# Semiclassical states for coupled nonlinear Schrödinger equations with a
critical frequency

Taiyong Chen  ,  Yahui Jiang  ,  Marco Squassina  and  Jianjun Zhang
School of Mathematics
China University of Mining and Technology
Xuzhou, 221116, China<EMAIL_ADDRESS>
School of Mathematics
China University of Mining and Technology
Xuzhou, 221116, China<EMAIL_ADDRESS>
Dipartimento di Matematica e Fisica,
Universita Cattolica del Sacro Cuore,
Via della Garzetta 48, 25133, Brescia, Italy<EMAIL_ADDRESS>
College of Mathematica and Statistics
Chongqing Jiaotong University
Chongqing 400074, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the the coupled nonlinear Schrödinger system

$\displaystyle\begin{cases}-\varepsilon^{2}\Delta u+a(x)u=\mu_{1}u^{3}+\beta
v^{2}u\ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\\ -\varepsilon^{2}\Delta
v+b(x)v=\mu_{2}v^{3}+\beta u^{2}v\ \ \ \ \ \mbox{in}\
\mathbb{R}^{N},\end{cases}$

where $1\leq N\leq 3$, $\mu_{1},\mu_{2},\beta>0$, $a(x)$ and $b(x)$ are
nonnegative continuous potentials, and $\varepsilon>0$ is a small parameter.
We show that the existence of positive ground state solutions for this system
and exhibit the concentration behaviour as $\varepsilon\rightarrow 0$ when
$a(x)$ and $b(x)$ achieve 0 with a homogeneous behaviour or vanish on the
nonempty open set with smooth boundary.

###### Key words and phrases:

Nonlinear Schrödinger systems, semiclassical limit, critical frequency,
variational methods.

###### 2000 Mathematics Subject Classification:

35B05 35J50

(1) Corresponding author<EMAIL_ADDRESS>
(2) M. Squassina is member of the Gruppo Nazionale per l’Analisi Matematica,
la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di
Alta Matematica (INdAM). J. J. Zhang was partially supported by
NSFC(No.11871123).

##  1\. Introduction

1.1 Background. Consider the following two-component coupled nonlinear
Schrödinger equations (also known as Gross-Pitaevskii equations):

(1.1)
$\displaystyle\begin{cases}-i\hbar\frac{\partial}{\partial_{t}}\Phi_{1}+V_{1}(x)\Phi_{1}=\frac{\hbar^{2}}{2m}\Delta\Phi_{1}+\mu_{1}|\Phi_{1}|^{2}\Phi_{1}+\beta|\Phi_{2}|^{2}\Phi_{1},\
\ \ \mbox{in}\ \mathbb{R}^{N},\ t>0,\\\
-i\hbar\frac{\partial}{\partial_{t}}\Phi_{2}+V_{2}(x)\Phi_{2}=\frac{\hbar^{2}}{2m}\Delta\Phi_{2}+\mu_{2}|\Phi_{2}|^{2}\Phi_{2}+\beta|\Phi_{1}|^{2}\Phi_{2},\
\ \ \mbox{in}\ \mathbb{R}^{N},\ t>0,\end{cases}$

where $N\leq 3$, $i$ is the imaginary unit, $\hbar$ is the Plank constant,
$\mu_{1},\mu_{2}>0$ and $\beta\neq 0$ is a coupling constant. System
($\mathcal{C}_{\varepsilon}$) has applications in many physical problems such
as in nonlinear optics, continuum mechanics, plasma physics and Bose-Einstein
condensates theory for multispecies Bose-Einstein condensates. In the context
of Bose-Einstein condensates, $\Phi_{j}(x,t)(j=1,2)$ are the corresponding
condensate amplitudes; $V_{j}(x)$ are the traping potentials; $\mu_{j}$ and
$\beta$ are the intraspecies and interspecies scattering lengths. The
physically realistic spatial dimension are $1\leq N\leq 3$. When $N=2$,
problem (1.1) arises in the Hartree-Fock theory for a double condenstate,
$i.e.$, a binary mixture of Bose-Einstein condensates in two different
hyperfine states $|1\rangle$ and $|2\rangle$ (see[1, 2]). The sign of $\beta$
determines whether the interactions of states are repulsive and attractive.
The interaction is attractive if $\beta>0$, and the interaction is repulsive
if $\beta<0$, where the two states are in strong competition.

A standing wave solution of (1.1) is of the following form:

(1.2) $\displaystyle\Phi_{1}(x,t)=e^{-iE_{t}/\hbar}u(x)\ \ \mbox{and}\ \
\Phi_{2}(x,t)=e^{-iE_{t}/\hbar}v(x).$

Substituting (1.2) into (1.1), then the system is reduced to the following
elliptic system:

(1.3) $\displaystyle\begin{cases}-\frac{\hbar^{2}}{2m}\Delta
u+(V_{1}(x)-E)u=\mu_{1}u^{3}+\beta v^{2}u,\ \ \ \mbox{in}\ \mathbb{R}^{N},\\\
-\frac{\hbar^{2}}{2m}\Delta v+(V_{2}(x)-E)v=\mu_{2}v^{3}+\beta u^{2}v,\ \ \ \
\mbox{in}\ \mathbb{R}^{N}.\\\ \end{cases}$

Renaming the parameters by

$\displaystyle\varepsilon=\sqrt{\frac{\hbar^{2}}{2m}},\ a(x)=V_{1}(x)-E,\
b(x)=V_{2}(x)-E,$

we obtain the folowing elliptic system:

($\mathcal{C}_{\varepsilon}$)
$\displaystyle\begin{cases}-\varepsilon^{2}\Delta u+a(x)u=\mu_{1}u^{3}+\beta
v^{2}u\ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\\ -\varepsilon^{2}\Delta
v+b(x)v=\mu_{2}v^{3}+\beta u^{2}v\ \ \ \ \ \mbox{in}\
\mathbb{R}^{N},\end{cases}$

Before going further, we point out that the system
($\mathcal{C}_{\varepsilon}$) also possesses a trivial solution $(0,0)$ and
semi-trivial solutions of type $(u,0)$ or $(0,v)$. A solution $(u,v)$ of
($\mathcal{C}_{\varepsilon}$) is nontrivial if $u\not\equiv 0$ and
$v\not\equiv 0$. A solution $(u,v)$ with $u>0$ and $v>0$ is called a positive
solution.

When $\varepsilon=1$, $a(x)$ and $b(x)$ are positive constants, the existence,
multiplicity, bifurcation, concentration behavior of positive solutions of
($\mathcal{C}_{\varepsilon}$) have been entensively studied, see Refs.[3, 4,
5, 7, 8, 6, 9, 11, 10, 12, 13]. Sign and size of $\beta$ are important in the
study of constant coefficient system. Particularly, the interesting paper [12]
proved that there exists $0<\bar{\beta_{1}}<\bar{\beta_{2}}$ such that
$\mathcal{C}_{\varepsilon}$ has nontrivial positive solutions for
$0<\beta<\bar{\beta_{1}}$ or $\beta>\bar{\beta_{2}}$. For the non-constant
potentials case, according to Bohr’s corresponding principle the classical
description of the mechanical physics should be recovered by considering the
semiclassical limit, i.e. when the Planck constant is sent to zero. Therefore
beside proving existence of solutions to $\mathcal{C}_{\varepsilon}$, it is
physically relevant to study the asymptotic limit as $\varepsilon\rightarrow
0$ of such solutions.

The semiclassical limit is well understood for the nonlinear Schrödinger
equation

$\displaystyle-\varepsilon^{2}\Delta
u_{\varepsilon}+Vu_{\varepsilon}=|u_{\varepsilon}|^{p-1}u_{\varepsilon}\ \ \
\mbox{in}\ \mathbb{R}^{N}.$

The semi-classical states of the above equations have different limiting
behaviors depend upon $\inf_{\mathbb{R}^{N}}V>0$ or
$\inf_{\mathbb{R}^{N}}V=0$, the latter is regarded as the $critical$
$frequency$ case. When $\inf_{\mathbb{R}^{N}}V>0$, P. Rabinowitz [14] proved
that the above problem has a least energy solution for $\varepsilon>0$ small
in the case that

$\inf_{\mathbb{R}^{N}}V<\liminf_{|x|\rightarrow\infty}V(x).$

In case $\inf_{\mathbb{R}^{N}}V>0$, solutions concentrating at critical points
of the potential $V$ have been construted by topological and variational
methods [15, 16, 14, 17] and many others. Motivations can be dated back to
Floer snd Weinstein in there pioneering work [15]. However, the semi-classical
solutions exhibit quite different characteristic features that the energy
governed by the shape of the potential $V$ near the zero set of $V$ in the
critical frequency case $\inf_{\mathbb{R}^{N}}V=0$ [18, 19].

The semi-classical case ($\mathcal{C}_{\varepsilon}$) with trapping potentials
$a(x)$ and $b(x)$ has been studied in some years. Lin and Wei[20] studied
$\mathcal{C}_{\varepsilon}$ by analyzing least energy non-trivial vector
solutions. They studied both of attractive interaction and replusive
interaction. Especially, when $\beta>0$, they showed the existence of a least
energy. In [22], for small $\beta>0$, Montefusco, Pellacci and Squassina
exhibited the existence of nonnegative ground state solutions of
($\mathcal{C}_{\varepsilon}$) concentrating around the local minimum points of
the potentials, which are in the same region. Ikoma and Tanaka [23] also
considered the case $\beta>0$. They connected the solutions of
$\mathcal{C}_{\varepsilon}$ with the limiting system with
$\lambda_{1}=a(x_{0})$ and $\lambda_{2}=b(x_{0})$ for a fixed
$x_{0}\in\mathbb{R}^{N}$. Assume that there exists an open bounded set
$\Lambda\subset\mathbb{R}^{N}$ such that

$\displaystyle\mathop{\inf}\limits_{x_{0}\in\Lambda}m(x_{0})<\mathop{\inf}\limits_{x_{0}\in\partial\Lambda}m(x_{0}),$

where $m_{x_{0}}$ is the ground state energy level of the limiting system.
When $\beta>0$ is relatively small, they constructed a family of solutions of
($\mathcal{C}_{\varepsilon}$) which concentrates to a positive vector
solution. For the large $\beta>0$ case, Shi and Wang [25] showed the positive
ground state solution of ($\mathcal{C}_{\varepsilon}$) and concentrate near
the minimum points of potential functions. They also showed that multiple
positive concentration solutions exist when the topological structure of the
set of minimum points satisfies certain condition. We also refer to [20, 21,
24] for the study of $\mathcal{C}_{\varepsilon}$ when $\beta<0$.

Note that in all works mentioned above, they all assumed that $a(x)$ and
$b(x)$ are positive bounded away from 0. In this paper, we consider the semi-
classical in the critical frequency case

$\displaystyle\mathop{\inf}\limits_{x\in\mathbb{R}^{N}}a(x)=\mathop{\inf}\limits_{x\in\mathbb{R}^{N}}b(x)=0.$

For the large $\beta>0$ case, Chen and Zou [26] assumed $a(x)$ and $b(x)$ are
non-negative and may vanish at someplace and decay to $0$ at infinity. They
proved the existence of positive solutions for $\mathcal{C}_{\varepsilon}$
which concentrated around local minima of the potentials as
$\varepsilon\rightarrow 0$ under suitable conditions on the behaviour of
$a(x)$ and $b(x)$. Lucia and Tang [27] obtained synchronized solutions which
trapped near zero sets of the potentials for $n-$coupled system under some
conditions. In [28], Tang and Wang obtained multi-scale segregated positive
solutions which are trapped near the nondegenerate critical points as well as
the zero set of potentials by using Lyapunov-Scjmidt reduction method. Tang
and Xie [29] studied $a(x)$ and $b(x)$ admits several common isolated
connected components by using spectral analysis. They obtained the existence
synchronized positive vector solutions which are trapped in a neighbourhood of
the zero set of potentials and also the local maximum points of potentials for
$\varepsilon$ small.

1.2 Motivation. In this paper, inspired by Byeon and Wang [18], we want to
study the asymptotic behavior of positive solutions of
($\mathcal{C}_{\varepsilon}$) in the critical frequency and $\beta>0$
sufficiently large case. We consider the coupled system with the potentials
$a(x)$ and $b(x)$ achieve $0$ with homogeneous behaviour or vanish on the
closure of an open set but remains bounded away from $0$ at infity. Throughout
the paper we assume that

$(f_{1})$ $\mu_{1},\mu_{1},\beta>0$.

$(f_{2})$ $a\in C(\mathbb{R}^{N},\mathbb{R})$ satisfying

$0=\inf_{x\in\mathbb{R}^{N}}a(x)<\liminf_{|x|\rightarrow\infty}a(x).$

$(f_{3})$ $b\in C(\mathbb{R}^{N},\mathbb{R})$ satisfying

$0=\inf_{x\in\mathbb{R}^{N}}b(x)<\liminf_{|x|\rightarrow\infty}b(x).$

One of the difficulties in the study of $\mathcal{C}_{\varepsilon}$ is that it
has semi-trivial solutions of type $(u,0)$ and $(0,v)$, where $u(x)$ where
$v(x)$ solves

(1.4) $\displaystyle-\varepsilon^{2}\Delta u+a(x)u=\mu_{1}u^{3},\ \ \mbox{in}\
\mathbb{R}^{N}.$ (1.5) $\displaystyle-\varepsilon^{2}\Delta
v+b(x)v=\mu_{2}v^{3},\ \ \mbox{in}\ \mathbb{R}^{N}.$

In the case of a single perturbation NLS equation

$\displaystyle-\varepsilon^{2}\Delta u+V(x)u=u^{3},\ \ \mbox{in}\
\mathbb{R}^{N},$

has a ground state solution [18]. Here, we borrow an idea of [12] to overcome
this difficulty. Let $(U,0)$ and $(0,V)$ denote the positive ground state
solutions of (1.4) and (1.5).

1.3 Main results. Let $\mathcal{A}=\\{x\in\mathbb{R}^{N}|a(x)=0\\}$,
$\mathcal{B}=\\{x\in\mathbb{R}^{N}|b(x)=0\\}$,
$\Omega=\mathcal{A}\cap\mathcal{B}$ is a non-empty set and
$\Sigma=\mathcal{A}\cup\mathcal{B}$. Set

$\displaystyle\beta_{0}=\max\left\\{\mu_{2}\frac{\int_{\mathbb{R}^{N}}V^{4}}{\int_{\mathbb{R}^{N}}U^{2}V^{2}},\mu_{1}\frac{\int_{\mathbb{R}^{N}}U^{4}}{\int_{\mathbb{R}^{N}}U^{2}V^{2}}\right\\},$
$\displaystyle\beta_{1}=\max\\{\mu_{1},\mu_{2}\\},$
$\displaystyle\beta_{2}=\max\left\\{\mu_{2}\frac{\int_{\mathcal{B}}V_{0}^{4}}{\int_{\Omega}U_{0}^{2}V_{0}^{2}},\mu_{1}\frac{\int_{\mathcal{A}}U_{0}^{4}}{\int_{\Omega}U_{0}^{2}V_{0}^{2}}\right\\},$

where $U_{0}$ denotes the positive solutions of $-\Delta u=\mu_{1}u^{3}\
\mbox{in}\ \mathcal{A}$, $V_{0}$ denotes the positive solutions of $-\Delta
v=\mu_{2}v^{3}\ \mbox{in}\ \mathcal{B}$.

######  Definition 1.1.

A function $W:\mathbb{R}^{N}\rightarrow\mathbb{R}$ is called a positive
$\gamma-$homogeneous function if $W(y)>0$ for $y\neq 0$ and
$W(ty)=t^{\gamma}W(y)$ for any $t\in[0,+\infty)$ and $y\in\mathbb{R}^{N}$.

Let $\widehat{\beta}=\max\\{\beta_{0},\beta_{1},\beta_{2}\\}$, we are ready to
state our main result.

######  Theorem 1.2.

Let $1\leq N\leq 3$ and $\beta>\widehat{\beta}$. Assume that
$(f_{1})$-$(f_{3})$ hold and there exists $\gamma>0$ such that for every
$x\in\Omega$, one of the following two cases holds

  * $(i)$

$\mathop{\lim}\limits_{z\rightarrow x}\frac{a(z)}{|z-x|^{\gamma}}=+\infty;$

  * $(ii)$

there exists a positive $\gamma$-homogeneous function $W\in C(\mathbb{R}^{N})$
such that

$\displaystyle\mathop{\lim}\limits_{z\rightarrow
x}\frac{a(z)-W(z-x)}{|z-x|^{\gamma}}=0.$

Denote by $\\{x_{i}\in\Omega:i\in\mathcal{I}\\}$ the set of all points, where
Case $(ii)$ hold. Additionally, if
$\\{x_{i}\in\Omega:i\in\mathcal{I}\\}\not=\emptyset$ and for any
$i\in\mathcal{I}$,

$\displaystyle\mathop{\lim}\limits_{z\rightarrow
x_{i}}\frac{b(z)-W(z-x_{i})}{|z-x_{i}|^{\gamma}}=0,$

then

  * $(1)$

for sufficiently small $\varepsilon>0$, system ($\mathcal{C}_{\varepsilon}$)
has a positive ground state solution $(u_{\varepsilon},v_{\varepsilon})$;

  * $(2)$

there exist $x_{*}\in\\{x_{i};i\in\mathcal{I}\\}$ and a positive
$\gamma$-homogeneous function $W_{x_{*}}\in C(\mathbb{R}^{N})$ such that

$\displaystyle\mathop{\lim}\limits_{z\rightarrow
x_{*}}\frac{a(z)-W_{x_{*}}(z-x_{*})}{|z-x_{*}|^{\gamma}}=0\ \mbox{and}\
\mathop{\lim}\limits_{z\rightarrow
x_{*}}\frac{b(z)-W_{x_{*}}(z-x_{*})}{|z-x_{*}|^{\gamma}}=0;$

  * $(3)$

as $\varepsilon\rightarrow 0$,

$\displaystyle\varepsilon^{-\frac{\gamma}{\gamma+2}}\left(u_{\varepsilon}(\varepsilon^{\frac{2}{\gamma+2}}\cdot+x_{*}),v_{\varepsilon}(\varepsilon^{\frac{2}{\gamma+2}}\cdot+x_{*})\right)\rightarrow(w,\phi)\
\mbox{strongly in}\ H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N});$

  * $(4)$

$\displaystyle\mathop{\lim}\limits_{\varepsilon\rightarrow
0}\frac{I_{\varepsilon}(u_{\varepsilon},v_{\varepsilon})}{\varepsilon^{\frac{2}{\gamma+2}(N+2\gamma)}}=\mathcal{E}_{W_{x_{*}}}=\mathop{\inf}\limits_{i\in\mathcal{I}}\mathcal{E}_{W_{i}}.$

Consider the limit problem of $\mathcal{C}_{\varepsilon}$ with homogeneous
potentials

$\displaystyle\begin{cases}-\Delta u+W(x)u=\mu_{1}u^{3}+\beta v^{2}u\ \ \ \
\mbox{in}\ \mathbb{R}^{N},\\\ -\Delta v+W(x)v=\mu_{2}v^{3}+\beta u^{2}v\ \ \ \
\ \mbox{in}\ \mathbb{R}^{N},\end{cases}$

whose corresponding functional $\mathcal{J}_{W}(u,v)\in
C^{1}(H_{W}(\mathbb{R}^{N})\times H_{W}(\mathbb{R}^{N}))$ is defined by

$\displaystyle\mathcal{J}_{W}(u,v)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+W(x)u^{2}+|\nabla
v|^{2}+W(x)v^{2}-\frac{1}{4}\int_{\mathbb{R}^{N}}\mu_{1}v^{4}+2\beta
u^{2}v^{2}+\mu_{2}v^{4}.$

on the Hilbert space $H_{W}(\mathbb{R}^{N})$ obtained by completion of the set
of smooth functions $C^{\infty}_{0}(\mathbb{R}^{N})$ endowed with the norm

$\displaystyle\|u\|_{H_{W}}=\Big{(}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+Wu^{2}\Big{)}^{\frac{1}{2}}.$

Since the potential $W$ is coercive, one can show that (see Section 4)

$\displaystyle\mathcal{E}_{W}:=\inf\\{\mathcal{J}_{W}(u,v)|(u,v)\in
H_{W}(\mathbb{R}^{N})\times
H_{W}(\mathbb{R}^{N})\setminus\\{(0,0)\\},\langle\mathcal{J}_{W}^{\prime}(u,v),(u,v)\rangle=0\\},$

could be attained by a positive ground state solution $(w,\phi)$.

Next we consider the case that the potential $a$ and $b$ vanish on the
nonempty open set with smooth boundary for the nonlinear Schrödinger system.

######  Theorem 1.3.

Let $1\leq N\leq 3$. Assume that $(f_{1})-(f_{3})$ hold and
$\beta>\widehat{\beta}$. Let $\mathcal{A}$ and $\mathcal{B}$ be nonempty open
set with smooth boundary. Then, for sufficiently small $\varepsilon>0$, the
system ($\mathcal{C}_{\varepsilon}$) has a positive groundstate solution
$(u_{\varepsilon},v_{\varepsilon})$. Moreover, there exists a ground state
solution $(w,z)\in H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N})$ of the
following problem

$\displaystyle\begin{cases}-\Delta w=\mu_{1}w^{3}+\beta z^{2}w,\ \ \ \ x\in\
\mathcal{A},\\\ -\Delta z=\mu_{2}z^{3}+\beta w^{2}z,\ \ \ \ \ \ x\in\
\mathcal{B},\\\ w\in H_{0}^{1}(\mathcal{A}),\ \ z\in
H_{0}^{1}(\mathcal{B}).\end{cases}$

such that, as $\varepsilon\rightarrow 0$,

$\displaystyle\varepsilon^{-\frac{1}{2}}(u_{\varepsilon},v_{\varepsilon})\rightarrow(w,z)\
\ \mbox{in}\ H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N}).$

For the proof of our results, we shall consider an equivalent system to
($\mathcal{C}_{\varepsilon}$). By the scaling $a(x)=a(\varepsilon x)$,
$b(x)=b(\varepsilon x)$, we then deduce that

($\mathcal{P}_{\varepsilon}$) $\displaystyle\begin{cases}-\Delta
u+a(\varepsilon x)u=\mu_{1}u^{3}+\beta uv^{2}\ \ \ \ x\in\mathbb{R}^{3},\\\
-\Delta v+b(\varepsilon x)v=\mu_{1}v^{3}+\beta u^{2}v\ \ \ \
x\in\mathbb{R}^{3},\end{cases}$

Thus, our theorem for ($\mathcal{C}_{\varepsilon}$) are equivalent to the
results for ($\mathcal{P}_{\varepsilon}$). Hence, in the sequel we focus on
the system ($\mathcal{P}_{\varepsilon}$).

The rest of this paper is organized as follows. In section 2, we introduce a
variational setting of our problem and present some preliminary results about
the extended problem. Section 3 is devoted to the study of positive ground
state solutions of the limit problem ($\mathcal{P}_{\varepsilon}$). Section 4
is devoted to the proof of Theorem 1.2 and Theorem 1.3.

##  2\. Preliminaries

To prove the main results, we use the following notations:

$\bullet$ $\mathbb{H}:=H^{1}(\mathbb{R}^{N})\times H^{1}(\mathbb{R}^{N})$ with
norm

$\displaystyle\|(u,v)\|_{\mathbb{H}}=(\|u\|_{H^{1}}^{2}+\|v\|_{H^{1}}^{2})^{1/2};$

$\bullet$ $\mathbb{L}^{q}(\mathbb{R}^{N}):=L^{q}(\mathbb{R}^{N})\times
L^{q}(\mathbb{R}^{N})$;

$\bullet$
$\mathbb{L}^{q}_{loc}(\mathbb{R}^{N}):=L^{q}_{loc}(\mathbb{R}^{N})\times
L^{q}_{loc}(\mathbb{R}^{N})$;

$\bullet$
$\mathbb{C}^{\infty}_{0}(\mathbb{R}^{N}):=C^{\infty}_{0}(\mathbb{R}^{N})\times
C^{\infty}_{0}(\mathbb{R}^{N})$;

$\bullet$ $a_{\infty}:=\mathop{\liminf}\limits_{|x|\rightarrow\infty}a(x)$ and
$b_{\infty}:=\mathop{\liminf}\limits_{|x|\rightarrow\infty}b(x)$;

$\bullet$ $2^{*}=6$ when $N=3$, and $2^{*}=+\infty$ when $N=1,2$.

For $\varepsilon>0$, we define the rescaled weighted Sobolev space
$H_{a,\varepsilon}$ by

$\displaystyle H_{a,\varepsilon}=\left\\{u\in
H^{1}(\mathbb{R}^{N})|\int_{\mathbb{R}^{N}}a(\varepsilon
x)u^{2}<+\infty\right\\},$

endowed with the norm

$\displaystyle\|u\|_{a,\varepsilon}=\Big{(}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+a(\varepsilon x)u^{2}\Big{)}^{\frac{1}{2}}.$

Define $E:=H_{a,\varepsilon}\times H_{b,\varepsilon}$ with norm

$\displaystyle\|(u,v)\|_{E}=(\|u\|_{a,\varepsilon}^{2}+\|v\|_{b,\varepsilon}^{2})^{1/2}.$

We set

$\displaystyle F(u,v)=\int_{\mathbb{R}^{N}}\mu_{1}u^{4}+2\beta
u^{2}v^{2}+\mu_{2}v^{4}.$

The energy functional for ($\mathcal{P}_{\varepsilon}$) is defined by

$\displaystyle I_{\varepsilon}(u,v):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+a(\varepsilon x)u^{2}+|\nabla v|^{2}+b(\varepsilon
x)v^{2}-\frac{1}{4}F(u,v).$

for $(u,v)\in E$.

We first illustrate the space $E$ can be embedded continuously into the
Sobolev space $\mathbb{H}$ for fixed $\varepsilon>0$, even though the
potential $a$ and $b$ have a nontrivial set of zeros.

######  Lemma 2.1.

Let $a:\mathbb{R}^{N}\rightarrow[0,+\infty)$. If
$\liminf_{|x|\rightarrow\infty}a(x)>0$ then for every $\varepsilon>0$, there
exists a constant $C_{\varepsilon}>0$ such that for $u\in
H_{a,\varepsilon}(\mathbb{R}^{N})$, $u\in H^{1}(\mathbb{R}^{N})$ and

$\displaystyle\int_{\mathbb{R}^{N}}|\nabla u|^{2}+|u|^{2}\leq
C_{\varepsilon}\int_{\mathbb{R}^{N}}|\nabla u|^{2}+a(\varepsilon x)u^{2}.$

Proof. Let $\nu=\frac{1}{2}\liminf_{|x|\rightarrow\infty}a(x)>0$. By
definition of the limit, for any fixed $\varepsilon>0$, there exists
$R_{\varepsilon}>0$ such that $a(\varepsilon x)\geq\nu$ for
$x\in\mathbb{R}^{N}\setminus B_{\frac{R_{\varepsilon}}{2}}$. (Here and in the
sequel, we use the notation $B_{r}(o)$ to denote the ball in $\mathbb{R}^{N}$
with radius $r$ and centred at $o$ and $B_{r}=B_{r}(0)$.) By integration, we
have

(2.1) $\displaystyle\int_{\mathbb{R}^{N}\setminus
B_{\frac{R_{\varepsilon}}{2}}}u^{2}\leq\frac{1}{\nu}\int_{\mathbb{R}^{N}\setminus
B_{\frac{R_{\varepsilon}}{2}}}a(\varepsilon x)u^{2}.$

We take a function $\psi_{\varepsilon}\in C^{\infty}(\mathbb{R}^{N})$ such
that $0\leq\psi_{\varepsilon}\leq 1$ in $\mathbb{R}^{N}$,
$\psi_{\varepsilon}(x)=1$ for $x\in B_{\frac{R_{\varepsilon}}{2}}$ and
$\psi_{\varepsilon}(x)=0$ for $x\in\mathbb{R}^{N}\setminus
B_{R_{\varepsilon}}$. Then, it follows from the classical Poincaré inequality
on the ball $B_{R_{\varepsilon}}$ that

(2.4)
$\displaystyle\begin{array}[]{ll}\int_{B_{\frac{R_{\varepsilon}}{2}}}|u|^{2}&\leq\int_{B_{R_{\varepsilon}}}|\psi_{\varepsilon}u|^{2}\leq
C_{1}\int_{B_{R_{\varepsilon}}}|\nabla(\psi_{\varepsilon}u)|^{2}\\\ &\leq
2C_{1}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+\frac{2C_{1}\|\nabla\psi_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{N})}^{2}}{\nu}\int_{B_{R_{\varepsilon}}\setminus
B_{\frac{R_{\varepsilon}}{2}}}a(\varepsilon x)u^{2},\end{array}$

where $C_{1}$ depends on $\varepsilon$. The conclusion then follows from the
combination of the inequalities (2.1) and (2.4). $\Box$

Similar result holds for $H_{b,\varepsilon}$. Lemma 2.1 implies that the space
$E$ embeds into $\mathbb{H}$ continuously. Moreover the space $E$ can be
continuously embedded in $\mathbb{L}^{q}(\mathbb{R}^{N})$ when
$q\in[2,2^{*}]$. It then follows that the functional $I_{\varepsilon}$ is well
defined in $E$ and is of class $C^{1}$.

A solution $(u,v)$ is a groundstate of the equation
($\mathcal{P}_{\varepsilon}$) is the least among all nontrival critical values
of $I_{\varepsilon}(u,v)$, namely, $(u,v)$ has the least energy among
nontrivial solutions. A natural method to search the groundstate is to
minimize the functional $I_{\varepsilon}(u,v)$ on the $Nehari$ $manifold$ of
the equation which is defined by

$\displaystyle\mathcal{N}_{\varepsilon}:=\left\\{(u,v)\in
E\setminus\left\\{(0,0)\right\\}|\langle
I_{\varepsilon}^{\prime}(u,v),(u,v)\rangle=0\right\\}.$

The corresponding groundstate energy is described as

$\displaystyle
c_{\varepsilon}:=\mathop{\inf}\limits_{(u,v)\in\mathcal{N}_{\varepsilon}}I_{\varepsilon}(u,v).$

######  Lemma 2.2.

For given $\varepsilon>0$, the groundstate energy $c_{\varepsilon}$ is
positive and $\mathcal{N}_{\varepsilon}$ is a manifold of class of $C^{1}$.
Moreover, if $(u,v)\in\mathcal{N}_{\varepsilon}$ is a critical point of
$I_{\varepsilon}|_{\mathcal{N}_{\varepsilon}}$, then
$I_{\varepsilon}^{\prime}(u,v)=0$.

Proof. We fix $\varepsilon>0$. For any $(u,v)\in
E\setminus\left\\{(0,0)\right\\}$, we have that

$\displaystyle t(u,v)\in\mathcal{N}_{\varepsilon}\Leftrightarrow
t^{2}\|(u,v)\|_{E}^{2}=t^{4}F(u,v).$

As a consequence, for all $(u,v)\in E\setminus\left\\{(0,0)\right\\}$, there
exists a unique $t>0$ such that $t(u,v)\in\mathcal{N}_{\varepsilon}$.
Moreover, since $F(u,v)$ is homogeneous with degree 4, there is $\rho>0$ such
that

$\displaystyle\|(u,v)\|_{H_{\varepsilon}}^{2}\geq\rho>0,\ \ \
\forall(u,v)\in\mathcal{N}_{\varepsilon}.$

Hence, for any $(u,v)\in\mathcal{N}_{\varepsilon}$, we have

$\displaystyle
I_{\varepsilon}(u,v)=\big{(}\frac{1}{2}-\frac{1}{4}\big{)}\|(u,v)\|_{H_{\varepsilon}}^{2}\geq\frac{1}{4}\rho>0$

thus, $c_{\varepsilon}>0$. We define $\mathfrak{g}(u,v)_{\varepsilon}=\langle
I_{\varepsilon}^{\prime}(u,v),(u,v)\rangle$, for each
$(u,v)\in\mathcal{N}_{\varepsilon}$,

(2.5)
$\displaystyle\langle\mathfrak{g}_{\varepsilon}^{\prime}(u,v),(u,v)\rangle=-2\|(u,v)\|_{H_{\varepsilon}}^{2}\leq-2\rho<0,$

we infer that $\mathcal{N}_{\varepsilon}$ is a smooth complete manifold of
codimension 1 in $H_{\varepsilon}$. Moreover, if
$(u,v)\in\mathcal{N}_{\varepsilon}$ is a critical point of the restricted
functional $I_{\varepsilon}|_{\mathcal{N}_{\varepsilon}}$, then there exists
$\lambda_{\varepsilon}\in\mathbb{R}$, such that

(2.6) $\displaystyle
I_{\varepsilon}^{\prime}(u,v)=\lambda_{\varepsilon}\mathfrak{G}_{\varepsilon}^{\prime}(u,v).$

Then, we have

$\displaystyle\lambda_{\varepsilon}\langle\mathfrak{G}_{\varepsilon}^{\prime}(u,v),(u,v)\rangle=\langle
I_{\varepsilon}^{\prime}(u,v),(u,v)\rangle=0.$

We thus deduce by (2.5) that $\lambda_{\varepsilon}=0$ and the conclusions
follows then from (2.6). $\Box$

##  3\. Existence of positive ground state solutions for
($\mathcal{P}_{\varepsilon}$)

######  Lemma 3.1.

(Mountain-Pass geometry) Assume that $f(1)$ hold, the function
$I_{\varepsilon}(u,v)$ satisfies the following conditions
$(i)$ There exist a positive constant $r>0$ such that $I_{\varepsilon}(u,v)>0$
for $\|(u,v)\|_{E}=r$;
$(ii)$ There exist $(e_{1},e_{2})\in E$ with $\|(e_{1},e_{2})\|_{E}>r$ such
that $I_{\varepsilon}(e_{1},e_{2})<0$.

Proof.

$\displaystyle I_{\varepsilon}(u,v)$
$\displaystyle=\frac{1}{2}\|(u,v)\|_{E}^{2}-F(u,v);$
$\displaystyle\geq\frac{1}{2}\|(u,v)\|_{E}^{2}-C\|(u,v)\|_{E}^{4}.$

Hence, there exist $r>0$ such that

$\displaystyle\mathop{\inf}\limits_{\|(u,v)\|_{E}=r}I_{\varepsilon}(u,v)>0.$

For any $(u,v)\in E\setminus{(0,0)}$ and $t>0$,

$\displaystyle
I_{\varepsilon}(tu,tv)=\frac{t^{2}}{2}\|(u,v)\|_{H_{\varepsilon}}^{2}-\frac{t^{4}}{4}F(u,v),$

which implies that $I_{\varepsilon}(tu,tv)\rightarrow-\infty$ as
$t\rightarrow+\infty$. $\Box$

That is, $I_{\varepsilon}$ has a mountaion-pass structure. From Lemma 3.1, one
can apply the Ambrosetti-Rabinowitz Mountain-Pass Theorem[30] without
$(PS)_{c}$ condition, and it follows that for any small $\varepsilon>0$, there
exists a $(PS)_{c}$ sequence $\left\\{(u_{n},v_{n})\right\\}\subset E$ (with
$c=c_{\varepsilon}^{*}$ defined below) such that

$\displaystyle I_{\varepsilon}(u_{n},v_{n})\rightarrow
c_{\varepsilon}^{*}=\mathop{\inf}\limits_{\gamma\in\Gamma_{\varepsilon}}\mathop{\max}\limits_{0\leq
t\leq 1}I_{\varepsilon}(\gamma(t)),$ $\displaystyle
I_{\varepsilon}^{\prime}(u_{n},v_{n})\rightarrow 0,\ \ \mbox{as}\
n\rightarrow\infty,$

where $\Gamma_{\varepsilon}=\left\\{\gamma\in
C(E,\mathbb{R}):\gamma(0)=(0,0),I_{\varepsilon}(\gamma(1))<0\right\\}$. $\Box$

######  Lemma 3.2.

$c_{\varepsilon}=c_{\varepsilon}^{*}$.

Proof. Define

$\displaystyle c_{\varepsilon}^{**}$
$\displaystyle:=\mathop{\inf}\limits_{(u,v)\in
E\setminus\left\\{(0,0)\right\\}}\mathop{\max}\limits_{t\geq
0}I_{\varepsilon}(tu,tv).$

For any $(u,v)\in E\setminus\\{(0,0)\\}$, there exists a unique $t>0$ such
that $(tu,tv)\in\mathcal{N}_{\varepsilon}$, and

$\displaystyle I_{\varepsilon}(t_{u,v}u,t_{u,v}v)=\mathop{\max}\limits_{t\geq
0}I_{\varepsilon}(tu,tv),$

it follows that $c_{\varepsilon}=c_{\varepsilon}^{**}$. There exists $t_{0}>0$
large enough such that $I_{\varepsilon}(t_{0}u,t_{0}v)<0$, and

$\displaystyle c_{\varepsilon}^{*}\leq\mathop{\max}\limits_{t\geq
0}I_{\varepsilon}(tt_{0}u,tt_{0}v),$

$c_{\varepsilon}^{*}\leq c_{\varepsilon}^{**}=c_{\varepsilon}$ can be
obtained. To show $c_{\varepsilon}^{*}\geq c_{\varepsilon}$, we only need to
prove that for any $\gamma\in\Gamma_{\varepsilon}$,
$\gamma([0,1])\cap\mathcal{N}_{\varepsilon}\neq\emptyset$. When
$0<\|(u,v)\|_{E}\ll 1$,

$\displaystyle\int|\nabla u|^{2}+a(\varepsilon x)u^{2}+|\nabla
v|^{2}+b(\varepsilon x)v^{2}>\int\mu_{1}u^{4}+2\beta u^{2}v^{2}+\mu_{2}v^{4}.$

Since $I_{\varepsilon}(\gamma(0))=0$, $\gamma$ is a continuous in $t\in[0,1]$,
$I_{\varepsilon}(\gamma(t))>0$ for $0<t<1$. Thus, if set
$\gamma_{t}=(u_{t},v_{t})$,

$\displaystyle\int|\nabla u_{t}|^{2}+a(\varepsilon x)u_{t}^{2}+|\nabla
v_{t}|^{2}+b(\varepsilon x)v_{t}^{2}>\int\mu_{1}u_{t}^{4}+2\beta
u_{t}^{2}v_{t}^{2}+\mu_{2}v_{t}^{4}$

holds for sufficiently small $t$. If
$\gamma([0,1])\cap\mathcal{N}_{\varepsilon}=\emptyset$, then

$\displaystyle\int|\nabla u_{1}|^{2}+a(\varepsilon x)u_{1}^{2}+|\nabla
v_{1}|^{2}+b(\varepsilon x)v_{1}^{2}>\int\mu_{1}u_{1}^{4}+2\beta
u_{1}^{2}v_{1}^{2}+\mu_{2}v_{1}^{4}.$

This is a contradiction with $I_{\varepsilon}(\gamma(1))<0$. $\Box$

In order to obtain the existence of ground state solutions for
($\mathcal{P}_{\varepsilon}$), we prove some compactness lemma for the
functional $I_{\varepsilon}$ to analysis the Palais-Smale sequence properties
for the functional $I_{\varepsilon}$.

Now, we consider the limit problem

$\displaystyle\begin{cases}-\Delta u+\tau u=\mu_{1}u^{3}+\beta v^{2}u\ \ \ \
\mbox{in}\ \mathbb{R}^{N},\\\ -\Delta v+\theta v=\mu_{2}v^{3}+\beta u^{2}v\ \
\ \ \ \mbox{in}\ \mathbb{R}^{N},\end{cases}$

where $(\tau,\theta)\in(0,+\infty)\times(0,\infty)$. The corresponding energy
functional and and Nehari manifold is defined by

$\displaystyle I_{\tau\theta}(u,v)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+\tau u^{2}+|\nabla v|^{2}+\theta v^{2}-\frac{1}{4}F(u,v)$

and

$\displaystyle\mathcal{N}_{\tau\theta}=\\{(u,v)\in\mathbb{H}\setminus\\{(0,0)\\}|\langle
I_{\tau\theta}^{\prime}(u,v),(u,v)\rangle=0\\}.$

Define

$\displaystyle
c_{\tau\theta}=\mathop{\inf}\limits_{(u,v)\in\mathcal{N}_{\tau\theta}}I_{\tau\theta}(u,v).$

######  Lemma 3.3.

The map $c:(0,+\infty)\times(0,+\infty)\rightarrow\mathbb{R}$;
$(\tau,\theta)\mapsto c_{\tau\theta}$, is strictly increasing in $\tau$ and
$\theta$.

Proof. The proof is similar to the proofs of Lemma 2.3 in [25]. We omit the
details.

######  Lemma 3.4.

Assume that $f(2)-f(3)$ hold, and $\Omega\neq\varnothing$. For any fixed
$\varepsilon>0$, $I_{\varepsilon}$ satisfies $Palais-Smale$ condition at level
$c<c_{a_{\infty}b_{\infty}}$ when $a_{\infty}<+\infty,b_{\infty}<+\infty$ and
at any level when $a_{\infty}=b_{\infty}=+\infty$.

Proof. When $a_{\infty}=b_{\infty}=+\infty$, there exists compact embedding
$E\hookrightarrow\mathbb{L}^{q}$ for $q\in[2,2^{*})$, and the Palais-Smale
condition naturally holds at any level. So we only consider the case
$a_{\infty},b_{\infty}<+\infty$. Let $c<c_{a_{\infty}b_{\infty}}$,
$\\{(u_{n},v_{n})\\}\subset E$ be a sequence such that
$I_{\varepsilon}(u_{n},v_{n})\rightarrow c$, and
$I_{\varepsilon}^{\prime}(u_{n},v_{n})\rightarrow 0$. A standard argument
leads us to the fact that ${(u_{n},v_{n})}$ is bounded in $E$, which implies
that

$\displaystyle(u_{n},v_{n})\rightharpoonup(u,v)\ \mbox{weakly in}\ E,\
I_{\varepsilon}^{\prime}(u,v)=0,\ I_{\varepsilon}(u,v)\geq 0.$

Set $w_{n}=u_{n}-u,z_{n}=v_{n}-v$, and $(w_{n},z_{n})\rightharpoonup 0$. By
Brezis-Lieb Lemma [33], it follows that,

$\displaystyle
I_{\varepsilon}(w_{n},z_{n})=I_{\varepsilon}(u_{n},v_{n})-I_{\varepsilon}(u,v)+o(1)\leq
c+o(1),$ $\displaystyle I_{\varepsilon}^{\prime}(w_{n},z_{n})\rightarrow 0\ \
\mbox{as}\ n\rightarrow\infty.$

There exists $t_{n}>0$ such that
$t_{n}(w_{n},v_{n})\in\mathcal{N}_{a_{\infty},b_{\infty}}$, from which it
follows that

(3.1) $\displaystyle t_{n}^{2}\int_{\mathbb{R}^{N}}|\nabla
w_{n}|^{2}+a_{\infty}w_{n}^{2}+|\nabla
z_{n}|^{2}+b_{\infty}z_{n}^{2}=t_{n}^{4}\int_{\mathbb{R}^{N}}\mu_{1}w_{n}^{4}+2\beta
w_{n}^{2}z_{n}^{2}+\mu_{2}z_{n}^{4},$

Suppose that $(w_{n},z_{n})\nrightarrow(0,0)$ in $E$, we first claim that

(3.2) $\displaystyle\limsup_{n\rightarrow\infty}t_{n}\leq 1.$

Otherwise, we can assume that $t_{n}\geq 1+\delta$ for $\delta>0$. From
$I_{\varepsilon}^{\prime}(w_{n},z_{n})=o(1)$, we have that

(3.3) $\displaystyle\int_{\mathbb{R}^{N}}|\nabla w_{n}|^{2}+a(\varepsilon
x)w_{n}^{2}+|\nabla z_{n}|^{2}+b(\varepsilon
x)z_{n}^{2}=\int_{\mathbb{R}^{N}}\mu_{1}w_{n}^{4}+2\beta
w_{n}^{2}z_{n}^{2}+\mu_{2}z_{n}^{4}+o(1).$

According to $(\ref{5})$ and $(\ref{7})$ , we have

(3.4) $\displaystyle\int_{\mathbb{R}^{N}}(a_{\infty}-a(\varepsilon
x))w_{n}^{2}+(b_{\infty}-b(\varepsilon
x))z_{n}^{2}=(t_{n}^{2}-1)\int_{\mathbb{R}^{N}}\mu_{1}w_{n}^{4}+2\beta
w_{n}^{2}z_{n}^{2}+\mu_{2}z_{n}^{4}+o(1)$

For any $\varepsilon>0$, there exists $R=R(\varepsilon)$ such that

(3.5) $\displaystyle a(\varepsilon x)>a_{\infty}-\varepsilon\ \ \mbox{and}\ \
b(\varepsilon x)>b_{\infty}-\varepsilon\ \ \mbox{for}\ |x|\geq R.$

Since $(w_{n},z_{n})\rightarrow 0$ in $\mathbb{L}^{2}_{B_{R}(0)}$ and
$(\ref{9})$, there exists $C>0$ such that

$\displaystyle(t_{n}^{2}-1)\int_{\mathbb{R}^{N}}\mu_{1}w_{n}^{4}+2\beta
w_{n}^{2}z_{n}^{2}+\mu_{2}z_{n}^{4}\leq C\varepsilon+o(1)$

Since $(w_{n},z_{n})\nrightarrow(0,0)$, there exists a sequence
${y_{n}}\in\mathbb{R}^{N}$ and positive constants $r,\zeta>0$,

(3.6)
$\displaystyle\mathop{\liminf}\limits_{n\rightarrow\infty}\int_{B_{r}(y_{n})}(w_{n}^{2}+z_{n}^{2})\geq\zeta.$

Set $(\tilde{w}_{n}(x),\tilde{z}_{n}(x))=(u(x+y_{n}),v(x+y_{n}))$. Then
$\left\\{(\tilde{w}_{n},\tilde{z}_{n})\right\\}$ is bounded and
$(\tilde{w}_{n},\tilde{z}_{n})\rightharpoonup(u,v)$ in $E$,
$(\tilde{w}_{n},\tilde{z}_{n})\rightarrow(u,v)$ in
$\mathbb{L}_{loc}^{2}(\mathbb{R}^{N})$. Moreover, there exists a subset
$\Lambda\subset\mathbb{R}^{N}$ such that $(u,v)\neq(0,0)$ $a.e.$ in $\Lambda$,
then it follows from Fatou’s lemma that

$\displaystyle 0<[(1+\delta)^{2}-1]\int_{\Lambda}\mu_{1}u^{4}+2\beta
u^{2}v^{2}+\mu_{2}v^{4}\leq C\varepsilon,$

for any $\varepsilon>0$, which yields a contradiction. Thus (3.2) holds. Using
the fact that $t_{n}(w_{n},v_{n})\in\mathcal{N}_{a_{\infty},b_{\infty}}$, and
(3.2)

$\displaystyle c_{a_{\infty}b_{\infty}}$ $\displaystyle\leq
I_{a_{\infty}b_{\infty}}(t_{n}w_{n},t_{n}z_{n})-\frac{1}{4}\langle
I_{\infty}^{\prime}(t_{n}(w_{n},v_{n})),t_{n}(w_{n},v_{n})\rangle$
$\displaystyle=\frac{t_{n}^{2}}{4}\int_{\mathbb{R}^{N}}|\nabla
w_{n}|^{2}+a_{\infty}w_{n}^{2}+|\nabla z_{n}|^{2}+b_{\infty}z_{n}^{2}$
$\displaystyle\leq\frac{1}{4}\int_{\mathbb{R}^{N}}|\nabla
w_{n}|^{2}+a(\varepsilon x)w_{n}^{2}+|\nabla z_{n}|^{2}+b(\varepsilon
x)z_{n}^{2}+o(1)$ $\displaystyle=I_{\varepsilon}(w_{n},z_{n})+o(1)\leq c+o(1)$

Let $n\rightarrow\infty$ in the above inequality, we have
$c_{a_{\infty}b_{\infty}}\leq c$ which is impossible. $\Box$

######  Lemma 3.5.

$c_{\varepsilon}<c_{a_{\infty}b_{\infty}}$.

Proof. Let $(u,v)$ be the ground state solution for $I_{\tau\theta}$. Take
$\eta\in C_{0}^{\infty}(\mathbb{R}^{n},[0,1])$ such that $\eta(x)=1$, for
$|x|\leq\frac{1}{2}$, and $\eta(x)=0$, for $|x|\geq 1$. And define
$(u_{n},v_{n})=\eta(\frac{\cdot}{n})(u,v)$ and
$(w_{n},z_{n})=t_{n}(u_{n},v_{n})$ such that
$(w_{n},z_{n})\in\mathcal{N}_{\tau\theta}$. Since
$(u_{n},v_{n})\rightarrow(u,v)$ in $\mathbb{H}$ as $n\rightarrow\infty$, and
$(u_{n},v_{n})\in\mathcal{N}_{\tau\theta}$, it is easy to see that
$\lim_{n\rightarrow\infty}t_{n}=1$. Consequently, we have

$\displaystyle
I_{\tau\theta}(w_{n},z_{n})=I_{\tau\theta}(t_{n}(u_{n},v_{n}))\rightarrow
c_{\tau\theta}\leq c_{a_{\infty}b_{\infty}}.$

Now, we have
$\\{(w_{n},z_{n})\\}\subset\mathbb{C}_{0}^{\infty}(\mathbb{R}^{N})$,
$(w_{n},z_{n})\rightarrow(u,v)$ in $\mathbb{H}$ and
$(w_{n},z_{n})\in\mathcal{N}_{\tau\theta}$, which implies that

$\displaystyle I_{\tau\theta}(w_{n},z_{n})=\mathop{\max}\limits_{t\geq
0}I_{\tau\theta}(tw_{n},tz_{n}).$

By assumption $f(2)-f(3)$, for any $\tau,\theta>0$, there exists $R>0$ such
that $a(x)<\tau,b(x)<\theta$ for $|x|<R$, then

$\displaystyle a(\varepsilon x)<\tau,\ \ \ b(\varepsilon x)<\theta,\ \ \
\mbox{for}\ |x|<\frac{R}{\varepsilon}.$

Define

$\displaystyle\mathbb{H}_{R/\varepsilon}:=C_{0}^{\infty}(B_{R/\varepsilon}(0))\times
C_{0}^{\infty}(B_{R/\varepsilon}(0)).$ $\displaystyle
c_{\tau\theta}^{R/\varepsilon}:=\mathop{\inf}\limits_{(u,v)\in\mathbb{H}_{R/\varepsilon}\setminus\\{(0,0)\\}}\mathop{\max}\limits_{t\geq
0}I_{\varepsilon}^{\tau\theta}(tu,tv),$

then

$\displaystyle c_{\varepsilon}$ $\displaystyle=\mathop{\inf}\limits_{(u,v)\in
E\setminus\\{(0,0)\\}}\mathop{\max}\limits_{t\geq
0}I_{\varepsilon}(tu,tv)\leq\mathop{\inf}\limits_{(u,v)\in\mathbb{H}_{R/\varepsilon}\setminus\\{(0,0)\\}}\mathop{\max}\limits_{t\geq
0}I_{\varepsilon}(tu,tv)$
$\displaystyle\leq\mathop{\inf}\limits_{(u,v)\in\mathbb{H}_{R/\varepsilon}\setminus\\{(0,0)\\}}\mathop{\max}\limits_{t\geq
0}I_{\varepsilon}^{\tau\theta}(tu,tv)=c_{\tau\theta}^{R/\varepsilon},$

Fixed ${(w_{n},z_{n})}$ then for sufficiently small $\varepsilon>0$ such that
$R/\varepsilon>n/2$, we have $(w_{n},z_{n})\in\mathbb{H}_{R/\varepsilon}$ and

$\displaystyle
c_{\tau\theta}^{R/\varepsilon}=\mathop{\inf}\limits_{(u,v)\in\mathbb{H}_{R/\varepsilon}\setminus\\{(0,0)\\}}\mathop{\max}\limits_{t\geq
0}I_{\varepsilon}^{\tau\theta}(tu,tv)\leq\mathop{\max}\limits_{t\geq
0}I^{\tau\theta}(tw_{n},tz_{n})=I_{\tau\theta}(w_{n},z_{n})<c_{a_{\infty}b_{\infty}}.$

Then the conclusion follows $c_{\varepsilon}<c_{a_{\infty}b_{\infty}}$. $\Box$

######  Lemma 3.6.

Assume that $1\leq N\leq 3$ and $f(1)$ hold. Then for $\beta>\widehat{\beta}$,
$\mathcal{P}_{\varepsilon}$ has a positive least energy vector solution.

Proof. It is sufficient to prove that

(3.7) $\displaystyle
c_{\varepsilon}<\\{I_{\varepsilon}(u,0),I_{\varepsilon}(0,v)\\}.$

First, as in Lemma 3.3 of [12], we know that for $\beta\geq 0$,

$\displaystyle c_{\varepsilon}=\mathop{\inf}\limits_{(u,v)\in
E\setminus\\{(0,0)\\}}J_{\varepsilon}(u,v),$

where

$\displaystyle
J_{\varepsilon}(u,v)=\frac{(\|u\|_{a,\varepsilon}^{2}+\|v\|_{b,\varepsilon}^{2})^{2}}{4F(u,v)}.$

For $(s,t)\in\Gamma=\\{(s,t):s\geq 0,t\geq 0,(s,t)\neq(0,0)\\}$, we define a
function

$\displaystyle
f(s,t)=J_{\varepsilon}(\sqrt{s}U,\sqrt{t}V)=\frac{(s\|U\|_{a,\varepsilon}^{2}+t\|V\|_{b,\varepsilon}^{2})^{2}}{4\int_{\mathbb{R}^{N}}\mu_{1}s^{2}U^{4}+2\beta
stU^{2}V^{2}+\mu_{2}t^{2}V^{4}}.$

It follows that

$\displaystyle
f(s,0)=\frac{\|U\|_{a,\varepsilon}^{4}}{4\int_{\mathbb{R}^{N}}\mu_{1}U^{4}}=J_{\varepsilon}(U,0),$
$\displaystyle
f(0,t)=\frac{\|V\|_{b,\varepsilon}^{4}}{4\int_{\mathbb{R}^{N}}\mu_{1}V^{4}}=J_{\varepsilon}(0,V).$

Thus, to guarantee (3.7) it is sufficient to prove that $f$ does not attain
its minimum over $\Gamma$ on the lines $s=0$ or $t=0$. The function $f$ is a
fraction of two quadratic forms in $(s,t)$, and elementary analysis shows that
the quantity

$\displaystyle\frac{(as+bt)^{2}}{cs^{2}+2dst+et^{2}},\ \ \ a,b,c,d,e>0,$

does not attain its minimum in $\Gamma$ on the axes if and only if

(3.8) $\displaystyle ad-bc>0,\ \ \ \ \ bd-ae>0$

and then the minimum is attained for $s=ad-bc$ and $t=bd-ae$. Clearly, (3.8)
is equal to

$\displaystyle\beta>\mu_{2}\frac{\int_{\mathbb{R}^{N}}V^{4}}{\int_{\mathbb{R}^{N}}U^{2}V^{2}}\
\ \mbox{and}\ \
\beta>\mu_{1}\frac{\int_{\mathbb{R}^{N}}U^{4}}{\int_{\mathbb{R}^{N}}U^{2}V^{2}}.$

Thus, if $\beta>\beta_{0}$, then (3.7) holds. $\Box$

######  Proposition 3.7.

Let $1\leq N\leq 3$. Assume that $f(1)-f(3)$ hold. Then for
$\beta>\widehat{\beta}$ sufficiently large and small $\varepsilon>0$, the
system ($\mathcal{P}_{\varepsilon}$) has a positive groundstate solution.

Proof. From Lemma 3.1 the functional $I_{\varepsilon}$ satisfies a mountain
pass geometry. By Lemma 3.2 and Mountain Pass Theorem [30], there exists a
$(PS)_{c_{\varepsilon}^{*}}$ sequence $\\{(u_{n},v_{n})\\}\subset E$ such that
$I_{\varepsilon}(u_{n},v_{n})\rightarrow c_{\varepsilon}$ and
$I_{\varepsilon}^{\prime}(u_{n},v_{n})\rightarrow 0$. Then standard argument
shows that $\\{(u_{n},v_{n})\\}$ is bounded. Up to a subsequence, Lemma 3.5
implies that there exists $(u_{\varepsilon},v_{\varepsilon})$ such that
$(u_{n},v_{n})\rightarrow(u_{\varepsilon},v_{\varepsilon})$,
$c_{\varepsilon}=I_{\varepsilon}(u_{\varepsilon},v_{\varepsilon})$ and
$I_{\varepsilon}^{\prime}(u_{\varepsilon},v_{\varepsilon})=0$. Moreover, since
$\beta>\beta_{0}$, from Lemma 3.6, we know that $u_{\varepsilon}\not\equiv 0$
and $u_{\varepsilon}\not\equiv 0$. Thus we prove
$(u_{\varepsilon},v_{\varepsilon})\in\mathcal{N}_{\varepsilon}$. Finally, we
prove that $u_{\varepsilon},v_{\varepsilon}>0$. Since
$(|u_{\varepsilon}|,|v_{\varepsilon}|)\in\mathcal{N}_{\varepsilon}$ and
$c_{\varepsilon}=I_{\varepsilon}(|u_{\varepsilon}|,|v_{\varepsilon}|)$, we
deduce that $(|u_{\varepsilon}|,|v_{\varepsilon}|)$ is a nonnegative solution
of ($\mathcal{P}_{\varepsilon}$). Using the strong maximum principle we infer
that at least one of $|u_{\varepsilon}|,|v_{\varepsilon}|>0$. This completes
the proof. $\Box$

##  4\. Proof of Theorem 1.1 and Theorem 1.2.

######  Lemma 4.1.

[31] Let a function $V\in C(\mathbb{R}^{N})$ and $\gamma>0$. If $V\geq 0$ on
$\mathbb{R}^{N}$,

$\displaystyle\mathop{\underline{\lim}}\limits_{|x|\rightarrow\infty}V(x)>0,$

and if for each $x\in\mathbb{R}^{N}$

$\displaystyle\mathop{\underline{\lim}}\limits_{z\rightarrow
x}\frac{V(z)}{|z-x|^{\gamma}}>0,$

then there exist $l\in\mathbb{N}$, $a_{1},\cdots,a_{l}\in\mathbb{R}^{N}$,
$\mu>0$ and $\nu>0$ such that for each $x\in\mathbb{R}^{N}$,

$\displaystyle
V(x)\geq\min\left\\{\mu,\nu|x-a_{1}|^{\gamma},\cdots,\nu|x-a_{l}|^{\gamma}\right\\}.$

Next we consider the limit problem corresponding to the case of Theorem 1.2.
Suppose that $W$ is a continuous positive $\gamma$-homogeneous function.
Define $H_{W}=\\{u\in
H^{1}(\mathbb{R}^{N})|\int_{\mathbb{R}^{N}}W(x)u^{2}<+\infty\\}$ and
$E_{W}=H_{W}\times H_{W}$. $E_{W}$ embeds into $\mathbb{H}$ continuously and
into $\mathbb{L}^{q}(\mathbb{R}^{N})$, $q\in[2,2^{*})$ compactly.

$\displaystyle\mathcal{J}_{W}(u,v)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+W(x)u^{2}+|\nabla
v|^{2}+W(x)v^{2}-\frac{1}{4}\int_{\mathbb{R}^{N}}\mu_{1}v^{4}+2\beta
u^{2}v^{2}+\mu_{2}v^{4}.$

######  Lemma 4.2.

Let $\beta>\widehat{\beta}$ sufficiently large, then $\mathcal{E}_{W}$ ia
attained by a positive solution.

Proof. The proof is similar to proofs of Lemma 3.4, we omit it here.

For the sake of simplicity, in this section we set

$\displaystyle k=\frac{2}{2+\gamma}.$

######  Proposition 4.3.

Under the assumptions of Theorem 1.2, there holds

$\displaystyle\mathop{\limsup}\limits_{\varepsilon\rightarrow
0}\frac{c_{\varepsilon}}{\varepsilon^{k(N+2\gamma)}}\leq\mathcal{E}_{W_{x_{*}}}=\mathop{\inf}\limits_{i\in\mathcal{I}}\mathcal{E}_{W_{i}}.$

Proof. For $i\in I$, $x_{i}\in\Omega$, $W_{i}$ a positive $\gamma$-homogeneous
function such that

(4.1) $\displaystyle\mathop{\lim}\limits_{z\rightarrow
x_{i}}\frac{a(z)-W_{i}(z-x_{i})}{|z-x_{i}|^{\gamma}}=0\ \ \mbox{and}\ \
\mathop{\lim}\limits_{z\rightarrow
x_{i}}\frac{b(z)-W_{i}(z-x_{i})}{|z-x_{i}|^{\gamma}}=0,$

and let
$(\eta,\varphi)\in\mathbb{C}_{0}^{\infty}(\mathbb{R}^{N})\setminus\left\\{(0,0)\right\\}$,
we define
$\eta_{\varepsilon}(\mbox{resp.}\varphi_{\varepsilon}):\mathbb{R}^{N}\rightarrow\mathbb{R}$
by

$\displaystyle(\eta_{\varepsilon}(x),\varphi_{\varepsilon}(x))=\varepsilon^{\frac{k\gamma}{2}}\big{(}\eta(\frac{x-x_{i}}{\varepsilon^{k}}),\varphi(\frac{x-x_{i}}{\varepsilon^{k}})\big{)}.$

Computing directly, we have

$\displaystyle\int_{\mathbb{R}^{N}}\varepsilon^{2}|\nabla\eta_{\varepsilon}|^{2}+\int_{\mathbb{R}^{N}}\varepsilon^{2}|\nabla\varphi_{\varepsilon}|^{2}=\varepsilon^{k(N+2\gamma)}\int_{\mathbb{R}^{N}}|\nabla\eta|^{2}+|\nabla\varphi|^{2};$
$\displaystyle\int_{\mathbb{R}^{N}}a|\eta_{\varepsilon}|^{2}+b|\varphi_{\varepsilon}|^{2}=\varepsilon^{k(N+2\gamma)}\int_{\mathbb{R}^{N}}\frac{a(\varepsilon^{k}x+x_{i})}{\varepsilon^{k\gamma}}|\eta|^{2}+\frac{b(\varepsilon^{k}x+x_{i})}{\varepsilon^{k\gamma}}|\varphi|^{2};$
$\displaystyle\int_{\mathbb{R}^{N}}\mu_{1}\eta_{\varepsilon}^{4}+2\beta\eta_{\varepsilon}^{2}\varphi_{\varepsilon}^{2}+\mu_{2}\varphi_{\varepsilon}^{4}=\varepsilon^{k(N+2\gamma)}\int_{\mathbb{R}^{N}}\mu_{1}\eta^{4}+2\beta\eta^{2}\varphi^{2}+\mu_{2}\varphi^{4}.$

Since the function $W_{i}$ is $\gamma$-homogeneous and satisfies (4.1), we
have

(4.2) $\displaystyle\mathop{\lim}\limits_{\varepsilon\rightarrow
0}\frac{a(\varepsilon^{k}x+x_{i})}{\varepsilon^{k\gamma}}=\mathop{\lim}\limits_{\varepsilon\rightarrow
0}\frac{W_{i}(\varepsilon^{k}x)}{\varepsilon^{k\gamma}}=W_{i}(x)\ \
\mbox{and}\ \ \mathop{\lim}\limits_{\varepsilon\rightarrow
0}\frac{b(\varepsilon^{k}x+x_{i})}{\varepsilon^{k\gamma}}=W_{i}(x),$

uniformly for $x\in supp(\eta)\cap supp(\varphi)$. Thus by Lebesgue’s
dominated convergence theorem, we have

$\displaystyle\mathop{\lim}\limits_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon^{k(N+2\gamma)}}\int_{\mathbb{R}^{N}}a|\eta_{\varepsilon}|^{2}+b|\varphi_{\varepsilon}|^{2}=\int_{\mathbb{R}^{N}}W_{i}|\eta|^{2}+W_{i}|\varphi|^{2}.$

Choosing $t_{\varepsilon}>0$ such that $\langle
I_{\varepsilon}^{\prime}(t_{\varepsilon}\eta_{\varepsilon},t_{\varepsilon}\varphi_{\varepsilon}),(t_{\varepsilon}\eta_{\varepsilon},t_{\varepsilon}\varphi_{\varepsilon})\rangle=0$,
and we observe that $t_{\varepsilon}\rightarrow t_{*}\in(0,+\infty)$ as
$\varepsilon\rightarrow 0$. Moreover, there holds

$\displaystyle
t_{*}\int_{\mathbb{R}^{N}}|\nabla\eta|^{2}+W_{i}|\eta|^{2}+|\nabla\varphi|^{2}+W_{i}|\varphi|^{2}=t_{*}\int_{\mathbb{R}^{N}}\mu_{1}\eta^{4}+2\beta\eta^{2}\varphi^{2}+\mu_{2}\varphi^{4},$

which implies

$\displaystyle\mathop{\limsup}\limits_{\varepsilon\rightarrow
0}\frac{c_{\varepsilon}}{\varepsilon^{k(N+2\gamma)}}\leq\mathop{\lim}\limits_{\varepsilon\rightarrow
0}\frac{I_{\varepsilon}(t_{\varepsilon}\eta_{\varepsilon},t_{\varepsilon}\varphi_{\varepsilon})}{\varepsilon^{k(N+2\gamma)}}=\mathcal{J}_{W_{i}}(t_{*}\eta,t_{*}\varphi)=\mathop{\sup}\limits_{t\in(0,+\infty)}\mathcal{J}_{W_{i}}(t\eta,t\varphi).$

Then by density of the set of smooth functions
$\mathbb{C}_{0}^{\infty}(\mathbb{R}^{N})$ in the space $E_{W}$, we can take
the infimum with respect to $\eta$ and $\varphi$, we have

$\displaystyle\mathop{\limsup}\limits_{\varepsilon\rightarrow
0}\frac{c_{\varepsilon}}{\varepsilon^{k(N+2\gamma)}}\leq\mathcal{E}_{W_{i}},$

combining this with the arbitrary choice of $i\in\mathcal{I}$, we obtain the
conclusion. $\Box$

######  Lemma 4.4.

Let $(u,v)\in E$. Then there exists positive constant $C$ such that

$\displaystyle
C\int_{\mathbb{R}^{N}}|u|^{4}+|v|^{4}\leq\Big{(}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+\frac{a(\varepsilon^{k}x)}{\varepsilon^{k\gamma}}u^{2}\Big{)}^{2}+\Big{(}\int_{\mathbb{R}^{N}}|\nabla
v|^{2}+\frac{b(\varepsilon^{k}x)}{\varepsilon^{k\gamma}}v^{2}\Big{)}^{2}.$

Proof. Define
$a^{-1}[0,\lambda]=\\{x\in\mathbb{R}^{N};\frac{a(\varepsilon^{k}x)}{\varepsilon^{k\gamma}}\leq\lambda\\}$,
$a^{-1}[\lambda,+\infty)=\\{x\in\mathbb{R}^{N};\frac{a(\varepsilon^{k}x)}{\varepsilon^{k\gamma}}\geq\lambda\\}$.
Thus, we have

$\displaystyle\int_{\mathbb{R}^{N}}u^{4}\leq\int_{a^{-1}[0,\lambda]}u^{4}+\int_{a^{-1}[\lambda,+\infty)}u^{4}.$

Using Hölder’s inequality, Young’s inequality and Sobolev inequality, there
exists a positive constant $C$ such that

(4.3)
$\displaystyle\begin{split}\int_{a^{-1}[\lambda,+\infty)}u^{4}&\leq\Big{(}\int_{a^{-1}[\lambda,+\infty)}u^{2}\Big{)}^{2s}\Big{(}\int_{a^{-1}[\lambda,+\infty)}u^{t}\Big{)}^{\frac{4(1-s)}{t}}\\\
&\leq
C\Big{(}\int_{a^{-1}[\lambda,+\infty)}u^{2}\Big{)}^{2}+C\Big{(}\int_{a^{-1}[\lambda,+\infty)}u^{t}\Big{)}^{\frac{4}{t}}\\\
&\leq C\Big{(}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+\frac{a(\varepsilon^{k}x)}{\varepsilon^{k\gamma}}u^{2}\Big{)}^{2}.\end{split}$

By Lemma 4.1, if $\lambda\leq\mu$,

$\displaystyle\mbox{m}\\{x\in\mathbb{R}^{N}|a(\varepsilon^{k}x)\leq\lambda\varepsilon^{k\gamma}\\}\leq
l\frac{4\pi}{3}\left(\frac{\lambda}{\mu}\right)^{\frac{N}{\gamma}},$

Thus,

$\displaystyle\int_{a^{-1}[0,\lambda]}u^{4}\leq
C\Big{(}\int_{\mathbb{R}^{N}}|\nabla u|^{2}\Big{)}^{2},$

combining which with (4.3),

$\displaystyle
C\int_{\mathbb{R}^{N}}|u|^{4}\leq\Big{(}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}+\frac{a(\varepsilon^{k}x)}{\varepsilon^{k\gamma}}u^{2}\Big{)}^{2},$

can be obtained. Similar results to $v$, we get the conclusion. $\Box$

Proof of Theorem 1.1. Let $(u_{\varepsilon},v_{\varepsilon})$ be the positive
ground state of system ($\mathcal{P}_{\varepsilon}$) satisfying
$I_{\varepsilon}(u_{\varepsilon},v_{\varepsilon})=c_{\varepsilon}$ and define

$\displaystyle\omega_{\varepsilon}=\varepsilon^{-\frac{k\gamma}{2}}u_{\varepsilon}(\varepsilon^{k}x),$
$\displaystyle\phi_{\varepsilon}=\varepsilon^{-\frac{k\gamma}{2}}v_{\varepsilon}(\varepsilon^{k}x),$

we have

(4.4) $\displaystyle\begin{split}&\int_{\mathbb{R}^{N}}\varepsilon^{2}|\nabla
u_{\varepsilon}|^{2}=\varepsilon^{k(N+2\gamma)}\int_{\mathbb{R}^{N}}|\nabla\omega_{\varepsilon}|^{2};\\\
&\int_{\mathbb{R}^{N}}au_{\varepsilon}^{2}=\varepsilon^{k(N+2\gamma)}\int_{\mathbb{R}^{N}}\frac{a_{\varepsilon}(x)}{\varepsilon^{k\gamma}}|\omega_{\varepsilon}|^{2};\\\
&\int_{\mathbb{R}^{N}}\mu_{1}u_{\varepsilon}^{4}=\varepsilon^{k(N+2\gamma)}\int_{\mathbb{R}^{N}}\mu_{1}\omega_{\varepsilon}^{4};\\\
&\int_{\mathbb{R}^{N}}\beta
u_{\varepsilon}^{2}v_{\varepsilon}^{2}=\varepsilon^{k(N+2\gamma)}\int_{\mathbb{R}^{N}}\beta\omega_{\varepsilon}^{2}\phi_{\varepsilon}^{2},\\\
\end{split}$

similar results to $v_{\varepsilon}$, and

(4.5)
$\displaystyle-\triangle\omega_{\varepsilon}+\frac{a_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\omega_{\varepsilon}=\mu_{1}\omega_{\varepsilon}^{3}+\beta\omega_{\varepsilon}\phi_{\varepsilon}^{2};$
(4.6)
$\displaystyle-\triangle\phi_{\varepsilon}+\frac{b_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\phi_{\varepsilon}=\mu_{2}\phi_{\varepsilon}^{3}+\beta\omega_{\varepsilon}^{2}\phi_{\varepsilon},$

where $a_{\varepsilon}(x)=a(\varepsilon^{k}x)$,
$b_{\varepsilon}(x)=b(\varepsilon^{k}x)$. Consequently, there holds

(4.7)
$\displaystyle\int_{\mathbb{R}^{N}}|\nabla\omega_{\varepsilon}|^{2}+\frac{a_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\omega_{\varepsilon}^{2}+|\nabla\phi_{\varepsilon}|^{2}+\frac{b_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\phi_{\varepsilon}^{2}=\int_{\mathbb{R}^{N}}\mu_{1}\omega_{\varepsilon}^{4}+2\beta\omega_{\varepsilon}^{2}\phi_{\varepsilon}^{2}+\mu_{2}\phi_{\varepsilon}^{4},$

By proposition 4.3, we see deduce that, for small $\varepsilon>0$,

$\displaystyle\frac{1}{4}\int_{\mathbb{R}^{N}}\varepsilon^{2}|\nabla
u_{\varepsilon}|^{2}+au_{\varepsilon}^{2}+\varepsilon^{2}|\nabla
v_{\varepsilon}|^{2}+bv_{\varepsilon}^{2}\leq\varepsilon^{k(N+2\gamma)}m,$

combining which with (4.4) and (4.7), we know that

(4.8)
$\displaystyle\int_{\mathbb{R}^{N}}|\nabla\omega_{\varepsilon}|^{2}+\frac{a_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\omega_{\varepsilon}^{2}+|\nabla\phi_{\varepsilon}|^{2}+\frac{b_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\phi_{\varepsilon}^{2}\leq
4m.$

Now we claim that there exists $\\{x_{\varepsilon}\\}\in\mathbb{R}^{N}$ and
$x_{\varepsilon},\varrho>0$ such that

$\displaystyle\mathop{\underline{\lim}}\limits_{\varepsilon\rightarrow
0}\int_{B_{\varepsilon^{k}}(x_{\varepsilon})}|w_{\varepsilon}|^{q}+|z_{\varepsilon}|^{q}=\varrho>0.$

Assume by contradiction that there exists a subsequence
$\varepsilon\rightarrow 0$ such that for any $R>0$,

$\displaystyle\mathop{\lim}\limits_{\varepsilon\rightarrow
0}\mathop{\sup}\limits_{x\in\mathbb{R}^{N}}\int_{B_{R}(x)}|w_{\varepsilon}|^{q}+|\phi_{\varepsilon}|^{q}=0.$

Thus by the Lions Concentration-Compactness Principle [34],

$\displaystyle(w_{\varepsilon},\phi_{\varepsilon})\rightarrow(0,0)\ \
\mbox{in}\ \mathbb{L}^{q}(\mathbb{R}^{N})\ \ \mbox{for}\ \ q\in(2,2^{*}).$

Hence, we have

$\displaystyle\int_{\mathbb{R}^{N}}|\nabla\omega_{\varepsilon}|^{2}+\frac{a_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\omega_{\varepsilon}^{2}+|\nabla\phi_{\varepsilon}|^{2}+\frac{b_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\phi_{\varepsilon}^{2}=\int\mu_{1}w_{\varepsilon}^{4}+2\beta
w_{\varepsilon}^{2}\phi_{\varepsilon}^{2}+\mu_{2}\phi_{\varepsilon}^{4}\rightarrow
0\ \ \mbox{as}\ \ \varepsilon\rightarrow 0.$

By Lemma 4.4, there exists a positive constant $C$ such that

$\displaystyle\int_{\mathbb{R}^{N}}|\nabla\omega_{\varepsilon}|^{2}+\frac{a_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\omega_{\varepsilon}^{2}+|\nabla\phi_{\varepsilon}|^{2}+\frac{b_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\phi_{\varepsilon}^{2}\geq
C,$

which is a contradiction. Hence, there exists
$x_{\varepsilon}\in\mathbb{R}^{N}$ such that

(4.9)
$\displaystyle\int_{B_{1}(x_{\varepsilon})}|w_{\varepsilon}|^{q}+|\phi_{\varepsilon}|^{q}>0$

We claim that $\varepsilon^{k}x_{\varepsilon}\rightarrow x_{i}$,
$i\in\mathcal{I}$, otherwise

$\displaystyle\int_{\mathbb{R}^{N}}\frac{a_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\omega_{\varepsilon}^{2}+\frac{b_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\phi_{\varepsilon}^{2}\rightarrow\infty,$

which contradicts (4.8). According to (4.8) and (4.9),

$\displaystyle\mathop{\inf}\limits_{x\in
B_{1}(x_{\varepsilon})}\frac{a_{\varepsilon}(x)}{\varepsilon^{k\gamma}}+\mathop{\inf}\limits_{x\in
B_{1}(x_{\varepsilon})}\frac{b_{\varepsilon}(x)}{\varepsilon^{k\gamma}}\leq
4m.$

can be obtained. Thus there exists $x\in B_{1}(x_{\varepsilon})$ such that
$\frac{a(\varepsilon^{k}x)}{\varepsilon^{k\gamma}}\leq C$ and we deduce that
$\varepsilon^{k}x\rightarrow x_{*}$ for some $i\in\mathcal{I}$. For
$\varepsilon$ small, by Lemma 4.1, there holds

$\displaystyle\frac{\min_{1\leq i\leq
k}|\varepsilon^{k}x-x_{i}|^{\gamma}}{\varepsilon^{k\gamma}}\leq\frac{a(\varepsilon^{k}x)}{\varepsilon^{k\gamma}}\leq
C.$

Hence, for some $x_{*}\in\\{x_{i};i\in I\\}$ such that

$\displaystyle\frac{|\varepsilon^{k}x-x_{*}|}{\varepsilon^{k}}\leq C_{1},$

from which it follows that

$\displaystyle|x_{\varepsilon}-\frac{x_{*}}{\varepsilon^{k}}|\leq 1+C_{1}.$

So that there exists $R_{*}>0$, such that

$\displaystyle\int_{B_{R_{*}(\frac{x_{*}}{\varepsilon^{k}})}}|w_{\varepsilon}|^{2}+|\phi_{\varepsilon}|^{2}\geq\delta_{1}>0.$

Define
$(\tilde{w_{\varepsilon}},\tilde{\phi_{\varepsilon}})=(w_{\varepsilon}(x+\frac{x_{*}}{\varepsilon^{k}}),\phi_{\varepsilon}(x+\frac{x_{*}}{\varepsilon^{k}}))$,
there holds

$\displaystyle\int_{\mathbb{R}^{N}}|\nabla\tilde{\omega_{\varepsilon}}|^{2}+\frac{a_{\varepsilon}(x+\frac{x_{*}}{\varepsilon^{k}})}{\varepsilon^{k\gamma}}\tilde{\omega_{\varepsilon}}^{2}+|\nabla\tilde{\phi_{\varepsilon}}|^{2}+\frac{b_{\varepsilon}(x+\frac{x_{*}}{\varepsilon^{k}})}{\varepsilon^{k\gamma}}\tilde{\phi_{\varepsilon}}^{2}=\int_{\mathbb{R}^{N}}\mu_{1}\tilde{\omega_{\varepsilon}}^{4}+2\beta\tilde{\omega_{\varepsilon}}^{2}\tilde{\phi_{\varepsilon}}^{2}+\mu_{2}\tilde{\phi_{\varepsilon}}^{4},$

Clearly, $\\{(\tilde{w_{\varepsilon}},\tilde{\phi_{\varepsilon}})\\}$ is
bounded in $E$, consequently,
$(\tilde{w_{\varepsilon}},\tilde{\phi_{\varepsilon}})\rightharpoonup(w,\phi)$
in $E$,
$(\tilde{w_{\varepsilon}},\tilde{\phi_{\varepsilon}})\rightarrow(w,\phi)$ a.e.
$x\in\mathbb{R}^{N}$. We can easily see that $(w,\phi)\neq(0,0)$. Next, we
will prove $(w,\phi)$ is a critical point of $\mathcal{J}_{W_{i}}$. By Fatou’s
Lemma, we have

$\displaystyle\int_{\mathbb{R}^{N}}|\nabla
w|^{2}+W_{*}w^{2}+|\nabla\phi|^{2}+W_{*}\phi^{2}$
$\displaystyle\leq\mathop{\underline{\lim}}\limits_{\varepsilon\rightarrow
0}\int_{\mathbb{R}^{N}}|\nabla\tilde{w_{\varepsilon}}|^{2}+\frac{a_{\varepsilon}(x+\frac{x_{*}}{\varepsilon^{k}})}{\varepsilon^{k\gamma}}\tilde{w_{\varepsilon}}^{2}+|\nabla\tilde{\phi_{\varepsilon}}|^{2}+\frac{a_{\varepsilon}(x+\frac{x_{*}}{\varepsilon^{k}})}{\varepsilon^{k\gamma}}\tilde{\phi_{\varepsilon}}^{2}$
$\displaystyle\leq 4m.$

Multiplying by $\xi,\eta\in C_{0}^{\infty}(\mathbb{R}^{N})$ on equation (4.5)
and (4.6) and making $\varepsilon\rightarrow 0$, we deduce that

$\displaystyle\int_{\mathbb{R}^{N}}\nabla
w\nabla\xi+W_{*}w\xi+\nabla\phi\nabla\eta+W_{*}\phi\eta=\int_{\mathbb{R}^{N}}\mu_{1}w^{3}\xi+\beta\phi^{2}w\xi+\beta
w^{2}\phi\eta+\mu_{2}\phi^{3}\eta,$

then from the density of $\mathbb{C}_{0}^{\infty}(\mathbb{R}^{N})$ in
$E_{W_{*}}$, it follows that $\mathcal{J}_{W_{*}}(w,\phi)=0$. Since
$\beta>\widehat{\beta}$, then $\mathcal{J}_{W_{*}}$ is attained by a positive
solution $(w,\phi)$.

Finally, we show $\mathcal{J}_{W_{*}}(w,\phi)=\mathcal{E}_{W_{x_{*}}}$. There
naturally holds $\mathcal{J}_{W_{*}}(w,\phi)\geq\mathcal{E}_{W_{x_{*}}}$. We
deduce that

$\displaystyle\mathcal{J}_{W_{*}}$
$\displaystyle=\frac{1}{4}\int_{\mathbb{R}^{N}}|\nabla
w|^{2}+W_{*}w^{2}+|\nabla\phi|^{2}+W_{*}\phi^{2}$
$\displaystyle\leq\mathop{\liminf}\limits_{\varepsilon\rightarrow
0}\frac{1}{4}\int_{\mathbb{R}^{N}}|\nabla\tilde{w_{\varepsilon}}|^{2}+\frac{a_{\varepsilon}(x+\frac{x_{*}}{\varepsilon^{k}})}{\varepsilon^{k\gamma}}\tilde{w_{\varepsilon}}^{2}+|\nabla\tilde{\phi_{\varepsilon}}|^{2}+\frac{b_{\varepsilon}(x+\frac{x_{*}}{\varepsilon^{k}})}{\varepsilon^{k\gamma}}\tilde{\phi_{\varepsilon}}^{2}$
$\displaystyle=\mathop{\liminf}\limits_{\varepsilon\rightarrow
0}\frac{c_{\varepsilon}}{\varepsilon^{k(N+2\gamma)}}$
$\displaystyle\leq\mathcal{E}_{W_{x_{*}}}.$

Clearly $(\tilde{w_{\varepsilon}},\tilde{z}_{\varepsilon})\rightarrow(w,z)$
strongly in $\mathbb{L}^{q}(\mathbb{R}^{N})$ for $q\in[2,2^{*})$, we obtain
immediately
$(\tilde{w_{\varepsilon}},\tilde{z}_{\varepsilon})\rightarrow(w,z)$ strongly
in $\mathbb{H}$ as $\varepsilon\rightarrow 0$. $\Box$

This last section is devoted to the proof of Theorem 1.3 which covers the case
that the potential $a$ and $b$ vanish on the closure of smooth bounded open
set. Since $\beta>\widehat{\beta}$, the limit problem corresponding to the
case of Theorem 1.3 has a positive ground state solution, its proof is similar
to Lemma 3.4, we omit it here.

Proof of Theorem 1.2. We define the functional $F_{\varepsilon}(u,v)\in
C^{1}(E)$ for each $(u,v)\in E$ by

$\displaystyle F_{\varepsilon}(w,z)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
w|^{2}+\frac{a(x)}{\varepsilon^{2}}w^{2}+|\nabla
z|^{2}+\frac{b(x)}{\varepsilon^{2}}z^{2}-\frac{1}{4}F(w,z),$

Easily to verify that $(u(x),v(x))$ solves ($\mathcal{P}_{\varepsilon}$) if
and only if
$(w(x),z(x))=\varepsilon^{-1}(u(\frac{x}{\varepsilon}),v(\frac{x}{\varepsilon}))$
is a critical point of $F_{\varepsilon}$, moreover there holds the
ralationship

$\displaystyle F_{\varepsilon}(w,z)=\varepsilon^{N-4}I_{\varepsilon}(u,v).$

This implies that

$\displaystyle(w_{\varepsilon}(x),z_{\varepsilon}(x))=\varepsilon^{-1}\big{(}u_{\varepsilon}(\frac{x}{\varepsilon}),v_{\varepsilon}(\frac{x}{\varepsilon})\big{)}$

is a critical point of $F_{\varepsilon}$ with the least energy. We also
consider the functional $F_{\Sigma}\in C^{1}(E_{\Sigma})$ defined for each
$(u,v)\in E$ by

$\displaystyle F_{\Sigma}(w,z)=\frac{1}{2}\int_{\Sigma}|\nabla w|^{2}+|\nabla
z|^{2}-\frac{1}{4}\int_{\Sigma}\mu_{1}w^{4}+2\beta w^{2}z^{2}+\mu_{2}z^{4},$

where

$\displaystyle E_{\Sigma}=\\{(w,z)\in E|w(x)\in H_{0}^{1}(\mathcal{A}),z(x)\in
H_{0}^{1}(\mathcal{B})\\}.$

For any $(w,z)\in E_{\Sigma}$, we have $F_{\varepsilon}(w,z)=F_{\Sigma}(w,z)$,
and thus

$\displaystyle F_{\varepsilon}(w_{\varepsilon},z_{\varepsilon})$
$\displaystyle=\inf\\{F_{\varepsilon}(w,z);\ (w,z)\in
E\setminus\\{(0,0)\\},\langle F^{\prime}_{\varepsilon}(w,z),(w,z)\rangle=0\\}$
$\displaystyle\leq c_{\Sigma}=\inf\\{F_{\Sigma}(w,z);\ (w,z)\in
E_{\Sigma}\setminus\\{(0,0)\\},\langle
F_{\Sigma}^{\prime}(w,z),(w,z)\rangle=0\\}.$

We deduce that for every $\varepsilon>0$, we have

(4.10) $\displaystyle\int_{\mathbb{R}^{N}}|\nabla
w_{\varepsilon}|^{2}+\frac{a(x)}{\varepsilon^{2}}w_{\varepsilon}^{2}+|\nabla
z_{\varepsilon}|^{2}+\frac{b(x)}{\varepsilon^{2}}z_{\varepsilon}^{2}\leq
4c_{\Sigma}.$

We choose $\delta>0$ such that $a(x)>0$ for $x\not\in\mathcal{A}_{\delta}$.
Take $\phi\in C_{0}^{\infty}(\mathbb{R}^{N})$ such that $\phi(x)=1$ for
$x\in\mathcal{A}_{\delta}$, we have

(4.11)
$\displaystyle\begin{split}\int_{\mathbb{R}^{N}}|w_{\varepsilon}|^{2}&\leq\int_{\mathbb{R}^{N}}\phi^{2}|w_{\varepsilon}|^{2}+(1-\phi)^{2}|w_{\varepsilon}|^{2}\\\
&\leq C_{1}\int_{\mathbb{R}^{N}}|\nabla(\phi
w_{\varepsilon})|^{2}+\frac{a(x)}{\varepsilon^{2}}|w_{\varepsilon}|^{2}\\\
&\leq C_{2}\int_{\mathbb{R}^{N}}|\nabla
w_{\varepsilon}|^{2}+\frac{a(x)}{\varepsilon^{2}}|w_{\varepsilon}|^{2}.\end{split}$

Similar result holds for $z_{\varepsilon}$. Hence, there exists a positive
constant $C_{1}$ such that
$\|(w_{\varepsilon},z_{\varepsilon})\|_{\mathbb{H}}\leq C_{1}c_{\Sigma}$. It
follows that $(w_{\varepsilon},z_{\varepsilon})\rightharpoonup(w,z)$ in
$\mathbb{H}$ and $(w_{\varepsilon},z_{\varepsilon})\rightarrow(w,z)$ in
$\mathbb{L}_{loc}^{2}(\mathbb{R}^{N})$ as $\varepsilon\rightarrow 0$. For any
open set $D\in\mathbb{R}^{N}$ such that $\mathcal{A}\in D$, we assume that
$a(x)\geq a_{1}>0$ for $x\in\mathbb{R}^{N}\setminus D$. Then by $(4.10)$, it
follows that

$\displaystyle\int_{D}|w_{\varepsilon}|^{2}\leq\frac{\varepsilon^{2}}{a_{1}}\int_{D}\frac{a(x)}{\varepsilon^{2}}|w_{\varepsilon}|^{2}\rightarrow
0,\ \ \ \varepsilon\rightarrow 0.$

So $w_{\varepsilon}\rightarrow w$ strongly in $L^{2}(\mathbb{R}^{N})$. By
Gagliardo-Nirenberg-Sobolev inequality, $w_{\varepsilon}\rightarrow w$
strongly in $L^{q}(\mathbb{R}^{N})$ for $q\in[2,2^{*}]$. Moreover we also have
$w\in H_{0}^{1}(\mathcal{A})$. Analogously, $z_{\varepsilon}\rightarrow z$
strongly in $L^{q}(\mathbb{R}^{N})$ for $q\in[2,2^{*}]$ and $z\in
H_{0}^{1}(\mathcal{B})$. Using (4.11) and the fact that
$F_{\varepsilon}^{\prime}(w_{\varepsilon},z_{\varepsilon})=0$, we have

$\displaystyle\|(w_{\varepsilon},z_{\varepsilon})\|_{4}^{2}\leq
C\|(w_{\varepsilon},z_{\varepsilon})\|_{\mathbb{H}}^{2}\leq
C_{1}\|(w_{\varepsilon},z_{\varepsilon})\|_{4}^{4}.$

Hence, we deduce that $\|(w,z)\|_{4}>0$ and $(w,z)\neq(0,0)$.

For any test function $\eta,\varphi\in E$, we have

$\displaystyle 0$ $\displaystyle=\int_{\mathbb{R}^{N}}\nabla
w_{\varepsilon}\nabla\eta+\frac{a}{\varepsilon^{2}}w_{\varepsilon}\eta+\nabla
z_{\varepsilon}\nabla\varphi+\frac{b}{\varepsilon^{2}}z_{\varepsilon}\varphi-\int_{\mathbb{R}^{N}}\mu_{1}w_{\varepsilon}^{3}\eta+\beta
z_{\varepsilon}^{2}w_{\varepsilon}\eta+\beta
w_{\varepsilon}^{2}z_{\varepsilon}\varphi+\mu_{2}z_{\varepsilon}^{3}\varphi$
$\displaystyle=\int_{\Sigma}\nabla w_{\varepsilon}\nabla\eta+\nabla
z_{\varepsilon}\nabla\varphi-\int_{\Sigma}\mu_{1}w_{\varepsilon}^{3}\eta+\beta
z_{\varepsilon}^{2}w_{\varepsilon}\eta+\beta
w_{\varepsilon}^{2}z_{\varepsilon}\varphi+\mu_{2}z_{\varepsilon}^{3}\varphi,$

and hence

$\displaystyle\int_{\Sigma}\nabla w\nabla\eta+\nabla
z\nabla\varphi-\mu_{1}w^{3}\eta+\beta z^{2}w\eta+\beta
w^{2}z\varphi+\mu_{2}z^{3}\varphi=0.$

In view of the regularity assumptions on the set $\Sigma$ and classical
regularity theory, $(w,z)$ satisfies the announced system. We also have

$\displaystyle\mathop{\limsup}\limits_{\varepsilon\rightarrow
0}\int_{\mathbb{R}^{N}}|\nabla w_{\varepsilon}|^{2}+|\nabla
z_{\varepsilon}|^{2}$
$\displaystyle\leq\mathop{\lim}\limits_{\varepsilon\rightarrow
0}\int_{\mathbb{R}^{N}}|\nabla
w_{\varepsilon}|^{2}+\frac{a(x)}{\varepsilon^{2}}w_{\varepsilon}^{2}+|\nabla
z_{\varepsilon}|^{2}+\frac{b(x)}{\varepsilon^{2}}z_{\varepsilon}^{2}$
$\displaystyle=\mathop{\lim}\limits_{\varepsilon\rightarrow
0}F(w_{\varepsilon},z_{\varepsilon})$
$\displaystyle=F(w,z)=\int_{\mathbb{R}^{N}}|\nabla w|^{2}+|\nabla z|^{2}.$

This implies $(w_{\varepsilon},z_{\varepsilon})\rightarrow(w,z)$ strongly in
$\mathbb{H}$ and that $(w,z)$ is a ground state of the limiting problem. Since
$\beta>\widehat{\beta}$, $(w,z)$ is positive. Then the proof is completed.
$\Box$

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# On the additivity of the little cubes operads

Miguel Barata and Ieke Moerdijk

###### Abstract.

We give a new proof of Dunn’s additivity for the little $n$-cubes operads
$C_{n}$, which has the advantage of being considerably shorter than the ones
in [1, 2]. At the end we remark on how our proof can be adjusted to work for
the tensor product of a finite number of factors.

The purpose of this note is to provide a simpler proof of the additivity
property of the little $n$-cubes operads $C_{n}$, usually known as Dunn’s
additivity, the precise statement of which is given below in Theorem 1. Before
stating it, recall that a morphism $f\colon\mathbf{P}\to\mathbf{Q}$ of
uncoloured topological operads is a local $\Sigma$-equivalence if the map on
operations $f\colon\mathbf{P}(j)\to\mathbf{Q}(j)$ is a
$\Sigma_{j}$-equivariant homotopy equivalence for all $j\geq 0$. We will write
$\mathbf{P}\simeq\mathbf{Q}$ if there exists a local $\Sigma$-equivalence, or
a zig-zag of such, between $\mathbf{P}$ and $\mathbf{Q}$.

######  Theorem 1.

$C_{n}\otimes C_{m}\simeq C_{n+m}$ for $n,m\geq 1$, where
$\text{-}\otimes\text{-}$ denotes the Boardman-Vogt tensor product of
topological operads.

The first version of this statement was proven by Dunn in [1], where he showed
that $C_{1}^{\otimes n}\simeq C_{n}$. However, it is important to remark that
Theorem 1 is not an immediate consequence of Dunn’s result since in general
the Boardman-Vogt tensor product need not preserve local
$\Sigma$-equivalences: for instance, for the associative operad $\mathbf{Ass}$
we have that $\mathbf{Ass}\otimes\mathbf{Ass}$ is the commutative operad
$\mathbf{Comm}$; however $C_{1}\simeq\mathbf{Ass}$ and $C_{1}\otimes
C_{1}\simeq C_{2}$, the last one by Dunn’s additivity, even though
$C_{2}\not\simeq\mathbf{Comm}$. A proof of Theorem 1 is given in [2]
folllowing the general idea of Dunn’s proof. In this text we aim to give a
different proof of the additivity statement, in which we rely on the
description of the operations of the operad $C_{n}\otimes C_{m}$ as being
generated, under composition, by elementary ones of the form $p\otimes 1$ and
$1\otimes q$, where $p\in C_{n}(a),q\in C_{m}(b)$. In Remark 6 we comment on
how our methods can be generalized to the case of a finite tensor product
$C_{k_{1}}\otimes\cdots\otimes C_{k_{n}}$.

To begin with, let us describe how one can construct an operad map $\mu\colon
C_{n}\otimes C_{m}\to C_{n+m}$. Consider the morphisms

$\mu_{1}\colon C_{n}\rightarrow C_{n+m}\hskip 10.00002pt\mathrm{and}\hskip
10.00002pt\mu_{2}\colon C_{m}\rightarrow C_{n+m}$

induced by the inclusions (of spaces) of the first $n$ coordinates
$\mathbb{R}^{n}\to\mathbb{R}^{n+m}$ and of the last $m$ coordinates
$\mathbb{R}^{m}\to\mathbb{R}^{n+m}$ respectively. These are easily seen to
give rise to an operad map on the Boardman-Vogt tensor product $C_{n}\otimes
C_{m}$,

$\mu\colon C_{n}\otimes C_{m}\longrightarrow C_{n+m},$

whose image is a suboperad $C_{n}|C_{m}\subseteq C_{n+m}$ which is defined in
[2, Sec. 8]. It follows that $\mu$ factors through the operad inclusion
$C_{n}|C_{m}\subseteq C_{n+m}$ as an operad morphism $\mu^{\prime}\colon
C_{n}\otimes C_{m}\to C_{n}|C_{m}$ which is surjective on operations.

In [2, Lem. 12.6.] Brinkmeier proves that the operad of decomposable little
$(n+m)$-cubes $D_{n+m}$ (see [1, Def. 2.1.] for a definition of $D_{k}$)
includes into $C_{n}\lvert C_{m}$ and that this inclusion is a local
$\Sigma$-equivalence. Moreover, in [1, Prop. 2.3.] Dunn shows that
$D_{n+m}\subseteq C_{n+m}$ is also a local $\Sigma$-equivalence. We note that
the proofs of these results are rather simple and the required homotopies are
obtained by radially contracting each of the cubes $c\in C_{n+m}(j)$ to their
centers and observing that, after some moment in this contraction, $c$ will be
an element of $D_{n+m}(j)$. Putting these results together, it follows after
examining the commutative diagram

(1) ${C_{n}\otimes
C_{m}}$${C_{n+m}}$${C_{n}|C_{m}}$${D_{n}|D_{m}}$${D_{n+m}}$$\scriptstyle{\mu}$$\scriptstyle{\mu^{\prime}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

that to prove Theorem 1 it is sufficient to show that $\mu^{\prime}\colon
C_{n}\otimes C_{m}\to C_{n}|C_{m}$ is an isomorphism of topological operads.
This is the difficult step in both Dunn’s and Brinkmeier’s proofs and so our
purpose is to give a shorter and simpler proof that this map is indeed an
isomorphism.

We start with the following lemma, which will allow us to split up the
inductive step of the proof of the injective statement in Proposition 3 into
more tractable cases.

######  Lemma 2.

Let $n,m\in\mathbb{N}$ and consider an operation $c$ in $C_{n}\otimes C_{m}$
of arity $j>1$. Then $c$ can be written as

(2) $c=(p\otimes 1)\circ(t_{1},\ldots,t_{a})\hskip 10.00002ptor\hskip
10.00002ptc=(1\otimes q)\circ(s_{1},\ldots,s_{b}),$

where $p\in C_{n}(a),q\in C_{m}(b)$ with $a,b>1$ and $t_{i},s_{j}$ are
operations in $C_{n}\otimes C_{m}$ of positive arity.

###### Proof.

Since the operations in $C_{n}\otimes C_{m}$ are generated under composition
by elements of the form $p\otimes 1$ and $1\otimes q$, we can certainly write
$c$ in one of the forms shown in (2). Thus, it remains to show that our
constraints on the arities of $p,q,t_{i},s_{j}$ can be realized. In what
follows we will write $\ast$ for the nullary operation of the operads
$C_{n},C_{m},C_{n}\otimes C_{m}$; it should always be clear from the context
in which operad we are considering $\ast$ so no confusion will arise from this
notation.

If $c=(p\otimes 1)\circ(t_{1},\ldots,t_{a})$ but $t_{i}=\ast$ holds for some
index $i$, then letting $p^{\prime}$ be the operation $p\circ_{i}\ast\in
C_{n}(a-1)$ we see that

$c=(p^{\prime}\otimes
1)\circ\left(t_{1},\ldots,\hat{t}_{i},\ldots,t_{a}\right)$

where $\hat{t}_{i}$ denotes the omission of the $i^{th}$ input $t_{i}$. Hence,
we can assume that all the $t_{i}$’s aren’t the nullary operation. Moreover, a
simple adaption of this argument also shows that we can assume that
$s_{i}\neq\ast$ in expression (2).

As for the restriction $a>1$, we induct on the length of $c$, $\ell(c)$, which
we define to be the infimum of the number of elementary operations of the the
form $p\otimes 1$ and $1\otimes q$ needed to write $c$ using the composition
of operations in the operad. If $\ell(c)=1$ then $c$ is either $p\otimes 1$ or
$1\otimes q$, where $p$ and $q$ have arity $j>1$, and so we have nothing to
prove. For the induction step, the length of $c$ must be realized by an
expression of the form $c=(p\otimes 1)\circ t_{1}$ or $c=(1\otimes q)\circ
s_{1}$, where $p,q$ have arity 1 (otherwise we have nothing to prove) and
$\ell(t_{1}),\ell(s_{1})<\ell(c)$. If the former holds then, by our induction
hypothesis, we know that $t_{1}$ must be of the form

$t_{1}=(p^{\prime}\otimes 1)\circ(t_{1}^{\prime},\ldots,t_{k}^{\prime})\hskip
10.00002pt\mathrm{or}\hskip 10.00002ptt_{1}=(1\otimes
q^{\prime})\circ(s^{\prime}_{1},\ldots,s^{\prime}_{\ell}),$

where $p^{\prime},q^{\prime}$ have at least arity 2. For the leftmost case we
can use the relation $(p\otimes 1)\circ(p^{\prime}\otimes 1)=(p\circ
p^{\prime})\otimes 1$ to conclude the intended; for the other case we use the
interchange relation of the tensor product instead. The argument using the
expression $c=(1\otimes q)\circ s_{1}$ is completely parallel to this one.
This finishes our induction. ∎

######  Lemma 3.

Let $j\geq 1$. Then $\mu^{\prime}\colon(C_{n}\otimes
C_{m})(j)\to(C_{n}|C_{m})(j)$ is a continuous bijection.

###### Proof.

Since $C_{n}|C_{m}$ is just the image of $\mu$ and thus $\mu^{\prime}$ is
surjective at the level of operations, we only have to check that
$\mu^{\prime}$ is also injective.

For $j=1$ notice that due to the interchange relation, any element of
$(C_{n}\otimes C_{m})(1)$ is of the form $(p\otimes 1)\circ(1\otimes q)$ where
$p\in C_{n}(1)$ and $q\in C_{m}(1)$. Moreover, it is clear that
$\mu^{\prime}((p\otimes 1)\circ(1\otimes q))=\mu^{\prime}((p^{\prime}\otimes
1)\circ(1\otimes q^{\prime}))$ can happen if and only if
$p=p^{\prime},q=q^{\prime}$.

For the general case when $j>1$, suppose that $c_{1},c_{2}\in(C_{n}\otimes
C_{m})(j)$ satisfy $\mu^{\prime}(c_{1})=\mu^{\prime}(c_{2})$. By Lemma 2, it
is enough to consider the following three scenarios:

  1. (i)

We can write

$c_{1}=(p\otimes 1)\circ(t_{1},\ldots,t_{a})\hskip
10.00002pt\mathrm{and}\hskip 10.00002ptc_{2}=(\overline{p}\otimes
1)\circ(\overline{t_{1}},\ldots,\overline{t_{\overline{a}}}),$

where $p\in C_{n}(a),\overline{p}\in C_{n}(\overline{a})$ with
$a,\overline{a}>1$ and $t_{i},\overline{t_{j}}\neq\ast$.

  2. (ii)

We can write

$c_{1}=(1\otimes q)\circ(s_{1},\ldots,s_{b})\hskip
10.00002pt\mathrm{and}\hskip
10.00002ptc_{2}=(1\otimes\overline{q})\circ(\overline{s_{1}},\ldots,\overline{s_{\overline{b}}}),$

where $q\in C_{m}(b),\overline{q}\in C_{m}(\overline{b})$ with
$b,\overline{b}>1$ and $s_{i},\overline{s_{j}}\neq\ast$.

  3. (iii)

We can write

$c_{1}=(p\otimes 1)\circ(t_{1},\ldots,t_{a})\hskip
10.00002pt\mathrm{and}\hskip 10.00002ptc_{2}=(1\otimes
q)\circ(s_{1},\ldots,s_{b}),$

where $p\in C_{n}(a),q\in C_{m}(b)$ with $a,b>1$ and $t_{i},s_{j}\neq\ast$.

Proceeding inductively on $j$, suppose first that (i) holds. If
$p=\overline{p}$ then, by considering the equality

$\mu^{\prime}(p\otimes
1)\circ(\mu^{\prime}(t_{1}),\ldots,\mu^{\prime}(t_{a}))=\mu^{\prime}(p\otimes
1)\circ(\mu^{\prime}(\overline{t}_{1}),\ldots,\mu^{\prime}(\overline{t}_{a}))$

together with the injectivity of the composition map $\mu^{\prime}(p\otimes
1)\circ(-)$ in $C_{n+m}$, we conclude that
$\mu^{\prime}(t_{i})=\mu^{\prime}(\overline{t}_{i})$ for all $i=1,\ldots,a$.
As $t_{i}$ and $\overline{t}_{i}$ have arity strictly smaller than that of $c$
by our hypotheses in (i), the induction hypothesis gives us that
$t_{i}=\overline{t}_{i}$ and thus $c_{1}=c_{2}$.

If $p\neq\overline{p}$ then let us define a new operation
$p\wedge\overline{p}$ in $C_{n}$ for which we can find operations
$p_{1},\ldots,p_{k}$ and $\overline{p}_{1},\ldots,\overline{p}_{\overline{k}}$
in $C_{n}$ such that

(3)
$p\wedge\overline{p}=p\circ(p_{1},\ldots,p_{k})=\overline{p}\circ(\overline{p}_{1},\ldots,\overline{p}_{\overline{k}}).$

One way of constructing $p\wedge\overline{p}$ is to consider, for instance,
the elements $\mu^{\prime}(p\otimes 1)$ and $\mu^{\prime}(\overline{p}\otimes
1)$ in $C_{n}|C_{m}$. These define strips in the cube $]0,1[^{n+m}$ and we can
choose $p\wedge\overline{p}$ to be such that the strips of
$\mu^{\prime}((p\wedge\overline{p})\otimes 1)$ are simultaneously contained in
$\mu^{\prime}(p\otimes 1)$ and $\mu^{\prime}(\overline{p}\otimes 1)$ and
contain the cubes of $\mu^{\prime}(c_{1})$ in their interiors, for example.
It’s easy to verify that this satisfies (3).

It is geometrically clear from $\mu^{\prime}(c_{1})=\mu^{\prime}(c_{2})$ and
the paragraph above that we can find $p\wedge\overline{p}$ together with an
element $c_{0}=((p\wedge\overline{p})\otimes 1)\circ(u_{1},\ldots,u_{\ell})$
in $C_{n}\otimes C_{m}$ such that
$\mu^{\prime}(c_{0})=\mu^{\prime}(c_{1})=\mu^{\prime}(c_{2})$. Using (3) we
may rewrite $c_{0}$ as an element of the form

(4) $c_{0}=(p\otimes 1)\circ(u^{\prime}_{1},\ldots,u^{\prime}_{a}),$

where $u_{i}^{\prime}$ can’t be $\ast$, even though $u_{i}$ could be. Indeed,
if $u^{\prime}_{i}=\ast$ for some index $i$ then, since
$\mu^{\prime}(c_{0})=\mu^{\prime}(c_{1})$, it would follow that the $i^{th}$
strip of $\mu^{\prime}(c_{1})$ (as defined by the $i^{th}$ cube of $p$)
doesn’t contain any cubes in its interior, which contradicts the assumption
$t_{i}\neq\ast$. Thus we can apply the special case of (i) which we discussed
first to

$c_{0}=(p\otimes 1)\circ(u^{\prime}_{1},\ldots,u^{\prime}_{a})\hskip
10.00002pt\mathrm{and}\hskip 10.00002ptc_{1}=(p\otimes
1)\circ(t_{1},\ldots,t_{a})$

to conclude that $c_{0}=c_{1}$. If we instead argued using $\overline{p}$ and
wrote $c_{0}$ as $(\overline{p}\otimes
1)\circ(v_{1}^{\prime},\ldots,v_{b}^{\prime})$ in equation (4) then we would
obtain $c_{0}=c_{2}$. Piecing all of this together, we have shown that
$c_{1}=c_{2}$, finishing the proof for (i). The argument for (ii) is similar
to the one just given.

If (iii) holds then we claim that each $t_{i}$ must be of the form $(1\otimes
q)\circ(v_{1}^{i},\ldots,v_{b}^{i})$. Indeed, using our initial condition
$\mu^{\prime}(c_{1})=\mu^{\prime}(c_{2})$, we can find an element of this form
- call it $t_{i}^{\prime}$ for the time being - such that
$\mu^{\prime}(t^{\prime}_{i})=\mu^{\prime}(t_{i})$ in $C_{n}|C_{m}$. Since the
arity of $t_{i}$ is strictly less than that of $c_{1}$, we can apply the
induction hypothesis to conclude that $t_{i}=t^{\prime}_{i}$.

The figure below exemplifies, for $n=m=1$, how one can find $t_{1}^{\prime}$.
Here $\mu^{\prime}(c_{1})$ is made up of the orange-filled squares. The
configuration $p\otimes 1$ gives rise to the vertical strips
$\mu^{\prime}(p_{i}\otimes 1)$ in light-blue and the red horizontal ones
$\mu^{\prime}(1\otimes q_{j})$ are defined by the operation $q\otimes 1$.
Moreover, all our cubes must be contained in the intersection of these
rectangles, which are the smaller cubes with blue and red solid edges. The
element $\mu^{\prime}(t_{1})$, which corresponds, up to rescalling on the
horizontal direction, to the orange cubes inside the strip
$\mu^{\prime}(p_{1}\otimes 1)$, has a subdivision by the horizontal strips,
which can then be used to construct our desired $t_{1}^{\prime}$.

Using the interchange relation in $C_{n}\otimes C_{m}$ we see that

$\displaystyle c_{1}=(p\otimes 1)\circ(t_{1},\ldots,t_{a})$
$\displaystyle=(p\otimes 1)\circ\left((1\otimes
q)\circ(v_{1}^{1},\ldots,v_{b}^{1}),\ldots,(1\otimes
q)\circ(v_{1}^{a},\ldots,v_{b}^{a})\right)$ $\displaystyle=(1\otimes
q)\circ\left((p\otimes 1)\circ(v_{1}^{1},\ldots,v_{1}^{a}),\ldots,(p\otimes
1)\circ(v_{b}^{1},\ldots,v_{b}^{a})\right)$ $\displaystyle=(1\otimes
q)\circ(v_{1},\ldots,v_{b})$

where in the last step $v_{i}=(p\otimes 1)\circ(v_{i}^{1},\ldots,v_{i}^{a})$
for $i=1,\ldots,b$. In particular $v_{i}\neq\ast$ since otherwise the $i^{th}$
strip of $\mu^{\prime}(c_{1})$, as determined by the operation
$\mu^{\prime}(1\otimes q)$, would have no cubes in its interior, which
contradicts $s_{i}\neq\ast$. We are now in the position of applying case (ii)
to conclude that $c_{1}=c_{2}$, which finishes the proof of injectivity.

∎

######  Lemma 4.

Let $j\geq 1$. Then $\mu^{\prime}\colon(C_{n}\otimes
C_{m})(j)\to(C_{n}|C_{m})(j)$ is a proper map.

###### Proof.

Observe that if we have a surjective continuous map
$\pi\colon\mathbf{F}(j)\rightarrow(C_{n}\otimes C_{m})(j)$ then the properness
of $\nu=\mu^{\prime}\circ\pi$ implies the properness of $\mu^{\prime}$ since
then the equality $(\mu^{\prime})^{-1}(K)=\pi\left(\nu^{-1}(K)\right)$ holds
for any (compact) subset $K\subset(C_{n}|C_{m})(j)$.

With this in mind, let $\mathbf{F}$ be the free operad on the symmetric
collection $C_{n}\coprod C_{m}$; that is, we forget the operad structure on
$C_{n}\coprod C_{m}$ and consider it only as a collection of spaces equipped
with continuous actions by the symmetric groups. Then there is an operad map
$\mathbf{F}\to C_{n}\otimes C_{m}$ and for each $j\geq 0$ a commutative
diagram

${\mathbf{F}(j)}$${(C_{n}\otimes
C_{m})(j)}$${(C_{n}|C_{m})(j),}$$\scriptstyle{\nu}$$\scriptstyle{\mu^{\prime}\leavevmode\nobreak\
}$

where the top horizontal map is surjective. By the paragraph above, it
suffices to verify that $\nu$ is a proper map.

By the usual construction of the free operad, $\mathbf{F}(j)$ is a disjoint
union of products of copies of $C_{n}(k)$’s and $C_{m}(\ell)$’s appropriately
indexed, hence it is a metric space, as is $(C_{n}|C_{m})(j)$. Therefore, we
can check the properness of $\nu$ by seeing that this map preserves sequences
escaping to infinity (we say that a sequence $\\{x_{k}\\}_{k\geq 1}$ in a
topological space $X$ escapes to infinity if for every compact set $K\subseteq
X$ the set $\\{k\in\mathbb{N}:x_{k}\in K\\}$ is finite). This can be easily
checked by using the fact that a sequence in $(C_{n}|C_{m})(j)$ escapes to
infinity precisely when the volume of one of its cubes tends to 0. ∎

###### Proof of Theorem 1.

By diagram (1), it suffices to show that $\mu^{\prime}\colon(C_{n}\otimes
C_{m})(j)\to(C_{n}|C_{m})(j)$ is a homeomorphism for all arities $j\geq 0$.
For $j=0$ this is clear since the operads in question have a single nullary
operation, so we may assume that $j\geq 1$.

From Lemma 3 we get that $\mu^{\prime}$ is a continuous bijection so we only
have to check that $\mu^{\prime}$ is closed. This follows from Lemma 4,
together with the fact that a proper continuous map $f\colon X\to Y$ with a
locally compact and Hausdorff target is a closed map (with compact fibers). ∎

######  Remark 5.

We used Lemma 4 in the proof of Theorem 1 to show that $\mu^{\prime}$ has a
continuous inverse. In [2] a different proof of this fact is given, which
consists in mapping $C_{n}$ into an operad $\overline{C_{n}}$ where its spaces
of operations, as well as the spaces of
$\overline{C_{n}}\otimes\overline{C_{m}}$, are compact and Hausdorff, and
extending $\mu^{\prime}$ to a morphism between these newly-defined operads.
However, it is not clear to us how Brinkmeier’s definition of
$\overline{C_{n}}$ leads to an operad nor how to define
$\overline{C_{n}}\otimes\overline{C_{m}}$. This was also one of our
motivations for presenting this new proof of Dunn’s additivity.

######  Remark 6.

One can consider the more general case of additivity for the tensor product of
a finite number of operads of the form $C_{n}$. More precisely, in an
analogous way to what we did at the beginning, we can construct an operad map

$\mu\colon C_{k_{1}}\otimes C_{k_{2}}\otimes\cdots\otimes
C_{k_{n}}\longrightarrow C_{k_{1}+k_{2}+\cdots+k_{n}}$

and ask whether $\mu$ is a local $\Sigma$-equivalence $C_{k_{1}}\otimes
C_{k_{2}}\otimes\cdots C_{k_{n}}\simeq C_{k_{1}+k_{2}+\cdots+k_{n}}.$ As we
have previously mentioned, this doesn’t follow immediately from the case $n=2$
since the Boardman-Vogt tensor product in general won’t preserve local
$\Sigma$-equivalences. In spite of this, we claim that the arguments explained
above can be carried over to this generalized case where $n\geq 2$.

Indeed, one can upgrade the construction of the operads $C_{n}|C_{m}$ and
$D_{n}|D_{m}$ to a multivariable version $C_{k_{1}}|\cdots|C_{k_{n}}$ and
$D_{k_{1}}|\cdots|D_{k_{n}}$, see [2], to get a commutative diagram similar to
(1),

${C_{k_{1}}\otimes\cdots\otimes
C_{k_{n}}}$${C_{k_{1}+\cdots+k_{n}}}$${C_{k_{1}}|\cdots|C_{k_{n}}}$${D_{k_{1}}|\cdots|D_{k_{n}}}$${D_{k_{1}+\cdots+k_{n}}.}$$\scriptstyle{\mu}$$\scriptstyle{\mu^{\prime}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Therefore, it again suffices to show that $\mu^{\prime}$ is an isomorphism.
Lemma 2 and Lemma 4 can be easily generalized to this new context. For the
proof of Lemma 3, checking the injectivity of $\mu^{\prime}$ involves studying
an equation of the form $\mu^{\prime}(c_{1})=\mu^{\prime}(c_{2})$, where
$c_{1}$ and $c_{2}$ are configurations which can be written as

$c_{1}=(1\otimes\cdots\otimes p\otimes\cdots\otimes 1)\circ\mathbf{t}\hskip
10.00002pt\mathrm{and}\hskip 10.00002ptc_{1}=(1\otimes\cdots\otimes
q\otimes\cdots\otimes 1)\circ\mathbf{s}$

by the generalization of Lemma 2. Here $p$ and $q$ are, respectively,
operations in the factors $C_{k_{i}}$ and $C_{k_{j}}$ of the tensor product,
with $i$ and $j$ not necessarily equal. One can then proceed as before, by an
argument only involving the $i^{th}$ and $j^{th}$ tensor factors.

######  Remark 7.

Since the little 1-cubes operad $C_{1}$ coincides with its suboperad of
decomposable cubes $D_{1}$, the isomorphism of operads

$\mu^{\prime}\colon D_{1}^{n}=C_{1}^{\otimes n}\longrightarrow
C_{1}|\cdots|C_{1}=D_{n}$

that we obtained from the considerations in Remark 6 shows that the collection
of decomposable little $n$-cubes operads $\left\\{D_{n}\right\\}_{n\geq 1}$
satisfies the stronger form of additivity $D_{n}\otimes D_{m}\cong D_{n+m}$.
Notice that this statement also follows from Dunn’s result (in contrast to the
additivity of $C_{n}$) since the Boardman-Vogt tensor product will preserve
isomorphisms of operads.

## References

  * [1] G. Dunn.  Tensor product of operads and iterated loop spaces.  J. Pure Appl. Algebra, 50:237–258, 1988.
  * [2] M. Brinkmeier.  On Operads.  PhD thesis, Universitat Osnabrüeck, 2000.