# Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic
systems

Mónica Clapp111M. Clapp was supported by CONACYT (Mexico) through the grant
for the research project A1-S-10457. and Mayra Soares222M. Soares was
supported by UNAM-DGAPA (Mexico) through a postdoctoral fellowship.

###### Abstract

We study the existence and asymptotic behavior of solutions having positive
and sign-changing components to the singularly perturbed system of elliptic
equations

$\begin{cases}-\varepsilon^{2}\Delta
u_{i}+u_{i}=\mu_{i}|u_{i}|^{p-2}u_{i}+\sum\limits_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}-2}u_{i},\\\
u_{i}\in H^{1}_{0}(\Omega),\quad u_{i}\neq 0,\qquad
i=1,\ldots,\ell,\end{cases}$

in a bounded domain $\Omega$ in $\mathbb{R}^{N}$, with $N\geq 4$,
$\varepsilon>0$, $\mu_{i}>0$, $\lambda_{ij}=\lambda_{ji}<0$,
$\alpha_{ij},\beta_{ij}>1$, $\alpha_{ij}=\beta_{ji}$,
$\alpha_{ij}+\beta_{ij}=p\in(2,2^{*})$, and $2^{*}:=\frac{2N}{N-2}$.

If $\Omega$ is the unit ball we obtain solutions with a prescribed combination
of positive and nonradial sign-changing components exhibiting two different
types of asymptotic behavior as $\varepsilon\to 0$: solutions whose limit
profile is a rescaling of a solution with positive and nonradial sign-changing
components of the limit system

$\begin{cases}-\Delta
u_{i}+u_{i}=\mu_{i}|u_{i}|^{p-2}u_{i}+\sum\limits_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}-2}u_{i},\\\
u_{i}\in H^{1}(\mathbb{R}^{N}),\quad u_{i}\neq 0,\qquad
i=1,\ldots,\ell,\end{cases}$

and solutions whose limit profile is a solution of the uncoupled system, i.e.,
after rescaling and translation, the limit profile of the $i$-th component is
a positive or a nonradial sign-changing solution to the equation

$-\Delta u+u=\mu_{i}|u|^{p-2}u,\qquad u\in H^{1}(\mathbb{R}^{N}),\qquad u\neq
0.$

Keywords: Nonlinear elliptic system, singularly perturbed, weakly coupled,
competitive, positive and sign-changing spikes.

MSC2020: 35J57 (35B06, 35B25, 35B40, 47J30).

##  1 Introduction

We consider the following system of singularly perturbed elliptic equations

($\mathscr{S}_{\varepsilon,\Omega}$)
$\qquad\begin{cases}-\varepsilon^{2}\Delta
u_{i}+u_{i}=\mu_{i}|u_{i}|^{p-2}u_{i}+\sum\limits_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}-2}u_{i},\\\
u_{i}\in H^{1}_{0}(\Omega),\quad u_{i}\neq 0,\qquad
i=1,\ldots,\ell,\end{cases}$

where $\varepsilon>0$ is a small parameter, $\Omega$ is a bounded smooth
domain in $\mathbb{R}^{N}$ which contains the origin, $N\geq 2$, $\mu_{i}>0$,
$\lambda_{ij}=\lambda_{ji}<0$, $\alpha_{ij},\beta_{ij}>1$,
$\alpha_{ij}=\beta_{ji}$, $\alpha_{ij}+\beta_{ij}=p\in(2,2^{*})$, and $2^{*}$
is the critical Sobolev exponent (i.e., $2^{*}:=\frac{2N}{N-2}$ if $N\geq 3$
and $2^{*}:=\infty$ if $N=2$).

This system arises as a model for various physical phenomena, in particular in
the study of standing waves for a mixture of Bose–Einstein condensates of
$\ell$ different hyperfine states which overlap in space, see for example
[10]. Here we consider the case in which the interaction between particles in
the same state is attractive ($\mu_{i}>0$) and the interaction between
particles in any two different states is repulsive ($\lambda_{ij}<0$).

In their seminal paper [12] Lin and Wei described the behavior of positive
least energy solutions for the system ($\mathscr{S}_{\varepsilon,\Omega}$)
with cubic nonlinearity ($\alpha_{ij}=\beta_{ij}=2$) as $\varepsilon\to 0$.
They showed that each component is a spike, i.e., a rescaling of the positive
ground state solution to the problem

($\mathscr{P}_{i}$) $\begin{cases}-\Delta u+u=\mu_{i}|u|^{p-2}u,\\\ u\in
H^{1}(\mathbb{R}^{N}),\quad u\neq 0,\end{cases}$

and that, as $\varepsilon\to 0$, the centers of the spikes approach a sphere-
packing position in $\Omega$, i.e., a configuration of points maximizing the
distances among them and to the boundary of $\Omega$. Multiple positive
solutions and a numerical description of them is given in [25].

Our main objective is to study the existence and profile of solutions to
($\mathscr{S}_{\varepsilon,\Omega}$) some of whose components can be positive
while others change sign. It is reasonable to expect that there will be
solutions with sign-changing spikes, i.e., solutions whose sign-changing
components look like rescalings of a sign-changing solution to the problem
($\mathscr{P}_{i}$). The existence of nonradial sign-changing solutions to
($\mathscr{P}_{i}$) was first established by Bartsch and Willem in [2] for
dimensions $N=4$ and $N\geq 6$, and by Lorca and Ubilla for $N=5$ in [16],
taking advantage of some symmetry properties of $\mathbb{R}^{N}$. Other
solutions of this type were found in [7].

On the other hand, rescaling the components by
$\widetilde{u}_{i}(x):=u_{i}(\varepsilon x)$ the system
($\mathscr{S}_{\varepsilon,\Omega}$) becomes

$\begin{cases}-\Delta
u_{i}+u_{i}=\mu_{i}|u_{i}|^{p-2}u_{i}+\sum\limits_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}-2}u_{i},\\\
u_{i}\in H^{1}_{0}(\Omega_{\varepsilon}),\quad u_{i}\neq 0,\qquad
i=1,\ldots,\ell,\end{cases}$

in $\Omega_{\varepsilon}:=\\{x\in\mathbb{R}^{N}:\varepsilon x\in\Omega\\}$. As
$\varepsilon\to 0$ these domains cover the whole space $\mathbb{R}^{N}$. So it
is natural to ask whether the system ($\mathscr{S}_{\varepsilon,\Omega}$) has
a solution that, after rescaling, approaches a solution to the system

($\mathscr{S}_{\infty,\ell}$) $\quad\begin{cases}-\Delta
u_{i}+u_{i}=\mu_{i}|u_{i}|^{p-2}u_{i}+\sum\limits_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}-2}u_{i},\\\
u_{i}\in H^{1}(\mathbb{R}^{N}),\quad u_{i}\neq 0,\qquad
i=1,\ldots,\ell.\end{cases}$

As shown by Lin and Wei in [13, Theorem 1], this system does not have a ground
state solution. On the other hand, Sirakov showed in [23] that it does have a
positive least energy radial solution (i.e., every component $u_{i}$ is
positive and radial); see also [4, 26] and the references therein.

One might expect to obtain solutions with positive and sign-changing
components for the system ($\mathscr{S}_{\varepsilon,\Omega}$) whose limit
profile is a solution of the same type for the system
($\mathscr{S}_{\infty,\ell}$). For this last system with cubic nonlinearity
(hence, $N\leq 3$) Sato and Wang [22] established the existence of least
energy solutions of this type whose components are radial; see also [5, 14].
The following result provides solutions with nonradial sign-changing
components. It is proved in Section 3.

######  Theorem 1.1.

Let $N=4$ or $N\geq 6$. Then, for any given $0\leq m\leq\ell$, the system
($\mathscr{S}_{\infty,\ell}$) has a solution
$\mathbf{w}=(w_{1},\ldots,w_{\ell})$ whose first $m$ components
$w_{1},\ldots,w_{m}$ are positive and whose last $\ell-m$ components
$w_{m+1},\ldots,w_{\ell}$ are nonradial and change sign. Furthermore,
$\mathbf{w}$ satisfies

(1.1)
$\begin{cases}w_{i}(z_{1},z_{2},x)=w_{i}(\mathrm{e}^{\mathrm{i}\vartheta}z_{1},\mathrm{e}^{\mathrm{i}\vartheta}z_{2},gx)&\text{for
all \ }\vartheta\in[0,2\pi),\ g\in O(N-4),\ i=1,\ldots,\ell,\\\
w_{i}(z_{1},z_{2},x)=w_{i}(z_{2},z_{1},x)&\text{if \ }i=1,\ldots,m,\\\
w_{i}(z_{1},z_{2},x)=-w_{i}(z_{2},z_{1},x)&\text{if \
}i=m+1,\ldots,\ell,\end{cases}$

for all
$(z_{1},z_{2},x)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}\equiv\mathbb{R}^{N}$,
and it has least energy among all nontrivial solutions with these symmetry
properties.

The symmetries of the sign-changing components $w_{m+1},\ldots,w_{\ell}$ are
basically the same as those considered in [2]. So one may wonder whether one
can obtain a similar result for $N=5$ using the symmetries introduced in [16]
or [7]. As we shall see, this is not possible. A key role is played by the
space of fixed points of the group of symmetries involved, see Theorem 3.4.

For the singularly perturbed system ($\mathscr{S}_{\varepsilon,\Omega}$) in a
domain $\Omega$ having suitable symmetries we obtain solutions with a
prescribed combination of positive and sign-changing components exhibiting two
different types of asymptotic behavior as $\varepsilon\to 0$. We get solutions
whose limit profile is a rescaling of a solution to the limit system
($\mathscr{S}_{\infty,\ell}$) with positive and sign-changing components, and
solutions whose limit profile is a solution to the uncoupled system, i.e., the
limit profile of the $i$-th component is a rescaling of a positive or a sign-
changing solution to the problem ($\mathscr{P}_{i}$). To illustrate our
results, let us focus on the case where $\Omega$ is the open unit ball
$B_{1}(0)$ in $\mathbb{R}^{N}$ centered at the origin. For $\varepsilon>0$ and
$u\in H^{1}(\mathbb{R}^{N})$ let

$\|u\|_{\varepsilon}^{2}:=\frac{1}{\varepsilon^{N}}\displaystyle\int_{\mathbb{R}^{N}}\Big{[}\varepsilon^{2}|\nabla
u|^{2}+u^{2}\Big{]}\qquad\text{and}\qquad\|u\|:=\|u\|_{1}.$

The following two theorems are special cases of Theorem 4.1, which is stated
and proved in Section 4.

######  Theorem 1.2.

Let $N=4$ or $N\geq 6$, and $\Omega=B_{1}(0)$. Then, for any given $0\leq
m\leq\ell$ and any sequence $(\varepsilon_{k})$ of positive numbers converging
to zero, there exists a solution
$\widehat{\mathbf{u}}_{k}=(\widehat{u}_{1k},\ldots,\widehat{u}_{\ell k})$ to
the system $(\mathscr{S}_{\varepsilon_{k},\Omega})$ whose first $m$ components
are positive and whose last $\ell-m$ components are nonradial and change sign,
with the following limit profile:

There exists a fully nontrivial solution
${\mathbf{w}=(w_{1},\ldots,w_{\ell})}$ to the system
($\mathscr{S}_{\infty,\ell}$) such that, after passing to a subsequence,

$\lim_{k\to\infty}\|\widehat{u}_{ik}-w_{i}(\varepsilon_{k}^{-1}\ \cdot\
)\|_{\varepsilon_{k}}=0\qquad\text{for all \ }i=1,\ldots,\ell.$

The first $m$ components of $\mathbf{w}$ are positive, its last $\ell-m$
components are nonradial and change sign, and $\mathbf{w}$ satisfies (1.1).
Therefore,

$\lim_{k\to\infty}\displaystyle\sum_{i=1}^{\ell}\|\widehat{u}_{ik}\|_{\varepsilon_{k}}^{2}=\displaystyle\sum_{i=1}^{\ell}\|w_{i}\|^{2}=:\widehat{\mathfrak{c}}_{m}.$

######  Theorem 1.3.

Let $N\geq 5$ and $\Omega=B_{1}(0)$. Then, for any given $0\leq m\leq\ell$ and
any sequence $(\varepsilon_{k})$ of positive numbers converging to zero, there
exists a solution $\mathbf{u}_{k}=(u_{1k},\ldots,u_{\ell k})$ to the system
$(\mathscr{S}_{\varepsilon_{k},\Omega})$ whose first $m$ components are
positive and whose last $\ell-m$ components are nonradial and change sign,
with the following limit profile:

For each $i=1,\ldots,\ell$, there exist a sequence $(\xi_{ik})$ in $B_{1}(0)$
and a nontrivial solution $v_{i}$ to the problem ($\mathscr{P}_{i}$) such
that, after passing to a subsequence,

$\lim_{k\to\infty}\varepsilon_{k}^{-1}\mathrm{dist}(\xi_{ik},\partial
B_{1}(0))=\infty,\quad\lim_{k\to\infty}\varepsilon_{k}^{-1}|\xi_{ik}-\xi_{jk}|=\infty\text{
if }i\neq
j,\quad\displaystyle\lim_{k\to\infty}\|u_{ik}-v_{i}(\varepsilon_{k}^{-1}(\
\cdot\ -\xi_{ik}))\|_{\varepsilon_{k}}=0.$

The functions $v_{1},\ldots,v_{m}$ are positive and radial, while the
functions $v_{m+1},\ldots,v_{\ell}$ are sign-changing, nonradial and satisfy

(1.2)
$\begin{cases}v_{i}(z_{1},z_{2},x)=v_{i}(\mathrm{e}^{\mathrm{i}\vartheta}z_{1},\mathrm{e}^{\mathrm{i}\vartheta}z_{2},gx)&\text{for
all \ }\vartheta\in[0,2\pi),\ g\in O(N-4),\\\
v_{i}(z_{1},z_{2},x)=-v_{i}(z_{2},z_{1},x),\end{cases}$

for all
$(z_{1},z_{2},x)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}\equiv\mathbb{R}^{N}$,
$i=m+1,\ldots,\ell$. Furthermore,

$\lim_{k\to\infty}\displaystyle\sum_{i=1}^{\ell}\|u_{ik}\|_{\varepsilon_{k}}^{2}=\displaystyle\sum_{i=1}^{\ell}\|v_{i}\|^{2}=:\mathfrak{c}_{m},$

satisfies $\mathfrak{c}_{m}<\widehat{\mathfrak{c}}_{m}$, with
$\widehat{\mathfrak{c}}_{m}$ as in _Theorem 1.2_, if $N\geq 6$.

These last two results exhibit some interesting facts. The solutions given by
Theorem 1.3 are least energy solutions to the system
($\mathscr{S}_{\varepsilon,\Omega}$) having some specific symmetries described
in Example 2.1$(ii)$ below. Their components behave as expected in the
repulsive case $\lambda_{ij}<0$, i.e., they concentrate at points that are far
from each other and from the boundary of the ball. On the other hand, the
solutions given by Theorem 1.2 behave as shown by Lin and Wei [12, Theorem
1.1(2)] for the attractive case $\lambda_{ij}>0$, i.e., all components
concentrate at the origin. The solutions given by Theorem 1.2 enjoy the same
symmetries as those given by Theorem 1.3 (and more), but they have higher
energy. This shows that the sign of the interaction coefficient $\lambda_{ij}$
is not determinant in the segregation behavior of higher energy solutions.

Another interesting feature of the solutions given by Theorem 1.2 is that the
limit profiles of their components are coupled, i.e., they solve the system
($\mathscr{S}_{\infty,\ell}$), up to rescaling. A similar behavior can be seen
for least energy positive radial solutions to
($\mathscr{S}_{\varepsilon,\Omega}$) in the unit ball: their limit profile is
a positive least energy radial solution to the system
($\mathscr{S}_{\infty,\ell}$), see Theorem 4.3. As far as we know, these are
the first examples in the literature that show this kind of asymptotic
behavior.

To prove our results we follow the approach introduced in [7]. Using
concentration compactness techniques, we carry out a careful analysis of the
behavior of minimizing sequences for the system ($\mathscr{S}_{\infty,\ell}$)
that have a specific type of symmetries, described in Section 2. The
symmetries can be chosen to produce a change of sign by construction. As
$\varepsilon\to 0$ the components of the system concentrate at fixed points of
the group action. If the space of fixed points is trivial, all components will
necessarily concentrate at the origin, but they will move far away from each
other when the fixed-point space has positive dimension. A detailed
description is given in Theorem 3.3.

We note that for $N=3$ there are no symmetries with the properties required to
produce nonradial sign-changing solutions. For the single equation
($\mathscr{P}_{i}$) the existence of this kind of solutions was shown by
Musso, Pacard and Wei in [18] using the Lyapunov-Schmidt reduction procedure.

We wish to point out that solutions with sign-changing components for a system
related to the existence of optimal partitions for the eigenvalue problem were
obtained by Tavares and Terracini in [24]. For the system
$(\mathscr{S}_{1,\Omega})$ with fixed $\varepsilon=1$ and cubic nonlinearity
(hence, $N\leq 3$) in a bounded domain $\Omega$, existence and multiplicity of
sign-changing and semi-nodal solutions (some components change sign and others
are positive) have been established by several authors, see, e.g., [21, 22,
5].

This paper is organized as follows: in Section 2 we introduce the symmetric
variational setting for systems with positive and sign-changing components and
establish the existence of minimizers for the system
($\mathscr{S}_{\varepsilon,\Omega}$). In Section 3 we give a detailed
description of the symmetric minimizing sequences for the system
($\mathscr{S}_{\infty,\ell}$) and prove Theorem 1.1. In Section 4 we state and
prove a general result for the system ($\mathscr{S}_{\varepsilon,\Omega}$) and
derive Theorems 1.2 and 1.3 from it.

##  2 Minimizers with positive and sign-changing components

Let $G$ be a closed subgroup of the group $O(N)$ of linear isometries of
$\mathbb{R}^{N}$ and denote by $Gx:=\\{gx:g\in G\\}$ the $G$-orbit of
$x\in\mathbb{R}^{N}$. Let $\phi:G\to\mathbb{Z}_{2}:=\\{-1,1\\}$ be a
continuous homomorphism of groups with the following property:

  * $(A_{1})$

If $\phi$ is surjective, then there exists $x_{0}\in\mathbb{R}^{N}$ such that
$Kx_{0}\neq Gx_{0}$ where $K:=\ker\phi$.

Let $\Theta$ be an open subset of $\mathbb{R}^{N}$ which is _$G$ -invariant_,
i.e., $Gx\subset\Theta$ for every $x\in\Theta$. Then, a function
$u:\Theta\to\mathbb{R}$ is called _$G$ -invariant_ if it is constant on $Gx$
for every $x\in\Theta$ and will be called _$\phi$ -equivariant_ if

$u(gx)=\phi(g)u(x)\text{ \ for all \ }g\in G,\ x\in\Theta.$

Define

$H_{0}^{1}(\Theta)^{\phi}:=\\{u\in H_{0}^{1}(\Theta):u\text{ is
}\phi\text{-equivariant}\\}.$

Assumption $(A_{1})$ guarantees that $H_{0}^{1}(\Theta)^{\phi}$ has infinite
dimension, see [3]. If $\phi\equiv 1$ is the trivial homomorphism, then
$H_{0}^{1}(\Theta)^{\phi}$ is the space of $G$-invariant functions in
$H_{0}^{1}(\Theta)$. On the other hand, if $\phi$ is surjective, then every
nontrivial function $u\in H_{0}^{1}(\Theta)^{\phi}$ is nonradial and changes
sign. The following examples are of relevance to our main results.

######  Examples 2.1.

Consider the following examples of groups of isometries and homomorphims
satisfying $(A_{1})$:

  * $(i)$

For any group $G$ the trivial homomorphism $\phi\equiv 1$ is not surjective,
so $(A_{1})$ is trivially satisfied.

  * $(ii)$

Let $\Gamma$ be the group generated by
$\\{\mathrm{e}^{\mathrm{i}\vartheta}:\vartheta\in[0,2\pi)\\}\cup\\{\tau\\}$
acting on $\mathbb{R}^{N}$ by

$\mathrm{e}^{\mathrm{i}\vartheta}(z_{1},z_{2},x)=(\mathrm{e}^{\mathrm{i}\vartheta}z_{1},\mathrm{e}^{\mathrm{i}\vartheta}z_{2},x)\quad\text{and}\quad\tau(z_{1},z_{2},x)=(z_{2},z_{1},x)\qquad\forall\
(z_{1},z_{2},x)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}\equiv\mathbb{R}^{N},$

and let $\phi:\Gamma\to\mathbb{Z}_{2}$ be the homomorphism given by
$\phi(\mathrm{e}^{\mathrm{i}\vartheta}):=1$ and $\phi(\tau):=-1$. The kernel
of $\phi$ is the group
$K:=\\{\mathrm{e}^{\mathrm{i}\vartheta}:\vartheta\in[0,2\pi)\\}$ and the point
$x_{0}=(1,0,0)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}$ is such
that

$Kx_{0}=\\{(\mathrm{e}^{\mathrm{i}\vartheta},0,0):\vartheta\in[0,2\pi)\\}\quad\text{and}\quad\Gamma
x_{0}=\\{(\mathrm{e}^{\mathrm{i}\vartheta},0,0):\vartheta\in[0,2\pi)\\}\cup\\{(0,\mathrm{e}^{\mathrm{i}\vartheta},0):\vartheta\in[0,2\pi)\\},$

so $(A_{1})$ is satisfied.

  * $(iii)$

Let $G:=\Gamma\times O(N-4)$ with $\Gamma$ as in $(ii)$ and $g\in O(N-4)$
acting as

$g(z_{1},z_{2},x)=(z_{1},z_{2},gx)\qquad\forall\
(z_{1},z_{2},x)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}\equiv\mathbb{R}^{N}.$

Let $\phi:G\to\mathbb{Z}_{2}$ be the homomorphism given by
$\phi(\mathrm{e}^{\mathrm{i}\vartheta}):=1$, $\phi(\tau):=-1$ and $\phi(g):=1$
for $g\in O(N-4)$. Then
$K:=\ker\phi=\\{\mathrm{e}^{\mathrm{i}\vartheta}:\vartheta\in[0,2\pi)\\}\times
O(N-4)$ and, if
$x_{0}=(1,0,0)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}$,

$Kx_{0}=\\{(\mathrm{e}^{\mathrm{i}\vartheta},0,0):\vartheta\in[0,2\pi)\\}\quad\text{and}\quad
Gx_{0}=\\{(\mathrm{e}^{\mathrm{i}\vartheta},0,0):\vartheta\in[0,2\pi)\\}\cup\\{(0,\mathrm{e}^{\mathrm{i}\vartheta},0):\vartheta\in[0,2\pi)\\},$

so $(A_{1})$ is satisfied.

For the remaining of this section we fix a closed subgroup $G$ of $O(N)$ and,
for each $i=1,\ldots,\ell$, a continuous homomorphism
$\phi_{i}:G\to\mathbb{Z}_{2}$ satisfying $(A_{1})$.

If $\Theta$ is a $G$-invariant open subset of $\mathbb{R}^{N}$, we consider
the system

($\mathscr{S}^{\phi}_{\varepsilon,\Theta}$)
$\qquad\begin{cases}-\varepsilon^{2}\Delta
u_{i}+u_{i}=\mu_{i}|u_{i}|^{p-2}u_{i}+\displaystyle\sum\limits_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}-2}u_{i},\\\
u_{i}\in H^{1}_{0}(\Theta)^{\phi_{i}},\quad u_{i}\neq 0,\qquad
i=1,\ldots,\ell,\end{cases}$

with $\varepsilon>0$, $\mu_{i}>0$, $\lambda_{ij}=\lambda_{ji}<0$,
$\alpha_{ij},\beta_{ij}>1$, $\alpha_{ij}=\beta_{ji}$ and
$\alpha_{ij}+\beta_{ij}=p\in(2,2^{*})$.

As usual, we consider $H^{1}_{0}(\Theta)$ as a subspace of
$H^{1}(\mathbb{R}^{N})$, identifying a function $u\in H^{1}_{0}(\Theta)$ with
its trivial extension to $\mathbb{R}^{N}$. Set

$\mathcal{H}^{\ell}(\Theta):=H_{0}^{1}(\Theta)^{\phi_{1}}\times\cdots\times
H_{0}^{1}(\Theta)^{\phi_{\ell}},$

and denote an element in $\mathcal{H}^{\ell}(\Theta)$ by
$\mathbf{u}=(u_{1},\ldots,u_{\ell})$. For each $\varepsilon>0$ we define

$\|\mathbf{u}\|_{\ell,\varepsilon}:=\left(\displaystyle\sum_{i=1}^{\ell}\|u_{i}\|_{\varepsilon}^{2}\right)^{1/2},\quad\text{where
\
}\|u_{i}\|_{\varepsilon}^{2}:=\frac{1}{\varepsilon^{N}}\displaystyle\int_{\mathbb{R}^{N}}\Big{[}\varepsilon^{2}|\nabla
u_{i}|^{2}+u_{i}^{2}\Big{]}.$

If $\varepsilon=1$ we write $\|\mathbf{u}\|_{\ell}$ instead of
$\|\mathbf{u}\|_{\ell,1}$. Note that $\|\cdot\|_{\varepsilon}$ is a norm in
$H_{0}^{1}(\Theta)$ equivalent the standard one $\|\cdot\|:=\|\cdot\|_{1}$.
Therefore, $\|\cdot\|_{\ell,\varepsilon}$ is a norm in
$\mathcal{H}^{\ell}(\Theta)$ for every $\varepsilon>0$ and all of these norms
are equivalent.

Consider the functional
$\mathcal{J}^{\ell}_{\varepsilon}:\mathcal{H}^{\ell}(\Theta)\to\mathbb{R}$
given by

$\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u}):=\dfrac{1}{2}\displaystyle\sum_{i=1}^{\ell}\|u_{i}\|_{\varepsilon}^{2}-\dfrac{1}{p}\displaystyle\sum_{i=1}^{\ell}\dfrac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p}-\dfrac{1}{2}\displaystyle\sum\limits_{\begin{subarray}{c}i,j=1\\\
j\not=i\end{subarray}}^{\ell}\dfrac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\lambda_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}},$

which is of class $\mathcal{C}^{1}$. Since $\lambda_{ij}=\lambda_{ji}$,
$\beta_{ij}=\alpha_{ji}$ and $\alpha_{ij}+\beta_{ij}=p$, its partial
derivatives are given by

$\displaystyle\partial_{i}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u})v$
$\displaystyle=\dfrac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\Big{(}\varepsilon^{2}\nabla
u_{i}\cdot\nabla
v+u_{i}v\Big{)}-\dfrac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p-2}u_{i}v$
$\displaystyle\qquad-\dfrac{1}{2}\displaystyle\sum\limits_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\dfrac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}-2}u_{i}v-\dfrac{1}{2}\displaystyle\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\dfrac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\lambda_{ij}\alpha_{ji}|u_{i}|^{\alpha_{ji}-2}u_{i}v|u_{j}|^{\beta_{ji}}$
(2.1)
$\displaystyle=\frac{1}{\varepsilon^{N}}\Big{[}\displaystyle\int_{\mathbb{R}^{N}}(\varepsilon^{2}\nabla
u_{i}\cdot\nabla
v+u_{i}v)-\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p-2}u_{i}v-\displaystyle\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}-2}u_{i}v\Big{]},$

for $v\in H^{1}_{0}(\Theta)^{\phi_{i}}$ and $i=1,\ldots,\ell.$ So, by the
principle of symmetric criticality, the solutions to system
($\mathscr{S}^{\phi}_{\varepsilon,\Theta}$) are the critical points of
$\mathcal{J}^{\ell}_{\varepsilon}$ whose components $u_{i}$ are nontrivial.
They belong to the Nehari-type set

$\mathcal{N}^{\ell}_{\varepsilon}(\Theta):=\Big{\\{}\mathbf{u}\in\mathcal{H}^{\ell}(\Theta):u_{i}\neq
0,\ \partial_{i}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u})u_{i}=0,\ \forall\
i=1,\ldots,\ell\Big{\\}}.$

Note that

(2.2)
$\partial_{i}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u})u_{i}=\|u_{i}\|^{2}_{\varepsilon}-\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p}-\displaystyle\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}}.$

Define

$c_{\varepsilon}^{\ell}(\Theta):=\inf_{\mathbf{u}\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u}).$

From (2) one sees that

(2.3)
$\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u})=\dfrac{p-2}{2p}\displaystyle\sum_{i=1}^{\ell}\|u_{i}\|_{\varepsilon}^{2}=\dfrac{p-2}{2p}\|\mathbf{u}\|^{2}_{\ell,\varepsilon}\qquad\text{if}\
\mathbf{u}=(u_{1},u_{2},\ldots,u_{\ell})\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta).$

######  Remark 2.2.

_For $\mathbf{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{H}^{\ell}(\Theta)$ and
$\varepsilon>0$, define $\widetilde{u}_{i}(z):=u_{i}(\varepsilon z)$. It is
straightforward to verify that $\widetilde{u}_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{i}}$ and_

$\|u_{i}\|^{2}_{\varepsilon}=\|\widetilde{u}_{i}\|^{2},\qquad\frac{1}{\varepsilon^{N}}\displaystyle\int_{\mathbb{R}^{N}}|u_{i}|^{p}=\displaystyle\int_{\mathbb{R}^{N}}|\widetilde{u}_{i}|^{p},\qquad\frac{1}{\varepsilon^{N}}\displaystyle\int_{\mathbb{R}^{N}}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}}=\displaystyle\int_{\mathbb{R}^{N}}|\widetilde{u}_{j}|^{\alpha_{ij}}|\widetilde{u}_{i}|^{\beta_{ij}}.$

For $\mathbf{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{H}^{\ell}(\Theta)$ and
$\mathbf{t}=(t_{1},\ldots,t_{\ell})\in(0,\infty)^{\ell}$, we write
$\mathbf{t}\mathbf{u}:=(t_{1}u_{1},\ldots,t_{\ell}u_{\ell}).$ Our next results
provide some useful information about the Nehari-type set
$\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$.

######  Lemma 2.3.

The following statements hold true:

  * $(i)$

$\mathcal{N}^{\ell}_{\varepsilon}(\Theta)\not=\emptyset$ and there exists a
constant $c_{0}>0$, independent of $\varepsilon$, such that

$\|u_{i}\|_{\varepsilon}^{2}>c_{0}\quad\text{for every \
}\mathbf{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta),\
i=1,\ldots,\ell.$

Therefore $\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$ is a closed subset of
$\mathcal{H}^{\ell}(\Theta)$ and $0<c_{\varepsilon}^{\ell}(\Theta)<\infty$.

  * $(ii)$

For each $\mathbf{u}\in\mathcal{H}^{\ell}(\Theta)$ such that

(2.4)
$\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p}+\displaystyle\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}}>0\quad\forall\
i=1,\ldots,\ell,$

there exists a unique $\mathbf{t}_{\mathbf{u}}\in(0,\infty)^{\ell},$ such that
$\mathbf{t}_{\mathbf{u}}\mathbf{u}\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$.
Furthermore,

$\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{t}_{\mathbf{u}}\mathbf{u})=\max_{{\mathbf{t}}\in(0,\infty)^{\ell}}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{t}\mathbf{u}).$

###### Proof.

$(i)$ Since $\phi_{i}$ satisfies $(A_{1})$, for each $i=1,\ldots,\ell$ one can
choose $u_{i}\in\mathcal{C}_{c}^{\infty}(\Theta)\cap
H_{0}^{1}(\Theta)^{\phi_{i}}$, $u_{i}\not=0$, such that
$\mathrm{supp}(u_{i})\displaystyle\cap\mathrm{supp}(u_{j})=\emptyset$ for all
$j\not=i$, and then take $t_{i}>0$ such that $v_{i}:=t_{i}u_{i}$ satisfies
$\|v_{i}\|_{\varepsilon}^{2}=\varepsilon^{-N}\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|v_{i}|^{p}$.
Since ${\mathrm{supp}(v_{i})\cap\mathrm{supp}(v_{j})=\emptyset}$ for all
$j\not=i$, setting ${\mathbf{v}:=(v_{1},\ldots,v_{\ell})}$ we get that
$\partial_{i}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{v})v_{i}=0$ and
$v_{i}\not=0$ for all $i=1,\ldots,\ell$. Therefore
$\mathbf{v}\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$.

Let
$\mathbf{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$
and set $\widetilde{u}_{i}(z):=u_{i}(\varepsilon z)$. Then, since
$\lambda_{ij}<0$ for every pair $i\not=j$, from Remark 2.2 and Sobolev’s
embedding theorem we derive

$\|\widetilde{u}_{i}\|^{2}\leq\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|\widetilde{u}_{i}|^{p}\leq
C\mu_{i}\|\widetilde{u}_{i}\|^{p},$

with $C$ a positive constant, depending only on $N$ and $p$. This yields the
estimate in statement $(i)$ and completes its proof.

$(ii)$ Fix $\mathbf{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{H}^{\ell}(\Theta)$
and define $J^{\ell}_{\mathbf{u}}:(0,\infty)^{\ell}\to\mathbb{R}$ by

$J^{\ell}_{\mathbf{u}}(\mathbf{t}):=\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{t}\mathbf{u})=\displaystyle\sum_{i=1}^{\ell}a_{u,i}t_{i}^{2}-\displaystyle\sum_{i=1}^{\ell}b_{u,i}t_{i}^{p}+\displaystyle\sum\limits_{\begin{subarray}{c}i,j=1\\\
j\neq i\end{subarray}}^{\ell}d_{u,ij}t_{j}^{\alpha_{ij}}t_{i}^{\beta_{ij}},$

where $\mathbf{t}=(t_{1},\ldots,t_{\ell})\in(0,\infty)^{\ell}$,

$a_{u,i}:=\frac{1}{2}\|u_{i}\|^{2}_{\varepsilon},\quad
b_{u,i}:=\frac{1}{p\varepsilon^{N}}\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p},\quad
d_{u,ij}:=\frac{1}{2\varepsilon^{N}}\displaystyle\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{\ell}\displaystyle\int_{\mathbb{R}^{N}}|\lambda_{ij}|\,\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}}.$

If $u_{i}\neq 0$ for all $i=1,\ldots,\ell$, then, as

$t_{i}\,\partial_{i}J^{\ell}_{\mathbf{u}}(\mathbf{t})=\partial_{i}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{t}\mathbf{u})[t_{i}u_{i}],\qquad
i=1,\ldots,\ell,$

we have that $\mathbf{t}\mathbf{u}\in\mathcal{N}^{\ell}(\Theta)$ iff
$\mathbf{t}$ is a critical point of $J^{\ell}_{\mathbf{u}}$. Note that (2.4)
implies that $u_{i}\neq 0$ for all $i=1,\ldots,\ell$. Hence, $(ii)$ follows
from [8, Lemmas 2.1 and 2.2]. ∎

######  Theorem 2.4.

If there exists $\mathbf{u}\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$ such
that
$\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u})=c_{\varepsilon}^{\ell}(\Theta)$,
then $\mathbf{u}=(u_{1},\ldots,u_{\ell})$ is a solution to system
($\mathscr{S}^{\phi}_{\varepsilon,\Theta}$).

###### Proof.

The proof is practically the same as that of [8, Theorem 3.4(a)] with the
obvious modifications. ∎

######  Definition 2.5.

A solution to the system ($\mathscr{S}^{\phi}_{\varepsilon,\Theta}$) such that
$\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u})=c_{\varepsilon}^{\ell}(\Theta)$
will be called a least energy solution to
($\mathscr{S}^{\phi}_{\varepsilon,\Theta}$).

######  Remark 2.6.

_If $\phi_{i}\equiv 1$ for every $i$ in some subset $I$ of
$\\{1,\ldots,\ell\\}$ and $\mathbf{u}$ is a least energy solution to
($\mathscr{S}^{\phi}_{\varepsilon,\Theta}$), then, setting $v_{i}:=|u_{i}|$ if
$i\in I$ and $v_{i}:=u_{i}$ if $i\not\in I$, we have that $v_{i}\in
H^{1}_{0}(\Theta)^{\phi_{i}}$,
$\mathbf{v}=(v_{1},\ldots,v_{\ell})\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$
and
$\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{v})=\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u})$.
Hence, $\mathbf{v}$ is a least energy solution to
($\mathscr{S}^{\phi}_{\varepsilon,\Theta}$) whose $i$-th component is positive
for every $i\in I$._

######  Theorem 2.7 (Existence of minimizers).

If $\Theta$ is a bounded $G$-invariant domain in $\mathbb{R}^{N}$, then, for
each $\varepsilon>0$, the system ($\mathscr{S}^{\phi}_{\varepsilon,\Theta}$)
has a least energy solution.

###### Proof.

Fix $\varepsilon>0$ and let $\mathbf{u}_{k}=(u_{1k},\ldots,u_{\ell
k})\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$ satisfy
$\mathcal{J}_{\varepsilon}^{\ell}(\mathbf{u}_{k})\to
c_{\varepsilon}^{\ell}(\Theta)$. From (2.3) we see that $(u_{ik})$ is bounded
in $H_{0}^{1}(\Theta)$ for every $i=1,\ldots,\ell$. So, after passing to a
subsequence, $u_{ik}\rightharpoonup u_{i}$ weakly in
$H_{0}^{1}(\Theta)^{\phi_{1}}$ and $u_{ik}\to u_{i}$ strongly in
$L^{p}(\Theta)$. From (2.2) and Lemma 2.3 we get that

$0<c_{0}\leq\|u_{ik}\|^{2}_{\varepsilon}=\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\mu_{i}|u_{ik}|^{p}+\displaystyle\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\frac{1}{\varepsilon^{N}}\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u_{jk}|^{\alpha_{ij}}|u_{ik}|^{\beta_{ij}}$

and, passing to the limit, we obtain

$0<c_{0}\leq\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p}+\displaystyle\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}}\qquad\text{for
all \ }i=1,\ldots,\ell.$

Set $\mathbf{u}:=(u_{1},\ldots,u_{\ell})$. By Lemma 2.3 there exists
$\mathbf{t}=(t_{1},\ldots,t_{\ell})\in(0,\infty)^{\ell}$ such that
$\mathbf{t}\mathbf{u}\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$ and

$\displaystyle c_{\varepsilon}^{\ell}(\Theta)$
$\displaystyle\leq\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{t}\mathbf{u})\leq\displaystyle\liminf_{k\to\infty}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{t}\mathbf{u}_{k})\leq\displaystyle\lim_{k\to\infty}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u}_{k})=c_{\varepsilon}^{\ell}(\Theta),$

because $\mathbf{u}_{k}\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$ for all
$k\in\mathbb{N}$. Thus,
$\lim\limits_{k\to\infty}\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{t}\mathbf{u}_{k})=\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{t}\mathbf{u})$.
As $u_{ik}\to u_{i}$ strongly in $L^{p}(\Theta)$, this implies that
$\lim\limits_{k\to\infty}\|\mathbf{t}\mathbf{u}_{k}\|_{\ell,\varepsilon}^{2}=\|\mathbf{t}\mathbf{u}\|_{\ell,\varepsilon}^{2}$.
Since $\mathbf{u}_{k}\rightharpoonup\mathbf{u}$ weakly in
$\mathcal{H}^{\ell}(\Theta)$, we derive that $\mathbf{u}_{k}\to\mathbf{u}$
strongly in $\mathcal{H}^{\ell}(\Theta)$. As a consequence,
$\mathbf{u}\in\mathcal{N}^{\ell}_{\varepsilon}(\Theta)$ and
$\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{u})=c_{\varepsilon}^{\ell}(\Theta)$.
∎

##  3 The limit system

We fix a closed subgroup $G$ of $O(N)$ and, for each $i=1,\ldots,\ell$, a
continuous homomorphism $\phi_{i}:G\to\mathbb{Z}_{2}$. We assume throughout
this section that the $\phi_{i}$ satisfy $(A_{1})$ and that $G$ has the
following property:

  * $(A_{2})$

For every $x\in\mathbb{R}^{N}$, the $G$-orbit of $x$ is either infinite, or
$Gx=\\{x\\}$.

Our aim is to study the behavior of minimizing sequences for the functional
related to the system

($\mathscr{S}_{\infty,\ell}^{\phi}$) $\qquad\begin{cases}-\Delta
u_{i}+u_{i}=\mu_{i}|u_{i}|^{p-2}u_{i}+\displaystyle\sum\limits_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\lambda_{ij}\beta_{ij}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}-2}u_{i},\\\
u_{i}\in H^{1}(\mathbb{R}^{N})^{\phi_{i}},\quad u_{i}\neq 0,\qquad
1<i\leq\ell.\end{cases}$

We write $\|\cdot\|$ for the standard norm in $H^{1}(\mathbb{R}^{N})$. Set
$\mathcal{H}^{\ell}:=H^{1}(\mathbb{R}^{N})^{\phi_{1}}\times\cdots\times
H^{1}(\mathbb{R}^{N})^{\phi_{\ell}}$,
${\mathcal{J}^{\ell}_{\infty}:=\mathcal{J}_{1}^{\ell}:\mathcal{H}^{\ell}\to\mathbb{R}}$
and $\mathcal{N}^{\ell}_{\infty}:=\mathcal{N}_{1}^{\ell}(\mathbb{R}^{N})$ as
in the previous section, and let

$c^{\ell}_{\infty}:=\inf_{\mathbf{u}\in\mathcal{N}^{\ell}_{\infty}}\mathcal{J}^{\ell}_{\infty}(\mathbf{u}).$

For each $i=1,\ldots,\ell$, consider the problem

($\mathscr{P}_{i}^{\phi_{i}}$) $\qquad\begin{cases}-\Delta
u+u=\mu_{i}|u|^{p-2}u,\\\ u\in H^{1}(\mathbb{R}^{N})^{\phi_{i}},\quad u\neq
0.\end{cases}$

Its solutions are the critical points of the functional
$J_{i}:H^{1}(\mathbb{R}^{N})^{\phi_{i}}\to\mathbb{R}$ given by

$J_{i}(u):=\frac{1}{2}\displaystyle\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+u^{2})-\frac{1}{p}\displaystyle\int_{\mathbb{R}^{N}}|u|^{p},$

belonging to the Nehari manifold $\mathcal{N}_{i}:=\\{u\in
H^{1}(\mathbb{R}^{N})^{\phi_{i}}:u\neq 0,\ J_{i}^{\prime}(u)u=0\\}$. We set

$c_{i}:=\inf_{u\in\mathcal{N}_{i}}J_{i}(u).$

A solution $\mathbf{u}$ to ($\mathscr{S}_{\infty,\ell}^{\phi}$) satisfying
$\mathcal{J}_{\infty}^{\ell}(\mathbf{u})=c_{\infty}^{\ell}$ will be called a
_least energy solution to ( $\mathscr{S}_{\infty,\ell}^{\phi}$)_. Similarly, a
solution $u$ to ($\mathscr{P}_{i}^{\phi_{i}}$) satisfying $J_{i}(u)=c_{i}$
will be called a _least energy solution to ( $\mathscr{P}_{i}^{\phi_{i}}$)_.

The _$G$ -fixed-point space_

$\mathrm{Fix}(G):=\\{x\in\mathbb{R}^{N}:gx=x\text{ for all }g\in G\\}$

plays an important role on the existence or nonexistence of a least energy
solution to the system ($\mathscr{S}_{\infty,\ell}^{\phi}$). We start with the
following result.

######  Proposition 3.1.

The following statements hold true:

  * $(i)$

$c_{\infty}^{\ell}\geq\displaystyle\displaystyle\sum_{i=1}^{\ell}c_{i}.$

  * $(ii)$

If $\mathrm{Fix}(G)$ has positive dimension, then
$c_{\infty}^{\ell}=\displaystyle\displaystyle\sum_{i=1}^{\ell}c_{i}$.

  * $(iii)$

If $\ell\geq 2$ and
$c_{\infty}^{\ell}=\displaystyle\displaystyle\sum_{i=1}^{\ell}c_{i}$, then
$c_{\infty}^{\ell}$ is not attained.

###### Proof.

$(i):$ Let $\mathbf{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{N}_{\infty}^{\ell}$.
As $\lambda_{ij}<0$, from (2.2) and Lemma 2.3 $(i)$ we get that

$0<\|u_{i}\|^{2}\leq\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p}\qquad\text{for
all \ }i=1,\ldots,\ell.$

Hence, there exist $t_{1},\ldots,t_{\ell}\in(0,\infty)$ such that
$t_{i}u_{i}\in\mathcal{N}_{i}$. Setting $\mathbf{t}:=(t_{1},\ldots,t_{\ell})$
and applying Lemma 2.3 $(ii)$ we obtain

$\displaystyle\sum_{i=1}^{\ell}c_{i}\leq\displaystyle\sum_{i=1}^{\ell}J_{i}(t_{i}u_{i})\leq\mathcal{J}^{\ell}_{\infty}(\mathbf{t}\mathbf{u})\leq\mathcal{J}^{\ell}_{\infty}(\mathbf{u}),$

as claimed.

$(ii):$ For each $i=1,\ldots,\ell$, let $v_{i}\in\mathcal{N}_{i}$. As
$\mathrm{Fix}(G)$ has positive dimension we may choose
$\xi_{ik}\in\mathrm{Fix}(G)$ such that $|\xi_{ik}-\xi_{jk}|\to\infty$ as
$k\to\infty$ if $i\neq j$. Define $\mathbf{u}_{k}=(u_{1k},\ldots,u_{\ell k})$
by $u_{ik}(x):=v_{i}(x-\xi_{ik})$. Note that, as $\xi_{ik}\in\mathrm{Fix}(G)$,
we have that $u_{ik}\in H^{1}(\mathbb{R}^{N})^{\phi_{i}}$. Furthermore,
$u_{ik}\in\mathcal{N}_{i}$, $J_{i}(u_{ik})=J_{i}(v_{i})$ and

$\displaystyle\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u_{jk}|^{\alpha_{ij}}|u_{ik}|^{\beta_{ij}}=\displaystyle\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|v_{j}(\
\cdot\ +\xi_{ik}-\xi_{jk})|^{\alpha_{ij}}|v_{i}|^{\beta_{ij}}=o(1).$

Therefore,

$\displaystyle
0<\frac{2p}{p-2}c_{i}\leq\|u_{ik}\|^{2}=\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{ik}|^{p}+\displaystyle\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\displaystyle\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u_{jk}|^{\alpha_{ij}}|u_{ik}|^{\beta_{ij}}+o(1).$

Applying Lemma 2.3 $(ii)$ we see that, for $k$ large enough, there exists
$\mathbf{t}_{k}:=(t_{1k},\ldots,t_{\ell k})\in(0,\infty)^{\ell}$ such that
$\mathbf{t}_{k}\mathbf{u}_{k}\in\mathcal{N}_{\infty}^{\ell}$, and $t_{ik}\to
1$ for all $i=1,\ldots,\ell$. Hence,

$\displaystyle c_{\infty}^{\ell}$
$\displaystyle\leq\mathcal{J}_{\infty}^{\ell}(\mathbf{t}_{k}\mathbf{u}_{k})=\displaystyle\sum_{i=1}^{\ell}J_{i}(t_{ik}u_{ik})+o(1)=\displaystyle\sum_{i=1}^{\ell}J_{i}(v_{i})+o(1).$

This shows that $\displaystyle\displaystyle\sum\limits_{i=1}^{\ell}c_{i}\geq
c_{\infty}^{\ell}$ and from $(i)$ we obtain
$\displaystyle\displaystyle\sum\limits_{i=1}^{\ell}c_{i}=c_{\infty}^{\ell}$.

$(iii):$ Let $c_{\infty}^{\ell}=\displaystyle\sum\limits_{i=1}^{\ell}c_{i}$.
To show that this value is not attained, we argue by contradiction. Assume
that $\mathbf{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{N}^{\ell}_{\infty}$ and
$\mathcal{J}^{\ell}_{\infty}(\mathbf{u})=c_{\infty}^{\ell}$. There are two
possibilities. If for every $i$ there exists $j\neq i$ such that
$\displaystyle\displaystyle\displaystyle\int_{\mathbb{R}^{N}}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}}\neq
0$, then
$\displaystyle\|u_{i}\|^{2}<\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p}$
for all $i=1,\ldots,\ell$ and, hence, there exists $t_{i}\in(0,1)$ such that
$t_{i}u_{i}\in\mathcal{N}_{i}$ and

$c_{i}\leq
J_{i}(t_{i}u_{i})=\dfrac{p-2}{2p}\|t_{i}u_{i}\|^{2}<\dfrac{p-2}{2p}\|u_{i}\|^{2}.$

It follows that

$c_{\infty}^{\ell}=\mathcal{J}^{\ell}_{\infty}(\mathbf{u})=\frac{p-2}{2p}\displaystyle\sum_{i=1}^{\ell}\|u_{i}\|^{2}>\displaystyle\sum_{i=1}^{\ell}c_{i},$

contradicting our assumption. On the other hand, if there exists $i$ such that
$\displaystyle\displaystyle\displaystyle\int_{\mathbb{R}^{N}}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}}=0$
for all $j\neq i$, then

$\|u_{i}\|^{2}=\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{i}|^{p}\qquad\text{and}\qquad
J_{i}(u_{i})=c_{i}.$

Hence, $u_{i}$ is a nontrivial solution to the problem

$-\Delta w+w=\mu_{i}|w|^{p-2}w,\qquad w\in H^{1}(\mathbb{R}^{N}).$

But
$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}}=0$
also implies that $|u_{j}|^{\alpha_{ij}}|u_{i}|^{\beta_{ij}}=0$ a.e. in
$\mathbb{R}^{N}$. As $u_{j}\neq 0$ for all $j$, we have that $u_{i}=0$ in some
subset of positive measure of $\mathbb{R}^{N}$. This contradicts the unique
continuation principle. ∎

The following lemma will play an important role in the proof of Theorem 3.3.

######  Lemma 3.2.

For any given sequence $(y_{k})$ in $\mathbb{R}^{N}$, there exist a sequence
$(\zeta_{k})$ in $\mathbb{R}^{N}$ and a constant $C>0$ such that, up to a
subsequence,

$\mathrm{dist}(Gy_{k},\zeta_{k})\leq C\qquad\text{for all \ }k\in\mathbb{N},$

and

  * •

either $\zeta_{k}\in\mathrm{Fix}(G)$ for all $k\in\mathbb{N}$,

  * •

or there exist $g_{n}\in G$, $n\in\mathbb{N}$, such that
$\lim\limits_{k\to\infty}|g_{n_{1}}\zeta_{k}-g_{n_{2}}\zeta_{k}|=\infty$ for
any pair $n_{1},n_{2}\in\mathbb{N}$ with $n_{1}\neq n_{2}$.

###### Proof.

This is the statement of [6, Lemma 3.2] for a group $G$ satisfying $(A_{2})$.
∎

We are ready to prove the main result of this section.

######  Theorem 3.3.

Let $\ell\geq 2$ and $\mathbf{u}_{k}=(u_{1k},\ldots,u_{\ell
k})\in\mathcal{N}_{\infty}^{\ell}$ be such that
$\mathcal{J}_{\infty}^{\ell}(\mathbf{u}_{k})\to c_{\infty}^{\ell}$.

  * $(I)$

If $\mathrm{Fix}(G)=\\{0\\}$, then there exists a least energy solution
$\mathbf{w}=(w_{1},\ldots,w_{\ell})$ to the system
($\mathscr{S}_{\infty,\ell}^{\phi}$) such that, after passing to a
subsequence,

$\lim_{k\to\infty}\|u_{ik}-w_{i}\|=0\qquad\text{for all \ }i=1,\ldots,\ell,$

and $c_{\infty}^{\ell}>\displaystyle\displaystyle\sum_{i=1}^{\ell}c_{i}$.
Moreover, if $u_{ik}\geq 0$ for all $k\in\mathbb{N}$, then $w_{i}\geq 0$.

  * $(II)$

If $\mathrm{Fix}(G)$ has positive dimension, then, for each $i=1,\ldots,\ell$,
there exist a sequence $(\xi_{ik})$ in $\mathrm{Fix}(G)$ and a least energy
solution $v_{i}$ to the problem ($\mathscr{P}_{i}^{\phi_{i}}$) such that,
after passing to a subsequence,

$\lim_{k\to\infty}|\xi_{ik}-\xi_{jk}|=\infty\text{ if }i\neq
j,\qquad\lim_{k\to\infty}\|u_{ik}-v_{i}(\ \cdot\ -\xi_{ik})\|=0\quad\text{for
\ }i,j=1,\ldots,\ell,$

and $c_{\infty}^{\ell}=\displaystyle\sum_{i=1}^{\ell}c_{i}$. Moreover, if
$u_{ik}\geq 0$ for all $k\in\mathbb{N}$, then $v_{i}\geq 0$.

###### Proof.

We split the proof into three steps.

Step 1.  _The choice of the concentration points._

As $\mathbf{u}_{k}\in\mathcal{N}_{\infty}^{\ell}$ and $\lambda_{ij}<0$, from
Lemma 2.3 we get a constant $c_{0}>0$ such that

(3.1)
$\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{ik}|^{p}>c_{0}\qquad\text{for
all \ }k\in\mathbb{N}\text{ \ and \ }i=1,\ldots,\ell.$

Invoking [8, Theorem 3.3] and Ekeland’s variational principle [29, Theorem
8.5] we may assume without loss of generality that
$(\mathcal{J}_{\infty}^{\ell})^{\prime}(\mathbf{u}_{k})\to 0$ in the dual
space $(\mathcal{H}^{\ell})^{\prime}$. From (3.1) and Lions’ lemma [29, Lemma
1.21] we see that there exists $\theta>0$ such that

$\limsup_{k\to\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B_{1}(y)}\mu_{i}|u_{ik}|^{p}>2\theta>0\qquad\text{for
all \ }i=1,\ldots,\ell.$

So, passing to a subsequence, we may choose $y_{ik}\in\mathbb{R}^{N}$
satisfying

(3.2) $\int_{B_{1}(y_{ik})}\mu_{i}|u_{ik}|^{p}>\theta\qquad\text{for all \
}k\in\mathbb{N}\text{ \ and \ }i=1,\ldots,\ell.$

Applying Lemma 3.2 to $(y_{ik})$ we obtain a sequence $(\zeta_{ik})$ and a
constant $C$ such that $\mathrm{dist}(Gy_{ik},\zeta_{ik})\leq C$ for all $k$
and $i$. Since $|u_{ik}|$ is $G$-invariant, taking $g_{ik}\in G$ such that
$|g_{ik}y_{ik}-\zeta_{ik}|=\mathrm{dist}(Gy_{ik},\zeta_{ik})$ we get

(3.3)
$\int_{B_{C+1}(\zeta_{ik})}\mu_{i}|u_{ik}|^{p}\geq\int_{B_{1}(g_{ik}y_{ik})}\mu_{i}|u_{ik}|^{p}\geq\int_{B_{1}(y_{ik})}\mu_{i}|u_{ik}|^{p}>\theta>0.$

We claim that $\zeta_{ik}\in\mathrm{Fix}(G)$. Otherwise, by the alternative in
Lemma 3.2, for every $M\in\mathbb{N}$ we can choose $g_{i1},\ldots,g_{iM}\in
G$ such that

$|g_{in}\zeta_{ik}-g_{in^{\prime}}\zeta_{ik}|\geq 2(C+1)\qquad\text{for all
pairs \ }n\neq n^{\prime},\ \ n,n^{\prime}=1,\ldots,M,$

and for sufficiently large $k\in\mathbb{N}$. As a consequence,
$B_{C+1}(g_{in}\zeta_{ik})\cap B_{C+1}(g_{in^{\prime}}\zeta_{ik})=\emptyset$
and, so,

$\displaystyle\int_{\mathbb{R}^{N}}\mu_{i}|u_{ik}|^{p}\geq\displaystyle\sum_{n=1}^{M}\int_{B_{C+1}(g_{in}\zeta_{ik})}\mu_{i}|u_{ik}|^{p}=M\int_{B_{C+1}(\zeta_{ik})}\mu_{i}|u_{ik}|^{p}>M\theta,$

which is a contradiction, because $M$ is arbitrary and, as $(\mathbf{u}_{k})$
is bounded in $\mathcal{H}^{\ell}$, the sequence $(u_{ik})$ is bounded in
$L^{p}(\mathbb{R}^{N})$. This shows that $\zeta_{ik}\in\mathrm{Fix}(G)$.

After passing to a subsequence and reordering the components, there are
integers

${0=:\ell_{0}<\ell_{1}<\cdots<\ell_{n}:=\ell}$

such that, setting $I_{r}:=(\ell_{r},\ell_{r+1}]$ for $r=0,\ldots,n-1$, we
have

$\displaystyle(|\zeta_{ik}-\zeta_{jk}|)\text{ is bounded}$
$\displaystyle\text{if \ }i,j\in I_{r}\text{ for some }r,$
$\displaystyle|\zeta_{ik}-\zeta_{jk}|\to\infty$ $\displaystyle\text{if \ }i\in
I_{r}\text{ and }j\in I_{r^{\prime}}\text{ with }r\neq r^{\prime}.$

Now we replace $\zeta_{ik}$ by $\xi_{ik}:=\zeta_{\ell_{r+1}\,k}$ for every
$i\in I_{r}$, and we fix $R>0$ large enough so that
$B_{C+1}(\zeta_{ik})\subset B_{R}(\xi_{ik})$ for all $k\in\mathbb{N}$ and
$i=1,\ldots,\ell$. Then these new points satisfy

$\displaystyle\xi_{ik}\in\mathrm{Fix}(G),$
$\displaystyle\xi_{ik}=\xi_{jk}\text{ for all }k$ $\displaystyle\text{if \
}i,j\in I_{r}\text{ for some }r,$ $\displaystyle|\xi_{ik}-\xi_{jk}|\to\infty$
$\displaystyle\text{if \ }i\in I_{r}\text{ and }j\in I_{r^{\prime}}\text{ with
}r\neq r^{\prime},$

and from (3.3) we obtain

(3.4) $\int_{B_{R}(\xi_{ik})}\mu_{i}|u_{ik}|^{p}>\theta>0\qquad\text{for all \
}k\in\mathbb{N}\text{ \ and \ }i=1,\ldots,\ell.$

Note that, if $\mathrm{Fix}(G)=\\{0\\}$, then $\xi_{ik}=0$ for all $k$ and
$i$, and there is only one interval $I_{0}:=(0,\ell\,]$.

Step 2.  _A solution to a subsystem._

Define

$w_{ik}(x):=u_{ik}(x+\xi_{ik}).$

Observe that $w_{ik}\in H^{1}(\mathbb{R}^{N})^{\phi_{i}}$, because
$\xi_{ik}\in\mathrm{Fix}(G)$. Since the sequence $(w_{ik})$ is bounded in
$H^{1}(\mathbb{R}^{N})$, passing to a subsequence, we have that
$w_{ik}\rightharpoonup w_{i}$ weakly in $H^{1}(\mathbb{R}^{N})$, $w_{ik}\to
w_{i}$ in $L^{p}_{\mathrm{loc}}(\mathbb{R}^{N})$ and $w_{ik}\to w_{i}$ a.e. in
$\mathbb{R}^{N}$. Then, $w_{i}\geq 0$ if $u_{ik}\geq 0$ for all
$k\in\mathbb{N}$. As, by (3.4),

$\int_{B_{R}(0)}\mu_{i}|w_{ik}|^{p}=\int_{B_{R}(\xi_{ik})}\mu_{i}|u_{ik}|^{p}>\theta>0\qquad\text{for
all \ }k\in\mathbb{N}\text{ \ and \ }i=1,\ldots,\ell,$

we deduce that $w_{i}\neq 0$ for all $i=1,\ldots,\ell$. Furthermore,

(3.5) $\displaystyle\frac{p-2}{2p}\sum_{i=1}^{\ell}\|w_{i}\|^{2}$
$\displaystyle\leq\lim_{k\to\infty}\frac{p-2}{2p}\sum_{i=1}^{\ell}\|w_{ik}\|^{2}=\lim_{k\to\infty}\frac{p-2}{2p}\sum_{i=1}^{\ell}\|u_{ik}\|^{2}=\lim_{k\to\infty}\mathcal{J}_{\infty}^{\ell}(\mathbf{u}_{k})=c_{\infty}^{\ell}.$

Fix $r\in\\{0,\ldots,n-1\\}$, take $\xi_{ik}$ with $i\in I_{r}$ and define
$\mathbf{u}_{k}^{r}=(u^{r}_{1k},\ldots,u^{r}_{\ell k})$ by

$u^{r}_{jk}(x):=u_{jk}(x+\xi_{ik})=\begin{cases}w_{jk}(x)&\text{if \ }j\in
I_{r},\\\ w_{jk}(x+\xi_{ik}-\xi_{jk})&\text{if \ }j\not\in I_{r}.\end{cases}$

After passing to a subsequence, we have that $u^{r}_{jk}\rightharpoonup
u^{r}_{j}$ weakly in $H^{1}(\mathbb{R}^{N})$. Note that $u_{j}^{r}=w_{j}\neq
0$ if $j\in I_{r}$. Set $\mathbf{u}^{r}:=(u_{1}^{r},\ldots,u_{\ell}^{r})$. Let
$\varphi\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{N})$ and define
$\widehat{\varphi}_{k}(x):=\varphi(x-\xi_{ik})$. Then we have that
$\partial_{j}\mathcal{J}^{\ell}_{\infty}(\mathbf{u}_{k})\widehat{\varphi}_{k}=\partial_{j}\mathcal{J}^{\ell}_{\infty}(\mathbf{u}_{k}^{r})\varphi$
and, as $(\mathcal{J}_{\infty}^{\ell})^{\prime}(\mathbf{u}_{k})\to 0$, we
deduce that

$\displaystyle
0=\lim_{k\to\infty}\partial_{j}\mathcal{J}^{\ell}_{\infty}(\mathbf{u}_{k})\widehat{\varphi}_{k}=\lim_{k\to\infty}\partial_{j}\mathcal{J}^{\ell}_{\infty}(\mathbf{u}_{k}^{r})\varphi=\partial_{j}\mathcal{J}^{\ell}_{\infty}(\mathbf{u}^{r})\varphi\qquad\text{for
all \ }j=1,\ldots,\ell.$

This shows that $\mathbf{u}^{r}$ is a critical point of
$\mathcal{J}_{\infty}^{\ell}:\mathcal{H}^{\ell}\to\mathbb{R}$. Hence, its
nontrivial components give a fully nontrivial solution to the subsystem

(3.6) $\begin{cases}-\Delta
v_{m}+v_{m}=\mu_{m}|v_{m}|^{p-2}v_{m}+\sum\limits_{\begin{subarray}{c}j\in\widehat{I}_{r}\\\
j\not=m\end{subarray}}\lambda_{mj}\beta_{mj}|v_{j}|^{\alpha_{mj}}|v_{m}|^{\beta_{mj}-2}v_{m},\\\
v_{m}\in H^{1}(\mathbb{R}^{N})^{\phi_{m}},\qquad v_{m}\neq 0,\qquad
m\in\widehat{I}_{r},\end{cases}$

where $\widehat{I}_{r}:=\\{m:1\leq m\leq\ell,\ u^{r}_{m}\neq 0\\}$. Note that
$\widehat{I}_{r}\supset I_{r}$.

Step 3.  _The conclusion._

Now we distinguish two cases.

$(I):$ Let $\mathrm{Fix}(G)=\\{0\\}$. Then $\xi_{ik}=0$ and $w_{ik}=u_{ik}$
for all $k$, $\widehat{I}_{0}=\\{1,\ldots,\ell\\}$ and
${\mathbf{w}:=(w_{1},\ldots,w_{\ell})}$ is a fully nontrivial solution to the
system (3.6) with $r=0$. From (3.5) we derive

$\displaystyle
c_{\infty}^{\ell}\leq\mathcal{J}_{\infty}^{\ell}(\mathbf{w})=\frac{p-2}{2p}\displaystyle\sum_{i=1}^{\ell}\|w_{i}\|^{2}\leq\lim_{k\to\infty}\frac{p-2}{2p}\displaystyle\sum_{i=1}^{\ell}\|u_{ik}\|^{2}=c_{\infty}^{\ell}.$

As $u_{ik}\rightharpoonup w_{i}$ weakly in $H^{1}(\mathbb{R}^{N})$, this
implies that $u_{ik}\to w_{i}$ strongly in $H^{1}(\mathbb{R}^{N})$ and that
$\mathcal{J}_{\infty}^{\ell}(\mathbf{w})=c_{\infty}^{\ell}$. Then, by
Proposition 3.1$(iii)$, we have that
$c_{\infty}^{\ell}>\displaystyle\sum_{i=1}^{\ell}c_{i}$, as claimed.

$(II):$ Let $\mathrm{Fix}(G)$ have positive dimension. Since for each
$r\in\\{0,\ldots,n-1\\}$, the nontrivial components of $\mathbf{u}^{r}$
satisfy (3.6), multiplying the $i$-th equation by $w_{i}$ we get

(3.7)
$\|w_{i}\|^{2}=\int_{\mathbb{R}^{N}}\mu_{i}|w_{i}|^{p}+\sum\limits_{\begin{subarray}{c}j\in\widehat{I}_{r}\\\
j\not=i\end{subarray}}\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u^{r}_{j}|^{\alpha_{ij}}|w_{i}|^{\beta_{ij}}\leq\int_{\mathbb{R}^{N}}\mu_{i}|w_{i}|^{p}.$

Hence, there exists $t_{i}\in(0,1]$ such that $t_{i}w_{i}\in\mathcal{N}_{i}$
and, so,

$c_{i}\leq\frac{p-2}{2p}\|t_{i}w_{i}\|^{2}\leq\frac{p-2}{2p}\|w_{i}\|^{2}\qquad\text{for
every \ }i=1,\ldots,\ell.$

From (3.5) and Proposition 3.1 we derive

(3.8)
$\sum_{i=1}^{\ell}c_{i}\leq\frac{p-2}{2p}\sum_{i=1}^{\ell}\|w_{i}\|^{2}\leq\lim_{k\to\infty}\frac{p-2}{2p}\sum_{i=1}^{\ell}\|w_{ik}\|^{2}=c_{\infty}^{\ell}=\sum_{i=1}^{\ell}c_{i}.$

It follows that $\frac{p-2}{2p}\|w_{i}\|^{2}=c_{i}$ and $t_{i}=1$, so $w_{i}$
is a nontrivial least energy solution to the problem
($\mathscr{P}_{i}^{\phi_{i}}$) for each $i=1,\ldots,\ell$. Furthermore, (3.7)
yields

$\int_{\mathbb{R}^{N}}\lambda_{ij}\beta_{ij}|u^{r}_{j}|^{\alpha_{ij}}|w_{i}|^{\beta_{ij}}=0\qquad\text{for
every \ }j\neq i.$

Therefore, $u^{r}_{j}=0$ for every $j\neq i$ and, so,
$\widehat{I}_{r}=I_{r}=\\{i\\}$. This implies that
$|\xi_{ik}-\xi_{jk}|\to\infty$ for all $j\neq i$. Finally, as
$w_{ik}\rightharpoonup w_{i}$ weakly in $H^{1}(\mathbb{R}^{N})$, it follows
from (3.8) that $w_{ik}\to w_{i}$ strongly in $H^{1}(\mathbb{R}^{N})$. This
implies that $\displaystyle\lim_{k\to\infty}\|u_{ik}-w_{i}(\ \cdot\
-\xi_{ik})\|=0$, and the proof is complete. ∎

From Theorem 3.3 and Proposition 3.1$(ii)$-$(iii)$ we immediately obtain the
following result.

######  Theorem 3.4.

Let $\ell\geq 2$. The system ($\mathscr{S}_{\infty,\ell}^{\phi}$) has a least
energy solution if and only if $\mathrm{Fix}(G)=\\{0\\}$.

Theorem 1.1 follows from the previous one.

###### Proof of Theorem 1.1.

The group $G:=\Gamma\times O(N-4)$ introduced in Example 2.1$(iii)$ satisfies
$(A_{2})$ if either $N=4$ or $N\geq 6$, and $\mathrm{Fix}(G)=\\{0\\}$. For
$i=1,\ldots,m$ we set $\phi_{i}\equiv 1$, and for $i=m+1,\ldots,\ell$ we take
$\phi_{i}$ as in Example 2.1$(iii)$. Applying Theorem 3.4 we obtain a least
energy solution $\mathbf{w}=(w_{1},\ldots,w_{\ell})$ to the system
($\mathscr{S}_{\infty,\ell}$) satisfying (1.1). By Remark 2.6, we may take
$w_{1},\ldots,w_{m}$ to be positive. The symmetries ensure that the last
$\ell-m$ components are nonradial and change sign. ∎

To further emphasize the role of the fixed points, we prove the following
result.

######  Theorem 3.5.

If $\mathrm{Fix}(G)=\\{0\\}$, then the system
($\mathscr{S}_{\infty,\ell}^{\phi}$) has infinitely many solutions.

The key ingredient is the following fact.

######  Lemma 3.6.

If $G$ satisfies $(A_{2})$ and $\mathrm{Fix}(G)=\\{0\\}$, then the embedding
$H^{1}(\mathbb{R}^{N})^{\phi}\hookrightarrow L^{p}(\mathbb{R}^{N})$ is compact
for any homomorphism $\phi:G\to\mathbb{Z}_{2}$ and $p\in(2,2^{*})$.

###### Proof.

Let $K:=\ker\phi$. Since the $K$-orbit of every
$x\in\mathbb{R}^{N}\smallsetminus\\{0\\}$ is infinite, the embedding
$H^{1}(\mathbb{R}^{N})^{K}\hookrightarrow L^{p}(\mathbb{R}^{N})$ is compact,
where $H^{1}(\mathbb{R}^{N})^{K}$ denotes the space of $K$-invariant functions
in $H^{1}(\mathbb{R}^{N})$; see, e.g., [8, Lemma 4.3]. Since
$H^{1}(\mathbb{R}^{N})^{\phi}\subset H^{1}(\mathbb{R}^{N})^{K}$, our claim
follows. ∎

###### Proof of Theorem 3.5.

We note that the results in [8, Section 3] hold true if we take
$\mathcal{H}:=\mathcal{H}^{\ell}$. A standard argument using Lemma 3.6 shows
that $\mathcal{J}_{\infty}^{\ell}:\mathcal{H}^{\ell}\to\mathbb{R}$ satisfies
the Palais-Smale condition at every level $c\in\mathbb{R}$. As in [8, Section
3] we set

$\mathcal{U}:=\\{\mathbf{u}\in\mathcal{H}^{\ell}:\|u_{i}\|=1\text{ \ for \
}i=1,\ldots,\ell,\text{ \ and there exists \
}\mathbf{t}\in(0,\infty)^{\ell}\text{ \ such that \
}\mathbf{t}\mathbf{u}\in\mathcal{N}_{\infty}^{\ell}\\},$

and we define $\Psi:\mathcal{U}\to\mathbb{R}$ by
$\Psi(u):=\mathcal{J}_{\infty}^{\ell}(\mathbf{t}\mathbf{u})$ for
$\mathbf{t}\in(0,\infty)^{\ell}$ such that
$\mathbf{t}\mathbf{u}\in\mathcal{N}_{\infty}^{\ell}$. Such $\mathbf{t}$ is
unique.

It follows from [8, Theorem 3.3] that
$\Psi\in\mathcal{C}^{1}(\mathcal{U},\mathbb{R})$ and it satisfies the Palais-
Smale condition at every level $c\in\mathbb{R}$. Exactly the same argument
used to prove [8, Lemma 4.5] shows that $\mathrm{genus}(\mathcal{U})=\infty$
(one simply replaces $H_{0}^{1}(\Omega)^{G}$ with
$H^{1}(\mathbb{R}^{N})^{\phi_{i}}$). Now [8, Theorem 3.4] asserts that
($\mathscr{S}_{\infty,\ell}^{\phi}$) has an unbounded sequence of solutions. ∎

We observe that, while the last $\ell-m$ components of the solutions given by
the Theorem 3.5 change sign by construction, this theorem does not give any
information about the sign of the first $m$ components.

##  4 The asymptotic behavior of minimizers

As before, we fix a closed subgroup $G$ of $O(N)$ satisfying $(A_{2})$ and,
for each $i=1,\ldots,\ell$, a continuous homomorphism
$\phi_{i}:G\to\mathbb{Z}_{2}$ satisfying $(A_{1})$. Furthermore, we assume
throughout that $\Omega$ is a $G$-invariant bounded domain in $\mathbb{R}^{N}$
and that $0\in\Omega$.

Our aim is to prove the following result.

######  Theorem 4.1.

For any given sequence $(\varepsilon_{k})$ of positive numbers converging to
zero, there exists a least energy solution
$\mathbf{u}_{k}=(u_{1k},\ldots,u_{\ell k})$ to the system
$(\mathscr{S}^{\phi}_{\varepsilon_{k},\Omega})$ with the following limit
profile:

  * $(I)$

If $\mathrm{Fix}(G)=\\{0\\}$, then there exists a least energy solution
$\mathbf{w}=(w_{1},\ldots,w_{\ell})$ to the system
($\mathscr{S}_{\infty,\ell}^{\phi}$) such that, after passing to a
subsequence,

$\lim_{k\to\infty}\|u_{ik}-w_{i}(\varepsilon_{k}^{-1}\ \cdot\
)\|_{\varepsilon_{k}}=0\qquad\text{for all \ }i=1,\ldots,\ell,$

and $\displaystyle\lim_{\varepsilon\to
0}c_{\varepsilon}^{\ell}(\Omega)=c_{\infty}^{\ell}$. Moreover, $w_{i}\geq 0$
if $u_{ik}\geq 0$ for all $k\in\mathbb{N}$.

  * $(II)$

If $\mathrm{Fix}(G)$ has positive dimension, then, for each $i=1,\ldots,\ell$,
there exist a sequence $(\xi_{ik})$ in $\Omega\cap\mathrm{Fix}(G)$ and a least
energy solution $v_{i}$ to the problem ($\mathscr{P}_{i}^{\phi_{i}}$) such
that, after passing to a subsequence,

$\lim_{k\to\infty}\varepsilon_{k}^{-1}\mathrm{dist}(\zeta_{ik},\partial\Omega)=\infty,\quad\lim_{k\to\infty}\varepsilon_{k}^{-1}|\zeta_{ik}-\zeta_{jk}|=\infty\text{
if }i\neq j,\qquad\lim_{k\to\infty}\|u_{ik}-v_{i}(\varepsilon_{k}^{-1}(\
\cdot\ -\zeta_{ik}))\|_{\varepsilon_{k}}=0,$

for $i,j=1,\ldots,\ell$, and $\displaystyle\lim_{\varepsilon\to
0}c_{\varepsilon}^{\ell}(\Omega)=\displaystyle\sum_{i=1}^{\ell}c_{i}$.
Moreover, $v_{i}\geq 0$ if $u_{ik}\geq 0$ for all $k\in\mathbb{N}$.

We start with the proof of the following statement.

######  Lemma 4.2.

$\displaystyle\lim_{\varepsilon\to
0}c_{\varepsilon}^{\ell}(\Omega)=c_{\infty}^{\ell}$.

###### Proof.

Let
$\mathbf{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{N}^{\ell}_{\varepsilon}(\Omega)$
and define $\widetilde{u}_{i}(z):=u_{i}(\varepsilon z)$. By Remark 2.2,
$\widetilde{\mathbf{u}}=(\widetilde{u}_{1},\ldots,\widetilde{u}_{\ell})\in\mathcal{N}^{\ell}_{\infty}$
and
$\mathcal{J}_{\varepsilon}^{\ell}(\mathbf{u})=\mathcal{J}_{\infty}^{\ell}(\widetilde{\mathbf{u}})\geq
c_{\infty}^{\ell}$. Therefore,

(4.1) $c_{\infty}^{\ell}\leq\displaystyle\liminf_{\varepsilon\to
0}c_{\varepsilon}^{\ell}.$

To prove the opposite inequality, let
$\mathbf{w}=(w_{1},\ldots,w_{\ell})\in\mathcal{N}^{\ell}_{\infty}$. Fix $r>0$
such that $B_{r}(0)\subset\Omega$ and a radial cut-off function $\chi\in
C^{\infty}_{c}(\mathbb{R}^{N})$ satisfying $\chi(x)=1$ for
$|x|\leq\frac{r}{2}$ and $\chi(x)=0$ for $|x|>r$. For $\varepsilon>0$ define
$\mathbf{w}_{\varepsilon}=(w_{1\varepsilon},\ldots,w_{\ell\varepsilon})$ by

$w_{i\varepsilon}(x):=w_{i}\Big{(}\frac{x}{\varepsilon}\Big{)}\chi(x)\qquad
i=1,\ldots,\ell.$

Since $w_{i}\in H^{1}(\mathbb{R}^{N})^{\phi_{i}}$ and $\chi$ is radial, we
have that $w_{i\varepsilon}\in H^{1}(\mathbb{R}^{N})^{\phi_{i}}$. As
$\mathbf{w}\in\mathcal{N}^{\ell}_{\infty}$, from (2.2) and Lemma 2.3$(i)$ we
get

$0<c_{0}\leq\|w_{i}\|^{2}={\mu_{i}}|w_{i}|_{p}^{p}+\displaystyle\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}{\lambda_{ij}}\beta_{ij}|w_{j}|^{\alpha_{ij}}|w_{i}|^{\beta_{ij}}\qquad\text{for
all \ }i=1,\ldots,\ell.$

Note that $w_{i\varepsilon}\to w_{i}$ in $H^{1}(\mathbb{R}^{N})$ as
$\varepsilon\to 0$. Therefore, for sufficiently small $\varepsilon>0$, Lemma
2.3$(ii)$ yields
$\mathbf{t}_{\varepsilon}=(t_{1\varepsilon},\ldots,t_{\ell\varepsilon})\in(0,\infty)^{\ell}$
such that
$\mathbf{t}_{\varepsilon}\mathbf{w}_{\varepsilon}\in\mathcal{N}^{\ell}_{\varepsilon}(\Omega)$,
and $t_{i\varepsilon}\to 1$ as $\varepsilon\to 0$. Thus,

$c_{\varepsilon}^{\ell}(\Omega)\leq\mathcal{J}^{\ell}_{\varepsilon}(\mathbf{t}_{\varepsilon}\mathbf{w}_{\varepsilon})=\dfrac{p-2}{2p}\|\mathbf{t}_{\varepsilon}\mathbf{w}_{\varepsilon}\|_{\ell,\varepsilon}^{2}\to\dfrac{p-2}{2p}\|\mathbf{w}\|_{\ell}^{2}=\mathcal{J}^{\ell}_{\infty}(\mathbf{w})\quad\text{as}\quad\varepsilon\to
0,$

and, as a consequence,

(4.2) $\limsup_{\varepsilon\to 0}c_{\varepsilon}^{\ell}(\Omega)\leq
c_{\infty}^{\ell}.$

Inequalities (4.1) and (4.2) yield the desired identity. ∎

###### Proof of Theorem 4.1.

Applying Theorem 2.7 we obtain a least energy solution
$\mathbf{u}_{k}=(u_{1k},\ldots,u_{\ell k})$ to the system
$(\mathscr{S}^{\phi}_{\varepsilon_{k},\Omega})$ for every $k\in\mathbb{N}$.
Define
$\widetilde{\mathbf{u}}_{k}=(\widetilde{u}_{1k},\ldots,\widetilde{u}_{\ell
k})$ by taking $\widetilde{u}_{ik}(z):=u_{ik}(\varepsilon_{k}z)$. From Remark
2.2 and Lemma 4.2 we see that
$\widetilde{\mathbf{u}}_{k}\in\mathcal{N}_{\infty}^{\ell}$ and
$\mathcal{J}_{\infty}^{\ell}(\widetilde{\mathbf{u}}_{k})\to
c_{\infty}^{\ell}$. Now we distinguish two cases:

$(I)$ If $\mathrm{Fix}(G)=\\{0\\}$, applying Theorem 3.3 to
$\widetilde{\mathbf{u}}_{k}$ we obtain a least energy solution
$\mathbf{w}=(w_{1},\ldots,w_{\ell})$ to the system
($\mathscr{S}_{\infty,\ell}^{\phi}$) such that
$\|u_{ik}-w_{i}(\varepsilon_{k}^{-1}\ \cdot\
)\|_{\varepsilon_{k}}=\|\widetilde{u}_{ik}-w_{i}\|\to 0$. The identity
$\displaystyle\lim_{\varepsilon\to
0}c_{\varepsilon}^{\ell}(\Omega)=c_{\infty}^{\ell}$ was proved in Lemma 4.2.

$(II)$ If $\mathrm{Fix}(G)$ has positive dimension, Theorem 3.3 applied to
$\widetilde{\mathbf{u}}_{k}$ yields a sequence $(\xi_{ik})$ in
$\mathrm{Fix}(G)$ and a least energy solution $v_{i}$ to the problem
($\mathscr{P}_{i}^{\phi_{i}}$) for each $i=1,\ldots,\ell$, such that, setting
$\zeta_{ik}:=\varepsilon_{k}\xi_{ik}$, we get

$\lim_{k\to\infty}\varepsilon_{k}^{-1}|\zeta_{ik}-\zeta_{jk}|=\infty\text{ \
if \ }i\neq j,\qquad\text{and}\qquad\|u_{ik}-v_{i}(\varepsilon_{k}^{-1}(\
\cdot\ -\zeta_{ik}))\|_{\varepsilon_{k}}=\|\widetilde{u}_{ik}-v_{i}(\ \cdot\
-\xi_{ik})\|\to 0,$

for $i,j=1,\ldots,\ell$. We are left with showing that $\zeta_{ik}\in\Omega$
and

$\lim_{k\to\infty}\varepsilon_{k}^{-1}\mathrm{dist}(\zeta_{ik},\partial\Omega)=\infty\qquad\text{for
all \ }i=1,\ldots,\ell.$

To this end, we recall that the points $\xi_{ik}$ satisfy (3.4), i.e., there
exist $\theta,R>0$ such that

(4.3)
$\int_{B_{R}(\xi_{ik})}\mu_{i}|\widetilde{u}_{ik}|^{p}>\theta>0\qquad\text{for
all \ }k\in\mathbb{N}\text{ \ and \ }i=1,\ldots,\ell.$

Note that $\mathrm{supp}(\widetilde{u}_{ik})\subset\Omega_{\varepsilon_{k}}$,
where
$\Omega_{\varepsilon_{k}}:=\\{z\in\mathbb{R}^{N}:\varepsilon_{k}z\in\Omega\\}$.
So (4.3) implies that $\mathrm{dist}(\xi_{ik},\Omega_{\varepsilon_{k}})<R$ for
every $k\in\mathbb{N}$ and every $i$. A standard argument shows that
$\mathrm{dist}(\xi_{ik},\partial\Omega_{\varepsilon_{k}})\to\infty$, otherwise
$v_{i}$ would be zero in some open subset of $\mathbb{R}^{N}$, contradicting
the unique continuation principle. In particular, we have that
$\xi_{ik}\in\Omega_{\varepsilon_{k}}$ for all $k\in\mathbb{N}$. Rescaling
yields $\zeta_{ik}\in\Omega$ and
$\displaystyle\lim_{k\to\infty}\varepsilon_{k}^{-1}\mathrm{dist}(\zeta_{ik},\partial\Omega)=\infty$
for all $i=1,\ldots,\ell$, as claimed. This completes the proof. ∎

Next, we derive Theorems 1.2 and 1.3 from Theorem 4.1.

###### Proof of Theorem 1.2..

The group $G:=\Gamma\times O(N-4)$ in Example 2.1$(iii)$ satisfies $(A_{2})$
if $N=4$ or $N\geq 6$, and $\mathrm{Fix}(G)=\\{0\\}$. For $i=1,\ldots,m$ we
take $\phi_{i}\equiv 1$, and for $i=m+1,\ldots,\ell$ we take $\phi_{i}:=\phi$
as in Example 2.1$(iii)$. By Theorem 4.1 there exist a least energy solution
$\widehat{\mathbf{u}}_{k}=(\widehat{u}_{1k},\ldots,\widehat{u}_{\ell k})$ to
the system $(\mathscr{S}^{\phi}_{\varepsilon_{k},B_{1}(0)})$ and a least
energy solution $\mathbf{w}=(w_{1},\ldots,w_{\ell})$ to the system
($\mathscr{S}_{\infty,\ell}$) satisfying (1.1) such that, after passing to a
subsequence,

$\lim_{k\to\infty}\|\widehat{u}_{ik}-w_{i}(\varepsilon_{k}^{-1}\ \cdot\
)\|_{\varepsilon_{k}}=0\qquad\text{for all \ }i=1,\ldots,\ell,$

and
$\displaystyle\lim_{k\to\infty}\displaystyle\sum_{i=1}^{\ell}\|\widehat{u}_{ik}\|_{\varepsilon_{k}}^{2}=\lim_{\varepsilon\to
0}c_{\varepsilon}^{\ell}(B_{1}(0))=c_{\infty}^{\ell}=\displaystyle\sum_{i=1}^{\ell}\|w_{i}\|^{2}$.
By Remark 2.6, we may take $\widehat{u}_{1k},\ldots,\widehat{u}_{mk}$ to be
positive. Then, $w_{1},\ldots,w_{m}$ are also positive. The symmetries ensure
that the last $\ell-m$ components of $\widehat{\mathbf{u}}_{k}$ and
$\mathbf{w}$ are nonradial and change sign. ∎

###### Proof of Theorem 1.3..

The group $G:=\Gamma$ in Example 2.1$(ii)$ satisfies $(A_{2})$, and
$\mathrm{Fix}(G)=\\{0\\}\times\mathbb{R}^{N-4}$ has positive dimension if
$N\geq 5$. For $i=1,\ldots,m$ we take $\phi_{i}\equiv 1$, and for
$i=m+1,\ldots,\ell$ we take $\phi_{i}:=\phi$ as in Example 2.1$(ii)$. By
Theorem 4.1, there exist a least energy solution
$\mathbf{u}_{k}=(u_{1k},\ldots,u_{\ell k})$ to the system
$(\mathscr{S}^{\phi}_{\varepsilon_{k},B_{1}(0)})$ and, for each
$i=1,\ldots,\ell$, a sequence $(\zeta_{ik})$ in $B_{1}(0)\cap\mathrm{Fix}(G)$
and a least energy solution $w_{i}$ to the problem
($\mathscr{P}_{i}^{\phi_{i}}$) such that, after passing to a subsequence,

$\lim_{k\to\infty}\varepsilon_{k}^{-1}\mathrm{dist}(\zeta_{ik},\partial
B_{1}(0))=\infty,\quad\lim_{k\to\infty}\varepsilon_{k}^{-1}|\zeta_{ik}-\zeta_{jk}|=\infty\text{
if }i\neq j,\qquad\lim_{k\to\infty}\|u_{ik}-w_{i}(\varepsilon_{k}^{-1}(\
\cdot\ -\zeta_{ik}))\|_{\varepsilon_{k}}=0,$

for $i,j=1,\ldots,\ell$, and
$\displaystyle\lim_{k\to\infty}\displaystyle\sum_{i=1}^{\ell}\|u_{ik}\|_{\varepsilon_{k}}^{2}=\lim_{\varepsilon\to
0}c_{\varepsilon}^{\ell}(B_{1}(0))=\displaystyle\sum_{i=1}^{\ell}c_{i}=\displaystyle\sum_{i=1}^{\ell}\|w_{i}\|^{2}$.

By Remark 2.6, we may take $u_{1k},\ldots,u_{mk}$ to be positive. Then,
$w_{1},\ldots,w_{m}$ are also positive. It is well known that problem
($\mathscr{P}_{i}$) has a unique positive radial solution $v_{i}$ and that any
other positive solution is a translation of it [11]. Hence, there exists
$\vartheta_{i}\in\mathbb{R}^{N}$ such that $v_{i}(y)=w_{i}(y+\vartheta_{i})$
for every $y\in\mathbb{R}^{N}$ and $i=1,\ldots,m$.

Now take $i\in\\{m+1,\ldots,\ell\\}$. Then, $w_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi}$ with $\phi$ as in Example 2.1$(ii)$, i.e.,
$w_{i}$ satisfies

(4.4)
$w_{i}(z_{1},z_{2},x)=w_{i}(\mathrm{e}^{\mathrm{i}\vartheta}z_{1},\mathrm{e}^{\mathrm{i}\vartheta}z_{2},x)\text{
\ for all \ }\vartheta\in[0,2\pi),\qquad
w_{i}(z_{1},z_{2},x)=-w_{i}(z_{2},z_{1},x),$

for all $(z_{1},z_{2},x)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}$.
We claim that there exists $\vartheta_{i}\in\mathrm{Fix}(G)$ such that
$v_{i}(y):=w_{i}(y+\vartheta_{i})$ satisfies (1.2). To prove this claim,
observe that, if $\pi_{j}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is the reflection
on the hyperplane
$\\{(z,x_{1},\ldots,x_{N-4})\in\mathbb{C}^{2}\times\mathbb{R}^{N-4}:x_{j}=a\\}$,
$a\in\mathbb{R}$, then the function

$\bar{w}(z,x):=\begin{cases}w_{i}(z,x)&\text{if \ }x_{j}>a,\\\
(w_{i}\circ\pi_{j})(z,x)&\text{if \ }x_{j}<a,\end{cases}$

satisfies (4.4). With this remark and following Lopes’ argument [15] (see also
[29, Appendix C]) one shows that, for each $j=1,\ldots,N-4$, there exists
$\theta_{ij}\in\mathbb{R}$ such that $w_{i}$ is invariant with respect to the
reflection on the hyperplane $x_{j}=\theta_{ij}$. Setting
$\theta_{i}=(\theta_{i1},\ldots,\theta_{1\,N-4})$ we infer that the function
$v_{i}(z,x):=w_{i}(z,x+\theta_{i})$ satisfies $v_{i}(z,x)=v_{i}(z,gx)$ for all
$g\in O(N-4)$ and $(z,x)\in\mathbb{C}^{2}\times\mathbb{R}^{N-4}$. Hence,
setting $\vartheta_{i}=(0,\theta_{i})\in\mathbb{C}^{2}\times\mathbb{R}^{N-4}$,
we have that $v_{i}(y)=w_{i}(y+\vartheta_{i})$ satisfies (1.2), as claimed.

Let $\xi_{ik}:=\zeta_{ik}+\varepsilon_{k}\vartheta_{i}$, where $\vartheta_{i}$
are the points defined in the previous paragraphs, so that
$v_{i}(y):=w_{i}(y+\vartheta_{i})$ is radial if $i=1,\ldots,m$ and it
satisfies (1.2) if $i=m+1,\ldots,\ell$. Note that, as $\varepsilon_{k}\to 0$,
we have that $\xi_{ik}\in B_{1}(0)$ for $k$ large enough. From the
corresponding statements for $\zeta_{ik}$ and $w_{i}$ one immediately derives

$\lim_{k\to\infty}\varepsilon_{k}^{-1}\mathrm{dist}(\xi_{ik},\partial
B_{1}(0))=\infty,\quad\lim_{k\to\infty}\varepsilon_{k}^{-1}|\xi_{ik}-\xi_{jk}|=\infty\text{
if }i\neq j,\qquad\lim_{k\to\infty}\|u_{ik}-v_{i}(\varepsilon_{k}^{-1}(\
\cdot\ -\xi_{ik}))\|_{\varepsilon_{k}}=0,$

for $i,j=1,\ldots,\ell$, and
$\displaystyle\lim_{k\to\infty}\displaystyle\sum_{i=1}^{\ell}\|u_{ik}\|_{\varepsilon_{k}}^{2}=\lim_{\varepsilon\to
0}c_{\varepsilon}^{\ell}(B_{1}(0))=c_{\infty}^{\ell}=\displaystyle\sum_{i=1}^{\ell}c_{i}=\displaystyle\sum_{i=1}^{\ell}\|v_{i}\|^{2}$.

The inequality $\mathfrak{c}_{m}<\widehat{\mathfrak{c}}_{m}$ follows from
Proposition 3.1. ∎

Finally, regarding positive solutions in the the unit ball $B_{1}(0)$ in
$\mathbb{R}^{N}$, we derive the following result.

######  Theorem 4.3.

Let $N\geq 2$, $\Omega=B_{1}(0)$ and $(\varepsilon_{k})$ be a sequence of
positive numbers converging to zero. Then there exist two solutions
$\widehat{\mathbf{u}}_{k}=(\widehat{u}_{1k},\ldots,\widehat{u}_{\ell k})$ and
$\mathbf{u}_{k}=(u_{1k},\ldots,u_{\ell k})$ to the system
$(\mathscr{S}_{\varepsilon_{k},\Omega})$ whose components $\widehat{u}_{ik}$
and $u_{ik}$ are positive and satisfy:

  * •

$\widehat{u}_{ik}$ is radial and, after passing to a subsequence,

$\lim_{k\to\infty}\|\widehat{u}_{ik}-\widehat{\omega}_{i}(\varepsilon_{k}^{-1}\
\cdot\ )\|_{\varepsilon_{k}}=0\qquad\text{for all \ }i=1,\ldots,\ell,$

where
$\widehat{\boldsymbol{\omega}}_{k}=(\widehat{\omega}_{1k},\ldots,\widehat{\omega}_{\ell
k})$ is a least energy positive radial solution to the system
($\mathscr{S}_{\infty,\ell}$) in $\mathbb{R}^{N}$.

  * •

For each $i=1,\ldots,\ell$ there is a sequence $(\xi_{ik})$ in $B_{1}(0)$ such
that, after passing to a subsequence,

$\lim_{k\to\infty}\varepsilon_{k}^{-1}\mathrm{dist}(\zeta_{ik},\partial\Omega)=\infty,\quad\lim_{k\to\infty}\varepsilon_{k}^{-1}|\zeta_{ik}-\zeta_{jk}|=\infty\text{
if }i\neq j,\qquad\lim_{k\to\infty}\|u_{ik}-v_{i}(\varepsilon_{k}^{-1}(\
\cdot\ -\zeta_{ik}))\|_{\varepsilon_{k}}=0,$

for $i,j=1,\ldots,\ell$, where $v_{i}$ is the positive radial solution to the
problem ($\mathscr{P}_{i}$).

###### Proof.

The solutions $\widehat{\mathbf{u}}_{k}$ are obtained applying Theorem 4.1
with $G=O(N)$ and $\phi_{i}\equiv 1$ for all $i=1,\ldots,\ell$. The solutions
$\mathbf{u}_{k}=(u_{1k},\ldots,u_{\ell k})$ are obtained applying Theorem 4.1
to the trivial group $G=\\{1\\}$ and $\phi_{i}\equiv 1$ for all
$i=1,\ldots,\ell$. ∎

The solutions $\mathbf{u}_{k}$ were obtained and fully described in [12]. In
addition, as mentioned in the introduction, the existence of a least energy
radial solution to the system ($\mathscr{S}_{\infty,\ell}$) was shown by
Sirakov in [23]. Theorem 4.3 gives an alternative proof of this fact. Finally,
we point out that the existence of multiple positive solutions, both in
$\mathbb{R}^{N}$ and in bounded domains, has been established, for instance,
in [1, 9, 19, 27, 28]. Positive solutions of singularly perturbed systems with
potentials have been obtained, e.g., in [17, 20].

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Mónica Clapp and Mayra Soares
Instituto de Matemáticas

Universidad Nacional Autónoma de México

Circuito Exterior, Ciudad Universitaria

04510 Coyoacán, Ciudad de México, México

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