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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 49, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/49\hfil Systems with critical nonlinearities]
{Existence of least energy solutions to  coupled elliptic systems
with critical nonlinearities}

\author[G.-M. Wei, Y.-H. Wang\hfil EJDE-2008/49\hfilneg]
{Gong-Ming Wei, Yan-Hua Wang}  

\address{Gong-Ming Wei \newline
 Tin Ka-Ping College of Science,  University of
 Shanghai for Science and Technology, Shanghai, 200093,  China}
 \email{gmweixy@163.com}

\address{Yan-Hua Wang \newline
 Department of Applied Mathematics, Shanghai
 University of Finance and Economics, Shanghai, 200433, China}
\email{yhw@mail.shufe.edu.cn}

\thanks{Submitted October 18, 2007. Published April 4, 2008.}
\thanks{YHW is supported by grant 10726039 from the
National Natural Science Foundation of China}
\subjclass[2000]{35B33, 35J50}
\keywords{Least energy solutions; Nehari manifold;
critical exponent; \hfill\break\indent  coupled elliptic systems}

\begin{abstract}
 In this paper we study  the existence of  nontrivial solutions of
 elliptic systems with critical nonlinearities and subcritical
 nonlinear coupling interactions, under  Dirichlet or Neumann
 boundary conditions. These equations are motivated from solitary
 waves of nonlinear Schr\"odinger systems in physics.  Using minimax
 theorem and by estimates on  the least energy, we prove the
 existence of nonstandard least energy solutions, i.e. solutions with
 least energy and  each component is nontrivial.
\end{abstract}

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

\section{Introduction}

  In this paper, we consider the existence of least energy solutions
to  the  Dirichlet problem
\begin{equation}\label{e1.1}
\begin{gathered}
-\Delta u+\lambda_1u=\mu_1u^3+\beta u^{p-1}v^p\quad \text{in } \Omega\\
-\Delta v+\lambda_2v=\mu_2v^3+\beta u^pv^{p-1}\quad \text{in } \Omega\\
               u>0,\quad  v>0 \quad \text{in } \Omega\\
               u=0,\quad  v=0 \quad \text{on } \partial\Omega
\end{gathered}
\end{equation}
and  to the Neumann prolem
\begin{equation}\label{e1.2}
\begin{gathered}
-\Delta u+\lambda_1u=\mu_1u^3+\beta u^{p-1}v^p \quad\text{in } \Omega\\
-\Delta v+\lambda_2v=\mu_2v^3+\beta u^pv^{p-1}\quad\text{in }\Omega\\
               u>0,\quad  v>0 \quad \text{in }\Omega\\
  \frac{\partial u}{\partial\nu}=0,\quad
\frac{\partial v}{\partial\nu}=0 \quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^4$ is a smooth bounded domain,
$\lambda_i,\mu_i,\beta$ are constants, $\mu_i>0$, $i=1,2$,  and
$1<p<2$. Since the dimension of $\Omega$ is $N=4$,
$4=\frac{2N}{N-2}$ is the critical Sobolev exponent. Therefore there
are critical nonlinearities and coupling interaction terms in  the
elliptic systems. In this paper, we are interested in positive
solutions. The solutions of problems  \eqref{e1.1} and \eqref{e1.2}
are equivalent to positive solutions of
\begin{gather*}
 -\Delta u+\lambda_1u=\mu_1|u|^2u+\beta |u|^{p-2}|v|^pu\quad\text{in }\Omega\\
 -\Delta v+\lambda_2v=\mu_2|v|^2v+\beta |u|^p|v|^{p-2}v\quad\text{in }\Omega
\end{gather*}

   By a least energy solution we mean
a nontrivial  solution with the least energy
\begin{equation}\label{f1.1}
E(u,v)=\int_{\Omega}\frac{1}{2}(|\nabla u|^2+\lambda_1u^2+|\nabla
v|^2+\lambda_2v^2)-\frac{1}{4}(\mu_1u^4+\mu_2v^4)-\frac{\beta}{p}|uv|^p
\end{equation}
among all nontrivial  solutions of problem \eqref{e1.1} or
\eqref{e1.2}. By nontrivial solution of system we mean that at least
one of its components is nontrivial(nonzero function). Of course,
the least energy solution we are interested in is {\it
nonstandard}(a definition in \cite{S}); i.e., each of its components
is nontrivial.

  Recently, the existence and multiplicity of solutions of classical
  coupled nonlinear Schr\"odinger equations (CNLS)
  \begin{equation}\label{cnls}
  \begin{gathered}
 \Delta u-\lambda_1u+\mu_1u^3+\beta uv^2=0  \\
\Delta v-\lambda_2v+\mu_2v^3+\beta u^2v=0
 \end{gathered}
 \end{equation}
   has been investigated by
  several authors in the case of subcritical nonlinearities, we
  recall, among many others, Ambrosetti \& Cororado \cite{AC},
Lin \& Wei \cite{LW1,LW2}, Maia,
  Montefusco \& Pellacci \cite{MMP}, Sirakov \cite{S},
and  the author \cite{Wei}.
   These CNLS are  motivated by
  nonlinear optics and Bose-Einstein double condensates and have
  attracted  a considerable attention in the last years.
On the other hand, systems of one-dimensional NLS
\begin{gather*}
 i\phi_t+\phi_{xx}+\alpha_1|\phi|^{p_1-2}\phi+\alpha_0|\psi|^{p_0}
|\phi|^{p_0-2}\phi=\delta  \psi_{xx}   \\
 i\psi_t+\psi_{xx}+\alpha_2|\psi|^{p_2-2}\psi+\alpha_0
|\phi|^{p_0}|\psi|^{p_0-2}\psi=\delta
 \phi_{xx}
\end{gather*}
 where $\alpha_j\geq0(j=1,2), \alpha_0\in\mathbb{R}$ and
 $|\delta|<1$, appear in several branches of physics, such as in the
 study of interactions of waves with different polarizations or in
 the description of nonlinear modulations of two monochromatic
 waves. These systems have been studied in many physical
 literatures. See \cite{CZ} for more references.  Standing waves of the form
 $$
\phi(t,x)=e^{i\lambda_1t}u(x),\quad
 \psi(t,x)=e^{i\lambda_2t}v(x)
 $$
 satisfy
\begin{equation}\label{e1.4}
\begin{gathered}
 -u_{xx}+\delta v_{xx}+\lambda_1u=\alpha_1|u|^{p_1-2}u+\alpha_0|v|^{p_0}|u|^{p_0-2}u
 \\
 -v_{xx}+\delta u_{xx}+\lambda_2v=\alpha_2|v|^{p_2-2}v
+\alpha_0|u|^{p_0}|v|^{p_0-2}v
 \end{gathered}.
\end{equation}
 A natural question is to study the multidimensional accompanist  of
 \eqref{e1.4}. Therefore, the system we consider in this paper can be viewed as a
 generalization of \eqref{cnls} and a high dimensional case  of \eqref{e1.4}.

 Single elliptic equations with critical nonlinearities have been
 extensively studied by many authors, including the classical results
 of  Brezis-Nirenberg \cite{BN},  singular perturbation problem \cite{LN,Ni,NPT,Wxj} for Neumann problems,
 multi-peak solutions \cite{GG,MP}, concentration phenomena
 \cite{MP1,Ni}, and so on. In \cite{MP2}, the authors constructed concentrated solutions for elliptic systems
 with critical nonlinearities and weakly  coupling interactions. To
 the author's knowledge, there are few results on Schr\"odinger type
 systems with critical nonlinearities and strong coupling
 interactions. This is another motivation of this paper.


Let $\lambda_1(\Omega)$ be the first eigenvalue of $-\Delta$ in
$H_0^1(\Omega)$. The main results of this paper are as follows.

\begin{theorem}\label{th1.1}
Assume $-\lambda_1(\Omega)<\lambda_1$, $\lambda_2<0$.
Then problem \eqref{e1.1} has a nonstandard
least energy solution for sufficiently large $\beta$.
\end{theorem}

\begin{theorem}\label{th1.2}
Assume $\lambda_1$, $\lambda_2$ are sufficiently large
(but independent of $\beta$). Then problem \eqref{e1.2} has a nonstandard
 least energy solution for sufficiently large $\beta$.
\end{theorem}

 The proof relies on a variational approach based on the well-known
 Mountain-Pass Theorem. It can be viewed as adaptation of an
 an approach which is now classical for CNLS. The compactness is
 recovered by imposing that $\beta$ is sufficiently large so that
 the mountain-pass min-max value $c$ satisfies a suitable inequality
 involving the best constant $S$ in the Sobolev embedding 
 $H^1(\mathbb{R}^4)\hookrightarrow L^4(\mathbb
 R^4)$. This ensures that the Palais-Smale condition holds at the level $c$. To prove that the least energy
is nonstandard, we use the semitrivial solutions $(U_1,0)$ and
$(0,U_2)$ as comparison functions, where $U_i$'s are the positive
least energy solutions of the equation
$$
-\Delta U_i+\lambda_iU_i=\mu_iU_i^3\quad \text{in}\quad\Omega .
$$
In the sequel we use the following notation.
$$
\|u\|_{\lambda_1}^2:=\int_\Omega|\nabla u|^2+\lambda_1u^2,\quad
\|v\|_{\lambda_2}^2:=\int_\Omega|\nabla v|^2+\lambda_2v^2,\quad
|u|_q^q:=\int_\Omega |u|^q.
$$

\section{Dirichlet problem}

Let $X=H^1_0(\Omega)\times H^1_0(\Omega)$ and
$$
c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}E(\gamma(t))
$$
where $\Gamma=\{\gamma\in
C([0,1],X)|\gamma(0)=0,E(\gamma(1))<0\}$.
Then (e.g. \cite{Wi}) $c>0$ and
\begin{equation} \label{f2.0}
c=\inf_{(u,v)\in X, (u,v)\neq(0,0)}\max_{t>0}E(tu,tv)\\
= \inf_{(u,v)\in\mathcal N}E(u,v)
\end{equation}
where
$$
\mathcal N=\{(u,v)\in
X\setminus\{(0,0)\}|\|u\|_{\lambda_1}^2+\|v\|_{\lambda_2}^2
=\mu_1|u|_4^4+\mu_2|v|_4^4+2\beta|uv|_p^p\}.
$$

\subsection*{Existence of nontrivial solution}
Using the mountain pass theorem, we first prove the existence of
nontrivial solution. In this section, we always assume that
\begin{equation}\label{f2.01}
-\lambda_1(\Omega)<\lambda_1<0,\quad
-\lambda_1(\Omega)<\lambda_2<0
\end{equation}
where $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta$ in
$H_0^1(\Omega)$.

\begin{theorem}\label{th2.1}
 Assume that condition \eqref{f2.01} holds, there exists  a nontrivial least
energy solution for problem \eqref{e1.1} for sufficiently large
$\beta$.
\end{theorem}

\begin{proof}
 By the mountain pass lemma (e.g. \cite{Wi}), there exists a minimizing
 sequence $(u_n,v_n)\in X$ such that as $n\to\infty$
 \begin{equation}\label{f2.1}
 E(u_n,v_n)\to c,\quad E'(u_n,v_n)\to0\quad \text{in }\ X'.
 \end{equation}
We  assume that $(u_n,v_n)$ is nonnegative; otherwise we consider
$(|u_n|,|v_n|)$.
 It is routine to prove that
$\|u_n\|_{\lambda_1}^2+\|v_n\|_{\lambda_2}^2$ is bounded and
\begin{equation} \label{f2.2}
\begin{gathered}
\frac{1}{2}(1-\frac{1}{p})(\|u_n\|_{\lambda_1}^2+\|v_n\|_{\lambda_2}^2)
+\frac{1}{2}(\frac{1}{p}-\frac{1}{2})(\mu_1|u_n|_4^4
+\mu_2|v_n|_4^4)=c+o(1),\\
\frac{1}{4}(\mu_1|u_n|_4^4+\mu_2|v_n|_4^4)+(1-\frac{1}{p})\beta|u_nv_n|_p^p
=c+o(1).
\end{gathered}
\end{equation}
Going  to a subsequence, if necessary, there exists $(u,v)\in X$ such
that
\begin{equation} \label{f2.02}
\begin{gathered}
u_j\rightharpoonup u,\quad  v_j\rightharpoonup v,
   \quad\text{in }H_0^1(\Omega),\\
u_j\to u,\quad v_j\to v, \quad\text{in }L^2(\Omega),\\
u_j\to u,\quad  v_j\to v,\ \  \text{a.e. in }  \Omega ,\\
u_j^3\rightharpoonup u^3,\quad  v_j^3\rightharpoonup v^3,
\quad\text{in }
L^{4/3}(\Omega)\\
u_j^{p-1}v_j^p\rightharpoonup u^{p-1}v^p,\quad
u_j^pv_j^{p-1}\rightharpoonup u^pv^{p-1}, \quad\text{in }
L^{2/p}(\Omega).
\end{gathered}
\end{equation}

  It is easy to see that  $(u,v)$ is a nonnegative solution of equations
\eqref{e1.1} and has nonnegative energy; i.e.,
\begin{equation}\label{f2.6}
E'(u,v)=0,\quad E(u,v)\geq0.
\end{equation}
Set
$\sigma_n=u_n-u$, $\tau_n=v_n-v$, $\gamma_n=u_nv_n-uv$.
 By Br\'ezis-Lieb theorem and \eqref{f2.02},
\begin{equation}\label{f2.11}
\begin{gathered}
|u_n|_4^4=|u|_4^4+|\sigma_n|_4^4+o(1),\\
|v_n|_4^4=|v|_4^4+|\tau_n|_4^4+o(1),\\
|u_nv_n|_p^p=|uv|_p^p+o(1).
\end{gathered}
\end{equation}
By a direct computation and \eqref{f2.1}, \eqref{f2.11},
\begin{equation} \label{f2.3}
\begin{aligned}
E(u_n,v_n)&=E(u,v)+\frac{1}{2}(|\nabla\sigma_n|_2^2+|\nabla\tau_n|_2^2)
  -\frac{1}{4}(\mu_1|\sigma_n|_4^4+\mu_2|\tau_n|_4^4)+o(1)\\
  &=c+o(1),
\end{aligned}
\end{equation}
with
\begin{equation} \label{f2.4}
\begin{aligned}
o(1)&=(E'(u_n,v_n),(u_n,v_n))\\
&=(E'(u,v),(u,v))+|\nabla\sigma_n|_2^2+|\nabla\tau_n|_2^2
  -(\mu_1|\sigma_n|_4^4+\mu_2|\tau_n|_4^4)\\
&=|\nabla\sigma_n|_2^2+|\nabla\tau_n|_2^2-(\mu_1|\sigma_n|_4^4
 +\mu_2|\tau_n|_4^4).
\end{aligned}
\end{equation}
 Assuming that $|\nabla\sigma_n|_2^2+|\nabla\tau_n|_2^2\to b$,
by  \eqref{f2.4},
\begin{equation}\label{f2.5}
\mu_1|\sigma_n|_4^4+\mu_2|\tau_n|_4^4\to b.
\end{equation}
 If $b=0$, the proof is done.
 Now we assume that $b>0$.
By the Sobolev imbedding theorem,
 \begin{equation*}
|\nabla\sigma_n|_2^2>S|\sigma_n|_4^2,\quad
 |\nabla\tau_n|_2^2>S|\tau_n|_4^2.
\end{equation*}
Hence
\begin{equation} \label{f2.16}
\begin{aligned}
(|\nabla\sigma_n|_2^2+|\nabla\tau_n|_2^2)^2
&\geq S^2(|\sigma_n|_4^2+|\tau_n|_4^2)^2\\
&\geq \frac{S^2}{\max\{\mu_1,\mu_2\}}(\mu_1|\sigma_n|_4^4
+\mu_2|\tau_n|_4^4)
\end{aligned}
\end{equation}
 Let $n\to\infty$ on both side of \eqref{f2.16}, we have
\begin{equation}\label{f2.8}
b^2>\frac{S^2}{\max\{\mu_1,\mu_2\}}b,\quad\text{i.e. }
b>\frac{S^2}{\max\{\mu_1,\mu_2\}}.
\end{equation}
 From \eqref{f2.6}, \eqref{f2.3} and \eqref{f2.5}, we have
\begin{equation}
\label{f2.7} b\leq4c .
\end{equation}
Therefore,
\begin{equation}\label{f2.8b}
4c>\frac{S^2}{\max\{\mu_1,\mu_2\}}.
\end{equation}
When $\beta$ is sufficiently large, this is a contradiction with the
following lemma. This completes the proof.
 \end{proof}

\begin{lemma} \label{le2.1} As $\beta\to\infty$, $c\to0$.
\end{lemma}

\begin{proof}
Fix a nontrivial $W\in H^1_0(\Omega)$.
 There exists $t_0>0$ such that  $(t_0W,t_0W)\in\mathcal N$. Indeed,
\begin{equation}\label{f2.9}
t_0=(\frac{\|W\|_{\lambda_1}^2
+\|W\|_{\lambda_2}^2}{t_0^{4-2p}(\mu_1+\mu_2)|W|_4^4
+2\beta|W|_{2p}^{2p}})^{\frac{1}{2(p-1)}}
 \leq O(\frac{1}{\beta^{1/2(p-1)}})\end{equation}
 as $\beta\to\infty$.
Hence
\begin{equation} \label{f2.10}
\begin{aligned}
 c&\leq E(t_0W,t_0W)\\
 &= \frac{1}{2}(1-\frac{1}{p})t_0^2(\|W\|_{\lambda_1}^2
   +\|W\|_{\lambda_2}^2)+\frac{1}{2}(\frac{1}{p}-\frac{1}{2})t_0^4(\mu_1+\mu_2)|W|_4^4\\
  &\leq O(\frac{1}{\beta^{1/(p-1)}}).
\end{aligned}
\end{equation}
\end{proof}

\subsection*{Nontrivial solution is  nonstandard}

In this subsection, we will show that the nontrivial least energy
solution in subsection 2.1 is nonstandard.

\begin{theorem}\label{th2.2}
The solution obtained in Theorem \ref{th2.1} is nonstandard.
\end{theorem}

\begin{proof}
From Br\'ezis-Nirenberg's  theorem(\cite{BN}, see also \cite[Theorem
1.45]{Wi}), there exists nontrivial solution $W_i\in H^1_0(\Omega)$
 for
\begin{equation}
-\Delta W+\lambda_iW=\mu_iW^3,\quad i=1,2.
\end{equation}
In fact, the $W_i$'s are  mountain pass solutions and hence they are
least energy solutions with respective energies
\begin{equation}
I_i=\frac{1}{4}\|W_i\|_{\lambda_i}^2=\frac{\mu_i}{4}|W_i|_4^4,\quad
i=1,2.
\end{equation}
From the proof of Lemma \ref{le2.1}, \eqref{f2.9} and \eqref{f2.10},
for sufficiently large $\beta$, we have
\begin{equation}
c<\min\{I_1,I_2\}.
\end{equation}
This implies that, for  sufficiently large $\beta$, any nontrivial
solution with the least energy  must be nonstandard.
\end{proof}

\begin{proof}[Proof of Theorem \ref{th1.1}]
By the maximum principle, any
nonstandard  nonnegative solution of  equations \eqref{e1.1} is
positive. Combining this with Theorem \ref{th2.1} and Theorem
\ref{th2.2}, we complete the proof.
\end{proof}

\section{Neumann problem}

In this section,  we  assume that $\lambda_1,\lambda_2$ are
sufficiently large as in \cite{Wxj}, but not independent of $\beta$.
Using the same procedure as in section 2,  we come to prove
existence of nonstandard solution of problem \eqref{e1.2}.

 In this section,  except we set $X=H^1(\Omega)\times H^1(\Omega)$
and let $W_i$ be the positive  least energy solution of problem
\eqref{e3.1}, we use the same notations and definitions $c,\Gamma,
\mathcal N$ as in section 2.

\begin{proof}[Proof of Theorem \ref{th1.2}]
We follow the same procedure as
in section 2 and we only give a sketch of the proof.

\noindent {\it Claim 1.} The least energy $c\to 0$ as $\beta\to\infty$.
The proof is the same as Lemma \ref{le2.1}.

\noindent {\it Claim 2.} Any least energy solution is  nonstandard
for sufficiently large $\beta$.
 From \cite[Theorem 3.1]{Wxj}, for $i=1,2$, problem
\begin{equation}\label{e3.1}
-\Delta W+\lambda_iW=\mu_iW^3\quad\text{in }\Omega,\quad
\frac{\partial W}{\partial\nu}=0\quad\text{on }\partial\Omega
\end{equation}
 possesses a positive solution $W_i$ for $\lambda_i$ suitably large.
In fact, the nonconstant solution in \cite{Wxj} is a mountain pass
and hence a least energy solution. Assume $I_i,i=1,2$ are their
corresponding least energies. By the  proof of Theorem \ref{th2.2},
we have $c<\min\{I_1,I_2\}$. So nontrivial least  energy solutions
are nonstandard.

\noindent{\it Claim 3.} Existence of nontrivial solution.
 Assume that $\{(u_j,v_j\}_{j=1}^\infty$ is a nonnegative minimizing
sequence for the mountain pass energy $c$, i.e.
\begin{equation}
E(u_j,v_j)\to c,\quad E'(u_j,v_j)\to0\quad\text{in } X'.
 \end{equation}

The same procedure as in section 2 \eqref{f2.2} implies that as
$j\to\infty$
\begin{equation} \label{f3.5}
\frac{1}{2}(1-\frac{1}{p})(\|u_j\|_{\lambda_1}^2
 +\|v_j\|_{\lambda_2}^2)+\frac{1}{2}(\frac{1}{p}
 -\frac{1}{2})(\mu_1|u_j|_4^4+\mu_2|v_j|_4^4)=c+o(1),
\end{equation}
Hence   $\{\|u_j\|_{\lambda_1}\}$ and $\{\|v_j\|_{\lambda_2}\}$ are
bounded sequences. Going if necessary to a subsequence, there exists
$(u,v)\in X$ such that
\begin{gather*}
u_j\rightharpoonup u,\quad  v_j\rightharpoonup v,
\quad\text{in }H^1(\Omega),\\
u_j\to u,\quad v_j\to v, \quad\text{in }L^2(\Omega),\\
u_j\to u,\quad  v_j\to v, \quad \text{a.e. in }  \Omega ,\\
u_j^3\rightharpoonup u^3,\quad  v_j^3\rightharpoonup v^3,
\quad\text{in }
L^{4/3(\Omega)}\\
u_j^{p-1}v_j^p\rightharpoonup u^{p-1}v^p,\quad
u_j^pv_j^{p-1}\rightharpoonup u^pv^{p-1}, \quad\text{in }
L^{2/p}(\Omega).
\end{gather*}
Hence $(u,v)$ is nonnegative and satisfies the equations in
\eqref{e1.2}.

\noindent{\bf\it Claim:} $(u,v)\neq(0,0)$.
Otherwise, $(u_j,v_j)\rightharpoonup(0,0)$ in $H^1(\Omega)\times
H^1(\Omega)$, $(u_j,v_j)\to(0,0)$ in $L^2(\Omega)\times
L^2(\Omega)$, and $u_jv_j\to0$ in $L^{p}(\Omega)$.
 From \cite[Lemma 2.1, page 289]{Wxj}, for any $\varepsilon>0$, as
$j\to\infty$
\begin{gather}
S_\varepsilon|u_j|_4^2\leq |\nabla u_j|_2^2+o(1),\label{f3.3}\\
S_\varepsilon|v_j|_4^2\leq |\nabla v_j|_2^2+o(1)\label{f3.4}
\end{gather}
where $S_\varepsilon=(2^{-1/2}S-\varepsilon)(1+\varepsilon)^{-1}$,
$S$ is the Sobolev constant.

Assume that $|\nabla u_j|_2^2+|\nabla v_j|_2^2\to b$.
Since $(E'(u_j,v_j),(u_j,v_j))\to0$ and $|u_jv_j|_p^p\to0$,
\begin{equation}\label{f3.6}
|\nabla u_j|_2^2+|\nabla v_j|_2^2=\mu_1|u_j|_4^4+\mu_2|v_j|_4^4+o(1).
\end{equation}
 It follows that
\begin{equation}\label{f3.7}
 E(u_j,v_j)=\frac{1}{4}(|\nabla u_j|_2^2+|\nabla v_j|_2^2)+o(1)
=\frac{1}{4}(\mu_1|u_j|_4^4+\mu_2|v_j|_4^4)+o(1)\to
c>0.
\end{equation}
 If $b=0$, this is a contradiction with \eqref{f3.7}. If $b>0$,
by \eqref{f3.3},\eqref{f3.4}, we have
\begin{equation}
(|\nabla u_j|_2^2+|\nabla v_j|_2^2)^2\geq
S_\varepsilon^2(|u_j|_4^2+|v_j|_4^2)^2+o(1)\geq
C_\varepsilon(\mu_1|u_j|_4^4+\mu_2|v_j|_4^4)+o(1)
\end{equation}
where $C_\varepsilon=\frac{S_\varepsilon^2}{\max\{\mu_1,\mu_2\}}$.
 From \eqref{f3.6},
\begin{equation}
b^2\geq C_\varepsilon b,\quad \text{i.e., }
  b\geq C_\varepsilon.
\end{equation} This is a
contradiction with \eqref{f3.7} and  Claim 1. Hence  $(u,v)$ is
nontrivial.
The same procedure as in \cite[page 9]{LNT} (also \cite{Wxj})
implies that the solution is positive. This completes the proof.
\end{proof}

\subsection*{Remarks} (1)  The arguments developed in this paper also work
in dimension $N>4$ with critical nonlinearities. For simplification
in writing, we consider only the case $N=4$.

(2) According to the proof developed in the paper we cannot exclude
the possibility that  the solutions of Neumann problem are
nonconstant for some suitably chosen large $\beta$.

\subsection*{Acknowledgements} The author would like to thank the
anonymous referees for their useful suggestions and helpful comments,
which improve the original manuscript.

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