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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 67, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/67\hfil Inhomogeneous biharmonic elliptic systems]
{Radial solutions for inhomogeneous biharmonic elliptic systems}

\author[R. Demarque, N. H. Lisboa \hfil EJDE-2018/67\hfilneg]
{Reginaldo Demarque, Narciso da Hora Lisboa}

\address{Reginaldo Demarque \newline
Departamento de Ci\^encias da Natureza,
Universidade Federal Fluminense,
Rio das Ostras, RJ, 28895-532, Brazil}
 \email{r.demarque@gmail.com}

 \address{Narciso da Hora Lisboa \newline
 Departamento de Ci\^encias Exatas,
Universidade Estadual de Montes Claros,
 Montes Claros, MG,  39401-089, Brazil}
\email{narciso.lisboa@unimontes.br}

\thanks{Submitted June 16, 2017. Published March 14, 2018.}
\subjclass[2010]{35J50, 31A30}
\keywords{Biharmonic operator; elliptic systems; existence of solutions;
\hfill\break\indent radial  solution; mountain pass theorem}

\begin{abstract}
 In this article we obtain weak radial solutions for the
 inhomogeneous elliptic system
 \begin{gather*}
 \Delta^2u+V_{1}(| x| )| u|^{q-2}u=Q(| x| )F_{u}(u,v)\quad\text{in }\mathbb{R}^N, \\
 \Delta^2v+V_2(| x| )| v|^{q-2}v=Q(| x| )F_{v}(u,v)\quad\text{in }\mathbb{R}^N, \\
 u,v\in D_0^{2,2}(\mathbb{R}^N),\quad N\geq 5,
 \end{gather*}
 where $\Delta^2$ is the biharmonic operator,  $V_i$,
 $ Q\in C^{0 }((0,+\infty ),[0,+\infty ))$, $i=1,2$, are radially
 symmetric potentials, $1<q<N$, $q\neq 2$, and $F$ is a $s$-homogeneous function.
 Our approach relies on an application of the Symmetric Mountain Pass
 Theorem and a compact embedding result proved in \cite{Demarque Miyagaki}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction} \label{intro}

In this article concerns  the existence of nontrivial
 solutions for the inhomogeneous biharmonic elliptic system
 \begin{equation}\label{S}
\begin{gathered}
 \Delta^2u+V_{1}(| x| )| u|^{q-2}u=Q(| x| )F_{u}(u,v)\quad\text{in } \mathbb{R}^N, \\
 \Delta^2v+V_2(| x| )| v|^{q-2}v=Q(| x| )F_{v}(u,v)\quad\text{in } \mathbb{R}^N, \\
 u,v\in D_0^{2,2}(\mathbb{R}^N),\quad N\geq 5,
 \end{gathered}
  \end{equation}
 where $\Delta^2$ is the biharmonic operator,  $V_i$,
$ Q\in C^{0 }((0,+\infty ),[0,+\infty ))$, $i=1$, $2$, are radially
 symmetric potentials, $1<q<N$, $q\neq 2$, and $F$ is a $s$-homogeneous function satisfying the 
following assumptions:
\begin{itemize} %[label=($V$),ref=$V$]
\item[(A1)]  $V_i\in C^{0}((0,+\infty ),[0,+\infty ))$, such that
 \begin{equation} \label{V}
 \liminf_{r\to +\infty }\frac{V_i(r)}{r^{a}}>0, \quad
 \liminf_{r\to 0}\frac{V_i(r)}{r^{a_0}}>0,
 \end{equation}
 for some real numbers $a$ and $a_0$.

% [label=($Q$),ref=$Q$]
\item[(A2)]  $Q\in C^{0}((0,+\infty ),[0,+\infty ))$, is such that
\begin{equation} \label{Q}
 \limsup_{r\to +\infty }\frac{Q(r)}{r^{b}}<\infty, \quad
 \limsup_{r\to 0}\frac{Q(r)}{r^{b_0}}<\infty,
 \end{equation}
 for some real numbers $b$ and $b_0$.

\item[(A3)] %\label{F0} 
$F\in C^{1}(\mathbb{R}\times \mathbb{R},\mathbb{R} )$ is a homogeneous 
function of degree $s$, with $s>\max \{2,q\}$.

%[label=($F_1$),ref=$F_1$]
 \item[(A4)]  There exists $C>0$ such that
 \begin{equation} \label{F1}
\begin{gathered}
 | F_{u}(u,v)| \leq C(| u|^{s-1}+| v|^{s-1})\text{, }(u,v)\in  \mathbb{R}^2, \\
 | F_{v}(u,v)| \leq C(| u|^{s-1}+| v|^{s-1})\text{, }(u,v)\in  \mathbb{R}^2. 
 \end{gathered}
 \end{equation}

% [label=($F_2$),ref=$F_2$] \label{F2}
\item[(A5)]  $F(u,v)>0$, $\forall $ $u$, $v>0$.
 \end{itemize}


Nonlinear elliptic problems of fourth order without singularities in bounded
domains have been extensively studied by several authors, see 
\cite{Bernis Garcia Azorero Peral,Gazzola Grunau Sweers, Wang Shen, Xiong Shen},
 and references therein.

For application or motivation, we note that, when 
$\Omega \subset \mathbb{R}^N$ is a bounded domain, the problem
\begin{gather*}
\Delta^2u+c\Delta u=f(x,u)\quad\text{in }\Omega , \\
u=\Delta u=0\quad \text{on }\partial \Omega ,
\end{gather*}
which arises in the study of traveling waves in suspension bridges (see
\cite{Chen McKenna, Lazer McKenna, McKenna Walter}) and in the
study of the static deflection of an elastic plate in a fluid.

For studies on the existence and multiplicity of solutions for nonlinear 
biharmonic problems in unbounded domains, the
reader is referred to \cite{Furusho Kusano, Noussair Swanson Yang, Swanson} 
in the radial case, and to \cite{Allegretto Yu} in the non-radial sub-(sup) 
linear case. Maximum principle results for biharmonic equation in
unbounded domains are obtained in
\cite{Stavrakakis Sweers}. Also for unbounded domains, nontrivial solutions and
multiplicity results are obtained in \cite{Alves JM do O, Alves JM do
 O Miyagaki,Alves JM do O Miyagaki1,Chabrowski J do O,Noussair Swanson Yang1,
Wang Shen1} and in references therein.
Additional results in the scalar case may be found in 
\cite{Su Wang Willem, Su Tian,Su Tian 1,Zhang Tang,Zhao Wang}.


Elliptic systems may be used to describe the multiplicative
chemical reactions catalyzed by catalyst. For the existence of nontrivial 
solutions to nonvariational systems, potential systems and Hamiltonian systems
 including critical exponents case see, for instance,
\cite{Alves Figueiredo,Han,Liu Han,Lou,Zhao Wang}. See also \cite{Alves,Alves Soares,Carriao Lisboa
 Miyagaki,Carriao Lisboa Miyagaki1,Carriao Miyagaki,Lu Xiao,Morais Souto}.

For results for fourth-order equations with singular potential see
 \cite{Noussair Swanson Yang} and \cite{Alves JM do O Miyagaki1}.
Alves et. al. in \cite{Alves JM do O Miyagaki1} proved the existence of 
solutions to the problem
\begin{gather*}
\Delta^2u +V(x)|u|^{q-1}u=|u|^{2^\ast-2}u,  \quad\text{in } \Omega\subset \mathbb{R}^N\\
u\in D^{2,2}_0(\Omega), \quad N\geq 5,
\end{gather*}
where $1\leq q\leq 2^\ast-1$ and $V=V(x)$ is a potential that changes sign 
and has singularities in $\Omega$.
Wang and Shen in \cite{Wang Shen1} proved existence of sign-changing solutions 
for the problem
\begin{gather*}
\Delta^2u =\lambda \frac{|u|^{2^{\ast\ast(s)}-2}u}{|x|^s}+\beta a(x)|u|^{r-2}u,
 \quad\text{in } \Omega\subset \mathbb{R}^N\\
u\in D^{2,2}_0(\Omega), \quad  N\geq 5,
\end{gather*}
motivated by the Hardy-Rellich's inequality
$$
\bar{\lambda}\int_{\mathbb{R}^N}\frac{u^2}{|x|^4}dx\leq \int_{\mathbb{R}^N}|\Delta u|^2dx,
$$
as improved in their work. Radial solutions for the biharmonic equation
$$
\Delta^2u+V(|x|)|u|^{q-2}u=Q(|x|)f(u), 
$$
were obtained in \cite{Carriao Demarque Miyagaki}, when $q=2$, and 
in \cite{Demarque Miyagaki}, when $q\neq 2$. Motivated by the work of 
Alves \cite{Alves} for elliptic systems,
a natural question is whether or not the results of \cite{Demarque Miyagaki} 
can be extended to the elliptic system \eqref{S}. Here we answer this
question in the affirmative when (A3) and (A4) are satisfied.

Before stating our results, we to introduce some notation. 
Let $D_0^{2,2}(\mathbb{R}^N)$ be the closure of
$C_0^{\infty }(\mathbb{R}^N)$ under the norm $\| \Delta u\| _2$ and
$D_{0,r}^{2,2}(\mathbb{R}^N)$ the set of radially symmetric functions in
 $D_0^{2,2}(\mathbb{R}^N)$.

For $p\geq 1$ and a function $\nu :\mathbb{R}^N\to\mathbb{R}$ define
\begin{equation*}
L^p(\mathbb{R}^N;\nu )=\big\{u:\mathbb{R}^N\to\mathbb{R}\: u
\text{ is Lebesgue measurable and }\int_{ \mathbb{R}^N}\nu (x)| u|^pdx<+\infty
\big\}
\end{equation*}
endowed with the norm
\begin{equation*}
\| u\| _{p,\nu }:=\Big(\int_{ \mathbb{R}^N}\nu (x)| u|^pdx\Big)^{1/p}.
\end{equation*}
Define the Banach space
$X_{V_i}:=D_0^{2,2}(\mathbb{R}^N)\cap L^p(\mathbb{R}^N;V_i)$,
with the norm
\begin{equation*}
\| u \| _{V_i}=\| \Delta u \|_2+\| u \| _{p,V_i}
\end{equation*}
and $X_{V_i,r}$ the set of radially symmetric functions in $X_{V_i}$, 
$i=1, 2$.


We consider the product space $X:=X_{V_{1}}\times X_{V_2}$ endowed
with the norm
\begin{equation*}
\| (u,v)\| :=\Big(\int_{ \mathbb{R}^N}(| \Delta u|^2
+| \Delta v|^2)dx\Big)^{1/2}+{\Big(\int_{ \mathbb{R}^N}(V_{1}}(| x| )| u|^q+{V_2}
(| x| )| v|^q)dx\Big)^{1/q},
\end{equation*}
and $X_r:=X_{V_{1},r}\times X_{V_2,r}$.
Also, we endow the space $L^{s}(\mathbb{R}^N;Q)\times L^{s}(\mathbb{R}^N;Q)$ 
endowed with the norm
\begin{equation*}
\| (u,v)\| _{s,Q}={\Big(\int_{ \mathbb{R}^N}Q}(| x| )(| u|^{s}
+|v|^{s})dx\Big)^{1/s}.
\end{equation*}

Let $\alpha^{\ast }:=\frac{N-4}{2}+\frac{q-1}{q}(a+N)$ and 
$\alpha_0^{\ast }:=\frac{N-4}{2}+\frac{q-1}{q}(a_0+N)$.
 Now, as in \cite{Demarque Miyagaki}, we define some indexes that will appear in our
results.

The bottom indices are defined as
\begin{equation*}
s_{\ast }:=\begin{cases}
q, & b\leq a\text{, }b\leq -N\text{ or }b\geq -N+\frac{q(N-4)}{2}
-\varepsilon, \\
\frac{2(N+b+\varepsilon )}{N-4}, 
& b\leq a\text{ and }-N<b<-N+\frac{q(N-4)}{2} -\varepsilon, \\
q+\frac{q(b-a)}{\alpha^{\ast }}, 
& b>a\geq -N+\frac{q(N-4)}{2}, \\
q+\frac{2(b-a)}{N-4}, 
& b>a, b>-N\text{ and }-N+\frac{q(N-4)}{2}-\varepsilon
<a<-N+\frac{q(N-4)}{2}, \\
\frac{2(N+b+\varepsilon )}{N-4}, 
& b>a, b>-N\text{ and }a\leq -N+\frac{q(N-4)}{2}
-\varepsilon,\\
q+\frac{2(b-a)}{N-4}, 
 & a<b\leq -N,
\end{cases}
\end{equation*}
and the top indices are defined as
\begin{equation*}
s^{\ast }:=\begin{cases}
\frac{2(N+b_0-\varepsilon )}{N-4}, 
& a_0\geq b_0>-N\text{ or }b_0\geq a_0\geq -N+\frac{q(N-4)}{2}+\varepsilon,\\
q+\frac{2(b_0-a_0)}{N-4}, 
& b_0\geq a_0 \text{ and }-N+\frac{q(N-4)}{2}\leq a_0<-N+\frac{q(N-4)}{2}
 +\varepsilon, \\
q+\frac{q(b_0-a_0)}{\alpha _0^{\ast }}, 
& b_0\geq a_0\text{ and }-N-\frac{q(N-4)}{2(q-1)}<a_0<-N+\frac{q(N-4)}{2}, \\
+\infty, 
& b_0\geq a_0 \text{ and }a\leq -N-\frac{q(N-4)}{2(q-1)}.
\end{cases}
\end{equation*}
Consider also $s_{\ast \ast }:=q+\frac{q(b_0-a_0)}{\alpha _0^{\ast }}$, with 
$b_0\leq a_0<-N-\frac{q(N-4)}{2(q-1)}$.

Our main result is the following.

\begin{theorem}\label{main-th}
Let {\rm (A1)--(A5)} be satisfied. If $s_{\ast }<s<s^{\ast }$, then  system
\eqref{S} has a nontrivial solution $(u,v)\in X_r$ by which we
mean
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}^N}(\Delta u.\Delta \varphi
+\Delta v.\Delta \psi )dx+{\int_{ \mathbb{R}^N}(V_{1}}(| x| )| u|^{q-2}u\varphi
+{V_2}(| x| )| v|^{q-2}v\psi )dx
 \\
&=\int_{\mathbb{R}^N}Q (| x| )(\varphi F_{u}(u,v)+\psi F_{v}(u,v))dx\,,
\end{aligned} \label{1}
\end{equation}
for all $(\varphi ,\psi )\in X$. Moreover, if $F(u,v)=F(-(u,v))$
and there exists $\eta >0$ such that $F(u,v)\geq \eta (| u|^{s}+| v|^{s})$,
for all $(u,v)\in \mathbb{R}^2$, then system \eqref{S} has infinitely many radial
 solutions $(u,v)\in X_r$, $i=1,2$.
\end{theorem}

The proof of Theorem \ref{main-th} will be given using arguments similar to 
those developed in \cite{Demarque Miyagaki}. First we define an Euler 
functional $I:X_r\to \mathbb{R}$ associated with the equation \eqref{1}. 
Then, we obtain a Principle of Symmetric
 Criticality result, which yields that the critical points of 
$I$ are solutions of the system. Finally, we prove that this functional 
has the mountain pass geometry and apply the Symmetric Mountain Pass 
Theorem to obtain the result.


\section{Existence results}

In this section we will prove our main result. To do this, we will divide 
the proof in some lemmas. Firstly, let us present the following embedding 
theorem established in \cite{Demarque Miyagaki}.

\begin{theorem}\label{th-DM}
 Let $V_i$, $i= 1, 2$, and $Q$
 be functions satisfying \eqref{V} and \eqref{Q}. If $s_{\ast }<s^{\ast }$,
 then the embedding
 \begin{equation*}
 X_{V_i,r}\hookrightarrow L^{s}(\mathbb{R}^N;Q),
 \end{equation*}
 is continuous for all $s_{\ast }\leq s\leq s^{\ast }$ when 
$s^{\ast }<\infty $, $s_{\ast }\leq s<\infty $ when $s^{\ast }=\infty $ or 
$\max \{s_{\ast }$, $s_{\ast \ast }\}\leq s<\infty $. Furthermore, the embedding is
 compact for all $s_{\ast }<s<s^{\ast }$ or $\max \{s_{\ast }$, 
$s_{\ast \ast }\}<s<\infty $.
\end{theorem}

Now let us define the Euler functional $I:X_r\to \mathbb{R}$ by
\begin{align*}
I(u,v) 
&=\frac{1}{2}\int_{ \mathbb{R}^N}(| \Delta u|^2+| \Delta v|^2)dx
+\frac{1}{q}{\int_{ \mathbb{R}^N}(V_{1}}(| x| )| u|^q+{V_2}(| x| )| v|^q)dx\\
&\quad -{\int_{ \mathbb{R}^N}Q}(| x| )F(u,v)\,dx.
\end{align*}

By conditions (A1)--(A4)  and the continuous
embeddings obtained in Theorem \ref{th-DM}, we have that 
$I\in C^{1}(X_r;\mathbb{R})$ with Fr\'{e}chet derivative in
$(u,v)\in X_r$ given by
\begin{align*}
&\langle I'(u,v),(\varphi ,\psi )\rangle \\
&=\int_{ \mathbb{R}^N}(\Delta u.\Delta \varphi +\Delta v.\Delta \psi )dx
+{\int_{ \mathbb{R}^N}(V_{1}}(| x| )| u|^{q-2}u\varphi 
+{V_2}(| x| )| v|^{q-2}v\psi )dx \\
&\quad -{\int_{ \mathbb{R}^N}Q}(| x| )(\varphi F_{u}(u,v)+\psi F_{v}(u,v))dx,
\end{align*}
for all $(\varphi ,\psi )\in X_r$.

The proof of the next lemma follows the arguments presented in 
\cite{Demarque Miyagaki}.

 \begin{lemma}\label{lem2.2}
 Every critical point of the functional $I:X_r\to  \mathbb{R}$ satisfies \eqref{1}.
 \end{lemma}

\begin{proof}
 Let $(u,v)\in X_r$ be a critical point of $I$. Given
$\left( \varphi ,\psi \right) \in X$, define
 \begin{equation}
 \bar{\varphi}(r):=\frac{1}{| \partial B_r| }
 \int_{\partial B_r}\varphi (\xi )dS(\xi ),\quad
\bar{\psi}(r):= \frac{1}{| \partial B_r| }\int_{\partial
 B_r}\psi (\xi )dS(\xi ), \label{2}
 \end{equation}
where $\partial B_r$ denotes the sphere of center 0 and radius $r$ and
$ | \partial B_r| $ denotes its Lebesgue measure.

 Proceeding as in the proof of the mean-value formulas for Laplace's equation
 (see \cite{Evans}), using polar coordinates in $ \mathbb{R}^N$ and divergence 
theorem, we conclude that
 \begin{gather*}
 \frac{d}{dr}\bar{\varphi}(r)=\frac{r}{N| B_r| }
 \int_{B_r}\Delta \varphi (\xi )d\xi , \\
\frac{d}{dr}\bar{ \psi}(r)=\frac{r}{N| B_r| }\int_{B_r}\Delta \psi (\xi )d\xi ,\\
 \frac{d^2}{dr^2}\bar{\varphi}(r)=-\frac{N-1}{r}\frac{d}{dr}\bar{\varphi}
 (r)+\frac{1}{| \partial B_r| }\int_{\partial
 B_r}\Delta \varphi (\xi )dS(\xi ) \\
 \frac{d^2}{dr^2}\bar{\psi}(r)=-\frac{N-1}{r}\frac{d}{dr}\bar{\psi}(r)+
 \frac{1}{| \partial B_r| }\int_{\partial
 B_r}\Delta \psi (\xi )dS(\xi ).
 \end{gather*}
 Since $\Delta \bar{\varphi}=\frac{d^2}{dr^2}\bar{\varphi}+\frac{N-1
 }{r}\frac{d}{dr}\bar{\varphi}$ and 
$\Delta \bar{\psi}=\frac{d^2}{dr^2} \bar{\psi}+\frac{N-1}{r}\frac{d}{dr}\bar{\psi}$, 
we obtain
 \begin{equation}
 \Delta \bar{\varphi}=\frac{1}{| \partial B_r| }
 \int_{\partial B_r}\Delta \varphi (\xi )dS(\xi )\text{ and\ }\Delta
 \bar{\psi}=\frac{1}{| \partial B_r| }
 \int_{\partial B_r}\Delta \psi (\xi )dS(\xi ). \label{3}
 \end{equation}
From this we see that $(\bar{\varphi},\bar{\psi})\in X_r$ and then
 \begin{align*}
&\langle I'(u,v),(\bar{\varphi},\bar{\psi})\rangle \\
&=\int_{ \mathbb{R}^N}(\Delta u.\Delta \bar{\varphi}+\Delta v.\Delta \bar{\psi})dx 
+\int_{ \mathbb{R}^N}({V_{1}}(| x| )| u|^{q-2}u\bar{ \varphi}
+{V_2}(| x| )| v|^{q-2}v \bar{\psi})dx \\
&\quad -\int_{ \mathbb{R}^N}{Q}(| x| )(\bar{\varphi}F_{u}(u,v)+\bar{\psi}
 F_{v}(u,v))dx=0.
 \end{align*}
Therefore, using polar coordinates in $ \mathbb{R}^N$ and Fubini's Theorem 
again and the identities \eqref{2} and \eqref{3} we obtain result.
\end{proof}

Before we prove the Palais-Smale condition for the functional $I$, we need 
to make some remarks about assumptions of the function $F$.

 \begin{remark} \label{remark} \rm
\begin{itemize}
 \item[(a)] Since $F$ is a $C^{1}$ homogeneous
 function of degree $s$, then $sF(u,v)=uF_{u}(u,v)+vF_{v}(u,v)$ and $\nabla
 F $ is a homogeneous function of degree $s-1$.

 \item[(b)] From $(F_{1})$, $(a)$ and the Young inequality we have $
 | F(u,v)| \leq C(| u|^{s}+| v|^{s})$ for all $(u,v)\in \mathbb{R}^2$.

 \item[(c)] Our prototype of $F$ is $F(u,v)=(a|
 u| +b| v| )^{s}+c| u|^{\alpha }| v|^{\beta }$, $u$, $v\in \mathbb{R} $; 
$a$, $b$, $c>0$ and $\alpha +\beta =s$, with $\alpha$, $\beta >1$.
\end{itemize}
 \end{remark}


\begin{lemma}\label{lem2.3}
The functional $I:X_r\to \mathbb{R}$ satisfies the Palais-Smale condition.
\end{lemma}

\begin{proof}
 Let  $\{(u_n,v_n)\}$  be  a  sequence  $X_r$ such  that
$I'(u_n,v_n)\to 0$ and $I(u_n,v_n)\to c$, as
$ n\to +\infty $. We shall see that $\{(u_n,v_n)\}$ is bounded in
$ X$. Indeed, since $I'(u_n,v_n)\to 0$, we have
$ \| I'(u_n,v_n)\| <1$ for all $n$ sufficiently
 large, and so, $| \langle I'(u_n,v_n),(u_n,v_n)\rangle |
\leq \| (u_n,v_n)\| $. Since $\{I(u_n,v_n)\}$ is convergent
 sequence, there exists a positive constant $C_0$ such that 
$| I(u_n,v_n)| \leq C_0$. In this case, from $(F_0)$ and the
 Remark $(a)$, we have
\begin{equation}
 \begin{aligned}
&C_0+\frac{1}{s}\| (u_n,v_n)\| \\
& \geq I(u_n,v_n)- \frac{1}{s}\langle I'(u_n,v_n),(u_n,v_n)\rangle  \\
& \geq C{\Big[}\int_{ \mathbb{R}^N}(| \Delta u_n|^2+| \Delta
 v_n|^2)dx
+\int_{ \mathbb{R}^N}(V_{1}(| x| )| u_n|^q+V_2(| x| )| v_n|^q)dx{\Big] },
\end{aligned}\label{4}
\end{equation}
where $C_0$ and $C$ are positive constants. To conclude that
$ \{(u_n,v_n)\}$ is bounded, we will split our arguments in the cases:
$ 1<q<2$ and $q>2$.
\smallskip

\noindent\textbf{Case $q>2$.}
Suppose $\{(u_n,v_n)\}$ is unbounded. Then, up to a subsequence,
$ \| (u_n,v_n)\| \to +\infty $, as $n\to +\infty $. From \eqref{4}
 we see that
\begin{equation}
 \begin{aligned}
&\frac{C_0}{\| (u_n,v_n)\|^2}+\frac{1}{s}\frac{1}{
 \| (u_n,v_n)\| } \\
&\geq \frac{C}{\|  (u_n,v_n)\|^2}{\Big[}
 \int_{ \mathbb{R}^N}(| \Delta u_n|^2+| \Delta v_n|^2)dx \\
&\quad +\int_{ \mathbb{R}^N}(V_{1}(| x| )| u_n|^q+V_2(| x| )| v_n|^q)dx{\Big] },
\end{aligned} \label{5}
\end{equation}
 for some positive constants $C_0$ and $C$.

If $\big\{\int_{ \mathbb{R}^N}(V_{1}(| x| )| u_n|^q+V_2(| x| )| v_n|^q)dx
\big\}$ is an unbounded sequence, then, up to a subsequence,
 $$
\int_{\mathbb{R}^N}(V_{1}(| x| )| u_n|^q+V_2(| x| )| v_n|^q)dx\to
 +\infty , \quad \text{as } n\to +\infty.
$$
 This implies that ${\int_{\mathbb{R}^N}(V_{1}(| x| )| u_n|^q
+V_2(| x| )| v_n|^q)dx>1}$,
 for $n$ sufficiently large. Consequently, since $q>2$, we obtain
 \begin{equation*}
 \| (u_n,v_n)\|^2\leq 2\Big[\int_{ \mathbb{R}^N}(| \Delta u_n|^2+| \Delta
 v_n|^2)dx
+\int_{ \mathbb{R}^N}(V_{1}(| x| )| u_n|^q+V_2(| x| )| v_n|^q)dx\Big] .% \label{6}
 \end{equation*}
Combining this with \eqref{5} we deduce that
 \begin{equation*}
 \frac{C_0}{\| (u_n,v_n)\|^2}+\frac{1}{s}\frac{1}{
 \| (u_n,v_n)\| }\geq \frac{C}{2},
 \end{equation*}
 for some constants $C_0$, $C>0$ and for $n$ sufficiently large. 
So we obtain a contradiction.

On the other hand, if $\big\{\int_{\mathbb{R}^N}(V_{1}(| x| )| u_n|^q+V_2(| x| )
| v_n|^q)dx\big\}$  is bounded, we conclude that
 $$
\int_{\mathbb{R}^N}(| \Delta u_n|^2+| \Delta v_n|^2)dx\to
+\infty,\quad \text{as } n\to +\infty,
$$
 up to a subsequence. Using \eqref{5}
 we see that, for some positive constants $C_0$ and $C$,
 \begin{align*}
 \frac{C_0}{\| (u_n,v_n)\|^2}+\frac{1}{s}\frac{1}{
 \| (u_n,v_n)\|^2}
&\geq C\frac{\| \Delta  u_n\| _2^2+\| \Delta v_n\| _2^2
 +\| u_n\| _{q,V_{1}}^q  +\| v_n\| _{q,V_2}^q}{\| (u_n,v_n)\|^2} \\
 &=C\frac{1+\frac{\| u_n\| _{q,V_{1}}^q+\| v_n\| _{q,V_2}^q}{\| \Delta u_n\|
 _2^2+\| \Delta v_n\| _2^2}}{{\Big[1+}\frac{({{ \| u_n\| _{q,V_{1}}^q
 +\| v_n\| _{q,V_2}^q)}}^{1/q}}{{(}\| \Delta u_n\| _2^2
 +\| \Delta v_n\| _2^2{)}^{1/2}}{\Big]}^2}
 \to C,
 \end{align*}
as $n\to +\infty $. Here we also have a contradiction.
\smallskip


\noindent \textbf{Case $1<q<2$.}
If $\big\{\int_{\mathbb{R}^N}(| \Delta u_n|^2+| \Delta v_n|^2)dx\big\}$ 
is an unbounded sequence, then, up to a  subsequence,
 $$
\int_{\mathbb{R}^N}(| \Delta u_n|^2+| \Delta v_n|^2)dx\to +\infty,\quad
\text{as } n\to +\infty. 
$$
 As a consequence $\int_{ \mathbb{R}^N}(| \Delta u_n|^2+| \Delta v_n|^2)dx>1$, 
for $n$ sufficiently large. Since $1<q<2$, it  follows that
 \begin{align*}
&\| (u_n,v_n)\|^q  \\
&\leq 2^q{\Big[\Big( \int_{ \mathbb{R}^N}(| \Delta u_n|^2
 +| \Delta v_n|^2)dx\Big)^{\frac{q}{2}}+\int_{ \mathbb{R}^N}(V_{1}(| x| )| u_n|
^q+V_2(| x| )| v_n|^q)dx\Big]} \\
&\leq 2^q{\Big[\int_{ \mathbb{R}^N}(| \Delta u_n|^2+| \Delta v_n|^2)dx
 +\int_{ \mathbb{R}^N}(V_{1}(| x| )| u_n|^q+V_2(| x| )| v_n|^q)dx\Big]} .
 \end{align*}
 Using this and \eqref{4} we conclude that, for some positive constants 
$ C_0$ and $C$,
 \begin{equation*}
 C_0+\frac{1}{s}\| (u_n,v_n)\| \geq \frac{C}{2^q}%
 \| (u_n,v_n)\|^q,
 \end{equation*}
for $n$ sufficiently large. But this is a contradiction.

 Now, if $\big\{\int_{\mathbb{R}^N}(| \Delta u_n|^2+| \Delta v_n|^2)dx\big\}$ 
is bounded, then, up to a subsequence,
 $$
{\int_{\mathbb{R}^N}(V_{1}(| x| )| u_n|^q+V_2(| x| )| v_n|^q)dx\to +\infty },\text{ as } n\to +\infty.$$
 Hence, using \eqref{4} again, we obtain
 \begin{align*}
 \frac{C_0}{\| (u_n,v_n)\|^q}+\frac{1}{s}\frac{1}{%
 \| (u_n,v_n)\|^{q-1}} 
&\geq C\frac{\| \Delta u_n\| _2^2+\| \Delta v_n\| _2^2
 +\| u_n\| _{q,V_{1}}^q+\| v_n\| _{q,V_2}^q}{\| (u_n,v_n)\|^q} \\
&= C\frac{\frac{\| \Delta u_n\| _2^2+\| \Delta v_n\| _2^2}{\| u_n\| _{q,V_{1}}^q
 +\| v_n\| _{q,V_2}^q}+1}{{\Big[}\frac{ (\| \Delta u_n\| _2^2+\| \Delta
 v_n\| _2^2)^{1/2}}{(\| u_n\| _{q,V_{1}}^q+\| v_n\| _{q,V_2}^q)^{1/q}}+1{\Big]}^q}
 \to C,
\end{align*}
as $n\to +\infty $. We have again a contradiction
 and, therefore, $\{(u_n,v_n)\}$ is bounded in $X_r$. Consequently,
$\{u_n\}$ and $ \{v_n\}$ are also bounded in $X_{r,V_{1}}$ and $X_{r,V_2}$, 
respectively.
 Using the fact that $X_{r,V_i}$, $i\in \{1$, $2\}$, is reflexive, we
 conclude that there exist $u\in $ $X_{r,V_{1}}$ and $v\in X_{r,V_2}$ such
 that $u_n\rightharpoonup u$ weakly in $X_{r,V_{1}}$ and 
$ v_n\rightharpoonup v$ weakly in $X_{r,V_2}$, as $n\to +\infty $ , up 
to subsequences. Hence $(u_n,v_n)\rightharpoonup (u,v)$ weakly in $X_r$, 
as $n\to +\infty $, up to a subsequence. Since $X_{r,V_i}$ is  compactly 
imbedded in $L^{s}( \mathbb{R}^N;Q)$, $i\in \{1$, $2\}$ 
(see Theorem \ref{th-DM}),  we deduce that $u_n\to u$ and $v_n\to v$ strongly in 
$ L^{s}( \mathbb{R}^N;Q)$, as $n\to +\infty $. As a consequence, $u_n\to u$
 and $v_n\to v$ a.e. in $ \mathbb{R}^N$, as $n\to +\infty $.

Now we shall prove that
\begin{equation*}
 \langle I'(u_n,v_n),(\varphi ,\psi )\rangle
 \to \langle I'(u,v),(\varphi ,\psi )\rangle,
\end{equation*}
for all $(\varphi ,\psi )\in X_r$, as $n\to +\infty $.

For $(\varphi ,\psi )\in X_r$, we define
 \begin{equation*}
 F_{(\varphi ,\psi )}(u,v):=\int_{ \mathbb{R}^N}[\Delta u\Delta \varphi 
+\Delta v\Delta \psi ]dx.
 \end{equation*}
Note that $F_{(\varphi ,\psi )}\in X'_r$ and
 \begin{equation*}
 \langle F_{(\varphi ,\psi )}'(u,v),(z,w)\rangle
 =\int_{ \mathbb{R}^N}[\Delta z\Delta \varphi +\Delta w\Delta \psi ]dx,
 \end{equation*}
 for all $(z,w)\in X_r$. Since $(u_n,v_n)\rightharpoonup (u,v)$ weakly in 
$ X_r $, as $n\to +\infty $, we deduce that 
$F_{(\varphi ,\psi )}(u_n,v_n)\to F_{(\varphi ,\psi )}(u,v)$ strongly in
$X_r$, as $ n\to +\infty $, for all $(\varphi ,\psi )\in X_r$, that is,
 \begin{equation}
 \int_{ \mathbb{R}^N}[\Delta u_n\Delta \varphi +\Delta v_n\Delta \psi ]dx\to
 \int_{ \mathbb{R}^N}[\Delta u\Delta \varphi +\Delta v\Delta \psi ]dx, \label{7}
 \end{equation}
as $n\to +\infty $, for all $(\varphi ,\psi )\in X_r$.

We consider $(\varphi ,\psi )\in X_r$ and define
 \begin{gather*}
g_n:=(V_{1})^{\frac{q-1}{q}}| u_n|^{q-2}u_n, \quad
h_n:=(V_2)^{\frac{q-1}{q}}| v_n|^{q-2}v_n, \\
g:=(V_{1})^{\frac{q-1}{q}}| u|^{q-2}u, \quad 
h:=(V_2)^{\frac{q-1}{q}}| v|^{q-2}v.
 \end{gather*}
 So, $g_n\to g$ and $h_n\to h$ a.e. in $ \mathbb{R}^N$. Moreover, 
$\{g_n\}$ and $\{h_n\}$ are bounded in $L^{q/(q-1)}( \mathbb{R}^N)$. 
It follows from Br\'{e}zis and Lieb lemma \cite{Brezis Lieb} (see
 also \cite[Lemma 4.8]{Kavian}) that
 \begin{equation*}
\int_{ \mathbb{R}^N}g_n\varphi dx\to \int_{ \mathbb{R}^N}g\varphi dx
\quad\text{and}\quad 
\int_{\mathbb{R}^N}h_n\psi dx\to \int_{\mathbb{R}^N}h\psi dx,
 \end{equation*}
as $n\to +\infty $, for all $\varphi,\psi \in L^q(
 \mathbb{R}
^N)$. In particular, given $(\varphi ,\psi )\in X_r$, we have 
$ (V_{1})^{1/q}\varphi $, $(V_2)^{1/q}\psi \in L^q( \mathbb{R}^N)$, so that,
 \begin{equation*}
\int_{ \mathbb{R}^N}g_n(V_{1}(| x| ))^{1/q}\varphi dx\to
 \int_{ \mathbb{R}^N}g(V_{1}(| x| ))^{1/q}\varphi dx
 \end{equation*}
and
 \begin{equation*}
 \int_{ \mathbb{R}^N}h_n(V_2(| x| ))^{1/q}\psi dx\to
 \int_{ \mathbb{R}^N}h(V_2(| x| ))^{1/q}\psi dx,
 \end{equation*}
as $n\to +\infty $. Hence,
 \begin{equation*}
 \int_{ \mathbb{R}^N}V_{1}(| x| )| u_n|^{q-2}u_n\varphi dx\to 
 \int_{ \mathbb{R}^N}V_{1}(| x| )| u|^{q-2}u\varphi dx
 \end{equation*}
 and
 \begin{equation*}
 \int_{ \mathbb{R}^N}V_2(| x| )| v_n|^{q-2}v_n\psi dx\to 
 \int_{ \mathbb{R}^N}V_2(| x| )| v|^{q-2}v\psi dx ,
 \end{equation*}
as $n\to +\infty $. Consequently,
\begin{equation}
 \begin{aligned}
& \int_{\mathbb{R}^N}(V_{1}(| x| )| u_n|
^{q-2}u_n\varphi +V_2(| x| )| v_n|^{q-2}v_n\psi )dx \\
&\to \int_{ \mathbb{R}^N}(V_{1}(| x| )| u|^{q-2}u\varphi
 +V_2(| x| )| v|^{q-2}v\psi )dx, \label{8}
 \end{aligned}
\end{equation}
 as $n\to +\infty $, for all $(\varphi ,\psi )\in X_r$.

We define $K:X_r\to X_r'$ by
 \begin{equation*}
 \langle K(u,v),(\varphi ,\psi )\rangle :=\int_{
 \mathbb{R}^N}Q(| x| )[\varphi F_{u}(u,v)+\psi F_{v}(u,v)]dx.
 \end{equation*}
First, we prove that
 \begin{equation*}
 \| F_{u}(u_n,v_n)-F_{u}(u,v)\| _{\frac{s}{s-1} ,Q}\to 0
\quad\text{and}\quad
\| F_{v}(u_n,v_n)-F_{v}(u,v)\| _{\frac{s}{s-1},Q}\to 0 ,
 \end{equation*}
as $n\to +\infty $. 
 By using (A4), we will deduce that
 \begin{equation*}
 Q(| x| )| F_{u}(u_n,v_n)-F_{u}(u,v)|^{\frac{s}{s-1}}\leq h(x)
 \end{equation*}
 a.e. $x\in  \mathbb{R}^N$, for some function $h\in L^{1}( \mathbb{R}^N)$. 
In fact, since $\| Q^{\frac{1}{s} }u_n-Q^{1/s}u\| _{s}\to 0$ and 
$\| Q^{\frac{1}{s} }v_n-Q^{1/s}v\| _{s}\to  0$, as $n\to +\infty $,
 we conclude that
 \begin{itemize}
 \item[(1)]  $Q^{1/s}u_n\to Q^{1/s}u$ and $Q^{1/s}v_n\to Q^{1/s}v$
 a.e. in $ \mathbb{R}^N$, as $n\to +\infty $;

 \item[(2)] $| Q^{1/s}u_n| \leq h_{1}$ and $| Q^{\frac{1}{s} }v_n| \leq h_2$ a.e. in 
$ \mathbb{R}^N$, where $h_{1}$, $h_2\in L^{1}( \mathbb{R}^N)$.
 \end{itemize}
Hence, for some positive constant $C$,
 \begin{align*}
&Q(| x| )| F_{u}(u_n,v_n)-F_{u}(u,v)|^{\frac{s}{s-1}} \\
&\leq 2^{\frac{s}{s-1}}Q(| x| )\Big(|
 F_{u}(u_n,v_n)|^{\frac{s}{s-1}}+| F_{u}(u,v)|^{\frac{s}{s-1}}\Big) \\
&\leq C\Big(Q(| x| )| u_n|^{s}+Q(| x| )| v_n|^{s}+Q(| x| )| u|^{s}
 +Q(| x| )| v|^{s}\Big) 
\leq h(x),
 \end{align*}
 where $h(x)=C\Big((h_{1}(x))^{s}+(h_2(x))^{s}+Q(| x| )| u|^{s}+Q(| x| )
| v|^{s}\Big)$.

Since $Q(| x| )| F_{u}(u_n,v_n)-F_{u}(u,v)|^{\frac{s}{s-1}}\to 0$ a.e. in 
$ \mathbb{R}^N$, as $n\to +\infty $, we see, by the Dominated Convergence 
Theorem of Lebesgue, that 
$ \| F_{u}(u_n,v_n)-F_{u}(u,v)\| _{\frac{s}{s-1} ,Q}\to 0$, as 
$n\to +\infty $. Similarly, $\| F_{v}(u_n,v_n)-F_{v}(u,v)\| _{\frac{s}{s-1},Q}\to 0$,
 as $n\to +\infty $.

 On the other hand, using H\"older's inequality and the continuous
 embedding $X_{r,Vi}\hookrightarrow L^{s}( \mathbb{R}^N,Q)$, $i\in \{1$, $2\}$, 
we have, for all $(\varphi ,\psi )\in X_r$,
 \begin{align*}
 & | \langle K(u_n,v_n)-K(u,v),(\varphi ,\psi )\rangle | \\
 & \leq \int_{ \mathbb{R}^N}Q(| x|) | F_{u}(u_n,v_n)-F_{u}(u,v)| | \varphi | dx\\
&\quad +\int_{ \mathbb{R}^N}Q(| x|) |  F_{v}(u_n,v_n)-F_{v}(u,v)| | \psi | dx
 \\
& \leq \| F_{u}(u_n,v_n)-F_{u}(u,v)\| _{\frac{s}{s-1} ,Q}\| \varphi \| _{s,Q}
 +\| F_{v}(u_n,v_n)-F_{v}(u,v)\| _{\frac{s}{s-1},Q}\| \psi
 \| _{s,Q} \\
& \leq C\| F_{u}(u_n,v_n)-F_{u}(u,v)\| _{\frac{s}{s-1} ,Q}\| (\varphi ,\psi )\|\\
&\quad +C\| F_{v}(u_n,v_n)-F_{v}(u,v)\| _{\frac{s}{s-1} ,Q}\| (\varphi ,\psi )\| ,
 \end{align*}
for some positive constant $C$. Using this we see that
 \begin{align*}
\| K(u_n,v_n)-K(u,v)\| _{X_r'} 
&\leq C[\| F_{u}(u_n,v_n)-F_{u}(u,v)\| _{\frac{s}{s-1} ,Q}\\
&\quad +\| F_{v}(u_n,v_n)-F_{v}(u,v)\| _{\frac{s}{s-1} ,Q}]\to 0,
\end{align*}
as $n\to +\infty $, so that, we obtain 
$\langle K(u_n,v_n)-K(u,v),(\varphi ,\psi )\rangle \to 0$, as 
$ n\to +\infty $; that is,
 \begin{equation*}
 \langle K(u_n,v_n),(\varphi ,\psi )\rangle \to
 \langle K(u,v),(\varphi ,\psi )\rangle ,
 \end{equation*}
 as $n\to +\infty $, for all $(\varphi ,\psi )\in X_r$. Consequently,
 \begin{equation}
\begin{aligned}
&\int_{ \mathbb{R}^N}Q(| x|) [\varphi  F_{u}(u_n,v_n)+\psi F_{v}(u_n,v_n)]dx \\
&\to \int_{ \mathbb{R}^N}Q(| x|) [\varphi F_{u}(u,v)+\psi  F_{v}(u,v)]dx,
\end{aligned} \label{9}
\end{equation}
as $n\to +\infty $, for all $(\varphi ,\psi )\in X_r$. Moreover, since
 $({u_n,v_n})\rightharpoonup (u,v)$ in $X_r$ and 
$K({u_n,v_n} )\to K(u,v)$ in $X_r'$, as $n\to +\infty $, it
follows that
 \begin{equation}
 \langle K(u_n,v_n),(u_n,v_n)\rangle \to
 \langle K(u,v),(u,v)\rangle , \label{10}
 \end{equation}
as $n\to +\infty $; that is,
 \begin{equation*}
\int_{ \mathbb{R}^N}Q(| x|) [F_{u}(u_n,v_n)u_n+F_{v}(u_n,v_n)v_n]dx
\to \int_{ \mathbb{R}^N}Q(| x|) [F_{u}(u,v)u+F_{v}(u,v)v]dx,
 \end{equation*}
as $n\to +\infty $. Combining \eqref{7}, \eqref{8} and \eqref{9} we obtain
\begin{equation*}
 \langle I'(u_n,v_n),(\varphi ,\psi )\rangle
 \to \langle I'(u,v),(\varphi ,\psi )\rangle ,
\end{equation*}
as $n\to +\infty $, for all $(\varphi ,\psi )\in X_r$. Hence, as 
$ I'(u_n,v_n)\to 0$, as $n\to +\infty $, we
deduce that $I'(u,v)=0$. This implies
\begin{align*}
 0 &=\langle I'(u,v),(u,v)\rangle \\
 &=\int_{ \mathbb{R}^N}[| \Delta u|^2+| \Delta v|^2]dx
+{\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u|^q+{V_2} (| x| )| v|^q]dx
 -\langle K(u,v),(u,v)\rangle .
 \end{align*}
Therefore,
 \begin{equation}
 \langle K(u,v),(u,v)\rangle =\int_{ \mathbb{R}^N}[| \Delta u|^2+| \Delta v|^2]dx
+{\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u|^q+{V_2} (| x| )| v|^q]dx. \label{11}
 \end{equation}
 On the other hand,
\begin{equation}
 \begin{aligned}
&\int_{ \mathbb{R}^N}[| \Delta u_n|^2+| \Delta v_n|^2]dx
 +{\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u_n|^q+{V_2 }(| x| )| v_n|^q]dx \\
&=\langle I'(u_n,v_n),(u_n,v_n)\rangle
 +\langle K(u_n,v_n),(u_n,v_n)\rangle .
\end{aligned} \label{12}
\end{equation}
 From \eqref{10}, \eqref{11} and \eqref{12} we have
\begin{equation}
\begin{aligned}
&\int_{ \mathbb{R}^N}[| \Delta u_n|^2+| \Delta v_n|^2]dx
 +{\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u_n|^q+{V_2  }(| x| )| v_n|^q]dx\\
&\to \int_{ \mathbb{R}^N}[| \Delta u|^2+| \Delta v|^2]dx
+{\int_{ \mathbb{R} ^N}[V_{1}}(| x| )| u|^q+{V_2}  (| x| )| v|^q]dx,
\end{aligned} \label{13}
\end{equation}  as $n\to +\infty $.
As before, from the Brezis-Lieb Lemma, we can show that
 \begin{gather*}
 {\int_{ \mathbb{R}^N}V_{1}}(| x| )| u_n|^qdx
 -{ \int_{ \mathbb{R}^N}V_{1}}(| x| )| u_n-u|^qdx
\to {\int_{ \mathbb{R}^N}V_{1}}(| x| )| u|^qdx, \\
 {\int_{ \mathbb{R}^N}V_2}(| x| )| v_n|^qdx
 -{ \int_{ \mathbb{R}^N}V_2}(| x| )| v_n-v|^qdx
 \to {\int_{ \mathbb{R}^N}V_2}(| x| )| v|^qdx, \\
 {\int_{ \mathbb{R}^N}}| \Delta u_n|^2dx
 -{\int_{ \mathbb{R}^N}}| \Delta (u_n-u)|^2dx
\to {\int_{ \mathbb{R}^N}}| \Delta u|^2dx, \\
 {\int_{ \mathbb{R}^N}}| \Delta v_n|^2dx
-{\int_{ \mathbb{R}^N}}| \Delta (v_n-v)|^2dx
\to {\int_{ \mathbb{R}^N}}| \Delta v|^2dx,
 \end{gather*}
 as $n\to +\infty $. This implies
\begin{equation}
 \begin{aligned}
&{\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u_n|^q+{V_2 }(| x| )| v_n|^q]dx
-{\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u_n-u|^q +{V_2}(| x| )| v_n-v|^q]dx\\
&\to {\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u|^q+{V_2}(| x| )| v|^q]dx
\end{aligned}\label{14}
\end{equation}
and
\begin{equation}
\begin{aligned}
&{\int_{ \mathbb{R}^N}[}| \Delta u_n|^2+| \Delta v_n|^2]dx
-{\int_{ \mathbb{R}^N}[}| \Delta (u_n-u)|^2+| \Delta(v_n-v)|^2]dx\\
&\to {\int_{ \mathbb{R}^N}[}| \Delta u|^2+| \Delta v|^2]dx,
\end{aligned}\label{15}
\end{equation}
as $n\to +\infty $. By \eqref{13}, \eqref{14} and \eqref{15},
we obtain
\begin{equation*}
{\int_{ \mathbb{R}^N}[}| \Delta (u_n-u)|^2+| \Delta(v_n-v)|^2]dx
+{\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u_n-u|^q+{ V_2}(| x| )| v_n-v|^q]dx\to 0,
\end{equation*}
as $n\to +\infty $. As a consequence, we deduce that
\begin{align*}
 \| (u_n,v_n)-(u,v)\|
&={\Big(\int_{ \mathbb{R}^N}(}| \Delta (u_n-u)|^2
 +| \Delta (v_n-v)|^2)dx\Big)^{1/2} \\
&\quad +{\Big({\int_{ \mathbb{R}^N}(V_{1}}(| x| )| u_n-u|^q
 +{ V_2}(| x| )| v_n-v|^q)dx\Big)}^{1/q}
 \to 0,
\end{align*}
as $n\to +\infty $. Therefore, $(u_n,v_n)\to (u,v)$ in $X_r $, as $n\to +\infty $.
\end{proof}


 \begin{lemma}[Geometry of the Mountain Pass Theorem]\label{lem2.4}
 The functional $I:X_r\to
 \mathbb{R}$ satisfies the following conditions:
 \begin{itemize}
 \item[(a)] $I(0,0)=0$ and there exist $c>0$,
$ \rho >0$ such that $I(u,v)\geq c$ for $\| (u,v)\| =\rho $;

 \item[(b)] There exists $(u,v)\in X_r$, with $ \| (u,v)\| >\rho$, 
such that $I(u,v)<0$.
 \end{itemize}
 \end{lemma}

\begin{proof}
 First  we note  that $I(0,0)=0$. Now, taking $q_0:=\max \{2,q\}$ and 
using the Remark \ref{remark} item (b), we conclude that
\begin{equation}
 \begin{aligned}
 I(u,v)
&\geq \frac{1}{q_0}{\Big[\int_{ \mathbb{R}^N}[}| \Delta u|^2
 +| \Delta v|^2]dx+{\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u|^q
 +{V_2} (| x| )| v|^q]dx{\Big]} \\
&\quad -C{\int_{ \mathbb{R}^N}Q(| x|) }[| u|^{s}+| v|^{s}{]dx},
\end{aligned} \label{16}
\end{equation}
 for some constant $C>0$. By the continuous embedding $X_{r,V_i}
 \hookrightarrow L^{s}( \mathbb{R}^N;Q)$, $i=1$, $2$, we deduce that
 \begin{equation*}
 \int_{ \mathbb{R}^N}Q(| x| )| u|^{s}dx\leq C\| (u,v)\|^{s}, \quad
\int_{ \mathbb{R}^N}Q(| x| )| v|^{s}dx\leq C\| (u,v)\|^{s},
 \end{equation*}
for some positive constant $C$. This and \eqref{16} implies that
 \begin{equation}
\begin{aligned}
I(u,v) 
&\geq \frac{1}{q_0}{\Big[\int_{ \mathbb{R}^N}[}| \Delta u|^2+| \Delta v|^2]dx
+{\int_{ \mathbb{R}^N}[V_{1}}(| x| )| u|^q+{V_2} (| x| )| v|^q]dx{\Big]} \\
&\quad {-C}\| (u,v)\|^{s},
\end{aligned} \label{17}
 \end{equation}
for some constant $C>0$. For $0<\| (u,v)\| <1$, we have
\begin{equation}
 \begin{aligned}
&\| (u,v)\|^{q_0} \\
& \leq 2^{q_0}{\Big[\Big(\int_{ \mathbb{R}^N}(}| \Delta u|^2
 +| \Delta v|^2)dx\Big)^{q_0/2}
 +{\Big(\int_{ \mathbb{R}^N}(V_{1}}(| x| )| u|^q
 +{V_2} (| x| )| v|^q]dx\Big)^{q_0/q}{\Big]} \\
&\leq 2^{q_0}{\Big[\int_{ \mathbb{R}^N}(}| \Delta u|^2
 +| \Delta v|^2)dx+{\int_{ \mathbb{R}^N}(V_{1}}(| x| )| u|^q
 +{V_2} (| x| )| v|^q)dx{\Big]}.
\end{aligned} \label{18}
\end{equation}
 Combining \eqref{17} with \eqref{18} we obtain
 \begin{equation*}
 I(u,v)\geq \frac{1}{q_02^{q_0}}\| (u,v)\|^{q_0}-C\| (u,v)\|^{s},
 \end{equation*}
for some positive constant $C$ and for $0<\| (u,v)\| <1$.
So, there exist $0<\rho <1$ sufficiently small and $c>0$ such that
$ I(u,v)\geq c>0$ for all $(u,v)\in X$, with
$\| (u,v)\| =\rho $. This completes the proof of (a).

Fixing $ (u_0,v_0)\in X_r$ such that $F(u_0,v_0)>0$, we have, for all $t>0$,
 \begin{align*}
I(t(u_0,v_0))&=\frac{t^2}{2}\int_{ \mathbb{R}^N}[| \Delta u_0|^2
 +| \Delta v_0|^2]dx +\frac{t^q}{q}\int_{ \mathbb{R}^N}[V_{1}(| x| )(u_0)^q
 +V_2(| x| )(v_0)^q]dx \\
&\quad -t^{s}\int_{ \mathbb{R}^N}Q(| x| )F(u_0,v_0)dx.
 \end{align*}
This implies 
 \begin{equation*}
 I(t(u_0,v_0))\to -\infty \quad \text{as }t\to +\infty.
 \end{equation*}
Thus, for $t>0$, sufficiently large, $\| t(u_0,v_0)\| >\rho $ and 
$I(t(u_0,v_0))<0$. Therefore, $(b)$ follows. This completes
 the proof.
\end{proof}

Finally, we can prove our main result.

\begin{proof}[Proof of Theorem \ref{main-th}]
As a consequence of Lemma \ref{lem2.3} and Lemma \ref{lem2.4}, 
we conclude, by using the Mountain Pass Theorem, due to Ambrosetti-Rabinowitz 
\cite{Amb Rabinowitz},
that there exists a sequence $\{ (u_n,v_n)\} $ in $X$ so that
\begin{equation*}
I(u_n,v_n)\to c>0\text{ and }I'(u_n,v_n)\to 0,
\end{equation*}
as $n\to +\infty $. By Lemma \ref{lem2.3}, $(u_n,v_n)\to (u,v)$
in $X_r$, as $n\to +\infty $, up to a subsequence. 
In view of $I\in C^{1}(X_r, \mathbb{R} )$, it follows that
\begin{equation*}
I(u_n,v_n)\to I(u,v)\quad \text{and}\quad I'(u_n,v_n)\to I'(u,v),
\end{equation*}
as $n\to +\infty $. This implies that $I'(u,v)=0$ and $
I(u,v)=c\neq 0$, that is, $(u,v)\in X_r$ is a nontrivial critical point of $I$.
By Lemma \ref{lem2.2}, we conclude that $(u,v)$ is a radial solution for the
system \eqref{S} in the sense of equation $\eqref{1}$.

Our next goal is to apply the Symmetric Mountain Pass Theorem 
\cite[Theorem 6.5]{Struwe} to complete the proof of Theorem \ref{main-th}. 
So, we need to show that $I$ satisfies the following conditions:
\begin{itemize}
\item[(a)] $I(-(u,v))=I(u,v)$, for all $(u,v)\in X_r$;

\item[(b)] For any nontrivial finite dimensional subspace $U\subset X_r$,
 there exists $R>0$ such that $I(u,v)\leq 0$ for all $(u,v)\in U$, with 
$ \| (u,v)\| \geq R$.
\end{itemize}
Since $F(u,v)=F(-(u,v))$, $(a)$ occurs.

Now, suppose that (b) is not true. Therefore, there exists a nontrivial
finite dimensional  subspace $U\subset X_r$  and  a sequence 
$ \{(u_n,v_n)\}$  in $U$  such that $\| (u_n,v_n)\|\to +\infty $, as 
$n\to +\infty $, and $I(u_n,v_n)>0$, for all $n\in \mathbb{N}$. 
Since $U$ has finite dimension, all norms are equivalent on $U$. In this
case, since $F(u,v)\geq \eta (| u|^{s}+|v|^{s})$ for all 
$(u,v)\in \mathbb{R}^2$, we obtain
\begin{align*}
\int_{ \mathbb{R}^N}Q(| x| )F(u_n,v_n)dx
&\geq \eta \int_{ \mathbb{R}^N}Q(| x| )(| u_n|^{s}+| v_n|^{s})dx\\
&=\eta \| (u_n,v_n)\| _{s,Q}^{s}
 \geq C\| (u_n,v_n)\|^{s},
\end{align*}
for some positive constant $C$. Since $s>\max \{2,q\}=q_0$ and 
$\|(u_n,v_n)\| \to +\infty $, as $n\to +\infty $,
we deduce that
\begin{align*}
 I(u_n,v_n)
& \leq \frac{1}{2}\int_{ \mathbb{R}^N}[| \Delta u_n|^2
 +| \Delta v_n|^2]dx+\frac{1}{q}{\int_{ \mathbb{R} ^N}[V_{1}}(| x| )| u_n|^q+{V_2
 }(| x| )| v_n|^q]dx \\
&\quad -C\| (u_n,v_n)\|^{s} \\
&\leq \frac{1}{2}\| (u_n,v_n)\|^2+\frac{1}{q}
 \| (u_n,v_n)\|^q-C\| (u_n,v_n)\|^{s} \\
 &\leq \Big(\frac{1}{2}+\frac{1}{q}\Big)\|
 (u_n,v_n)\|^{q_0}-C\| (u_n,v_n)\|^{s},
\end{align*}
for $n$ sufficiently large and for some positive constant $C$. As a
consequence,
\begin{equation*}
\lim_{n\to +\infty }I(u_n,v_n)=-\infty;
\end{equation*}
that is, there exists $n$, sufficiently large, such that $I(u_n,v_n)<0$,
which is a contradiction. This completes the proof of (b).

So, by Symmetric Mountain Pass Theorem, there exists an unbounded sequence
of critical values for $I$, which corresponds to the existence of a sequence
of nontrivial critical points for $I$. Consequently, by Lemma \ref{lem2.2}, 
equation \eqref{1} holds, which completes the proof of Theorem \ref{main-th}.
\end{proof}

\subsection*{Acknowledgments}
We wish to thank Professor Olimpio Miyagaki for the suggestions made 
on an earlier version of this article, and for encouraging us to publish it. 
We also like to thank the referee for his/her  valuable comments and
suggestions that certainly improved this article.

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\end{document}