.
$$
Dividing both sides of this inequality by $R^{1/2}$ and passing to the limit
as $R\to +\infty$  we get a contradiction.
Assume now $4\frac{\alpha+1}{\beta\alpha-1}\neq 1$. Then
$$
\delta_0 (R-r_0)<
  c\frac{\beta\alpha-1}{\alpha(\beta-4)-5}(R^\frac{\alpha(\beta-4)-5}
  {\beta\alpha-1}-r_0^\frac{\alpha(\beta-4)-5}{\beta\alpha-1}) +C.
$$
Dividing both sides of the previous inequality by $R^\frac{2}{3}$, we get
\begin{equation}
\label{potr}
\delta_0 (R^\frac{1}{3}-r_0R^{-\frac{2}{3}})\leq  C\frac{\beta\alpha-1}
{\alpha(\beta-4)-5}(R^\frac{\alpha(\beta-12)-13}{3\beta\alpha-3}
-r_0^\frac{\alpha(\beta-4)-5}{\beta\alpha-1}R^{-\frac{2}{3}})
+C R^{-\frac{2}{3}}.
\end{equation}
Since $\beta\leq 9$ for $N\geq 5$, again we obtain a contradiction, thus
proving the lemma. \mbox{ }\hfill$\square$

\begin{lem} \label{intb1}
Let $(u,v)$ be a non-negative $C^4(\mathbb{R}^N)$ solution of system
{\em (\ref{sist1})}, with  $1\leq\alpha,\beta\leq \frac{N+4}{N-4}$ and
$\alpha\beta\neq 1$. Then
$$
|x|^{4-N} v^\alpha(x)\quad\mbox{and}\quad
|x|^{4-N} u^\beta(x)\in L^1(\mathbb{R}^N\setminus B_1(0)).
$$
\end{lem}
For the proof of this lemma we proceed as in \cite[Proposition 3.5]{xu},
thanks to Lemma \ref{seqri}.

\section{Proof of Theorem \ref{thm2}}

Let $(u,v)$ be a $C^4(\mathbb{R}^N)$ non-negative, nontrivial solution of
system (\ref{sist1}). By Theorem \ref{txu} and the maximum principle,
we have
\begin{equation} \label{u,v>0}
u(x)>0\quad\mbox{and}\quad v(x)>0\quad \mbox{in }  \mathbb{R}^N.
\end{equation}
So  $(u,v)$ is a  positive solution of system (\ref{sist1}).
We introduce their Kelvin transforms
$$
u^*(x)= |x|^{4-N}u(\frac{x}{|x|^2}),\quad v^*(x)=|x|^{4-N}v(\frac{x}{|x|^2}),
$$
for $x \in \mathbb{R}^N\setminus\{ 0\}$. Then
$$
\Delta u^*(x)=|x|^{-N}\Delta u (\frac{x}{|x|^2})
-\frac{4}{|x|^{N-2}}(\nabla u(\frac{x}{|x|^2}), \frac{x}{|x|^2})
-2(N-4)|x|^{2-N}u (\frac{x}{|x|^2})
$$
and  at infinity,
$$
\Delta u^*(x)=-2(N-4)|x|^{2-N} u_0 -\sum_{j=1}^{N}\frac{a_j}{|x|^{N}}x_j
+O(\frac{1}{|x|^N}),
$$
where $u_0= u(0)$, $a_i=\frac{\partial u}{\partial x_i}(0)$.
Consequently, for large $|x|$, $\Delta u^*(x)<0$. Similarly to
$\Delta v^*$. Without loss of generality, we assume that
\begin{equation}
\label{lulv*pos}
\Delta u^*(x)\leq 0 \quad\mbox { and }\quad \Delta v^*(x)\leq0,
\quad\forall x \in \mathbb{R}^N\setminus B_1(0).
\end{equation}
In order to prove that $\Delta u^*(x)$ and $\Delta v^*(x)$ are non-positive
in $B_1(0) \setminus \{ 0\}$, we begin with an auxiliary lemma.

\begin{lem} \label{liu}
Let $f\in C^2 (\overline{B}_1(0)\setminus\{ 0\})$ be  such that
$f=0$ on $\partial B_1(0)$ and
\begin{equation} \label{idint}
\int_{B_1(0)} f\Delta \varphi\,dx  \geq 0,
\end{equation}
for all $\varphi \in C^2 ({B}_1(0))\cap C^1 (\overline{B}_1(0))$ such
that $\varphi =0$ on $\partial B_1(0)$ and $\varphi \geq 0$ in $B_1(0)$.
Then $f\leq 0$ in $B_1(0) \setminus \{ 0\}$.
\end{lem}

\paragraph{Proof.} Suppose by contradiction that $f>0$ over some ball
$B\subset B_1(0)\setminus \{ 0\}$. Fix any nonzero, non-negative function
$\psi \in {\cal D}(B)$ and denote by $\tilde\psi$ its  extension by zero.
Consider $\varphi \in  C^2 ({B}_1(0))\cap C^1 (\overline{B}_1(0))$,
$\varphi \geq 0$, such that $-\Delta \varphi =\tilde \psi$ in $B_1(0)$,
$\varphi=0$ on $\partial B_1(0)$. By the maximum principle, $\varphi\geq 0$.
Then
$$
0< \int_B \psi f\,dx =\int_{B_1(0)} \tilde \psi f\,dx
=-\int_{B_1(0)}f\Delta \varphi\,dx.
$$
This contradicts (\ref{idint}).\hfill$\square$ \vskip5pt

The argument for the proof of the next lemma was partially taken from
\cite{lin}. Since we could not find a precise reference for the
complete proof, we present here a proof pointed to  us by Prof. J.Q. Liu,
to whom we acknowledge.

\begin{lem} \label{lapu*lapv*}
{\it Let $(u,v)$ be a $C^4(\mathbb{R}^N)$ positive  solution of  system
{\em (\ref{sist1})}. Then $(u^*,v^*)$ satisfies
\begin{equation} \label{sistkt}
\begin{gathered}
\Delta^2 u^*  =|x|^{\alpha(N-4)-(N+4)}(v^*)^\alpha \\
\Delta^2 v^*  =|x|^{\beta(N-4)-(N+4)}(u^*)^\beta\\
\end{gathered}
\end{equation}
in $\mathbb{R}^N \setminus\{ 0\}$. Moreover
\begin{equation}
\label{maxp}
\Delta u^*(x)<0\quad \mbox{and}\quad\Delta v^*(x)<0,
\end{equation}
for all  $x\in \mathbb{R}^N \setminus\{ 0\}$.}
\end{lem}

\paragraph{Proof.}
An easy computation allows us to establish (\ref{sistkt}).
 From (\ref{lulv*pos}) we are led to prove the second conclusion in
 $B_1(0)\setminus \{ 0\}$.
Let $w \in  C^2 ({B}_1(0))\cap C^1 (\overline{B}_1(0))$ be such that
$\Delta w =0$ in $B_1(0)$, $w=\Delta u^*$ on $\partial B_1(0)$.
By the maximum principle, $w\leq 0$ in $B_1(0)$.
 Let $\varphi \in C^2 ({B}_1(0))\cap C^1 (\overline{B}_1(0))$ be  such
that $\varphi =0$ on $\partial B_1(0)$ and $\varphi \geq 0$ in $B_1(0)$ and,
for each  $\varepsilon >0$, let $\eta_\varepsilon\in{\cal D}(\mathbb{R}^N)$
be such that for $\quad i=1,2,3,4$,
$$
\eta_\varepsilon(x)=1 \quad\mbox{for } |x|\geq 2\varepsilon, \quad
\eta_\varepsilon(x)=0 \quad\mbox{for } |x|\leq \varepsilon,\quad
|D^i\eta_\varepsilon | \leq c\varepsilon^{-i},
$$
where c is a positive constant independent of $\varepsilon$.
From now on  we denote  $B_s=B_s(0)$, for $s=R, 2\varepsilon,\varepsilon $.
Multiplying  the first equation  of (\ref{sistkt}) by
$\varphi\eta_\varepsilon$, we get
\begin{eqnarray*}
\int_{B_1} \varphi\eta_\varepsilon|x|^{\alpha(N-4)-(N+4)}(v^*)^\alpha (x)\,dx
 &=&
\int_{B_1}\varphi\eta_\varepsilon \Delta^2 u^*\,dx\\
& =&\int_{B_1}\varphi\eta_\varepsilon \Delta (\Delta u^* -w)\,dx.
\end{eqnarray*}
After two integrations by parts in the last integral and observing that both
$\varphi\eta_\varepsilon$ and $(\Delta u^* -w)$ vanish on the boundary of
$B_1(0)$,
we obtain
\begin{multline}
\label{mulequ}
\int_{B_1} \varphi\eta_\varepsilon|x|^{\alpha(N-4)-(N+4)}(v^*)^\alpha (x)\,dx \\
= \int_{B_1}\{\eta_\varepsilon\Delta \varphi+2(\nabla\varphi,
\nabla\eta_\varepsilon) +\varphi \Delta\eta_\varepsilon\}(\Delta u^*-w)\,dx.
\end{multline}
Let $\psi=2(\nabla\varphi,\nabla\eta_\varepsilon)
+\varphi \Delta\eta_\varepsilon $. So $\psi=0$ for $|x|\leq \varepsilon$
and  $|x|\geq 2\varepsilon$. Then, for small $\varepsilon$,
\begin{align*}
\int_{B_1}\psi\Delta u^*\,dx  =& \int_{B_1} u^*\Delta\psi \,dx =
\int_{B_{2\varepsilon}\setminus B_{\varepsilon} }u^*\Delta\psi \,dx \\
=&\int_{B_{2\varepsilon}\setminus B_{\varepsilon} }
|x|^{\frac{\beta(N-4)-(N+4)}{\beta}} u^*(x)|x|^{\frac{(N+4)-\beta(N-4)}{\beta}}
\Delta\psi\,dx \\
\leq&
 \Big(\int_{B_{2\varepsilon}\setminus  B_{\varepsilon} } |x|^{\beta(N-4)-(N+4)}
 (u^*)^\beta(x)\,dx\Big)^{1/\beta} \\
& \times\Big(\int_{B_{2\varepsilon}\setminus B_{\varepsilon} }
 |x|^{\frac{(N+4)-\beta(N-4)}{\beta-1}}|\Delta\psi|^\frac{\beta}{\beta-1}\,dx
 \Big)^{(\beta-1)/\beta}.
\end{align*}
On the other hand,
\begin{align*}
\int_{B_1 } |x|^{\beta(N-4)-(N+4)} (u^*)^\beta (x)dx
=& \int_{B_1 } |x|^{-(N+4)} u^\beta(\frac{x}{|x|^2})dx\\
=&\int_{\mathbb{R}^N\setminus B_1} |y|^{-(N+4)} u^\beta(y)dy.
\end{align*}
Thus, from Lemma \ref{intb1} there exists a constant $c$ such that
$$
|\int_{B_{2\varepsilon}\setminus B_{\varepsilon}}\psi\Delta u^* \,dx| \leq
c\Big(\int_{B_{2\varepsilon}\setminus B_{\varepsilon} }
|x|^{\frac{(N+4)-\beta(N-4)}{\beta-1}}|\Delta\psi|^\frac{\beta}{\beta-1}\,dx
\Big)^{(\beta-1)/\beta}.
$$
Since  $|\Delta \psi|\leq c\varepsilon^{-4}$ for $\varepsilon$ small, we get
$$
|\int_{B_{2\varepsilon}\setminus B_{\varepsilon}}\psi\Delta u^* \,dx|
\leq c \varepsilon^{4/\beta}\to 0\quad \mbox{as } \varepsilon \to 0.
$$
Then, passing to the limit in (\ref{mulequ}) yields
$$
\int_{B_1} \Delta \varphi(\Delta u^*-w)\,dx = \int_{B_1} \varphi
|x|^{\alpha(N-4)-(N+4)}(v^*)^\alpha (x)\,dx>0.
$$
From Lemma \ref{liu} we conclude that $\Delta u^*\leq w\leq 0$ in
$B_1(0)\setminus \{ 0 \}$. Since $\Delta^2 u^*>0$, by the maximum principle
we must have $\Delta u^*<0$ in $\mathbb{R}^N\setminus \{ 0 \}$.
One can proceed similarly with $\Delta v^*$.\hfill$\square$ \smallskip

 We now apply the moving planes method. We start by considering planes
parallel to $x_1=0$, coming from $-\infty$.
From now on, for $x\in \mathbb{R}^N$ we write $x=(x_1,x')$ where
$x':=(x_2,\cdots,x_N)\in \mathbb{R}^{N-1}$. For each $\lambda$ we define
$$
\Sigma_\lambda :=\{ x:=(x_1,x')\in \mathbb{R}^N: x_1< \lambda\},\quad
T_\lambda :=\partial \Sigma_\lambda.
$$
 For $x=(x_1,x')\in\Sigma_\lambda $, let $x^\lambda:= (2\lambda-x_1, x')$ be
 the reflected point with respect to $T_\lambda$. We also consider
$$
e_\lambda := (2\lambda,0)\quad\mbox{and}\quad
\widetilde {\Sigma}_\lambda  := \Sigma_\lambda\setminus \{ e_\lambda\}.
$$
 Finally, for $x \in \widetilde {\Sigma}_\lambda$ we define
 $U_\lambda (x)=u^*(x^\lambda)-u^*(x)$ and
 $V_\lambda (x)=v^*(x^\lambda)-v^*(x)$.

In what follows  we take $\lambda\leq 0$. Using the invariance of the
Laplacian under a reflection together with the mean value theorem and  the
fact that $|x^\lambda|\leq |x|$, it follows from (\ref{sistkt}) that
\begin{equation} \label{sistbimp}
\begin{gathered}
\Delta^2 U_\lambda  \geq c(x,\lambda) V_\lambda (x)\\
\Delta^2 V_\lambda \geq \hat c(x,\lambda) U_\lambda (x),\\
\end{gathered}
\end{equation}
for $x \in \widetilde {\Sigma}_\lambda$, where
$c(x;\lambda)=\alpha|x|^{\alpha(N-4)-(N+4)}(\psi(x,\lambda))
^{\alpha-1}$, with
$\psi(x;\lambda)$  a real number between $v^*(x^\lambda)$ and $v^*(x)$,
 and similarly
$$
\hat{c}(x;\lambda)=\beta|x|^{\beta(N-4)-(N+4)}(\hat{\psi}(x,\lambda))
^{\beta-1},
$$
with $\hat {\psi}(x;\lambda)$  a real number between $u^*(x^\lambda)$ and
$u^*(x)$. From (\ref{u,v>0}) we conclude that both $c(x;\lambda)$ and
$\hat{c}(x;\lambda)$ are  positive.

Our next goal is to see that we can start the process of the moving planes.
For that matter we begin with some auxiliary facts.

\paragraph{Definition.}
Let $m \in \mathbb{N}$.  We say that a $C^2$ function  in a neighborhood of
infinity  $f$ has a {\em harmonic asymptotic expansion} at infinity if:
\begin{equation} \label{asyexp}
\begin{gathered}
f(x)= \frac{1}{|x|^{m}}(a_0+\sum_{i=1}^{m+2}\frac{a_ix_i}{|x|^2})+
O(\frac{1}{|x|^{m+2}}),\\
f_{x_i}(x) =-ma_0\frac{x_i}{|x|^{m+2}}+O(\frac{1}{|x|^{m+2}}),\\
f_{x_i,x_j}(x) =O(\frac{1}{|x|^{m+2}})
\end{gathered}
\end{equation}
where $a_i\in \mathbb{R}$, for $i=0,\ldots,m+2$. \smallskip

We observe that both  $u^*$ and $\,v^*$ have harmonic asymptotic expansions
at infinity, with $m=N-4$ and $a_0 >0$. Also $(-\Delta u^*)$ and
$(-\Delta v^*)$ have harmonic asymptotic expansions at infinity with
$m=N-2$ and $a_0 >0$.

\begin{lem} \label{cgs1}
Let $f$ be a function  in a neighborhood of infinity satisfying the
asymptotic expansion {\em (\ref{asyexp})}, with $a_0>0$. Then there exist
constants $\tilde\lambda_1<0$, $R_1>0$ such that, if
$\lambda\leq \tilde\lambda_1$, then
$$
f(x)<f(x^\lambda)\quad\hbox{for}\quad x_1<\lambda,\quad
x\not\in B_{R_1}(e_\lambda).
$$
\end{lem}

\begin{lem} \label{cgs2}
Let $f$ be a $C^2$ positive solution of
$-\Delta f=F(x)\quad \hbox{in}\quad |x|>R,
$ where $f$ has a harmonic asymptotic expansion (\ref{asyexp}) at
infinity, with $a_0>0$. Suppose that, for some negative $\lambda_0$
and for every $(x_1,x')$ with $x_1<\lambda_0$,
$$
f(x_1, x')<f(2\lambda_0-x_1, x') \quad\mbox{and}\quad
F(x_1, x')\leq F(2\lambda_0-x_1, x').
$$
Then there exist $\varepsilon >0$, $S>R$ such that
\begin{itemize}
\item[(i)] $f_{x_1}(x_1, x')>0$ in $|x_1-\lambda_0|<\varepsilon,\,|x|>S$,
\item[(ii)] $f(x_1, x')<f(2\lambda-x_1, x')$ in
$x_1<\lambda_0-\frac{1}{2}\varepsilon<\lambda$,  $|x|>S$,
\end{itemize}
for all $x\in \Sigma_\lambda$, $\lambda\geq \lambda_1$ with
$|\lambda_1-\lambda_0|<c_0\varepsilon$, where $c_0$ is a
positive number depending on $\lambda_0$ and $f$.
\end{lem}
We refer the reader to \cite[Lemmas 2.3 and 2.4]{cgs} for the proof.

\begin{prop}
There exists $\lambda_1<0$ such that if $\lambda\leq \lambda_1$ then
$\Delta U_\lambda(x)<0$, $\Delta V_\lambda(x)<0$, $U_\lambda(x)>0$ and
$ V_\lambda(x)>0$ in $\widetilde {\Sigma}_\lambda$.
\end{prop}

\paragraph{Proof.} From Lemma \ref{cgs1},  there exists $\tilde\lambda_1<0$
and $R_1>0$ such that $\Delta U_\lambda(x)<0$, $\Delta V_\lambda(x)<0$,
$U_\lambda(x)>0$ and $ V_\lambda(x)>0$ in
$\widetilde {\Sigma}_\lambda\setminus B_{R_1}(e_\lambda) $,  for all
$\lambda \leq\tilde\lambda_1$.

By Lemma \ref{lapu*lapv*}, $\Delta u^*(x) <0$ in $\mathbb{R}^N\setminus\{0\}$.
Since
$$
\Delta (\Delta u^*)(x) =|x|^{\alpha(N-4)-(N+4)}(v^*)^\alpha (x)>0,
$$
Lemma \ref{lemfife} allows us to conclude that
$$
\Delta u^*(x) \leq M(R_1):=\max \{ \Delta u^*(y):\,|y|=R_1 \},\quad
 \forall x: 0<|x|<R_1.
$$
If $x \in B_{R_1}(e_\lambda)\setminus (e_\lambda)$ then
$|x-e_\lambda|=|x^\lambda|< R_1$. So, for
$x\in B_{R_1}(e_\lambda)\setminus (e_\lambda)$ we have
$\Delta u^*(x^\lambda) \leq M(R_1)$. From the fact that $\Delta u^*(x)\to 0$
as $|x|\to +\infty$, we conclude that there exists  $R_2>0$ such that
$\Delta u^*(x) > M(R_1)/2$, for $|x|>R_2$.
Let $\bar\lambda_1:=\min\{-R_1,-R_2\}$. Then, for all $\lambda<\bar\lambda_1$,
$$
\Delta U_\lambda(x) = \Delta u^*(x^\lambda)-\Delta u^*(x)<M(R_1)
-\frac{M(R_1)}{2}<0,
$$
for $x \in B_{R_1}(e_\lambda)\setminus (e_\lambda)$.
 Similarly, let  $\bar\lambda_2,\bar \lambda_3,\bar\lambda_4<0$ be such that
$$
\Delta V_\lambda(x)  = \Delta v^*(x^\lambda)-\Delta v^*(x)
<M'(R_1)-M'(R_1)/2 <0, \mbox { for } x \in B_{R_1}(e_\lambda)
$$
for all $\lambda<\bar\lambda_2$,
$$
U_\lambda(x)  =   u^*(x^\lambda)- u^*(x)>m(R_1)-m(R_1)/2 > 0
$$
 for $x \in B_{R_1}(e_\lambda)$ and all $\lambda<\bar\lambda_3$,
$$
V_\lambda(x) =  v^*(x^\lambda)- v^*(x)>m'(R_1)-m'(R_1)/2 >0
$$
for  $x \in B_{R_1}(e_\lambda)$ and all $\lambda<\bar\lambda_4$,
where
\begin{gather*}
M'(R_1):=\max \{\Delta v^*(y): |y|=R_1\},\quad
m(R_1):=\min \{ u^*(y):|y|=R_1\},\\
 m'(R_1):=\min \{v^*(y): |y|=R_1\}.
\end{gather*}
 By choosing $\lambda_1=\min\{\bar\lambda_1,\bar\lambda_2,\bar\lambda_3,
 \bar\lambda_4,\tilde\lambda_1\}$  we get the conclusion.
 \hfill$\square$\smallskip

Let
$\lambda_0:=\sup \{\lambda<0\,: \Delta U_\lambda(x)<0,\, \Delta V_\lambda(x)<0,
\,U_\lambda(x)>0,\mbox{ and }  V_\lambda(x)>0 \mbox{ in }
 \widetilde {\Sigma}_\lambda\}$.

\paragraph{Remark.}
By continuity, we have $\Delta U_{\lambda_0}(x)\leq 0$,
$\Delta V_{\lambda_0}(x)\leq 0$, $U_{\lambda_0}(x)\geq 0$ and
$ V_{\lambda_0}(x)\geq 0 $ in $\widetilde {\Sigma}_{\lambda_0}$.

\begin{lem} \label{uv0}
$U_{\lambda_0}\equiv 0$ if and only if $V_{\lambda_0}\equiv 0$.
\end{lem}

\paragraph{Proof.}
If $U_{\lambda_0} \not\equiv 0 $ and $V_{\lambda_0}\equiv 0 $, by
(\ref{sistbimp}) we have $ \hat c(x,\lambda_0)U_{\lambda_0}(x) \leq 0$.
Since $\hat c(x,\lambda_0)> 0$, then $U_{\lambda_0} \leq 0$.
Since also $U_{\lambda_0}>0$, this is a contradiction.\hfill$\square$

\begin{prop} \label{la0<0}
If $\lambda_0<0$ then $U_{\lambda_0}\equiv 0 $ and
$V_{\lambda_0}\equiv 0 $.
\end{prop}

\paragraph{Proof.}
Suppose by contradiction that the conclusion of the proposition is not true.
By Lemma \ref{uv0} we conclude that  $U_{\lambda_0} \not\equiv 0 $ and
$V_{\lambda_0} \not\equiv 0$.
Since
\begin{gather*}
\Delta U_{\lambda_0} \leq 0 \quad \mbox{in } \widetilde {\Sigma}_{\lambda_0}\\
 U_{\lambda_0} \geq 0,\quad U_{\lambda_0} \not\equiv 0 \quad \mbox{in }
 \widetilde {\Sigma}_{\lambda_0} \\
U_{\lambda_0}=0 \quad \mbox{on } T _{\lambda_0}
\end{gather*}
and since $U_{\lambda_0}(x)\to 0$ when $|x|\to \infty$, by the maximum principle we have that $U_{\lambda_0}(x)>0$ in $\widetilde {\Sigma}_{\lambda_0}$. By the same arguments we can prove that  $ V_{\lambda_0}(x)>0 \mbox{ in } \widetilde {\Sigma}_{\lambda_0}$. Then
\begin{equation}
\label{uopos}
\Delta^2 U_{\lambda_0}\geq c(x,\lambda_0)V_{\lambda_0}>0 \quad \mbox{ in }\widetilde {\Sigma}_{\lambda_0}
\end{equation}
and, as a consequence,
\begin{gather*}
\Delta^2 U_{\lambda_0} > 0 \quad \mbox{in } \widetilde {\Sigma}_{\lambda_0}\\
\Delta U_{\lambda_0} \leq 0 \quad \mbox{in } \widetilde {\Sigma}_{\lambda_0}\\
\Delta U_{\lambda_0}=0 \quad \mbox{on } T _{\lambda_0}.
\end{gather*}
Since $\Delta U_{\lambda_0}(x)\to 0$ when $|x|\to 0$, by the maximum principle
 we must have  $\Delta U_{\lambda_0}(x)<0$ in $\widetilde {\Sigma}_{\lambda_0}$.
Using the Hopf maximum principle  we obtain that
$$
\frac{\partial\Delta U_{\lambda_0} }{\partial\nu}(x)>0 \quad\mbox{on }
T_{\lambda_0},
$$
where $\nu$ is the outward unit normal to $\widetilde {\Sigma}_{\lambda_0}$.
We will  prove that this is impossible.

From the definition of  $\lambda_0$, there exists a sequence of real numbers
$\lambda_n\searrow \lambda_0$ and a sequence of points in
$\widetilde {\Sigma}_{\lambda_n}$ where $\Delta U_{\lambda_n}$ or
$\Delta V_{\lambda_n} $ is positive  or $U_{\lambda_n}$ or  $ V_{\lambda_n} $
is negative.

If $\Delta U_{\lambda_n}(x) >0$ for some $x\in \widetilde {\Sigma}_{\lambda_n}$,
 then
$$
c_1:=\sup_{\widetilde {\Sigma}_{\lambda_n}} \Delta U_{\lambda_n} >0.
$$
We shall see that this supreme is attained. Let
$$
c_2:=\max_{\partial B_{\frac{\lambda_0}{2}}(e_{\lambda_0})}
\Delta U_{\lambda_0} <0.
$$
With $0<r<\lambda_0/2$, we define
$g(x)=\Delta U_{\lambda_0}(x)+c_2(|x-e_{\lambda_0}|^{2-N}r^{N-2}-1)$, for
$x \in B_{\lambda_0 /2}(e_{\lambda_0})\setminus B_r(e_{\lambda_0} )$.
It is  easy to see that
$g(x)\leq 0$ in $\partial (B_{\lambda_0/2}(e_{\lambda_0})\setminus
B_r(e_{\lambda_0}))$. Then we have
\begin{gather*}
\Delta g = \Delta^2 U_{\lambda_0}>0  \quad
\mbox{in } B_{\lambda_0 /2}(e_{\lambda_0})\setminus B_r(e_{\lambda_0})  \\
  g\leq 0 \quad \mbox{on } \partial (B_{\lambda_0 /2}(e_{\lambda_0})
  \setminus B_r(e_{\lambda_0})).
\end{gather*}
By the maximum principle, $g(x)\leq 0$ for all
$x\in B_{\lambda_0 /2}(e_{\lambda_0})\setminus B_r(e_{\lambda_0})$.
Since $r$ is arbitrary, we conclude that
$$
\Delta U_{\lambda_0}(x) \leq c_2<0  \quad\mbox{in }
\dot B_{\lambda_0 /2}(e_{\lambda_0}),
$$
where $\dot B_s(e_{\lambda_0})$ denotes the punctured ball
$ B_{s}(e_{\lambda_0})\setminus\{ e_{\lambda_0}\}$, for $s> 0$.
By continuity, we have
$$
\Delta U_{\lambda_n}(x) \leq \frac{c_2}{2}<0 \quad\mbox{in }
\dot B_{\lambda_0 /4}(e_{\lambda_n}),
$$
for large $n$. As  $\Delta U_{\lambda_n}(x)\to 0$ when $|x|\to +\infty$,
there exists  $r_n$ such that, for all $|x|\geq r_n$,
$\Delta U_{\lambda_n}(x) <c_1/2$. Thus
$$
\sup_{\widetilde {\Sigma}_{\lambda_n}} \Delta U_{\lambda_n}
= \sup\big\{\Delta U_{\lambda_n} (x): x \in
(\widetilde {\Sigma}_{\lambda_n}\setminus B_{\lambda_0 /4}(e_{\lambda_n}))
\cap B_{r_n}(0)\big\}.
$$
Then there exists a sequence
$(x_n) \subset (\widetilde {\Sigma}_{\lambda_n}\setminus
B_{\lambda_0 /4}(e_{\lambda_n}))\cap B_{r_n}(0)$ such that
$$
\sup_{\widetilde {\Sigma}_{\lambda_n}} \Delta U_{\lambda_n}
= \Delta U_{\lambda_n}(x_n)>0.
$$
Hence
\begin{equation} \label{max}
\nabla (\Delta U_{\lambda_n})(x_n) = 0 \quad\mbox{and}\quad
\Delta( \Delta U_{\lambda_n})(x_n) \leq 0.
\end{equation}
It follows from Lemma \ref{cgs1} that $(x_n)$ is bounded.
Thus, up to a subsequence, $x_n\to x_0$ with
$x_0 \in {\overline {\widetilde \Sigma}}_{\lambda_0} $.
Passing (\ref{max}) to the limit  we obtain
\begin{equation} \label{limmax}
\nabla (\Delta U_{\lambda_0})(x_0) = 0 \quad\mbox{and}\quad
\Delta( \Delta U_{\lambda_0})(x_0) \leq 0.
\end{equation}
Thus from (\ref{uopos}) and (\ref{limmax}) we conclude that
$x_0 \in T_{\lambda_0} $. Since
$$
0<\frac{\partial\Delta U_{\lambda_0} }{\partial\nu}(x_0)
=\frac{\partial\Delta U_{\lambda_0} }{\partial x_1} (x_0)=0,
$$
we have a contradiction.

The case when  $\Delta V_{\lambda_n}$ takes positive values and the
cases when $ U_{\lambda_n} $ and $V_{\lambda_n}$ take negative values are
 proved similarly. \hfill$\square$

\paragraph{Proof of Theorem \ref{thm2} completed.}
 {\bf I)} In applying  the moving planes method, we must consider two cases.
\begin{itemize}
\item[(a)]
If $\lambda_0<0$, by Proposition \ref{la0<0} we have $U_{\lambda_0}\equiv0$
and $V_{\lambda_0}\equiv 0$, so $u^*(x)$ and $v^*(x)$ are symmetric with
respect to the plane $T_{\lambda_0}$. Since the bilaplacian is invariant for
dilations, from (\ref{sistkt}) we get a contradiction. So $u=v=0$ in
$\mathbb{R}^N$.
\item[(b)] If $\lambda_0=0$ then $U_0(x)\geq 0$ and $V_0(x)\geq 0$ in
$\widetilde {\Sigma}_{0}$,  i.e.
\begin{equation} \label{ineq1}
u(-x_1,x')\geq u(x_1,x') \mbox{ and } v(-x_1,x')\geq v(x_1,x') \quad    \mbox{ for } x_1\leq 0.
\end{equation}
\indent Defining $\bar u (x_1,x'):= u(-x_1,x)$ and
$\bar v (x_1,x'):= v(-x_1,x)$, we have
\begin{gather*}
\Delta^2 \bar u  =\bar v^\alpha\\
\Delta^2 \bar v  =\bar u^\beta
\end{gather*}
in $\mathbb{R}^N$.
By performing the latter procedure, we deduce the existence of a
corresponding value $\bar\lambda_0\leq 0$. If $\bar\lambda_0<0 $
then $\bar u =\bar v=0$ and consequently $u=v=0$. If $\bar\lambda_0=0 $
then
$$
\bar u(-x_1,x')\geq \bar u(x_1,x') \quad\mbox{and}\quad \bar v(-x_1,x')
\geq \bar v(x_1,x')\quad \mbox{ for } x_1\leq 0.
$$
By  (\ref{ineq1}) we conclude that $u$ and $v$ are radially symmetric with
respect to the origin.
\end{itemize}
 We can perform the Kelvin transform with respect to any point, thus $u$ and
 $v$ are radially  symmetric with respect to any point. This implies that
 $u$ and $v$ are constant functions. From system (\ref{sist1}), we get $u=v=0$.

\noindent{\bf II)} We proceed like Figueiredo and Felmer in \cite{fife}.
\hfill$\square$

\paragraph{Acknowledgments.}
The authors would like to  thank  Miguel Ramos for useful conversations and
for introducing them to this problem. A. R. Domingos was partially
supported by FCT . Y. Guo benefitted from post-doc scholarship from CMAF-UL.

\begin{thebibliography}{11}  \frenchspacing

\bibitem {cgs} {\sc L. Caffarelli, B. Gidas, J. Spruck, }
{\it Asymptotic symmetry and local behaviour os semilinear elliptic
equations with critical Sobolev growth,} Comm. Pure App. Math.
{\bf XLII}, 1989, 271-297.

\bibitem{fife}{\sc D.G. de Figueiredo, P.L. Felmer, }
{\it A Liouville-type theorem
for elliptic system,} Ann. Sc. Norm. Sup. Pisa {\bf XXI} (1994), 387-397.

\bibitem{fimi}{\sc D.G. de Figueiredo, E. Mitidieri, }{\it Maximum principles
for linear elliptic systems,} Rend. Inst. Mat. Univ. Trieste {\bf XXII}
(1990), 36-66.

\bibitem{gnn}{\sc B. Gidas, W.-M. ni, L. Niremberg,  }
{\it Symmetry and related properties via the maximum principle,}
Commun. Math. Phys. {\bf 68} (1979), 209-243.

\bibitem{gs}{\sc B. Gidas, J. Spruck,}
{\it Global and local behavior of positive solutions of nonlinear elliptic
 equations}, Comm. Pure Appl. Math  {\bf 34} (1981), 525-598.

\bibitem{lin}{\sc C.-S. Lin, }
{\it  A classification of solutions of a conformally invariant fourth order
equation in $\mathbb{R}^N$}, Comment. Math Helv. {\bf 73} (1998), 206-231.

\bibitem{mi}{\sc E. Mitidieri, }
{\it A Rellich type identity and applications, }
Comm.  Partial Differential Equations {\bf 18 } (1993), 125-151.

\bibitem{pw}{\sc M. Protter, H. Weinberger, }
{\it ``Maximum principles in differential equations''}, Prentice-Hall,
Inc. 1967.

\bibitem{sz}{\sc J. Serrin, H. Zou, }
{\it Non-existence of positive solutions of
Lane-Emden Systems, } Differential and Integral Equations  {\bf 9} (1996),
635-653.

\bibitem{s}{\sc M.A. Souto, }
{\it ``Sobre a exist\^encia de solu\c c\~oes positivas para sistemas
cooperativos n\~ao lineares'',} PhD thesis, Unicamp (1992).

\bibitem{xu}{\sc X. Xu, }
{\it Uniqueness theorem for the entire positive solutions of biharmonic
equations in $\mathbb{R}^N$, } Proc.  Royal Soc.
Edinburgh {\bf 130A} (2000), 651-670.

\end{thebibliography}

\noindent\textsc{Ana  Rute Domingos}\\
CMAF, Universidade de Lisboa\\
Av. Prof. Gama Pinto, 2\\
P-1649-003 Lisboa, Portugal\\
e-mail: rute@ptmat.ptmat.fc.ul.pt \smallskip

\noindent\textsc{Yuxia Guo}\\
Department of Mathematics\\
Tsinghua University\\
Beijing, 100084, China.\\
 e-mail: yxguo@yahoo.com

\end{document}