# Energy estimates for seminodal solutions to an elliptic system with mixed
couplings

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

###### Abstract

We study the system of semilinear elliptic equations

$-\Delta
u_{i}+u_{i}=\displaystyle\sum_{j=1}^{\ell}\beta_{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\qquad
u_{i}\in H^{1}(\mathbb{R}^{N}),\qquad i=1,\ldots,\ell,$

where $N\geq 4$, $1<p<\frac{N}{N-2}$, and the matrix $(\beta_{ij})$ is
symmetric and admits a block decomposition such that the entries within each
block are positive or zero and all other entries are negative.

We provide simple conditions on $(\beta_{ij})$, which guarantee the existence
of fully nontrivial solutions, i.e., solutions all of whose components are
nontrivial.

We establish existence of fully nontrivial solutions to the system having a
prescribed combination of positive and nonradial sign-changing components, and
we give an upper bound for their energy when the system has at most two
blocks.

We derive the existence of solutions with positive and nonradial sign-changing
components to the system of singularly perturbed elliptic equations

$-\varepsilon^{2}\Delta
u_{i}+u_{i}=\displaystyle\sum_{j=1}^{\ell}\beta_{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\qquad
u_{i}\in H^{1}_{0}(B_{1}(0)),\qquad i=1,\ldots,\ell,$

in the unit ball, exhibiting two different kinds of asymptotic behavior:
solutions whose components decouple as $\varepsilon\to 0$, and solutions whose
components remain coupled all the way up to their limit.

Keywords: Nonlinear elliptic system, weakly coupled, mixed cooperation and
competition, positive and sign-changing components, singularly perturbed
elliptic system.

MSC2020: 35J47 (35A15, 35B06, 35B25, 35B40)

##  1 Introduction

Consider the system of nonlinear elliptic equations

(1.1) $\begin{cases}-\Delta
u_{i}+u_{i}=\displaystyle\sum_{j=1}^{\ell}\beta_{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\\\
u_{i}\in H^{1}(\mathbb{R}^{N}),\qquad i=1,\ldots,\ell,\end{cases}$

where $N\geq 4$, $\beta_{ij}\in\mathbb{R}$, and $1<p<\frac{N}{N-2}$.

This system arises as a model for various physical phenomena, for instance, in
the study of standing waves for a mixture of Bose-Einstein condensates of
$\ell$ different hyperfine states which overlap in space; see [11]. The
coefficient $\beta_{ij}$ represents the interaction force between the
components $u_{i}$ and $u_{j}$. It can be attractive ($\beta_{ij}>0$),
repulsive ($\beta_{ij}<0$) or null ($\beta_{ij}=0$). We make the following
assumptions:

  * $(B_{1})$

The matrix $(\beta_{ij})$ is symmetric and admits a block decomposition as
follows: For some $1\leq q\leq\ell$ there exist
$0=\ell_{0}<\ell_{1}<\dots<\ell_{q-1}<\ell_{q}=\ell$ such that, if we set

$\displaystyle I_{h}:=\\{i\in\\{1,\dots,\ell\\}:\ell_{h-1}<i\leq\ell_{h}\\},$
$\displaystyle\mathcal{I}_{h}:=I_{h}\times
I_{h},\quad\text{and}\quad\mathcal{K}_{h}:=\big{\\{}(i,j)\in I_{h}\times
I_{k}:k\in\\{1,\ldots,q\\}\smallsetminus\\{h\\}\big{\\}},$

then $\beta_{ii}>0,\ \beta_{ij}\geq 0\text{ for all }(i,j)\in\mathcal{I}_{h},$
and $\beta_{ij}<0\text{ for all }(i,j)\in\mathcal{K}_{h},\quad h=1,\ldots,q$.

  * $(B_{2})$

For each $h=1,\ldots,q$, the graph whose set of vertices is $I_{h}$ and whose
set of edges is $E_{h}:=\\{\\{i,j\\}:i,j\in I_{h},\ i\neq j,\ \beta_{ij}>0\\}$
is connected.

According to the decomposition given by $(B_{1})$, a solution
$\boldsymbol{u}=(u_{1},\ldots,u_{\ell})$ to (1.1) may be written in block-form
as

$\boldsymbol{u}=(\overline{u}_{1},\ldots,\overline{u}_{q})\qquad\text{with \
}\overline{u}_{h}=(u_{\ell_{h-1}+1},\ldots,u_{\ell_{h}}).$

$\boldsymbol{u}$ is said to be _nontrivial_ if at least one of its components
$u_{i}$ is different from zero and it is said to be _fully nontrivial_ if
every component $u_{i}$ is different from zero. We shall call it _block-wise
nontrivial_ if at least one component in each block $\overline{u}_{h}$ is
nontrivial.

If $q=1$ the system is called _cooperative_ and it is called _competitive_ if
$q=\ell$. These kinds of systems have been studied extensively. Systems with
mixed couplings were already investigated in the seminal paper by Lin and Wei
[13], and more recently, e.g., in [3, 7, 10, 17, 18, 20, 21, 22, 23, 25].

Our main interest is to study the existence of fully nontrivial solutions to
the system (1.1) with a prescribed set of positive and sign-changing
components - referred to in the literature as _seminodal_ or _semipositive_
solutions -, and to give an upper bound for their energy.

We stress that obtaining an upper bound for the least energy of a solution to
the system to (1.1) is interesting even in the case when all components are
positive. Indeed, as shown by Lin and Wei in [13] a positive least energy
solution does not exist when $q\geq 2$; see also Proposition 2.5 below. On the
other hand, in [19] Sirakov established the existence of solutions with
positive radial components, while sign-changing and seminodal solutions have
been exhibited, e.g., in [5, 8, 14, 15, 16, 24]. But nothing is said about
their energy. For a single equation an upper bound for the least energy of a
sign-changing solution is given in [9]. Here we improve that upper bound; see
Corollary 1.2 below.

We write $\|\,\cdot\,\|$ for the usual norm in the Sobolev space
$H^{1}(\mathbb{R}^{N})$ and, abusing notation, we set

$\|\boldsymbol{u}\|^{2}:=\sum_{i=1}^{\ell}\|u_{i}\|^{2}\qquad\text{for \
}\boldsymbol{u}=(u_{1},\ldots,u_{\ell})\in(H^{1}(\mathbb{R}^{N}))^{\ell}.$

For each $h=1,\ldots,q$ set
$\mathbb{R}^{I_{h}}:=\\{\overline{s}=(s_{\ell_{h-1}+1},\ldots,s_{\ell_{h}}):s_{i}\in\mathbb{R}\text{
for all }i\in I_{h}\\}$ and define

(1.2)
$\mu_{h}:=\inf_{\begin{subarray}{c}\overline{s}\in\mathbb{R}^{I_{h}}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \overline{s}\neq
0\end{subarray}}\left(\frac{\displaystyle\sum_{i\in
I_{h}}s_{i}^{2}}{\Big{(}\displaystyle\sum\limits_{{(i,j)\in\mathcal{I}_{h}}}\beta_{ij}|s_{i}|^{p}|s_{j}|^{p}\Big{)}^{\frac{2}{2p}}}\right)^{\frac{p}{p-1}}.$

Our first result reads as follows.

######  Theorem 1.1.

Assume $(B_{1})$, $(B_{2})$ and

  * $(B_{3})$

If $q\geq 2$ then, for every $h\in\\{1,\ldots,q\\}$ such that
$\ell_{h}-\ell_{h-1}\geq 2$, the inequality

$\Big{(}\min_{\\{i,j\\}\in
E_{h}}\beta_{ij}\Big{)}\left[\frac{\displaystyle\min_{h=1,\ldots,q}\displaystyle\max_{i\in
I_{h}}\beta_{ii}}{\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}}\right]^{\frac{p}{p-1}}>\,C_{q}\sum_{(i,j)\in\mathcal{K}_{h}}|\beta_{ij}|$

holds true, where $C_{q}$ is a positive constant independent of
$(\beta_{ij})$.

If $q\geq 2$ assume also that $N\neq 5$. Let $\\{1,\ldots,q\\}=Q^{+}\cup
Q^{-}$ with $Q^{+}\cap Q^{-}=\emptyset$. Then there exists a fully nontrivial
solution $\boldsymbol{w}=(\overline{w}_{1},\ldots,\overline{w}_{q})$ to the
system (1.1) with the following properties:

  * $(a)$

Every component of $\overline{w}_{h}$ is positive if $h\in Q^{+}$ and every
component of $\overline{w}_{h}$ is nonradial and changes sign if $h\in Q^{-}$.

  * $(b)$

If $q\geq 2$ then

$\|\boldsymbol{w}\|^{2}>\sum_{h\in Q^{+}}\mu_{h}\,\|\omega\|^{2}+2\sum_{k\in
Q^{-}}\mu_{k}\,\|\omega\|^{2}.$

  * $(c)$

If $q=1$, i.e., if the system is cooperative, then

$\|\boldsymbol{w}\|^{2}=\mu_{1}\,\|\omega\|^{2}\text{ \ if \
}Q^{+}=\\{1\\}\text{ \ and \
}\|\boldsymbol{w}\|^{2}<10\,\mu_{1}\,\|\omega\|^{2}\text{ \ if \
}Q^{-}=\\{1\\}.$

  * $(d)$

If $q=2$, then

$\|\boldsymbol{w}\|^{2}<\begin{cases}\min\limits_{\begin{subarray}{c}h,k\in
Q^{+}\\\ h\neq k\end{subarray}}(\mu_{k}+6\mu_{h})\,\|\omega\|^{2}&\text{if \
}Q^{+}=\\{1,2\\},\\\ (\mu_{k}+12\mu_{h})\,\|\omega\|^{2}&\text{if \ }k\in
Q^{+}\text{ and }h\in Q^{-},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
12(\mu_{1}+\mu_{2})\,\|\omega\|^{2}&\text{if \ }Q^{-}=\\{1,2\\},\end{cases}$

where $\omega$ is the unique positive radial solution to the equation

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

To prove this result we introduce suitable symmetries that produce, by
construction, a change of sign on some of the components. Then, using
concentration compactness techniques, we carry out a careful analysis of the
effect of the symmetries on the behavior of symmetric minimizing sequences for
the system (1.1); see Theorem 2.7. This approach was taken in [8, Theorem 1.1]
to establish existence of solutions with a prescribed number of positive and
sign-changing components for the competitive system (1.1) with $q=\ell$.
However, the symmetries used there are too restrictive and do not allow energy
bounds to be obtained. In this paper we consider general symmetries, admitting
finite orbits. This makes it more difficult to show existence, due to the lack
of compactness of the variational functional. On the other hand, the
compactness condition given in Corollary 2.8 immediately gives energy
estimates. But showing that this condition is fulfilled requires, in turn,
precise knowledge about the asymptotic decay of the components of the system.
This information does not seem to be available so far. This is why the upper
energy bounds in Theorem 1.1 are given only for a system with at most two
blocks; see Remark 5.2.

Regarding our assumptions on the matrix $(\beta_{ij})$, condition $(B_{1})$ is
enough to guarantee the existence of a block-wise nontrivial solution (see
Theorem 2.12), while $(B_{2})$ and $(B_{3})$ ensure that this solution is
fully nontrivial. These last two conditions are weaker than the condition
stated in [7, Theorem 1.2]. Indeed, the latter forces $\beta_{ij}>0$ if
$i,j\in I_{h}$ for every $h=1,\ldots,q$. Note that if this is the case, the
graph defined in $(B_{2})$ is complete and, therefore, connected. So the
condition in [7, Theorem 1.2] implies $(B_{2})$ and $(B_{3})$ but not the
other way around. The constant $C_{q}$ that appears in $(B_{3})$ depends on
the symmetries we have chosen to obtain Theorem 1.1 and it is explicitly given
in its proof.

The following result is a special case, for $\ell=1$, of Theorem 1.1. It
improves [9, Theorem 1.1].

######  Corollary 1.2.

For $N\geq 4$ the equation

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

has a nonradial sign-changing solution $\widehat{\omega}$ such that
$\|\widehat{\omega}\|^{2}<10\,\|\omega\|^{2}$, where $\omega$ is the unique
positive radial solution to this equation.

The system (1.1) is the limit, as $\varepsilon\to 0$, of the system of
singularly perturbed elliptic equations

($\mathscr{S}_{\varepsilon,\mathbb{B}}$)
$\qquad\begin{cases}-\varepsilon^{2}\Delta
u_{i}+u_{i}=\displaystyle\sum_{j=1}^{\ell}\beta_{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\\\
u_{i}\in H^{1}_{0}(\mathbb{B}),\qquad i=1,\ldots,\ell,\end{cases}$

where $\varepsilon>0$, $\mathbb{B}$ is the unit ball in $\mathbb{R}^{N}$,
$N\geq 4$, $1<p<\frac{N}{N-2}$, and the matrix $(\beta_{ij})$ satisfies
assumptions $(B_{1})$ and $(B_{2})$.

In [12] Lin and Wei described the limit profile of positive least energy
solutions to this system. They showed that their components blow up and
decouple as $\varepsilon\to 0$. Each component is a rescaling of the positive
ground state of the stationary Schrödinger equation in $\mathbb{R}^{N}$ and
the concentration points of the components approach a sphere-packing position
as $\varepsilon\to 0$. The analysis of the effect of symmetries on the
behavior of symmetric minimizing sequences for the system (1.1) given by
Theorem 2.7 allows to exhibit solutions for
$(\mathscr{S}_{\varepsilon,\mathbb{B}})$ that blow up with two different types
of limit profiles: solutions whose blocks decouple as $\varepsilon\to 0$ and
solutions whose blocks remain coupled all the way up to their limit. The
following results extend and complete those obtained in [8] for competitive
systems.

######  Theorem 1.3 (Solutions with decoupling blocks).

Assume $(B_{1})$ and $(B_{2})$ and let $N\geq 5$. Given a partition
$\\{1,\ldots,q\\}=Q^{+}\cup Q^{-}$ with $Q^{+}\cap Q^{-}=\emptyset$ and a
sequence of positive numbers $(\varepsilon_{n})$ with $\varepsilon_{n}\to 0$,
there exists a fully nontrivial solution
${\boldsymbol{u}}_{n}=(\overline{u}_{1n},\ldots,\overline{u}_{qn})$ to the
system $(\mathscr{S}_{\varepsilon_{n},\mathbb{B}})$ such that the components
of $\overline{u}_{hn}$ are positive if $h\in Q^{+}$, are nonradial and change
sign if $h\in Q^{-}$, and have the following limit profile: For each
$h=1,\ldots,q$, there exist a sequence $(\zeta_{hn})$ in $\mathbb{B}$ and a
least energy fully nontrivial solution $\overline{v}_{h}$ to the cooperative
system

($\mathscr{S}_{h}$) $\qquad\begin{cases}-\Delta
v_{i}+v_{i}=\displaystyle\sum\limits_{j\in
I_{h}}\beta_{ij}|v_{j}|^{p}|v_{i}|^{p-2}v_{i},\\\ v_{i}\in
H^{1}(\mathbb{R}^{N}),\qquad i\in I_{h},\end{cases}$

such that, after passing to a subsequence,

  * $(a)$

the components of $\overline{v}_{h}$ are positive if $h\in Q^{+}$ and they are
nonradial and change sign if $h\in Q^{-}$,

  * $(b)$

$\displaystyle\lim_{n\to\infty}\varepsilon_{n}^{-1}\mathrm{dist}(\zeta_{hn},\partial\mathbb{B})=\infty$
and
$\displaystyle\lim_{n\to\infty}\varepsilon_{n}^{-1}|\zeta_{hn}-\zeta_{kn}|=\infty$
if $h\neq k$, $h,k=1,\ldots,q$,

  * $(c)$

$\displaystyle\lim_{n\to\infty}\left\|\widetilde{u}_{in}-v_{i}\right\|=0$ for
every $i=1,\ldots,\ell$, where
$\widetilde{u}_{in}(z):=u_{in}(\varepsilon_{n}z+\zeta_{hn}),$

  * $(d)$

$\|\overline{v}_{h}\|^{2}=\mu_{h}\,\|\omega\|^{2}$ if $h\in Q^{+}$, and
$\|\overline{v}_{h}\|^{2}<10\,\mu_{h}\,\|\omega\|^{2}$ if $h\in Q^{-}$.

######  Theorem 1.4 (Solutions with persisting coupling).

Assume $(B_{1})-(B_{3})$ and let $N=4$ or $N\geq 6$. Given a partition
$\\{1,\ldots,q\\}=Q^{+}\cup Q^{-}$ with $Q^{+}\cap Q^{-}=\emptyset$ and a
sequence of positive numbers $(\varepsilon_{n})$ with $\varepsilon_{n}\to 0$,
there exists a fully nontrivial solution
${\boldsymbol{u}}_{n}=(\overline{u}_{1n},\ldots,\overline{u}_{qn})$ to the
system $(\mathscr{S}_{\varepsilon_{n},\mathbb{B}})$ such that the components
of $\overline{u}_{hn}$ are positive if $h\in Q^{+}$ and they are nonradial and
change sign if $h\in Q^{-}$, and $\boldsymbol{u}_{n}$ has the following limit
profile: There exists a fully nontrivial solution
$\boldsymbol{w}=(\overline{w}_{1},\ldots,\overline{w}_{q})$ to the system
(1.1) such that, after passing to a subsequence,

  * $(a)$

the components of $\overline{w}_{h}$ are positive if $h\in Q^{+}$ and they are
nonradial and change sign if $h\in Q^{-}$,

  * $(b)$

$\displaystyle\lim_{n\to\infty}\left\|\widetilde{u}_{in}-w_{i}\right\|=0$ for
every $i=1,\ldots,\ell$, where
$\widetilde{u}_{in}(z):=u_{in}(\varepsilon_{n}z),$

  * $(c)$

If $q=2$, then $\boldsymbol{w}$ satisfies the energy estimates given in
_Theorem_ 1.1$(c)$.

This paper is organized as follows: In Section 2 we introduce the symmetric
variational setting, give an accurate description of the effect of symmetries
on the behavior of symmetric minimizing sequences for the system (1.1), and
derive a condition for the existence of least energy block-wise nontrivial
symmetric solutions. Section 3 is devoted to showing that such solutions are
fully nontrivial. In Section 4 we establish existence of positive and nodal
solutions for cooperative systems and estimate their energy. Similar results
for positive, nodal and seminodal solutions to a system with two blocks are
obtained in Section 5. In Section 6 we prove the results concerning the system
of singularly perturbed equations.

##  2 The profile of minimizing sequences

Throughout this section we assume $(B_{1})$. 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:

  * $(A_{\phi})$

If $\phi$ is surjective, then there exists $\zeta\in\mathbb{R}^{N}$ such that
$(\ker\phi)\zeta\neq G\zeta$,

where $\ker\phi:=\\{g\in G:\phi(g)=1\\}$. A function
$u:\mathbb{R}^{N}\to\mathbb{R}$ is called _$G$ -invariant_ if it is constant
on $Gx$ for every $x\in\mathbb{R}^{N}$ and will be called _$\phi$
-equivariant_ if

$u(gx)=\phi(g)u(x)\text{ \ for all \ }g\in G,\ x\in\mathbb{R}^{N}.$

Define

$H^{1}(\mathbb{R}^{N})^{\phi}:=\\{u\in H^{1}(\mathbb{R}^{N}):u\text{ is
}\phi\text{-equivariant}\\}.$

Assumption $(A_{\phi})$ guarantees that $H^{1}(\mathbb{R}^{N})^{\phi}$ has
infinite dimension, see [2]. If $\phi\equiv 1$ is the trivial homomorphism,
then $H^{1}(\mathbb{R}^{N})^{\phi}$ is the space of $G$-invariant functions in
$H^{1}(\mathbb{R}^{N})$. On the other hand, if $\phi$ is surjective, then
every nontrivial function $u\in H^{1}(\mathbb{R}^{N})^{\phi}$ is nonradial and
changes sign.

If $K$ is a closed subgroup of $G$ we write $\phi|K:K\to\mathbb{Z}_{2}$ for
the restriction of $\phi$ to $K$. Note that $\phi|K$ satisfies $(A_{\phi|K})$
if $\phi$ satisfies $(A_{\phi})$.

Recall the notation in assumption $(B_{1})$. Fix a closed subgroup $G$ of
$O(N)$ and, for each $h=1,\ldots,q$, fix a continuous homomorphism
$\phi_{h}:G\to\mathbb{Z}_{2}$ satisfying $(A_{\phi_{h}})$. Set
$\boldsymbol{\phi}:=(\phi_{1},\ldots,\phi_{q})$ and define
$\phi_{i}:=\phi_{h}$ for all $i\in I_{h}$.

If $Q=\\{h_{1},\ldots,h_{m}\\}$ is a nonempty ordered subset of
$\\{1,\ldots,q\\}$ and $K$ is a closed subgroup of $G$, we set
$I_{Q}:=\displaystyle\bigcup_{h\in Q}I_{h}$ and consider the system

($\mathscr{S}^{\boldsymbol{\phi}|K}_{Q}$) $\qquad\begin{cases}-\Delta
u_{i}+u_{i}=\displaystyle\sum_{j\in
I_{Q}}\beta_{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\\\ u_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{i}|K},\qquad i\in I_{Q}.\end{cases}$

Define

$\displaystyle\mathcal{H}^{\boldsymbol{\phi}|K}_{Q}$
$\displaystyle:=\prod_{i\in Q}H^{1}(\mathbb{R}^{N})^{\phi_{i}|K},$

where the factors are ordered following the natural order of the indices $i\in
Q$. So a point in $\mathcal{H}^{\boldsymbol{\phi}|K}_{Q}$ will be written as

$\boldsymbol{u}=(\overline{u}_{h_{1}},\ldots,\overline{u}_{h_{m}})\qquad\text{with
\ }\overline{u}_{h}=(u_{\ell_{h-1}+1},\ldots,u_{\ell_{h}}).$

Abusing notation, we denote their norms by

$\|\overline{u}_{h}\|:=\Big{(}\displaystyle\sum_{i\in
I_{h}}\|u_{i}\|^{2}\Big{)}^{1/2}\qquad\text{and}\qquad\|\boldsymbol{u}\|:=\Big{(}\displaystyle\sum_{h\in
Q}\|\overline{u}_{h}\|^{2}\Big{)}^{1/2},$

where $\|u_{i}\|$ is the standard norm of $u_{i}$ in $H^{1}(\mathbb{R}^{N})$.

Let
$\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}:\mathcal{H}^{\boldsymbol{\phi}|K}_{Q}\to\mathbb{R}$
be the functional given by

$\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}(\boldsymbol{u}):=\frac{1}{2}\displaystyle\sum_{i\in
I_{Q}}\|u_{i}\|^{2}-\frac{1}{2p}\displaystyle\sum_{i,j\in
I_{Q}}\beta_{ij}\displaystyle\int_{\mathbb{R}^{N}}|u_{i}|^{p}|u_{j}|^{p}.$

This functional is of class $\mathcal{C}^{1}$ and its critical points are the
solutions to the system ($\mathscr{S}^{\boldsymbol{\phi}|K}_{Q}$). The block-
wise nontrivial ones belong to the set

$\mathcal{N}^{\boldsymbol{\phi}|K}_{Q}:=\\{\boldsymbol{u}\in\mathcal{H}^{\boldsymbol{\phi}|K}_{Q}:\|\overline{u}_{h}\|\neq
0\text{ \ and \
}\partial_{\overline{u}_{h}}\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}(\boldsymbol{u})\overline{u}_{h}=0\text{
\ for all \ }h\in Q\\}.$

Note that, for each $h\in Q$

$\displaystyle\partial_{\overline{u}_{h}}\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}(\boldsymbol{u})\overline{u}_{h}=\|\overline{u}_{h}\|^{2}-\displaystyle\sum_{k\in
Q}\,\displaystyle\sum_{(i,j)\in I_{h}\times
I_{k}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{i}|^{p}|u_{j}|^{p},$

and that

$\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}(\boldsymbol{u})=\dfrac{p-1}{2p}\|\boldsymbol{u}\|^{2}\qquad\text{if
\ }\boldsymbol{u}\in\mathcal{N}^{\boldsymbol{\phi}|K}_{Q}.$

Set

$c^{\boldsymbol{\phi}|K}_{Q}:=\inf_{\boldsymbol{u}\in\mathcal{N}^{\boldsymbol{\phi}|K}_{Q}}\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}(\boldsymbol{u}).$

######  Notation 2.1.

If $Q=\\{h\\}$ we omit the curly brackets and write

$(\mathscr{S}^{\boldsymbol{\phi}|K}_{h}),\qquad\mathcal{H}^{\boldsymbol{\phi}|K}_{h},\qquad\mathcal{J}^{\boldsymbol{\phi}|K}_{h},\qquad\mathcal{N}^{\boldsymbol{\phi}|K}_{h},\qquad
c^{\boldsymbol{\phi}|K}_{h}.$

If $Q=\\{1,\ldots,q\\}$ we omit the subscript $Q$ and write

$(\mathscr{S}^{\boldsymbol{\phi}|K}),\qquad\mathcal{H}^{\boldsymbol{\phi}|K},\qquad\mathcal{J}^{\boldsymbol{\phi}|K},\qquad\mathcal{N}^{\boldsymbol{\phi}|K},\qquad
c^{\boldsymbol{\phi}|K}.$

If $K=G$ we write $\boldsymbol{\phi}$ instead of $\boldsymbol{\phi}|G$, if
$\phi|K\equiv 1$ we replace the superscript $\phi|K$ by $K$, and if $K$ is the
trivial group we omit the superscript.

For
$\boldsymbol{u}=(\overline{u}_{h_{1}},\ldots,\overline{u}_{h_{m}})\in\mathcal{H}^{\boldsymbol{\phi}|K}_{Q}$
and $\boldsymbol{s}:=(s_{1},\ldots,s_{m})\in(0,\infty)^{m}$ we set

$\boldsymbol{s}\boldsymbol{u}:=(s_{1}\overline{u}_{h_{1}},\ldots,s_{m}\overline{u}_{h_{m}}).$

Our next result provides helpful information about the Nehari-type set
$\mathcal{N}^{\boldsymbol{\phi}|K}_{Q}$.

######  Lemma 2.2.

The following statements hold true:

  * $(i)$

$\mathcal{N}^{\boldsymbol{\phi}|K}_{Q}\neq\emptyset$ and there exists
$d_{0}>0$ such that

$\min_{1\leq i\leq m}\|\overline{u}_{h_{i}}\|^{2}>d_{0}\quad\text{for every \
}\boldsymbol{u}=(\overline{u}_{h_{1}},\ldots,\overline{u}_{h_{m}})\in\mathcal{N}^{\boldsymbol{\phi}|K}_{Q}.$

Therefore $\mathcal{N}^{\boldsymbol{\phi}|K}_{Q}$ is a closed subset of
$\mathcal{H}^{\boldsymbol{\phi}|K}_{Q}$ and $c^{\boldsymbol{\phi}|K}_{Q}>0$.

  * $(ii)$

If the coordinates of $\boldsymbol{u}\in\mathcal{H}^{\boldsymbol{\phi}|K}_{Q}$
satisfy

(2.1) $\sum_{k\in Q}\,\sum_{(i,j)\in I_{h}\times
I_{k}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{i}|^{p}|u_{j}|^{p}>0$

for every $h\in Q$, then there exists a unique
$\boldsymbol{s}_{\boldsymbol{u}}\in(0,\infty)^{m},$ such that
$\boldsymbol{s}_{\boldsymbol{u}}\boldsymbol{u}\in\mathcal{N}^{\boldsymbol{\phi}|K}_{Q}$.
Furthermore,

$\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}(\boldsymbol{s}_{\boldsymbol{u}}\boldsymbol{u})=\max_{{\boldsymbol{s}}\in(0,\infty)^{m}}\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}(\boldsymbol{s}\boldsymbol{u}).$

  * $(iii)$

There exists $d_{\boldsymbol{\phi}}>0$ independent of $(\beta_{ij})$ such that

$c^{\boldsymbol{\phi}}\leq
d_{\boldsymbol{\phi}}\Big{(}\min_{h=1,\ldots,q}\max_{i\in
I_{h}}\beta_{ii}\Big{)}^{-\frac{1}{p-1}}.$

###### Proof.

These results are obtained by adapting the proofs of [7, Lemma 2.2] and [8,
Lemma 2.3] in the obvious way. The constant in statement $(iii)$ is defined as

$d_{\boldsymbol{\phi}}:=\dfrac{p-1}{2p}\inf_{(v_{1},\ldots,v_{q})\in\mathcal{U}^{\boldsymbol{\phi}}}\sum_{h=1}^{q}\|v_{h}\|^{2},$

where $\mathcal{U}^{\boldsymbol{\phi}}:=\\{(v_{1},\ldots,v_{q}):v_{h}\in
H^{1}(\mathbb{R}^{N})^{\boldsymbol{\phi}_{h}},\ v_{h}\neq 0,\
\|v_{h}\|^{2}=\displaystyle\int_{\mathbb{R}^{N}}|v_{h}|^{2p},\
v_{h}v_{k}=0\text{ if }h\neq k\\}$. ∎

It is shown in [7, Lemma 2.4] that any minimizer of
$\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}$ on
$\mathcal{N}^{\boldsymbol{\phi}|K}_{Q}$ is a critical point of
$\mathcal{J}^{\boldsymbol{\phi}|K}_{Q}$, i.e., a block-wise nontrivial
solution to ($\mathscr{S}^{\boldsymbol{\phi}|K}_{Q}$). We call it a least
energy block-wise nontrivial solution of
($\mathscr{S}^{\boldsymbol{\phi}|K}_{Q}$).

######  Remark 2.3.

_If $\phi_{h}\equiv 1$ for every $h$ in some subset $\overline{Q}$ of $Q$ and
$\boldsymbol{u}$ is a least energy block-wise nontrivial solution to
($\mathscr{S}^{\boldsymbol{\phi}|K}_{Q}$), then $|u_{i}|\in
H^{1}(\mathbb{R}^{N})^{K}$ for every $i\in I_{\overline{Q}}$ and replacing
$u_{i}$ by $|u_{i}|$ we obtain a least energy block-wise nontrivial solution
to ($\mathscr{S}^{\boldsymbol{\phi}|K}_{Q}$) whose $i$-th component is
positive for every $i\in I_{\overline{Q}}$._

Let $G_{\xi}:=\\{g\in G:g\xi=\xi\\}$ be the $G$-isotropy subgroup of the point
$\xi\in\mathbb{R}^{N}$. Recall that the $G$-orbit $G\xi$ of $\xi$ is
$G$-homeomorphic to the homogeneous space $G/G_{\xi}$. So they have the same
cardinality, i.e., $|G/G_{\xi}|=|G\xi|$.

######  Lemma 2.4.

If $Q=Q_{1}\cup Q_{2}$ with $Q_{1}\cap Q_{2}=\emptyset$ and
$\xi_{1},\xi_{2}\in\mathbb{R}^{N}$ are such that $G\xi_{1}\neq G\xi_{2}$, then

$c^{\boldsymbol{\phi}}_{Q}\leq|G\xi_{1}|\,c^{\boldsymbol{\phi}|G_{\xi_{1}}}_{Q_{1}}+|G\xi_{2}|\,c^{\boldsymbol{\phi}|G_{\xi_{2}}}_{Q_{2}}.$

###### Proof.

To simplify notation, assume $Q_{1}=\\{1,\ldots,m\\}$ and
$Q_{2}=\\{m+1,\ldots,q\\}$ with $1\leq m<q$. The argument is easily adapted to
the general case.

If either $|G\xi_{1}|=\infty$ or $|G\xi_{2}|=\infty$, the statement is
obvious. So let us assume that both $G$-orbits are finite. Let
$\boldsymbol{u}_{1}=(\overline{u}_{1},\ldots,\overline{u}_{m})\in\mathcal{N}^{\boldsymbol{\phi}|G_{\xi_{1}}}_{Q_{1}}$
and
$\boldsymbol{u}_{2}=(\overline{u}_{m+1},\ldots,\overline{u}_{q})\in\mathcal{N}^{\boldsymbol{\phi}|G_{\xi_{2}}}_{Q_{2}}$.
Fix $r>0$ such that

$|z_{1}-z_{2}|>2r\qquad\text{if \ }z_{1},z_{2}\in G\xi_{1}\cup G\xi_{2}\text{
\ and \ }z_{1}\neq z_{2}.$

Let $\chi\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N})$ be a radial cut-off
function such that $\chi(x)=1$ if $|x|\leq\frac{r}{2}$ and $\chi(x)=0$ if
$|x|\geq r$, and for each $n\in\mathbb{N}$ define
$\boldsymbol{u}_{1n}:=(\overline{u}_{1n},\ldots,\overline{u}_{mn})$ and
$\boldsymbol{u}_{2n}:=(\overline{u}_{m+1\,n},\ldots,\overline{u}_{qn})$ by
setting

$\overline{u}_{hn}(x):=\chi\left(\frac{x-n\xi_{\nu}}{n}\right)\overline{u}_{h}(x-n\xi_{\nu})\qquad\text{for
\ }h\in Q_{\nu},\quad\nu=1,2.$

Then, $\overline{u}_{hn}\in\mathcal{H}^{\boldsymbol{\phi}|G_{\xi_{\nu}}}_{h}$
and, as

$\sum_{k\in Q_{\nu}}\sum_{(i,j)\in I_{h}\times
I_{k}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{i}|^{p}|u_{j}|^{p}=\|\overline{u}_{h}\|^{2}\geq
d_{0}>0,$

Lemma 2.2 yields
$\boldsymbol{t}^{1}_{n}=(t_{1n},\ldots,t_{mn})\in(0,\infty)^{m}$ and
$\boldsymbol{t}^{2}_{n}=(t_{m+1\,n},\ldots,t_{qn})\in(0,\infty)^{q-m}$ such
that
${\boldsymbol{t}^{1}_{n}\boldsymbol{u}_{1n}\in\mathcal{N}^{\boldsymbol{\phi}|G_{\xi_{1}}}_{Q_{1}}}$
and
$\boldsymbol{t}^{2}_{n}\boldsymbol{u}_{2n}\in\mathcal{N}^{\boldsymbol{\phi}|G_{\xi_{2}}}_{Q_{2}}$
for every $n$ sufficiently large, and $t_{in}\to 1$ as $n\to\infty$. Define

$\widetilde{u}_{in}(x):=\displaystyle\sum_{[g]\in
G/G_{\xi_{\nu}}}\phi_{i}(g)t_{hn}u_{in}(g^{-1}x)\qquad\text{if \ }i\in
I_{h},\text{ \ and \ }h\in Q_{\nu}.$

Then, $\widetilde{u}_{in}$ is well defined and
$\widetilde{u}_{in}(gx)=\phi_{i}(g)\widetilde{u}_{in}(x)$ for every $g\in G$,
$x\in\mathbb{R}^{N}$ and $i=1,\ldots,\ell$. Since
${\text{supp}(\overline{u}_{hn}\circ g^{-1})\subset B_{nr}(gn\xi_{\nu})}$ if
$h\in Q_{\nu}$ and the balls $B_{nr}(gn\xi_{\nu})$ with $[g]\in
G/G_{\xi_{\nu}}$ and $\nu=1,2$ are pairwise disjoint, we have that
$\widetilde{\boldsymbol{u}}_{n}:=(\widetilde{u}_{1n},\ldots,\widetilde{u}_{\ell
n})\in\mathcal{N}^{\boldsymbol{\phi}}$ and

$\displaystyle\sum_{i\in
I_{h}}\|\widetilde{u}_{in}\|^{2}=|G\xi_{\nu}|\,|t_{hn}|^{2}\|\overline{u}_{hn}\|^{2}\to|G\xi_{\nu}|\,\|\overline{u}_{h}\|^{2}\qquad\text{if
\ }h\in Q_{\nu},\ \nu=1,2.$

Hence,

$\displaystyle c^{\boldsymbol{\phi}}$
$\displaystyle\leq\displaystyle\lim_{n\to\infty}\mathcal{J}^{\boldsymbol{\phi}}(\widetilde{\boldsymbol{u}}_{n})=\displaystyle\lim_{n\to\infty}\frac{p-2}{2p}\displaystyle\sum_{h=1}^{q}\displaystyle\sum_{i\in
I_{h}}\|\widetilde{u}_{in}\|^{2}$
$\displaystyle=|G\xi_{1}|\frac{p-2}{2p}\displaystyle\sum_{h=1}^{m}\|\overline{u}_{h}\|^{2}+|G\xi_{2}|\frac{p-2}{2p}\displaystyle\sum_{h=m+1}^{q}\|\overline{u}_{h}\|^{2}$
$\displaystyle=|G\xi_{1}|\mathcal{J}^{\boldsymbol{\phi}|G_{\xi_{1}}}_{Q_{1}}(\boldsymbol{u}_{1})+|G\xi_{2}|\mathcal{J}^{\boldsymbol{\phi}|G_{\xi_{2}}}_{Q_{2}}(\boldsymbol{u}_{2}).$

As $\boldsymbol{u}_{1}\in\mathcal{N}^{\boldsymbol{\phi}|G_{\xi_{1}}}_{Q_{1}}$
and $\boldsymbol{u}_{2}\in\mathcal{N}^{\boldsymbol{\phi}|G_{\xi_{2}}}_{Q_{2}}$
were arbitrarily chosen, the proof is complete. ∎

Consider the _$G$ -fixed-point space_

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

######  Proposition 2.5.

The following statements hold true:

  * $(i)$

$c^{\boldsymbol{\phi}}\geq\displaystyle\sum_{h=1}^{q}c^{\boldsymbol{\phi}}_{h}$.

  * $(ii)$

If $\mathrm{Fix}(G)\neq 0$, then
$c^{\boldsymbol{\phi}}=\displaystyle\sum_{h=1}^{q}c^{\boldsymbol{\phi}}_{h}$.

  * $(iii)$

If $q\geq 2$ and
$c^{\boldsymbol{\phi}}=\displaystyle\sum_{h=1}^{q}c_{h}^{\boldsymbol{\phi}}$,
then $c^{\boldsymbol{\phi}}$ is not attained.

###### Proof.

$(i):$ Let
$\boldsymbol{u}=(\overline{u}_{1},\ldots,\overline{u}_{q})\in\mathcal{N}^{\boldsymbol{\phi}}$.
As $\beta_{ij}<0$ for $(i,j)\in\mathcal{K}_{h}$, we have

$0<\|\overline{u}_{h}\|^{2}\leq\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\int_{\Omega}\beta_{ij}|u_{i}|^{p}|u_{j}|^{p}\qquad\text{for
every \ }h=1,\ldots,q.$

Hence, there exist $(s_{1},\ldots,s_{q})\in(0,\infty)^{q}$ such that
$s_{h}\overline{u}_{h}\in\mathcal{N}^{\boldsymbol{\phi}}_{h}$. Setting
$\boldsymbol{s}:=(s_{1},\ldots,s_{q})$ and applying Lemma 2.2 we obtain

$\displaystyle\sum_{h=1}^{q}c^{\boldsymbol{\phi}}_{h}\leq\displaystyle\sum_{h=1}^{q}\mathcal{J}^{\boldsymbol{\phi}}_{h}(s_{h}\overline{u}_{h})\leq\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{s}\boldsymbol{u})\leq\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u}).$

Hence, $c^{\boldsymbol{\phi}}\geq
c^{\boldsymbol{\phi}}_{1}+\cdots+c^{\boldsymbol{\phi}}_{q}$, as claimed.

$(ii):$ As $\mathrm{Fix}(G)\neq 0$ we may choose $q$ different points
$\xi_{1},\ldots,\xi_{q}\in\mathrm{Fix}(G)$. Setting $Q_{h}:=\\{h\\}$ and
iterating Lemma 2.4 we obtain

$c^{\boldsymbol{\phi}}\leq\displaystyle\sum_{h=1}^{q}c_{h}^{\boldsymbol{\phi}}.$

The equality follows from $(i)$.

$(iii):$ We argue by contradiction. Assume that
$\boldsymbol{u}=(\overline{u}_{1},\ldots,\overline{u}_{q})\in\mathcal{N}^{\boldsymbol{\phi}}$
and $\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u})=c^{\boldsymbol{\phi}}$.
There are two possibilities. If for every $h$ there exists $k\neq h$ such that

$\displaystyle\sum_{(i,j)\in I_{h}\times
I_{k}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{i}|^{p}|u_{j}|^{p}\neq
0,$

then
$\|\overline{u}_{h}\|^{2}<\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{i}|^{p}|u_{j}|^{p}$
and, hence, there exists $s_{h}\in(0,1)$ such that
$s_{h}\overline{u}_{h}\in\mathcal{N}^{\boldsymbol{\phi}}_{h}$ and

$c^{\boldsymbol{\phi}}_{h}\leq\mathcal{J}^{\boldsymbol{\phi}}_{h}(s_{h}\overline{u}_{h})=\dfrac{p-1}{2p}\|s_{h}\overline{u}_{h}\|^{2}<\dfrac{p-1}{2p}\|\overline{u}_{h}\|^{2}\qquad\text{for
every \ }h=1,\ldots,q.$

It follows that

$c^{\boldsymbol{\phi}}=\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u})=\dfrac{p-1}{2p}\sum_{h=1}^{q}\|\overline{u}_{h}\|^{2}>\displaystyle\displaystyle\sum_{h=1}^{q}c^{\boldsymbol{\phi}}_{h},$

contradicting our assumption. On the other hand, if there exists $h$ such that

(2.2) $\displaystyle\sum_{(s,r)\in I_{h}\times
I_{k}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{sr}|u_{s}|^{p}|u_{r}|^{p}=0\qquad\text{for
every \ }k\neq h,$

then

$\|\overline{u}_{h}\|^{2}=\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{i}|^{p}|u_{j}|^{p}\qquad\text{and}\qquad\mathcal{J}^{\boldsymbol{\phi}}_{h}(\overline{u}_{h})=c^{\boldsymbol{\phi}}_{h}.$

Hence, $\overline{u}_{h}$ is a solution to the system
$(\mathscr{S}_{h}^{\boldsymbol{\phi}})$ with $\|\overline{u}_{h}\|\neq 0$. So
$u_{i}\neq 0$ for some $i\in I_{h}$ and it solves the equation

$-\Delta u_{i}+u_{i}=\displaystyle\sum_{j\in
I_{h}}\beta_{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\qquad u_{i}\in
H^{1}(\mathbb{R}^{N}).$

As $q\geq 2$ there exists $k\neq h$ and $j\in I_{k}$ such that $u_{j}\neq 0$,
because $\|\overline{u}_{k}\|\neq 0$. Since all $\beta_{sr}$ in the sum (2.2)
are strictly negative, it follows that $|u_{i}||u_{j}|=0$ a.e. in
$\mathbb{R}^{N}$. Hence $u_{i}=0$ in some subset of positive measure of
$\mathbb{R}^{N}$, contradicting the unique continuation principle. ∎

The following lemma will be used in the proof of Theorem 2.7.

######  Lemma 2.6.

For any given sequence $(y_{n})$ in $\mathbb{R}^{N}$, there exist a sequence
$(\zeta_{n})$ in $\mathbb{R}^{N}$ and a closed subgroup $K$ of $G$ such that,
up to a subsequence, the following statements hold true:

  * $(a)$

$\mathrm{dist}(Gy_{n},\zeta_{n})\leq C$ for all $n\in\mathbb{N}$ and some
positive constant $C$.

  * $(b)$

$G_{\zeta_{n}}=K$ for all $n\in\mathbb{N}$.

  * $(c)$

If $|G/K|<\infty$ then
$\lim\limits_{n\to\infty}|g\zeta_{n}-\overline{g}\zeta_{n}|=\infty$ for any
pair $g,\overline{g}\in G$ with $\overline{g}g^{-1}\notin K$.

  * $(d)$

If $|G/K|=\infty$ then, for each $n\in\mathbb{N}$, there exists $g_{n}\in G$
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 follows from [6, Lemma 3.2]. ∎

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

######  Theorem 2.7.

Let $\boldsymbol{u}_{n}=(u_{1n},\ldots,u_{\ell
n})\in\mathcal{N}^{\boldsymbol{\phi}}$ be such that
$\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u}_{n})\to
c^{\boldsymbol{\phi}}$. Then, after passing to a subsequence, there exist
$Q\subset\\{1,\ldots,q\\}$, a least energy block-wise nontrivial solution
$\boldsymbol{w}$ to the subsystem

($\mathscr{S}_{Q}^{\boldsymbol{\phi}}$) $\qquad\begin{cases}-\Delta
w_{i}+w_{i}=\displaystyle\sum_{j\in
I_{Q}}\beta_{ij}|w_{j}|^{p}|w_{i}|^{p-2}w_{i},\\\ w_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{i}},\qquad i\in I_{Q},\end{cases}$

whenever $Q\neq\emptyset$, and, for each $k\notin Q$, a sequence $(\xi_{kn})$
in $\mathbb{R}^{N}$, a closed subgroup $G_{k}$ of $G$, and a least energy
nontrivial solution $\overline{v}_{k}$ to the cooperative subsystem

($\mathscr{S}_{k}^{\boldsymbol{\phi}|G_{k}}$) $\qquad\begin{cases}-\Delta
v_{i}+v_{i}=\displaystyle\sum_{j\in
I_{k}}\beta_{ij}|v_{j}|^{p}|v_{i}|^{p-2}v_{i},\\\ v_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{i}|G_{k}},\qquad i\in I_{k},\end{cases}$

with the following properties:

  * $(i)$

If $i\in I_{Q}$, then $\displaystyle\lim_{n\to\infty}\|u_{in}-w_{i}\|=0$.

  * $(ii)$

If $k\notin Q$, then $G_{\xi_{kn}}=G_{k}$ for all $n\in\mathbb{N}$,
$|G/G_{k}|<\infty$,
$\displaystyle\lim\limits_{n\to\infty}|g\xi_{kn}-\overline{g}\xi_{kn}|=\infty$
for any $g,\overline{g}\in G$ with $\overline{g}g^{-1}\notin G_{k}$, and
$\displaystyle\lim_{n\to\infty}\mathrm{dist}(G\xi_{kn},G\xi_{k^{\prime}n})=\infty$
whenever $k^{\prime}\neq k$, $k^{\prime}\notin Q$.

  * $(iii)$

If $Q\neq\emptyset$ and $k\notin Q$, then
$\displaystyle\lim_{n\to\infty}|\xi_{kn}|=\infty$.

  * $(iv)$

If $i\in I_{k}$ and $k\notin Q$, then

$\displaystyle\lim_{n\to\infty}\left\|u_{in}-\displaystyle\sum_{[g]\in
G/G_{k}}\phi_{i}(g)(v_{i}\circ g^{-1})(\ \cdot\ -g\xi_{kn})\right\|=0.$

  * $(v)$

$c^{\boldsymbol{\phi}}=c_{Q}^{\boldsymbol{\phi}}+\displaystyle\sum_{k\notin
Q}|G/G_{k}|\,c^{\boldsymbol{\phi}|G_{k}}_{k}$, where
$c_{Q}^{\boldsymbol{\phi}}:=0$ if $Q=\emptyset$.

Moreover, if $u_{in}\geq 0$ for all $n\in\mathbb{N}$, then $w_{i}\geq 0$ if
$i\in I_{Q}$ and $v_{i}\geq 0$ if $i\notin I_{Q}$.

###### Proof.

We split the proof into three steps.

Step 1.  _The choice of the concentration points._

Since $\boldsymbol{u}_{n}\in\mathcal{N}^{\boldsymbol{\phi}}$ and
$\beta_{ij}<0$ for $(i,j)\in\mathcal{K}_{h}$ by assumption $(B_{1})$, Lemma
2.2 yields a constant $d_{0}>0$ such that

$\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}>d_{0}\qquad\text{for
all \ }n\in\mathbb{N}\text{ \ and \ }h=1,\ldots,q.$

Hence, there exist $i\in I_{h}$ and $c_{0}>0$ such that

$\displaystyle\int_{\mathbb{R}^{N}}\beta_{ii}|u_{in}|^{2p}>c_{0}\qquad\text{for
all \ }n\in\mathbb{N}.$

Invoking [7, Lemma 2.4] and Ekeland’s variational principle [26, Theorem 8.5]
we may assume without loss of generality that
$(\mathcal{J}^{\boldsymbol{\phi}})^{\prime}(\boldsymbol{u}_{n})\to 0$ in
$(\mathcal{H}^{\boldsymbol{\phi}})^{\prime}$. By Lions’ lemma [26, Lemma 1.21]
there exists $\theta>0$ such that, after passing to a subsequence,

$\sup_{y\in\mathbb{R}^{N}}\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\int_{B_{1}(y)}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}\geq\sup_{y\in\mathbb{R}^{N}}\int_{B_{1}(y)}\beta_{ii}|u_{in}|^{2p}>2\theta,$

and we may choose $y_{hn}\in\mathbb{R}^{N}$ satisfying

$\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\int_{B_{1}(y_{hn})}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}>\theta\qquad\text{for
all \ }n\in\mathbb{N}\text{ \ and \ }h=1,\ldots,q.$

Applying Lemma 2.6 to $(y_{hn})$ we obtain a sequence $(\zeta_{hn})$, a closed
subgroup $K_{h}=G_{\zeta_{hn}}$ and a constant $C>0$ such that
$\mathrm{dist}(Gy_{hn},\zeta_{hn})\leq C$ for all $n\in\mathbb{N}$ and
$h=1,\ldots,q$. Since $|u_{in}|$ is $G$-invariant, taking $g_{hn}\in G$ such
that $|g_{hn}y_{hn}-\zeta_{hn}|=\mathrm{dist}(Gy_{hn},\zeta_{hn})$ we get

(2.3)
$\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\int_{B_{C+1}(\zeta_{hn})}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}\geq\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\int_{B_{1}(g_{hn}y_{hn})}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}>\theta,$

for all $n\in\mathbb{N}$ and $h=1,\ldots,q$. We claim that $|G/K_{h}|<\infty$.
Otherwise, by Lemma 2.6$(d)$, for each $M\in\mathbb{N}$ we would be able to
choose $g_{h1},\ldots,g_{hM}\in G$ such that

$|g_{hm}\zeta_{hn}-g_{hm^{\prime}}\zeta_{hn}|\geq 2(C+1)\quad\text{for all
pairs \ }m\neq m^{\prime},\ \ m,m^{\prime}=1,\ldots,M,$

and large $n\in\mathbb{N}$. As a consequence, $B_{C+1}(g_{hm}\zeta_{hn})\cap
B_{C+1}(g_{hm^{\prime}}\zeta_{hn})=\emptyset$. Thus,

$\displaystyle\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}$
$\displaystyle\geq\displaystyle\sum_{m=1}^{M}\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\int_{B_{C+1}(g_{h\overline{m}}\zeta_{hn})}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}$
$\displaystyle=M\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\int_{B_{C+1}(\zeta_{hn})}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}>M\theta,$

which is a contradiction, because $M$ is arbitrary and, as
$(\boldsymbol{u}_{n})$ is bounded in $\mathcal{H}^{\boldsymbol{\phi}}$,
$(u_{in})$ is bounded in $L^{2p}(\mathbb{R}^{N})$. This shows that
$|G/K_{h}|<\infty$. Therefore, by Lemma 2.6$(c)$,

(2.4)
$\lim_{n\to\infty}|g\zeta_{hn}-\overline{g}\zeta_{hn}|=\infty\quad\text{for
all \ }g,\overline{g}\in G\text{ \ with \ }\overline{g}g^{-1}\notin K_{h},$

and every $h=1,\ldots,q$.

Let $Q_{1},\ldots,Q_{t}$ be nonempty and pairwise disjoint subsets of
$\\{1,\ldots,q\\}$ such that $Q_{1}\cup\cdots\cup Q_{t}=\\{1,\ldots,q\\}$ and,
after passing to a subsequence,

$\displaystyle(\mathrm{dist}(G\zeta_{hn},G\zeta_{kn}))\text{ is bounded}$
$\displaystyle\text{if \ }h,k\in Q_{r}\text{ for some }r,$
$\displaystyle\mathrm{dist}(G\zeta_{hn},G\zeta_{kn})\to\infty\ \ \text{as}\ \
n\to\infty$ $\displaystyle\text{if \ }h\in Q_{r}\text{ and }k\in
Q_{r^{\prime}}\text{ with }r\neq r^{\prime}.$

Choose $h_{r}\in Q_{r}$ and set $\xi_{rn}:=\zeta_{h_{r}n}$. Fix
$R>\mathrm{dist}(\xi_{rn},G\zeta_{hn})+C+1$ for every $n\in\mathbb{N}$ and
$h\in Q_{r}$. Then

$\displaystyle\mathrm{dist}(G\xi_{rn},G\xi_{r^{\prime}n})\to\infty\text{ \ as
\ }n\to\infty\qquad\text{if \ }r\neq r^{\prime},$

and from (2.3) we obtain

(2.5)
$\sum_{(i,j)\in\mathcal{I}_{h}}\int_{B_{R}(\xi_{rn})}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}>\theta>0\quad\text{for
all \ }n\in\mathbb{N},\quad h\in Q_{r}.$

Finally, set

$\Gamma_{r}:=G_{\xi_{rn}}=K_{h_{r}}\qquad r=1,\ldots,t.$

Note that if $\mathrm{Fix}(\Gamma_{r})=\\{0\\}$, then $\xi_{rn}=0$ for all
$n\in\mathbb{N}$ and, hence, $\mathrm{Fix}(\Gamma_{r^{\prime}})\neq\\{0\\}$
for each $r^{\prime}\neq r.$

Step 2.  _Finding a block-wise nontrivial solution to a subsystem._

Define

$w_{in}(x):=u_{in}(x+\xi_{rn})\quad\text{if \ }i\in I_{Q_{r}}.$

Observe that $w_{in}\in H^{1}(\mathbb{R}^{N})^{\phi_{i}|\Gamma_{r}}$ if $i\in
I_{Q_{r}}$. Since the sequence $(w_{in})$ is bounded in
$H^{1}(\mathbb{R}^{N})$, passing to a subsequence, we have that
$w_{in}\rightharpoonup w_{i}$ weakly in
$H^{1}(\mathbb{R}^{N})^{\phi_{i}|\Gamma_{r}}$, $w_{in}\to w_{i}$ in
$L^{2p}_{\mathrm{loc}}(\mathbb{R}^{N})$ and $w_{in}\to w_{i}$ a.e. in
$\mathbb{R}^{N}$. Hence, $w_{in}\geq 0$ if $u_{in}\geq 0$ for all
$n\in\mathbb{N}$. As, by (2.5),

$\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\int_{B_{R}(0)}\beta_{ij}|w_{in}|^{p}|w_{jn}|^{p}=\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\int_{B_{R}(\xi_{rn})}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}>\theta>0\quad\text{for
all \ }n\in\mathbb{N},\quad h\in Q_{r},$

we deduce that $w_{i}\neq 0$ for some $i\in I_{h}$. So the vector
$\overline{w}_{h}=(w_{\ell_{h-1}+1},\ldots,w_{\ell_{h}})$ is nonzero.

Fix $i\in I_{Q_{r}}$ and let $g_{1},\ldots,g_{m}\in G$ represent all the
different cosets in $G/\Gamma_{r}$. By (2.4),
$|g_{j}\xi_{rn}-g_{s}\xi_{rn}|\to\infty$ if $j\neq s$. Therefore,

$\phi_{i}(g_{j})\,(w_{in}\circ
g_{j}^{-1})-\sum_{s=j+1}^{m}\phi_{i}(g_{s})\,(w_{i}\circ
g_{s}^{-1})(\,\cdot\,-g_{s}\xi_{rn}+g_{j}\xi_{rn})\rightharpoonup\phi_{i}(g_{j})\,(w_{i}\circ
g_{j})$

weakly in $H^{1}(\mathbb{R}^{N})$. It follows that

$\displaystyle\Big{\|}\phi_{i}(g_{j})\,(w_{in}\circ
g_{j}^{-1})-\sum_{s=j+1}^{m}\phi_{i}(g_{s})\,(w_{i}\circ
g_{s}^{-1})(\,\cdot\,-g_{s}\xi_{rn}+g_{j}\xi_{rn})\Big{\|}^{2}$
$\displaystyle=\Big{\|}\phi_{i}(g_{j})\,(w_{in}\circ
g_{j}^{-1})-\sum_{s=j}^{m}\phi_{i}(g_{s})\,(w_{i}\circ
g_{s}^{-1})(\,\cdot\,-g_{s}\xi_{rn}+g_{j}\xi_{rn})\Big{\|}^{2}+\Big{\|}\phi_{i}(g_{j})\,(w_{i}\circ
g_{j})\Big{\|}^{2}+o_{n}(1).$

Since $u_{in}$ is $\phi_{i}$-equivariant, the change of variable
$y=z-g_{j}\xi_{rn}$ yields

$\displaystyle\Big{\|}u_{in}-\sum_{s=j+1}^{m}\phi_{i}(g_{s})\,(w_{i}\circ
g_{s}^{-1})(\,\cdot\,-g_{s}\xi_{rn})\Big{\|}^{2}=\Big{\|}u_{in}-\sum_{s=j}^{m}\phi_{i}(g_{s})\,(w_{i}\circ
g_{s}^{-1})(\,\cdot\,-g_{s}\xi_{rn})\Big{\|}^{2}+\|w_{i}\|^{2}+o_{n}(1),$

and iterating this identity we obtain

(2.6)
$\|u_{in}\|^{2}-\Big{\|}u_{in}-\sum_{s=1}^{m}\phi_{i}(g_{s})\,(w_{i}\circ
g_{s}^{-1})(\,\cdot\,-g_{s}\xi_{rn})\Big{\|}^{2}+m\|w_{i}\|^{2}+o_{n}(1)\quad\text{for
each \ }i\in I_{Q_{r}},$

with $m=|G/\Gamma_{r}|$.

Next, we fix $r\in\\{1,\ldots,t\\}$ and define

$u^{r}_{in}(x):=u_{in}(x+\xi_{rn}).$

Then, $u^{r}_{in}\in H^{1}(\mathbb{R}^{N})^{\phi_{i}|\Gamma_{r}}$ and, after
passing to a subsequence, $u^{r}_{in}\rightharpoonup u^{r}_{i}$ weakly in
$H^{1}(\mathbb{R}^{N})$. Note that $u^{r}_{i}=w_{i}$ if $i\in I_{Q_{r}}$.

For any
$\varphi\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N})^{\phi_{i}|\Gamma_{r}}$
define $\varphi_{n}(y):=\varphi(y-\xi_{rn})$ and set

$\widehat{\varphi}_{n}(y):=\sum_{[g]\in
G/\Gamma_{r}}\phi_{i}(g)\varphi_{n}(gy).$

Then $\widehat{\varphi}_{n}\in H^{1}(\mathbb{R}^{N})^{\phi_{i}}$. As
$\boldsymbol{u}_{n}\in\mathcal{H}^{\boldsymbol{\phi}}$ and
$(\mathcal{J}^{\boldsymbol{\phi}})^{\prime}(\boldsymbol{u}_{n})\to 0$ in
$(\mathcal{H}^{\boldsymbol{\phi}})^{\prime}$, we deduce that

$\displaystyle 0$
$\displaystyle=\lim_{n\to\infty}\partial_{i}\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u}_{n})\widehat{\varphi}_{n}=\lim_{n\to\infty}\sum_{[g]\in
G/\Gamma_{r}}\partial_{i}\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u}_{n})[\phi_{i}(g)(\varphi_{n}\circ
g)]$
$\displaystyle=\lim_{n\to\infty}|G/\Gamma_{r}|\partial_{i}\mathcal{J}^{{\boldsymbol{\phi}}|\Gamma_{r}}(\boldsymbol{u}_{n}^{r})\varphi=|G/\Gamma_{r}|\partial_{i}\mathcal{J}^{{\boldsymbol{\phi}}|\Gamma_{r}}(\boldsymbol{u}^{r}){\varphi}\quad\text{for
all \ }i=1,\ldots,\ell,$

where $\boldsymbol{u}^{r}=(\overline{u}_{1}^{r},\ldots,\overline{u}_{q}^{r})$
with $\overline{u}_{h}^{r}=(u^{r}_{\ell_{h-1}+1},\ldots,u^{r}_{\ell_{h}})$. So
$\overline{u}_{h}^{r}=\overline{w}_{h}\neq 0$ if $h\in{Q_{r}}$. The above
identity shows that $\boldsymbol{u}^{r}$ is a critical point of
$\mathcal{J}^{\boldsymbol{\phi}}:\mathcal{H}^{\boldsymbol{\phi}}\to\mathbb{R}$,
so its nontrivial block-wise components $\overline{u}_{h}^{r}$ provide a
nontrivial block-wise solution $\boldsymbol{w}^{r}$ to the system

($\mathscr{S}^{\boldsymbol{\phi}|\Gamma_{r}}_{\widehat{Q}_{r}}$)
$\begin{cases}-\Delta u_{i}+u_{i}=\displaystyle\sum_{j\in
I_{\widehat{Q}_{r}}}\beta_{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\\\ u_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{i}|\Gamma_{r}},\qquad i\in
I_{\widehat{Q}_{r}},\end{cases}$

where $\widehat{Q}_{r}:=\\{h:1\leq h\leq q,\ \overline{u}_{h}^{r}\neq
0\\}\supset Q_{r}$.

Step 3.  _The conclusion._

We distinguish two cases.

  * $(I):$

Assume $\mathrm{Fix}(\Gamma_{r})\neq\\{0\\}$ for every $r=1,\ldots,t$.

Since $\boldsymbol{w}^{r}$ is a block-wise nontrivial solution to
$(\mathscr{S}^{\boldsymbol{\phi}|\Gamma_{r}}_{\widehat{Q}_{r}})$, multiplying
the $i$-th equation by $w_{i}$ for $i\in I_{h}$ and $h\in Q_{r}$, and
integrating we get

(2.7) $\|w_{i}\|^{2}=\sum_{j\in
I_{\widehat{Q}_{r}}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|w_{i}|^{p}|u^{h}_{j}|^{p}\leq\displaystyle\sum_{j\in
I_{h}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|w_{i}|^{p}|w_{j}|^{p}.$

So there exists $t_{h}\in(0,1]$ such that
$\|t_{h}\overline{w}_{h}\|^{2}=\displaystyle\sum_{i,j\in
I_{h}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|t_{h}w_{i}|^{p}|t_{h}w_{j}|^{p}$.
Then, $t_{h}\overline{w}_{h}\in\mathcal{N}_{h}^{\boldsymbol{\phi}|\Gamma_{r}}$
and

(2.8)
$c^{\boldsymbol{\phi}|\Gamma_{r}}_{h}\leq\dfrac{p-1}{2p}\|t_{h}\overline{w}_{h}\|^{2}\leq\dfrac{p-1}{2p}\|\overline{w}_{h}\|^{2}.$

Since $\mathrm{Fix}(\Gamma_{r})\neq\\{0\\}$ for every $r$, Proposition
2.5$(ii)$ yields
$c_{Q_{r}}^{\boldsymbol{\phi}|\Gamma_{r}}=\displaystyle\sum_{h\in
Q_{r}}c_{h}^{\boldsymbol{\phi}|\Gamma_{r}}$. Applying Lemma 2.4 recursively
and using (2.6) and (2.8) we obtain

(2.9) $\displaystyle\sum_{r=1}^{t}\sum_{h\in
Q_{r}}|G/\Gamma_{r}|c_{h}^{\boldsymbol{\phi}|\Gamma_{r}}=\sum_{r=1}^{t}|G/\Gamma_{r}|\,c^{{\boldsymbol{\phi}}|\Gamma_{r}}_{Q_{r}}\geq
c^{\boldsymbol{\phi}}=\lim_{n\to\infty}\dfrac{p-1}{2p}\sum_{r=1}^{t}\sum_{h\in
Q_{r}}\|\overline{u}_{hn}\|^{2}$
$\displaystyle\qquad\geq\sum_{r=1}^{t}\sum_{h\in
Q_{r}}\dfrac{p-1}{2p}|G/\Gamma_{r}|\,\|\overline{w}_{h}\|^{2}\geq\sum_{r=1}^{t}\sum_{h\in
Q_{r}}\dfrac{p-1}{2p}|G/\Gamma_{r}|\,\|t_{h}\overline{w}_{h}\|^{2}\geq\sum_{r=1}^{t}\sum_{h\in
Q_{r}}|G/\Gamma_{r}|c_{h}^{\boldsymbol{\phi}|\Gamma_{r}}.$

Therefore, $t_{h}=1$ and from (2.7) we get that

$\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|w_{i}|^{p}|u^{h}_{j}|^{p}=0\qquad\text{for
every \ }j\in I_{\widehat{Q}_{r}}\smallsetminus I_{h}.$

Choosing $i\in I_{h}$ such that $w_{i}\neq 0$, as $\boldsymbol{w}^{r}$ solves
the system, we have that $w_{i}(x)\neq 0$ a.e. in $\mathbb{R}^{N}$. As a
consequence $u^{h}_{j}=0$ for every $j\in I_{\widehat{Q}_{r}}\smallsetminus
I_{h}$. This shows that $\widehat{Q}_{r}=Q_{r}$, and from (2.8) and (2.9) we
get

$c_{Q_{r}}^{\boldsymbol{\phi}|\Gamma_{r}}=\sum_{h\in
Q_{r}}c^{\boldsymbol{\phi}|\Gamma_{r}}_{h}=\dfrac{p-1}{2p}\sum_{h\in
Q_{r}}\|\overline{w}_{h}\|^{2}=\dfrac{p-1}{2p}\|\boldsymbol{w}^{r}\|^{2}.$

Therefore $c_{Q_{r}}^{\boldsymbol{\phi}|\Gamma_{r}}$ is attained. As
$\mathrm{Fix}(\Gamma_{r})\neq\\{0\\}$, Proposition 2.5 implies that each
$Q_{r}$ must consist of a single element. We relabel these sets as
$Q_{k}=\\{k\\}$ and the corresponding points $\xi_{rn}$ as $\xi_{kn}$ for each
$k\in\\{1,\ldots,q\\}$. From the way the sets $Q_{r}$ were chosen, we get
$\mathrm{dist}(G\xi_{hn},G\xi_{kn})\to\infty$ if $h\neq k$. Setting
$Q:=\emptyset$, $G_{k}:=\Gamma_{k}=G_{\xi_{kn}}$ and
$\overline{v}_{k}:=\overline{w}_{k}$, we have that $\overline{v}_{k}$ is a
least energy nontrivial solution to the cooperative system
($\mathscr{S}_{k}^{\boldsymbol{\phi}|G_{k}}$) and we derive from (2.4) and
(2.9) that these data satisfy $(ii)$ and $(v)$. Finally, from (2.6) and (2.9)
we obtain

$\lim_{n\to\infty}\Big{\|}u_{in}-\displaystyle\sum_{g\in
G/G_{h}}\phi_{i}(g)(w_{i}\circ g^{-1})(\ \cdot\
-g\xi_{hn})\Big{\|}^{2}=0\quad\text{if \ }i\in I_{h},$

as claimed in $(iv)$.

  * $(II):$

Assume $\mathrm{Fix}(\Gamma_{r_{0}})=\\{0\\}$ for some
$r_{0}\in\\{1,\ldots,t\\}$. To simplify notation, let $r_{0}=1$.

Then $\xi_{1n}=0$ for every $n\in\mathbb{N}$, $\Gamma_{1}=G$ and
$\mathrm{Fix}(\Gamma_{r})\neq\\{0\\}$ for all $r\neq 1$. Setting
$\overline{Q}_{r}:=Q_{r}\smallsetminus\widehat{Q}_{1}$, choosing
$t_{h}\in(0,1]$ as before if $h\in\overline{Q}_{r}$, and arguing as in $(I)$
we derive

(2.10)
$\displaystyle\sum_{r=2}^{t}\sum_{h\in\overline{Q}_{r}}|G/\Gamma_{r}|c_{h}^{\boldsymbol{\phi}|\Gamma_{r}}+c_{\widehat{Q}_{1}}^{\boldsymbol{\phi}}=\sum_{r=2}^{t}|G/\Gamma_{r}|\,c^{{\boldsymbol{\phi}}|\Gamma_{r}}_{\overline{Q}_{r}}+c_{\widehat{Q}_{1}}^{\boldsymbol{\phi}}\geq
c^{\boldsymbol{\phi}}$
$\displaystyle=\dfrac{p-1}{2p}\Big{(}\lim_{n\to\infty}\sum_{r=2}^{t}\sum_{h\in\overline{Q}_{r}}\|\overline{u}_{hn}\|^{2}+\sum_{h\in\widehat{Q}_{1}}\|\overline{u}_{hn}\|^{2}\Big{)}\geq\dfrac{p-1}{2p}\Big{(}\sum_{r=2}^{t}\sum_{h\in\overline{Q}_{r}}|G/\Gamma_{r}|\,\|\overline{w}_{h}\|^{2}+\sum_{h\in\widehat{Q}_{1}}\|\overline{u}_{h}^{1}\|^{2}\Big{)}$
$\displaystyle\geq\dfrac{p-1}{2p}\Big{(}\sum_{r=2}^{t}\sum_{h\in\overline{Q}_{r}}|G/\Gamma_{r}|\,\|t_{h}\overline{w}_{h}\|^{2}+\|\boldsymbol{w}^{1}\|^{2}\Big{)}\geq\sum_{r=2}^{t}\sum_{h\in\overline{Q}_{r}}|G/\Gamma_{r}|c_{h}^{\boldsymbol{\phi}|\Gamma_{r}}+c_{\widehat{Q}_{1}}^{\boldsymbol{\phi}}.$

It follows that $\overline{u}_{hn}\to\overline{u}^{1}_{h}\neq 0$ strongly in
$H^{1}(\mathbb{R}^{N})$ for all $h\in\widehat{Q}_{1}$. Since
$|\xi_{rn}|\to\infty$ for every $r\neq 1$ and the weak limit of
$\overline{u}_{hn}(\,\cdot\,+\xi_{rn})$ is nonzero for every $h\in Q_{r}$, we
must have $\widehat{Q}_{1}=Q_{1}$. Set $Q:=Q_{1}$ and
$\boldsymbol{w}:=\boldsymbol{w}^{1}$. As in case $(I)$ we see that $Q_{r}$
must consist of a single element if $r\neq 1$. We relabel these sets as
$Q_{k}=\\{k\\}$ and set $\xi_{kn}:=\zeta_{kn}$,
$G_{k}:=\Gamma_{k}=G_{\xi_{kn}}$ and $\overline{v}_{k}:=\overline{w}_{k}$ for
each $k\notin Q$. These data have the properties stated in Theorem 2.7.

The proof is complete. ∎

######  Corollary 2.8.

Under the assumptions of _Theorem 2.7_, the following statements hold true:

  * $(i)$

If $q\geq 2$ and $\mathrm{Fix}(G)\neq 0$ the system
$(\mathscr{S}^{\boldsymbol{\phi}})$ does not have a least energy block-wise
nontrivial solution.

  * $(ii)$

If for every $h\in\widehat{Q}:=\\{1,\ldots,q\\}$ one has that

(2.11)
$c^{\boldsymbol{\phi}}<c_{\widehat{Q}\smallsetminus\\{h\\}}^{\boldsymbol{\phi}}+|G\xi|\,c_{h}^{{\boldsymbol{\phi}}|G_{\xi}}\qquad\text{for
every \ }\xi\in\mathbb{R}^{N}\smallsetminus\\{0\\},$

then the system $(\mathscr{S}^{\boldsymbol{\phi}})$ has a least energy block-
wise nontrivial solution.

###### Proof.

$(i)$ is an immediate consequence of Proposition 2.5.

$(ii):$ Let $Q\subset\widehat{Q}$ be as in Theorem 2.7. If $Q=\widehat{Q}$,
the function $\boldsymbol{w}$ given by that theorem is a least energy block-
wise nontrivial solution to $(\mathscr{S}^{\boldsymbol{\phi}})$. If
$Q\neq\widehat{Q}$ then, for each $k\notin Q$, there are $(\xi_{kn})$, $G_{k}$
and $\overline{v}_{k}$ with the properties stated in Theorem 2.7. So we may
choose $n_{0}$ large enough so that $\xi_{k}:=\xi_{kn_{0}}$ satisfies
$G\xi_{k}\neq G\xi_{k^{\prime}}$ if $k,k^{\prime}\notin Q$ and $k\neq
k^{\prime}$ and, furthermore, $G\xi_{k}\neq\\{0\\}$ if $Q\neq\emptyset$ for
every $k\notin Q$. Then, as $G_{k}=G_{\xi_{k}}$, iterating Lemma 2.4 and using
the identity in Theorem 2.7$(v)$, we obtain

$c^{\boldsymbol{\phi}}=c_{Q}^{\boldsymbol{\phi}}+\displaystyle\sum_{k\notin
Q}|G/G_{k}|\,c^{{\boldsymbol{\phi}}|G_{k}}_{k}\geq
c_{\widehat{Q}\smallsetminus\\{h\\}}^{\boldsymbol{\phi}}+|G/G_{h}|\,c_{h}^{{\boldsymbol{\phi}}|G_{h}},$

where $h\notin Q$ is chosen so that $\xi_{h}\neq 0$ if $q\geq 2$. This
inequality stands in contradiction with (2.11) unless $\xi_{h}=0$, in which
case $q=1$, $Q=\emptyset,$ $G_{h}=G$, the system
$(\mathscr{S}^{{\boldsymbol{\phi}}|G_{h}}_{h})$ is
$(\mathscr{S}^{\boldsymbol{\phi}})$, and $\overline{v}_{h}$ is a least energy
block-wise nontrivial solution to $(\mathscr{S}^{\boldsymbol{\phi}})$. This
completes the proof. ∎

######  Remark 2.9.

_If the system is cooperative, i.e., if $q=1$, condition (2.11) implies that
$\mathrm{Fix}(G)=\\{0\\}$. This last condition is actually not necessary, see
[7, Theorem 1.6] and Section 4 below._

######  Corollary 2.10.

If $|G\xi|=\infty$ for every $\xi\in\mathbb{R}^{N}\smallsetminus\\{0\\}$, the
system $(\mathscr{S}^{\boldsymbol{\phi}})$ has a least energy block-wise
nontrivial solution.

###### Proof.

This is an immediate consequence of Corollary 2.8 $(ii)$. ∎

The following example shows that there are groups with infinite orbits
admitting a surjective homomorphism onto $\mathbb{Z}_{2}$ satisfying
assumption $(A_{\phi})$.

######  Example 2.11.

Let $G_{\infty}$ be the group generated by
$\\{\mathrm{e}^{\mathrm{i}\vartheta}:\vartheta\in[0,2\pi)\\}\cup\\{\tau\\}\cup
O(N-4)$, acting on a point
$(z_{1},z_{2},y)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}\equiv\mathbb{R}^{N}$
as follows:

$\displaystyle\mathrm{e}^{\mathrm{i}\vartheta}(z_{1},z_{2},y)$
$\displaystyle:=(\mathrm{e}^{\mathrm{i}\vartheta}z_{1},\mathrm{e}^{\mathrm{i}\vartheta}z_{2},y),\qquad\qquad\tau(z_{1},z_{2},y):=(z_{2},z_{1},y),$
$\displaystyle\alpha(z_{1},z_{2},y)$ $\displaystyle:=(z_{1},z_{2},\alpha
y)\quad\text{if \ }\alpha\in O(N-4).$

and let $\Theta:G_{\infty}\to\mathbb{Z}_{2}$ be the homomorphism satisfying
$\Theta(\mathrm{e}^{\mathrm{i}\vartheta})=1$, $\Theta(\tau)=-1$, and
$\Theta(\alpha)=1$ for every $\alpha\in O(N-4)$. Then $\Theta$ satisfies
$(A_{\Theta})$, and $|G_{\infty}\xi|=\infty$ for every
$\xi\in\mathbb{R}^{N}\smallsetminus\\{0\\}$ if $N=4$ or $N\geq 6$.

######  Theorem 2.12.

Let $N=4$ or $N\geq 6$, and $G:=G_{\infty}$ and $\Theta$ be as in _Example_
2.11. Let $\\{1,\ldots,q\\}=Q^{+}\cup Q^{-}$ with $Q^{+}\cap Q^{-}=\emptyset$
and set $\phi_{h}\equiv 1$ if $h\in Q^{+}$ and $\phi_{h}=\Theta$ if $h\in
Q^{-}$. Then there exists a least energy block-wise nontrivial solution
$\boldsymbol{w}=(\overline{w}_{1},\ldots,\overline{w}_{q})$ to the system
$(\mathscr{S}^{\boldsymbol{\phi}})$ such that every nontrivial component of
$\overline{w}_{h}$ is positive if $h\in Q^{+}$ and every nontrivial component
of $\overline{w}_{h}$ changes sign if $h\in Q^{-}$.

###### Proof.

The existence of a least energy block-wise nontrivial solution to
$(\mathscr{S}^{\boldsymbol{\phi}})$ is given by Corollary 2.10. By Remark 2.3
we may choose the nontrivial components $w_{i}$ to be positive if $i\in
I_{Q^{+}}$, and every nontrivial $w_{i}$ with $i\in I_{Q^{-}}$ changes sign
because $\phi_{h}$ is surjective for $h\in Q^{-}$. ∎

##  3 Fully nontrivial solutions

Fix a closed subgroup $G$ of $O(N)$ and, for each $h=1,\ldots,q$, fix a
continuous homomorphism $\phi_{h}:G\to\mathbb{Z}_{2}$ satisfying
$(A_{\phi_{h}})$. Set $\boldsymbol{\phi}:=(\phi_{1},\ldots,\phi_{q})$. Our aim
now is to give conditions guaranteeing that every least energy block-wise
nontrivial solution to the system $(\mathscr{S}^{\boldsymbol{\phi}})$ is fully
nontrivial. Recall we are always assuming $N\geq 4$, so $1<p<2$.

######  Lemma 3.1.

Assume $(B_{1})$. Let
$\boldsymbol{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{N}^{\boldsymbol{\phi}}$ be
such that some component $u_{i}=0$ and $i\in I_{h}$. Given $\varphi\in
H^{1}(\mathbb{R}^{N})^{\phi_{h}}$ and $\varepsilon>0$, define
$\boldsymbol{u}_{\varepsilon}=(u_{\varepsilon 1},\ldots,u_{\varepsilon\ell})$
by $u_{\varepsilon i}:=\varepsilon\varphi$ and $u_{\varepsilon j}:=u_{j}$ if
$j\neq i$. Then, there exist $\varepsilon_{0}>0$ and a $\mathcal{C}^{1}$-map
$\boldsymbol{t}:(-\varepsilon_{0},\varepsilon_{0})\to(0,\infty)^{q}$
satisfying
$\boldsymbol{t}(\varepsilon)\boldsymbol{u}_{\varepsilon}\in\mathcal{N}^{\boldsymbol{\phi}}$,
$\boldsymbol{t}(0)=(1,\ldots,1)$, $\boldsymbol{t}^{\prime}(0)=(0,\ldots,0)$
and

$\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{t}(\varepsilon)\boldsymbol{u}_{\varepsilon})-\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u})=\varepsilon^{p}\Big{(}-\frac{1}{p}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{\ell}\beta_{ij}\displaystyle\int_{\mathbb{R}^{N}}|\varphi|^{p}|u_{j}|^{p}+o(1)\Big{)}.$

###### Proof.

See [7, Lemmas 3.1 and 3.3]. ∎

Let $d_{\boldsymbol{\phi}}>0$ be as in Lemma 2.2 and set

$S^{\boldsymbol{\phi}}:=\min_{h=1,\ldots,q}\,\inf_{\begin{subarray}{c}v\in
H^{1}(\mathbb{R}^{N})^{\phi_{h}}\\\ v\neq
0\end{subarray}}\frac{\|v\|^{2}}{|v|_{2p}^{2}},$

where $|v|_{2p}$ denotes the norm of $v$ in $L^{2p}(\mathbb{R}^{N})$. Define

(3.1)
$C_{\boldsymbol{\phi}}:=\left(\frac{pd_{\boldsymbol{\phi}}}{(p-1)(S^{\boldsymbol{\phi}})^{\frac{p}{p-1}}}\right)^{p}.$

We stress that $C_{\boldsymbol{\phi}}$ depends on the chosen symmetries. We
shall consider the following assumption.

  * $(B_{3}^{\boldsymbol{\phi}})$

If $q\geq 2$ then, for every $h\in\\{1,\ldots,q\\}$ such that
$\ell_{h}-\ell_{h-1}\geq 2$, the following inequality holds true:

$\Big{(}\min_{\\{i,j\\}\in
E_{h}}\beta_{ij}\Big{)}\left[\frac{\displaystyle\min_{h=1,\ldots,q}\displaystyle\max_{i\in
I_{h}}\beta_{ii}}{\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}}\right]^{\frac{p}{p-1}}>C_{\boldsymbol{\phi}}\sum_{(i,j)\in\mathcal{K}_{h}}|\beta_{ij}|.$

If $i\in I_{h}$ we denote by $N_{h}(i):=\\{j\in I_{h}:j\neq i,\
\beta_{ij}>0\\}$ the neighborhood of the vertex $i$ in the graph whose set of
vertices is $I_{h}$ and whose set of edges is $E_{h}:=\\{\\{i,j\\}:i,j\in
I_{h},\ i\neq j,\ \beta_{ij}>0\\}$.

######  Lemma 3.2.

Assume $(B_{1})$ and $(B_{3}^{\boldsymbol{\phi}})$. Let
$\boldsymbol{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{N}^{\boldsymbol{\phi}}$ be
such that
$\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u})=c^{\boldsymbol{\phi}}$. If
$u_{j_{0}}\neq 0$ and $j_{0}\in I_{h}$, then $u_{i}\neq 0$ for every $i\in
N_{h}(j_{0})$.

###### Proof.

Arguing by contradiction, assume that $u_{i_{0}}=0$ for some $i_{0}\in
N_{h}(j_{0})$. Choose $i_{0}^{*}\in N_{h}(i_{0})$ such that
$|u_{i_{0}^{*}}|_{2p}\geq|u_{j}|_{2p}$ for every $j\in N_{h}(i_{0})$. As
$j_{0}\in N_{h}(i_{0})$ and $u_{j_{0}}\neq 0$, we have that
$|u_{i_{0}^{*}}|_{2p}>0$. Furthermore,

$S^{\boldsymbol{\phi}}|u_{i_{0}^{*}}|_{2p}^{2}\leq\|u_{i_{0}^{*}}\|^{2}\leq\sum_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}\displaystyle\int_{\mathbb{R}^{N}}|u_{i}|^{p}|u_{j}|^{p}\leq\sum_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}|u_{i}|_{2p}^{p}|u_{j}|_{2p}^{p}\leq\sum_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}|u_{i_{0}^{*}}|_{2p}^{2p}.$

Therefore,

(3.2) $\displaystyle\sum_{j\in
N_{h}(i_{0})}\beta_{i_{0}j}\displaystyle\int_{\mathbb{R}^{N}}|u_{i_{0}^{*}}|^{p}|u_{j}|^{p}$
$\displaystyle\geq\beta_{i_{0}i_{0}^{*}}|u_{i_{0}^{*}}|_{2p}^{2p}\geq\Big{(}\min_{\\{i,j\\}\in
E_{h}}\beta_{ij}\Big{)}|u_{i_{0}^{*}}|_{2p}^{2p}$
$\displaystyle\geq\Big{(}\min_{\\{i,j\\}\in
E_{h}}\beta_{ij}\Big{)}\left[\frac{S^{\boldsymbol{\phi}}}{\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}}\right]^{\frac{p}{p-1}}.$

On the other hand, as
$\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u})=\frac{p-1}{2p}\|\boldsymbol{u}\|^{2}=c^{\boldsymbol{\phi}}$,
from Lemma 2.2 we derive

(3.3)
$\displaystyle\Big{(}\displaystyle\int_{\mathbb{R}^{N}}|u_{i_{0}^{*}}|^{p}|u_{j}|^{p}\Big{)}^{\frac{2}{p}}$
$\displaystyle\leq|u_{i_{0}^{*}}|_{2p}^{2}|u_{j}|_{2p}^{2}\leq(S^{\boldsymbol{\phi}})^{-2}\|u_{i_{0}^{*}}\|^{2}\|u_{j}\|^{2}\leq\left((S^{\boldsymbol{\phi}})^{-1}\|\boldsymbol{u}\|^{2}\right)^{2}$
$\displaystyle\leq\left(\frac{pd_{\boldsymbol{\phi}}}{(p-1)S^{\boldsymbol{\phi}}}\Big{(}\min_{h=1,\ldots,q}\max_{i\in
I_{h}}\beta_{ii}\Big{)}^{-\frac{1}{p-1}}\right)^{2}.$

Set $\varphi:=u_{i_{0}^{*}}$. If $q=1$ then

$\sum_{\begin{subarray}{c}j=1\\\ j\neq
i_{0}\end{subarray}}^{\ell}\beta_{i_{0}j}\displaystyle\int_{\mathbb{R}^{N}}|\varphi|^{p}|u_{j}|^{p}=\sum_{j\in
N_{h}(i_{0})}\beta_{i_{0}j}\displaystyle\int_{\mathbb{R}^{N}}|u_{i_{0}^{*}}|^{p}|u_{j}|^{p}>0.$

If $q\geq 2$, inequalities (3.2) and (3.3) and assumption
$(B_{3}^{\boldsymbol{\phi}})$ yield

$\displaystyle\sum_{\begin{subarray}{c}j=1\\\ j\neq
i_{0}\end{subarray}}^{\ell}\beta_{i_{0}j}\displaystyle\int_{\mathbb{R}^{N}}|\varphi|^{p}|u_{j}|^{p}=\sum_{j\in
N_{h}(i_{0})}\beta_{i_{0}j}\displaystyle\int_{\mathbb{R}^{N}}|u_{i_{0}^{*}}|^{p}|u_{j}|^{p}-\sum_{\begin{subarray}{c}k=1\\\
k\neq h\end{subarray}}^{q}\sum_{j\in
I_{k}}|\beta_{i_{0}j}|\displaystyle\int_{\mathbb{R}^{N}}|u_{i_{0}^{*}}|^{p}|u_{j}|^{p}$
$\displaystyle\qquad\geq\Big{(}\min_{\\{i,j\\}\in
E_{h}}\beta_{ij}\Big{)}\left[\frac{S^{\boldsymbol{\phi}}}{\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}}\right]^{\frac{p}{p-1}}-\displaystyle\sum_{(i,j)\in\mathcal{K}_{h}}|\beta_{ij}|\left[\frac{pd_{\boldsymbol{\phi}}}{(p-1)S^{\boldsymbol{\phi}}}\right]^{p}\Big{(}\min_{h=1,\ldots,q}\max_{i\in
I_{h}}\beta_{ii}\Big{)}^{-\frac{p}{p-1}}>0.$

But then, for any $q$, Lemma 3.1 implies that
$\boldsymbol{t}(\varepsilon)\boldsymbol{u}_{\varepsilon}\in\mathcal{N}^{\boldsymbol{\phi}}$
and
$\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{t}(\varepsilon)\boldsymbol{u}_{\varepsilon})<\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u})=c^{\boldsymbol{\phi}}$
for $\varepsilon$ small enough. This is a contradiction. ∎

######  Theorem 3.3.

If $(B_{1})$, $(B_{2})$ and $(B_{3}^{\boldsymbol{\phi}})$ hold true, then
every least energy block-wise nontrivial solution to
$(\mathscr{S}^{\boldsymbol{\phi}})$ is fully nontrivial.

###### Proof.

Let $\boldsymbol{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{N}^{\boldsymbol{\phi}}$
be such that
$\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u})=c^{\boldsymbol{\phi}}$. Then,
for each $h\in\\{1,\ldots,q\\}$ there exists $j_{0}\in I_{h}$ such that
$u_{j_{0}}\neq 0$. Let $i\in I_{h}$. By assumption $(B_{2})$ there exist
$j_{1},\ldots,j_{m}=i\in I_{h}$ such that $j_{i}\in N_{h}(j_{i-1})$ for all
$i=1,\ldots,m$. Then Lemma 3.2 implies that $u_{i}\neq 0$. This shows that
every component of $\overline{u}_{h}$ is nontrivial for every $h$. ∎

Our aim now is to obtain some energy bounds for solutions with positive and
sign-changing components. We start with the cooperative case.

##  4 The cooperative system

Throughout this section we assume $(B_{1})$ and $(B_{2})$. If
$\phi:G\to\mathbb{Z}_{2}$ is a continuous homomorphism, we consider the
functional $J^{\phi}:H^{1}(\mathbb{R}^{N})^{\phi}\to\mathbb{R}$ given by

$J^{\phi}(w):=\frac{1}{2}\|w\|^{2}-\frac{1}{2p}\displaystyle\int_{\mathbb{R}^{N}}|w|^{2p}$

and the Nehari manifold

$\mathcal{M}^{\phi}:=\Big{\\{}w\in H^{1}(\mathbb{R}^{N})^{\phi}:w\neq 0,\
\|w\|^{2}=\displaystyle\int_{\mathbb{R}^{N}}|w|^{2p}\Big{\\}}$

associated to the problem

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

Set

$\mathfrak{c}^{\phi}:=\inf_{w\in\mathcal{M}^{\phi}}J^{\phi}(w).$

If $G$ is the trivial group, i.e., if there are no symmetries, we omit the
superscript from the notation and set

$\mathfrak{c}:=\inf_{w\in\mathcal{M}}J.$

It is well known that this value is attained at a positive radial solution
$\omega$ to the problem

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

We also consider the function
$F_{h}:\mathbb{R}^{\ell_{h}-\ell_{h-1}}\to\mathbb{R}$ given by

$F_{h}(\overline{s}):=\frac{1}{2}\displaystyle\sum_{i\in
I_{h}}s_{i}^{2}-\frac{1}{2p}\displaystyle\sum_{{(i,j)\in\mathcal{I}_{h}}}\beta_{ij}|s_{i}|^{p}|s_{j}|^{p},\qquad\overline{s}=(s_{\ell_{h-1}+1},\ldots,s_{\ell_{h}}),$

and the set

$M_{h}:=\Big{\\{}\overline{s}\in\mathbb{R}^{\ell_{h}-\ell_{h-1}}:\overline{s}\neq\overline{0},\
\displaystyle\sum_{i\in
I_{h}}s_{i}^{2}=\displaystyle\sum_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}|s_{i}|^{p}|s_{j}|^{p}\Big{\\}}.$

Note that

$\inf_{M_{h}}F_{h}=\dfrac{p-1}{2p}\mu_{h},$

with $\mu_{h}$ as defined in (1.2). Clearly, $\mu_{h}>0$ and it is attained at
some $\overline{t}=(t_{\ell_{h-1}+1},\ldots,t_{\ell_{h}})\in M_{h}.$

In the following statement we use the Notation 2.1.

######  Proposition 4.1.

The following statements hold true:

  * $(i)$

If $\overline{s}=(s_{\ell_{h-1}+1},\ldots,s_{\ell_{h}})\in M_{h}$ and
$w\in\mathcal{M}^{\phi_{h}}$, set
${\overline{s}w:=(s_{\ell_{h-1}+1}w,\ldots,s_{\ell_{h}}w)}$. Then
$\overline{s}w\in\mathcal{N}_{h}^{\boldsymbol{\phi}}$ and

$\mathcal{J}_{h}^{\boldsymbol{\phi}}(\overline{s}w)=\frac{2p}{p-1}F_{h}(\overline{s})J^{\phi_{h}}(w).$

  * $(ii)$

$c_{h}^{\boldsymbol{\phi}}=\mu_{h}\mathfrak{c}^{\phi_{h}}$.

  * $(iii)$

If $\mathfrak{c}^{\phi_{h}}$ is attained at $w\in\mathcal{M}^{\phi_{h}}$ and
$\mu_{h}$ is attained at $\overline{t}\in M_{h}$, then $\overline{t}w$ is a
least energy nontrivial solution to the system
$(\mathscr{S}_{h}^{\boldsymbol{\phi}})$, and it is fully nontrivial.

###### Proof.

$(i):$ For any $w\in H^{1}(\mathbb{R}^{N})^{\phi_{h}}\smallsetminus\\{0\\}$
and $\overline{s}\in\mathbb{R}^{\ell_{h}-\ell_{h-1}}\smallsetminus\\{0\\}$ we
have that

$\displaystyle\frac{\displaystyle\sum\limits_{i\in
I_{h}}\|s_{i}w\|^{2}}{\Big{(}\displaystyle\sum\limits_{(i,j)\in\mathcal{I}_{h}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|s_{i}w|^{p}|s_{j}w|^{p}\Big{)}^{\frac{2}{2p}}}=\frac{\displaystyle\sum\limits_{i\in
I_{h}}|s_{i}|^{2}}{\Big{(}\displaystyle\sum\limits_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}|s_{i}|^{p}|s_{j}|^{p}\Big{)}^{\frac{2}{2p}}}\frac{\|w\|^{2}}{\Big{(}\displaystyle\int_{\mathbb{R}^{N}}|w|^{2p}\Big{)}^{\frac{2}{2p}}}.$

If $\overline{s}\in M_{h}$ and $w\in\mathcal{M}^{\phi_{h}}$, by definition, it
follows that $\overline{s}w\in\mathcal{N}_{h}^{\boldsymbol{\phi}}$. Raising
each side of this identity to the power $\dfrac{p}{p-1}$ and multiplying it by
$\dfrac{p-1}{2p}$ we obtain

$\mathcal{J}_{h}^{\boldsymbol{\phi}}(\overline{s}w)=\dfrac{2p}{p-1}F_{h}(\overline{s})J^{\phi_{h}}(w),$

as claimed.

$(ii):$ From $(i)$ we get
$c_{h}^{\boldsymbol{\phi}}\leq\mu_{h}\mathfrak{c}^{\phi_{h}}$. To prove the
opposite inequality, set

$\displaystyle
S_{h}^{\boldsymbol{\phi}}:=\inf_{\begin{subarray}{c}\overline{u}_{h}\in\mathcal{H}_{h}^{\boldsymbol{\phi}}\\\
\|\overline{u}_{h}\|\neq
0\end{subarray}}\frac{\|\overline{u}_{h}\|^{2}}{\Big{(}\displaystyle\sum\limits_{{(i,j)\in\mathcal{I}_{h}}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{i}|^{p}|u_{j}|^{p}\Big{)}^{\frac{2}{2p}}},\qquad
S^{\phi_{h}}:=\inf_{\begin{subarray}{c}w\in{H^{1}(\mathbb{R}^{N})}^{\phi_{h}}\\\
w\neq
0\end{subarray}}\frac{\|w\|^{2}}{\left(\displaystyle\int_{\mathbb{R}^{N}}|w|^{2p}\right)^{\frac{2}{2p}}}.$

It is straightforward to verify that
$c_{h}^{\boldsymbol{\phi}}=\dfrac{p-1}{2p}(S_{h}^{\boldsymbol{\phi}})^{\frac{p}{p-1}}$
and $\mathfrak{c}^{\phi_{h}}=\dfrac{p-1}{2p}(S^{\phi_{h}})^{\frac{p}{p-1}}$.
If $(\overline{u}_{n})$ is a sequence in $\mathcal{N}_{h}^{\boldsymbol{\phi}}$
such that $\mathcal{J}_{h}^{\boldsymbol{\phi}}(\overline{u}_{n})\to
c_{h}^{\boldsymbol{\phi}}$, then

$\displaystyle S^{\phi_{h}}\displaystyle\sum\limits_{i\in
I_{h}}|u_{in}|_{2p}^{2}\leq\displaystyle\sum\limits_{i\in
I_{h}}\|u_{in}\|^{2}=\sum\limits_{(i,j)\in\mathcal{I}_{h}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}\leq\displaystyle\sum\limits_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}|u_{in}|_{2p}^{p}|u_{jn}|_{2p}^{p},$

where $|\,\cdot\,|_{2p}$ is the norm in $L^{2p}(\mathbb{R}^{N})$. Therefore,

$\displaystyle(\mu_{h})^{\frac{p-1}{p}}S^{\phi_{h}}\leq\frac{\displaystyle\sum\limits_{i\in
I_{h}}|u_{in}|_{2p}^{2}}{\Big{(}\displaystyle\sum\limits_{(i,j)\in\mathcal{I}_{h}}\beta_{ij}|u_{in}|_{2p}^{p}|u_{jn}|_{2p}^{p}\Big{)}^{\frac{2}{2p}}}S^{\phi_{h}}\leq\frac{\displaystyle\sum\limits_{i\in
I_{h}}\|u_{in}\|^{2}}{\Big{(}\displaystyle\sum\limits_{(i,j)\in\mathcal{I}_{h}}\displaystyle\int_{\mathbb{R}^{N}}\beta_{ij}|u_{in}|^{p}|u_{jn}|^{p}\Big{)}^{\frac{2}{2p}}}=S_{h}^{\boldsymbol{\phi}}+o_{n}(1).$

As a consequence, $\mu_{h}\mathfrak{c}^{\phi_{h}}\leq
c_{h}^{\boldsymbol{\phi}}$, as claimed.

$(iii):$ If follows from $(i)$ and $(ii)$ that
$\mathcal{J}^{\boldsymbol{\phi}}_{h}(\overline{t}w)=\mu_{h}\mathfrak{c}^{\phi_{h}}=c_{h}^{\boldsymbol{\phi}}$.
Hence, $\overline{t}w$ is a least energy nontrivial solution to
$(\mathscr{S}_{h}^{\boldsymbol{\phi}})$. By Theorem 3.3 it is fully
nontrivial. ∎

######  Corollary 4.2.

If $\phi_{h}\equiv 1$, then $c_{h}^{G}=\mu_{h}\mathfrak{c}$ and there exists
$\overline{s}=(s_{\ell_{h-1}+1},\ldots,s_{\ell_{h}})\in(0,\infty)^{\ell_{h}-\ell_{h-1}}$
such that $\overline{s}\omega\in\mathcal{N}_{h}^{G}$ and
$\mathcal{J}_{h}^{G}(\overline{s}\omega)=c_{h}^{G}$, where $\omega$ is the
positive radial solution to (4.2).

###### Proof.

Since $\omega$ is radial, it is $G$-invariant for any $G$. Hence,
$\omega\in\mathcal{M}^{G}$ and $J^{G}(\omega)=\mathfrak{c}^{G}=\mathfrak{c}$.
The value $\mu_{h}$ is attained at
$\overline{s}=(s_{\ell_{h-1}+1},\ldots,s_{\ell_{h}})\in M_{h}$ with $s_{i}\geq
0$, so Proposition 4.1 implies that
$\overline{s}\omega\in\mathcal{N}_{h}^{G}$,
$\mathcal{J}_{h}^{G}(\overline{s}\omega)=c_{h}^{G}=\mu_{h}\mathfrak{c}$ and
$\overline{s}\omega$ is fully nontrivial. Hence, $s_{i}>0$ for every $i$. ∎

The following groups will play an important role in the proofs of our main
results.

######  Examples 4.3.

Let $N\geq 4$. We write
$\mathbb{R}^{N}\equiv\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}$ and a
point in $\mathbb{R}^{N}$ as
$(z_{1},z_{2},y)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}$. For
$m\in\mathbb{N}$, let

$K_{m}:=\\{\mathrm{e}^{2\pi\mathrm{i}j/m}:j=0,\ldots,m-1\\},$

$G_{m}$ be the group generated by $K_{m}\cup\\{\tau\\}\cup O(N-4)$, acting on
$\mathbb{R}^{N}$ as

$\displaystyle\mathrm{e}^{2\pi\mathrm{i}j/m}(z_{1},z_{2},y)$
$\displaystyle:=(\mathrm{e}^{2\pi\mathrm{i}j/m}z_{1},\mathrm{e}^{2\pi\mathrm{i}j/m}z_{2},y),\qquad\qquad\tau(z_{1},z_{2},y):=(z_{2},z_{1},y),$
$\displaystyle\alpha(z_{1},z_{2},y)$ $\displaystyle:=(z_{1},z_{2},\alpha
y)\quad\text{if \ }\alpha\in O(N-4).$

and $\theta:G_{m}\to\mathbb{Z}_{2}$ be the homomorphism satisfying
$\theta(\mathrm{e}^{2\pi\mathrm{i}j/m})=1$, $\theta(\tau)=-1$, and
$\theta(\alpha)=1$ for every $\alpha\in O(N-4)$.

Let $G^{\prime}_{m}$ be the subgroup of $G_{m}$ generated by
$K_{m}\cup\\{\tau\\}$. Abusing notation we write
$\theta:G^{\prime}_{m}\to\mathbb{Z}_{2}$ for the restriction of $\theta$ to
$G_{m}^{\prime}$.

If $\phi_{h}\equiv 1$ we set
$\zeta:=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ and, for each $R>1$, we
define

$\widehat{\sigma}_{hR}(x):=\sum_{g\in K_{m}}\omega(x-Rg\zeta),\qquad
x\in\mathbb{R}^{N},$

and if $\phi_{h}=\theta$ we take $\zeta:=(1,0,0)$ and we define

$\widehat{\sigma}_{hR}(x):=\sum_{g\in
G_{m}^{\prime}}\phi_{h}(g)\,\omega(x-Rg\zeta),\qquad x\in\mathbb{R}^{N},$

where $\omega$ is the positive radial solution to (4.2). Note that
$\widehat{\sigma}_{hR}(gx)=\phi_{h}(g)\widehat{\sigma}_{hR}(x)$ for every
$g\in G_{m}$, $x\in\mathbb{R}^{N}$. Let $t_{hR}>0$ be such that

$\sigma_{hR}:=t_{hR}\widehat{\sigma}_{hR}\in\mathcal{M}^{\phi_{h}}.$

######  Lemma 4.4.

If $m\geq 2$ and $m\geq 5$ whenever $\phi_{h}=\theta$, then there exist
$C_{0},R_{0}>0$ such that

$J^{\phi_{h}}(\sigma_{hR})\leq|G^{\prime}_{m}\zeta|\,\mathfrak{c}-C_{0}\mathrm{e}^{-dR}\qquad\forall\
R\geq R_{0},$

where $d:=|\zeta-\mathrm{e}^{2\pi\mathrm{i}/m}\zeta|$.

###### Proof.

Consider the case $\phi_{h}=\theta$ and $\zeta=(1,0,0)$, and set
$G:=G_{m}^{\prime}$. The $G$-orbit of $\zeta$ is

$G\zeta=\\{\zeta_{1}^{+},\ldots,\zeta_{m}^{+},\zeta_{1}^{-},\ldots,\zeta_{m}^{-}\\}\quad\text{where
\
}\zeta_{j}^{+}:=\mathrm{e}^{2\pi\mathrm{i}j/m}\zeta,\quad\zeta_{j}^{-}:=\tau\zeta_{j}^{+},$

so

$\widehat{\sigma}_{hR}(x):=\sum_{j=1}^{m}\left(\omega(x-R\zeta_{j}^{+})-\omega(x-R\zeta_{j}^{-})\right).$

For $y\in\mathbb{R}^{N}$ set

$\Psi(y):=\displaystyle\int_{\mathbb{R}^{N}}\omega^{2p-1}(x)\omega(x-y)\,\mathrm{d}x.$

From the well known decay estimates for $\omega$ we deduce that

$\lim_{y\to\infty}\Psi(y)|y|^{\frac{N-1}{2}}\exp|y|=b>0.$

Hence, for large enough $R$,

$\frac{\Psi(Ry)}{\Psi(Rz)}\leq C\mathrm{e}^{-R(|y|-|z|)}\qquad\text{if \
}|y|\geq|z|.$

Here, and in what follows, $C$ denotes a positive constant independent of $R$.
Set

$\displaystyle\varepsilon_{R}^{+}:=\sum\limits_{\begin{subarray}{c}i,j=1\\\
i\neq
j\end{subarray}}^{m}\Psi(R\zeta_{i}^{+}-R\zeta_{j}^{+}),\qquad\varepsilon_{R}^{-}:=\sum\limits_{\begin{subarray}{c}i,j=1\\\
i\neq j\end{subarray}}^{m}\Psi(R\zeta_{i}^{-}-R\zeta_{j}^{-}),$
$\displaystyle\tilde{\varepsilon}_{R}:=\sum_{i,j=1}^{m}\Psi(R\zeta_{i}^{+}-R\zeta_{j}^{-}).$

Then, $\varepsilon_{R}^{+}=\varepsilon_{R}^{-}$. Since $m\geq 5$, we have that
$|\zeta_{i}^{+}-\zeta_{j}^{-}|>|\zeta_{1}^{+}-\zeta_{2}^{+}|=d$ for every
$i,j=1,\ldots,m$. Therefore,

$\frac{\tilde{\varepsilon}_{R}}{\varepsilon_{R}^{+}}\leq\sum_{i,j=1}^{m}\frac{\Psi(R\zeta_{i}^{+}-R\zeta_{j}^{-})}{\Psi(R\zeta_{1}^{+}-R\zeta_{2}^{+})}\leq
C\sum_{i,j=1}^{m}\mathrm{e}^{-R(|\zeta_{i}^{+}-\zeta_{j}^{-}|-d)}\leq
C\mathrm{e}^{-Ra},$

where $a:=\min\limits_{i,j=1,\ldots,m}|\zeta_{i}^{+}-\zeta_{j}^{-}|-d>0$. This
shows that $\tilde{\varepsilon}_{R}=o(\varepsilon_{R}^{+})$ as $R\to\infty$.
Since $\omega$ solves (4.2), for any $y,y^{\prime}\in\mathbb{R}^{N}$ we have
that

$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\left[\nabla\omega(x-y)\cdot\nabla\omega(x-y^{\prime})+\omega(x-y)\omega(x-y^{\prime})\right]\,\mathrm{d}x$
$\displaystyle\qquad=\displaystyle\int_{\mathbb{R}^{N}}\omega^{2p-1}(x-y)\omega(x-y^{\prime})\,\mathrm{d}x=\Psi(y^{\prime}-y).$

So, using [4, Lemma 4], we get

$\displaystyle\frac{\|\widehat{\sigma}_{hR}\|^{2}}{|\widehat{\sigma}_{hR}|_{2p}^{2}}$
$\displaystyle\leq\frac{2m\|\omega\|^{2}+2\varepsilon_{R}^{+}+o(\varepsilon_{R}^{+})}{\left(2m\|\omega\|^{2}+2(2p-1)\varepsilon_{R}^{+}+o(\varepsilon_{R}^{+})\right)^{\frac{2}{2p}}}\leq(2m\|\omega\|^{2})^{\frac{p-1}{p}}-C\varepsilon_{R}^{+}.$

As a consequence,

$J^{\phi}(\sigma_{hR})=\dfrac{p-1}{2p}\Big{(}\frac{\|\widehat{\sigma}_{hR}\|^{2}}{|\widehat{\sigma}_{hR}|_{2p}^{2}}\Big{)}^{\frac{p}{p-1}}\leq
2m\,\mathfrak{c}-C_{0}\mathrm{e}^{-dR},$

as claimed.

The proof for $\phi_{h}\equiv 1$ and
$\zeta=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ is similar but simpler. ∎

######  Proposition 4.5.

Let $G^{\prime}_{m}$ and $\theta$ be as in _Example 4.3_. If $m\geq 5$, then

  * $(a)$

the problem (4.1) has a least energy nontrivial solution $\widehat{\omega}$
and $2\,\mathfrak{c}<\mathfrak{c}^{\theta}<2m\,\mathfrak{c}$.

  * $(b)$

Let $\phi_{h}:=\theta$. Then the system
$(\mathscr{S}_{h}^{\boldsymbol{\phi}})$ has a least energy solution of the
form $(t_{\ell_{h-1}+1}\widehat{\omega},\ldots,t_{\ell_{h}}\widehat{\omega})$
with $t_{i}>0$ for all $i\in I_{h}$, and its energy satisfies

$2\,\mu_{h}\mathfrak{c}<c_{h}^{\boldsymbol{\phi}}<2m\,\mu_{h}\mathfrak{c}.$

###### Proof.

$(a):$ Fix $m\geq 5$ and set $G:=G_{m}$. Let
$\xi=(z_{1},z_{2},y)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}$. Then
$|G\xi|=m$ and $G_{\xi}=\\{1,\tau\\}$ if $z_{1}=z_{2}\neq 0$, and $|G\xi|=2m$
and $G_{\xi}=\\{1\\}$ if $z_{1}\neq z_{2}$. According to [9, Corollary 3.1],
$\mathfrak{c}^{\theta}$ is attained if

$\mathfrak{c}^{\theta}<|G\xi|\,\mathfrak{c}^{\theta|G_{\xi}}\quad\forall\
\xi=(z_{1},z_{2},y)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}\text{ \
with \ }(z_{1},z_{2})\neq(0,0).$

i.e., if
$\mathfrak{c}^{\theta}<\min\\{2m\mathfrak{c},\,m\mathfrak{c}^{\theta|\\{1,\tau\\}}\\}.$
Note that every $w\in\mathcal{M}^{\theta|\\{1,\tau\\}}$ changes sign and
$w^{+}:=\max\\{w,0\\}$ and $w^{-}:=\min\\{w,0\\}$ belong to $\mathcal{M}$.
Therefore,

$J^{\theta|\\{1,\tau\\}}(w)=\dfrac{p-1}{2p}\|w\|^{2}=\dfrac{p-1}{2p}\|w^{+}\|^{2}+\dfrac{p-1}{2p}\|w^{-}\|^{2}\geq
2\mathfrak{c}.$

As a consequence,
$\min\\{2m\mathfrak{c},\,m\mathfrak{c}^{\theta|\\{1,\tau\\}}\\}=2m\mathfrak{c}$.
Let $\sigma_{hR}\in\mathcal{M}^{\theta}$ be as in Example 4.3. By Lemma 4.4,

$\mathfrak{c}^{\theta}\leq J^{\theta}(\sigma_{hR})<2m\,\mathfrak{c}.$

It is well known that the energy of any sign-changing solution to (4.2) is
greater than $2\,\mathfrak{c}$. This proves $(a)$.

$(b):$ This statement is easily derived from $(a)$ and Proposition 4.1. ∎

######  Remark 4.6.

_We wish to stress that the previous proposition is valid (with $2m$ replaced
by $|G\zeta|$) for any group $G$ and any homomorphism
$\phi_{h}:G\to\mathbb{Z}_{2}$ for which there exist $\zeta\in\mathbb{R}^{N}$
and $g\in\ker\phi_{h}$ such that_

$\min\\{|\overline{g}\zeta-\zeta|:\overline{g}\in G,\
\phi_{h}(\overline{g})=-1\\}>|g\zeta-\zeta|>0.$

_Indeed, defining $\sigma_{hR}$ as in Example 4.3, it is easy to see that the
proof of Lemma 4.4 relies only on this property. _

##  5 Energy bounds for a system with two blocks

In this section we consider the system $(\mathscr{S}^{\boldsymbol{\phi}})$
with two blocks, i.e., $q=2$. As before, we assume $(B_{1})$ and $(B_{2})$. We
have the following result.

######  Lemma 5.1.

Let $N=4$ or $N\geq 6$, $q=2$, $G:=G_{m}$ and $\theta$ be as in _Examples_ 4.3
and, for $h=1,2$, let $\phi_{h}$ be either the trivial homomorphism or equal
to $\theta$. If $m\geq 6$, then

$c^{\boldsymbol{\phi}}<c_{k}^{\boldsymbol{\phi}}+|G\xi|\,c_{h}^{{\boldsymbol{\phi}}|G_{\xi}}\qquad\text{for
every \ }\xi\in\mathbb{R}^{N}\smallsetminus\\{0\\}\text{ \ and \ }k\neq h,\
h,k\in\\{1,2\\}.$

###### Proof.

Fix $m\geq 6$. Let
$\xi=(z_{1},z_{2},y)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}$. Then
$|G\xi|=\infty$ if $y\neq 0$, $|G\xi|=m$ and $G_{\xi}=\\{1,\tau\\}$ if
$z_{1}=z_{2}\neq 0$ and $y=0$, and $|G\xi|=2m$ and $G_{\xi}=\\{1\\}$ if
$z_{1}\neq z_{2}$ and $y=0$. As in the proof of Proposition 4.5 we see that
$\min\\{2m\mathfrak{c},\,m\mathfrak{c}^{\theta\,|\\{1,\tau\\}}\\}=2m\mathfrak{c}$,
so Proposition 4.1 yields

(5.1)
$\min_{\xi\in\mathbb{R}^{N}\smallsetminus\\{0\\}}|G\xi|\,c_{h}^{{\boldsymbol{\phi}}|G_{\xi}}=\min_{\xi\in\mathbb{R}^{N}\smallsetminus\\{0\\}}|G\xi|\mu_{h}\mathfrak{c}^{{\phi_{h}}|G_{\xi}}=\begin{cases}m\mu_{h}\mathfrak{c}&\text{if
\ }\phi_{h}\equiv 1,\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt2m\mu_{h}\mathfrak{c}&\text{if \ }\phi_{h}=\theta.\end{cases}$

Without loss of generality, assume $k=1$ and $h=2$. Let $w_{1}$ be a least
energy solution to the problem

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

i.e., $w_{1}$ is the positive radial solution $\omega$ to (4.2) if
$\phi_{1}\equiv 1$, and $w_{1}=\widehat{\omega}$ as in Proposition 4.5 if
$\phi_{1}=\theta$. By Corollary 4.2 and Proposition 4.5 there exists
$\overline{t}_{1}\in(0,\infty)^{\ell_{1}}$ be such that
$\overline{t}_{1}w_{1}\in\mathcal{N}_{1}^{\boldsymbol{\phi}}$ and
$\mathcal{J}_{1}^{\boldsymbol{\phi}}(\overline{t}_{1}w_{1})=c_{1}^{\phi}$.

Let $\zeta:=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$ if $\phi_{2}\equiv 1$
and $\zeta:=(1,0,0)$ if $\phi_{2}=\theta$, and for each $R>1$ define
$\sigma_{2R}\in\mathcal{M}^{\phi_{2}}$ as in Examples 4.3. Let
$\overline{t}_{2}\in M_{2}$ be such that $F_{2}(\overline{t}_{2})=\mu_{2}$.
Then $\overline{t}_{2}\sigma_{2R}\in\mathcal{N}_{2}^{\boldsymbol{\phi}}$, and
from Proposition 4.1 and Lemma 4.4 for $R$ large enough we get

$\displaystyle\mathcal{J}_{2}^{\boldsymbol{\phi}}(\overline{t}_{2}\sigma_{2R})=\mu_{2}J^{\phi_{2}}(\sigma_{2R})\leq\mu_{2}(|G\zeta|\mathfrak{c}-C_{0}\mathrm{e}^{-dR}),$

where $d:=|\zeta-\mathrm{e}^{2\pi\mathrm{i}/m}\zeta|$.

Define $\boldsymbol{w}_{R}=(w_{1R},\ldots,w_{\ell
R})=(\overline{w}_{1R},\overline{w}_{2R})$ with
$\overline{w}_{1R}:=\overline{t}_{1}w_{1}$ and
$\overline{w}_{2R}:=\overline{t}_{2}\sigma_{2R}$. Then $\boldsymbol{w}_{R}$
satisfies (2.1) if $R$ is large enough, so there exist $R_{0}>0$ and, for each
$R>R_{0}$, a pair $(s_{1R},s_{2R})\in[{1}/{2},2]^{2}$ such that
$\boldsymbol{u}_{R}:=(s_{1R}\overline{w}_{1R},s_{2R}\overline{w}_{2R})\in\mathcal{N}^{\boldsymbol{\phi}}$.
From the last statement in Lemma 2.2$(ii)$ we derive

$\displaystyle\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u}_{R})$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{\ell}\|s_{iR}w_{iR}\|^{2}-\frac{1}{2p}\sum_{i,j=1}^{\ell}\beta_{ij}\displaystyle\int_{\mathbb{R}^{N}}|s_{iR}w_{iR}|^{p}|s_{jR}w_{jR}|^{p}$
$\displaystyle\leq\mathcal{J}_{1}^{\boldsymbol{\phi}}(\overline{w}_{1R})+\mathcal{J}_{2}^{\boldsymbol{\phi}}(\overline{w}_{2R})-\frac{1}{2p}\sum_{i\in
I_{1},\,j\in
I_{2}}\beta_{ij}\displaystyle\int_{\mathbb{R}^{N}}|s_{1R}w_{iR}|^{p}|s_{2R}w_{jR}|^{p}$
$\displaystyle\leq
c_{1}^{\boldsymbol{\phi}}+|G\zeta|\mu_{2}\mathfrak{c}-C\mathrm{e}^{-Rd}-\frac{1}{2p}\sum_{i\in
I_{1},\,j\in
I_{2}}\beta_{ij}\displaystyle\int_{\mathbb{R}^{N}}|s_{1R}w_{iR}|^{p}|s_{2R}w_{jR}|^{p}.$

Since $\omega$ and $\widehat{\omega}$ solve (4.2) we have that

$|w_{iR}(x)|\leq C\mathrm{e}^{-|x|}\quad\forall\ i\in
I_{1}\qquad\text{and}\qquad|w_{jR}(x)|\leq
C\mathrm{e}^{-|x-R\zeta|}\quad\forall\ j\in I_{2},$

see [1, Corollary 2.4]. Therefore,

$\displaystyle\int_{\mathbb{R}^{N}}|w_{iR}|^{p}|w_{jR}|^{p}\leq
C\displaystyle\int_{\mathbb{R}^{N}}\mathrm{e}^{-p|x|}\,\mathrm{e}^{-p|x-R\zeta|}\,\mathrm{d}x\leq
C\mathrm{e}^{-Rp}\qquad\forall\ i\in I_{1},\ j\in I_{2}.$

Since $p>1\geq d$ when $m\geq 6$, it follows that

$c^{\boldsymbol{\phi}}\leq\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{u}_{R})<c_{1}^{\boldsymbol{\phi}}+|G\zeta|\mu_{2}\mathfrak{c}.$

This, together with (5.1), yields our claim. ∎

######  Remark 5.2.

_To extend the previous argument to a system with more than two blocks one
needs some information on the decay of the components of solutions to a
system. Namely, if we knew that a minimizer $(w_{1},\ldots,w_{\ell_{q-1}})$ of
$\mathcal{J}_{Q}^{\boldsymbol{\phi}}$ on $\mathcal{N}_{Q}^{\boldsymbol{\phi}}$
with $Q=\\{1,\ldots,q-1\\}$, $q\geq 3$, satisfies_

$|w_{i}(x)|\leq C_{\delta}\mathrm{e}^{-\delta|x|}\qquad\forall\ i\in I_{Q},$

_for every $\delta\in(0,1)$ and some constant $C_{\delta}$, then, taking
$w_{jR}$ to be a suitable multiple of $\sigma_{qR}$ for $j\in I_{q}$ and
choosing $\delta$ such that $p\delta>1$, we would have_

$\displaystyle\int_{\mathbb{R}^{N}}|w_{i}|^{p}|w_{jR}|^{p}\leq
C\displaystyle\int_{\mathbb{R}^{N}}\mathrm{e}^{-p\delta|x|}\,\mathrm{e}^{-p|x-R\zeta|}\,\mathrm{d}x\leq
C\mathrm{e}^{-Rp\delta}=o(\mathrm{e}^{-Rd})\qquad\forall\ i\in I_{Q},\ j\in
I_{q}.$

_This would allow us to extend Lemma 5.1 to any $q>2$. Information on the
decay of the components does not seem to be available so far._

######  Theorem 5.3.

Let $N=4$ or $N\geq 6$, $q=2$, $G:=G_{6}$ and $\theta$ be as in _Example_ 4.3
(with $m=6$). Let $\\{1,2\\}=Q^{+}\cup Q^{-}$ with $Q^{+}\cap Q^{-}=\emptyset$
and set $\phi_{h}\equiv 1$ if $h\in Q^{+}$ and $\phi_{h}=\theta^{\prime}$ if
$h\in Q^{-}$. Then there is a least energy block-wise nontrivial solution
$\boldsymbol{w}=(\overline{w}_{1},\overline{w}_{2})$ to the system
$(\mathscr{S}^{\boldsymbol{\phi}})$ such that all nontrivial components of
$\overline{w}_{h}$ are positive if $h\in Q^{+}$, all nontrivial components of
$\overline{w}_{h}$ change sign if $h\in Q^{-}$, and

(5.2)
$\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{w})<\begin{cases}\min\limits_{h\neq
k}(\mu_{k}+6\mu_{h})\mathfrak{c}&\text{if \ }Q^{+}=\\{1,2\\},\\\
(\mu_{k}+12\mu_{h})\mathfrak{c}&\text{if \ }k\in Q^{+}\text{ and }h\in
Q^{-},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
12(\mu_{1}+\mu_{2})\mathfrak{c}&\text{if \ }Q^{-}=\\{1,2\\}.\end{cases}$

###### Proof.

Lemma 5.1 with $m=6$, $\phi_{h}\equiv 1$ if $h\in Q^{+}$ and $\phi_{h}=\theta$
if $h\in Q^{-}$, together with Corollary 2.8 and Remark 2.3, yields the
existence of a solution $\boldsymbol{w}=(\overline{w}_{1},\overline{w}_{2})$
such that all nontrivial of $\overline{w}_{h}$ are positive if $h\in{Q^{+}}$,
all nontrivial components of $\overline{w}_{k}$ change sign if $k\in{Q^{-}}$,
and

$\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{w})=c^{\boldsymbol{\phi}}<c_{k}^{\boldsymbol{\phi}}+|G\xi|\,c_{h}^{{\boldsymbol{\phi}}|G_{\xi}}\qquad\text{for
every \ }\xi\in\mathbb{R}^{N}\smallsetminus\\{0\\}\text{ \ and \ }k\neq h,\
h,k\in\\{1,2\\}.$

The upper bounds for $\mathcal{J}^{\boldsymbol{\phi}}(\boldsymbol{w})$ now
follow from Corollary 4.2, Proposition 4.5 and equation (5.1). ∎

###### Proof of Theorem 1.1..

Let $\phi_{h}\equiv 1$ if $h\in Q^{+}$ and $\phi_{h}$ be equal to
$\theta:G^{\prime}_{m}\to\mathbb{Z}_{2}$ if $q=1$,
$\theta:G_{m}\to\mathbb{Z}_{2}$ if $q=2$ and
$\Theta:G_{\infty}\to\mathbb{Z}_{2}$ if $q\geq 3$, as in Examples 4.3 and
2.11, if $h\in Q^{-}$. Set $C_{q}:=C_{\boldsymbol{\phi}}$ as in (3.1). Then,
Corollary 4.2, Proposition 4.5, Theorem 5.3 and Theorem 2.12 establish
existence of a least energy block-wise nontrivial solution whose components
have the required signs, and provide the upper energy estimates for $q=1,2$.
Theorem 3.3 asserts that these solutions are fully nontrivial. The strict
lower energy estimate follows from Proposition 2.5, Corollary 4.2 and
Proposition 4.5. ∎

##  6 The singularly perturbed system

Let $G$ be a closed subgroup of $O(N)$, $\phi_{h}:G\to\mathbb{Z}_{2}$ be
continuous homomorphisms satisfying $(A_{\phi_{h}})$ for each $h\in Q$ and
define $\phi_{i}:=\phi_{h}$ for all $i\in I_{h}$. Let $\Omega$ be a
$G$-invariant bounded domain in $\mathbb{R}^{N}$ such that $0\in\Omega$. We
consider the system of singularly perturbed elliptic equations

$(\mathscr{S}_{\varepsilon,\Omega}^{\boldsymbol{\phi}})\qquad\begin{cases}-\varepsilon^{2}\Delta
u_{i}+u_{i}=\displaystyle\sum_{j=1}^{\ell}\beta_{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\\\
u_{i}\in H^{1}(\Omega)^{\phi_{i}},\qquad i=1,\ldots,\ell,\end{cases}$

where $\varepsilon>0$, $1<p<\frac{N}{N-2}$, $N\geq 4$, and the matrix
$(\beta_{ij})$ satisfies assumption $(B_{1})$.

Note that $\boldsymbol{u}=(u_{1},\ldots,u_{\ell})$ solves
$(\mathscr{S}_{\varepsilon,\Omega}^{\boldsymbol{\phi}})$ iff
$\widetilde{\boldsymbol{u}}=(\widetilde{u}_{1},\ldots,\widetilde{u}_{\ell})$
with $\widetilde{u}_{i}(z):=u_{i}(\varepsilon z)$ solves the system

$(\mathscr{S}_{\Omega_{\varepsilon}}^{\boldsymbol{\phi}})\qquad\begin{cases}-\Delta
v_{i}+v_{i}=\displaystyle\sum_{j=1}^{\ell}\beta_{ij}|v_{j}|^{p}|v_{i}|^{p-2}v_{i},\\\
v_{i}\in H^{1}(\Omega_{\varepsilon})^{\phi_{i}},\qquad
i=1,\ldots,\ell,\end{cases}$

where $\Omega_{\varepsilon}:=\\{z\in\mathbb{R}^{N}:\varepsilon z\in\Omega\\}$.

A least energy block-wise nontrivial solution to this system is defined as in
Section 2, replacing $\mathbb{R}^{N}$ with $\Omega_{\varepsilon}$. If
$\varepsilon_{n}\to 0$ and $\widetilde{\boldsymbol{u}}_{n}$ is a least energy
block-wise nontrivial solution to
$(\mathscr{S}_{\Omega_{\varepsilon_{n}}}^{\boldsymbol{\phi}})$, then, arguing
as in [8, Lemma 4.2], one sees that $(\widetilde{\boldsymbol{u}}_{n})$ is a
minimizing sequence for the functional $\mathcal{J}^{\boldsymbol{\phi}}$
associated to the system $(\mathscr{S}^{\boldsymbol{\phi}})$ on
$\mathcal{N}^{\boldsymbol{\phi}}$.

######  Proposition 6.1.

Assume $(B_{1})$ and $(B_{2})$. The following statements hold true:

  * $(i)$

For every $\varepsilon>0$ the system
$(\mathscr{S}_{\Omega_{\varepsilon}}^{\boldsymbol{\phi}})$ has a least energy
block-wise nontrivial solution.

  * $(ii)$

If $\mathrm{Fix}(G)\neq 0$ there exists $\varepsilon_{0}>0$ such that, if
$\varepsilon<\varepsilon_{0}$, every least energy block-wise nontrivial
solution to the system
$(\mathscr{S}_{\Omega_{\varepsilon}}^{\boldsymbol{\phi}})$ is fully
nontrivial.

###### Proof.

$(i):$ It follows from [7, Theorem 1.1] that
$(\mathscr{S}_{\Omega_{\varepsilon}}^{\boldsymbol{\phi}})$ has a least energy
blockwise nontrivial solution.

$(ii):$ Arguing by contradiction, assume that for any $n\in\mathbb{N}$ there
exist $\varepsilon_{n}\in(0,{1/n})$ and a least energy block-wise nontrivial
solution $\widetilde{\boldsymbol{u}}_{n}$ to
$(\mathscr{S}_{\Omega_{\varepsilon_{n}}}^{\boldsymbol{\phi}})$ which is not
fully nontrivial. Passing to a subsequence, we may choose $i\in I_{k}$ such
that $\widetilde{u}_{in}=0$ for all $n$.

As stated above, $(\widetilde{\boldsymbol{u}}_{n})$ is a minimizing sequence
for $\mathcal{J}^{\boldsymbol{\phi}}$ on $\mathcal{N}^{\boldsymbol{\phi}}$. If
$\mathrm{Fix}(G)\neq 0$ Proposition 2.5 states that no subsystem of
$(\mathscr{S}^{\boldsymbol{\phi}})$ having at least two blocks has a least
energy block-wise nontrivial solution. So we derive from Theorem 2.7 that,
after passing to a subsequence, there exist a sequence $(\xi_{kn})$ in
$\mathbb{R}^{N}$, a closed subgroup $G_{k}$ of $G$, and a least energy
nontrivial solution $\overline{v}_{k}$ to the cooperative system

$(\mathscr{S}_{k}^{\boldsymbol{\phi}|G_{k}})\qquad\begin{cases}-\Delta
v_{i}+v_{i}=\displaystyle\sum\limits_{j\in
I_{k}}\beta_{ij}|v_{j}|^{p}|v_{i}|^{p-2}v_{i},\\\ v_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{i}|G_{k}},\qquad i\in I_{k},\end{cases}$

such that $|G/G_{k}|<\infty$,
$\displaystyle\lim\limits_{n\to\infty}|g\xi_{kn}-\overline{g}\xi_{kn}|=\infty$
for any $g,\overline{g}\in G$ with $\overline{g}g^{-1}\notin G_{k}$, and

$\lim_{n\to\infty}\Big{\|}\widetilde{u}_{in}-\displaystyle\sum_{[g]\in
G/G_{k}}\phi_{i}(g)(v_{i}\circ g^{-1})(\ \cdot\ -g\xi_{kn})\Big{\|}=0.$

Since we are assuming that $\widetilde{u}_{in}=0$ for all $n$, this implies
that $v_{i}=0$, which is a contradiction because $\overline{v}_{k}$ is fully
nontrivial by Theorem 3.3. ∎

Recall Examples 4.3. Note that $\mathrm{Fix}(G^{\prime}_{m})\neq\\{0\\}$ if
$N\geq 5$. The following theorem establishes the existence of solutions whose
limit profile becomes uncoupled.

######  Theorem 6.2.

Assume $(B_{1})$ and $(B_{2})$. Let $N\geq 5$. Set
$G^{\prime}:=G^{\prime}_{5}$ and let $\Omega$ be a $G$-invariant bounded
domain in $\mathbb{R}^{N}$ such that $0\in\Omega$. Write
$\\{1,\ldots,q\\}=Q^{+}\cup Q^{-}$ with $Q^{+}\cap Q^{-}=\emptyset$ and let
$\phi_{h}\equiv 1$ if $h\in Q^{+}$ and $\phi_{h}=\theta$ if $h\in Q^{-}$.
Then, for any $\varepsilon_{n}>0$ with $\varepsilon_{n}\to 0$, after passing
to a subsequence, there exists a least energy fully nontrivial solution
$\widetilde{\boldsymbol{u}}_{n}=(\widetilde{u}_{1n},\ldots,\widetilde{u}_{\ell
n})$ to the system
$(\mathscr{S}_{\Omega_{\varepsilon_{n}}}^{\boldsymbol{\phi}})$ such that
$\widetilde{u}_{in}$ is positive if $i\in I_{Q^{+}}$ and $\widetilde{u}_{in}$
changes sign if $i\in I_{Q^{-}}$. Furthermore, for each $h=1,\ldots,q$, there
exist a sequence $(\xi_{hn})$ in $\mathbb{R}^{N}$ and a least energy fully
nontrivial solution $\overline{v}_{h}$ to the cooperative system

($\mathscr{S}_{h}^{\boldsymbol{\phi}}$) $\qquad\begin{cases}-\Delta
v_{i}+v_{i}=\displaystyle\sum\limits_{j\in
I_{h}}\beta_{ij}|v_{j}|^{p}|v_{i}|^{p-2}v_{i},\\\ v_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{h}},\qquad i\in I_{h},\end{cases}$

which satisfy

  * $(a)$

the components of $\overline{v}_{h}$ are positive if $h\in Q^{+}$ and they
change sign if $h\in Q^{-}$,

  * $(b)$

$\xi_{hn}\in\mathrm{Fix}(G^{\prime})$ for all $h$, and
$\displaystyle\lim_{n\to\infty}|\xi_{hn}-\xi_{kn}|=\infty$ if $h\neq k$,

  * $(c)$

$\displaystyle\lim_{n\to\infty}\left\|\widetilde{u}_{in}-v_{i}(\ \cdot\
-\xi_{hn})\right\|=0$ for every $i\in I_{h},$

  * $(d)$

$\|\overline{v}_{h}\|^{2}=\mu_{h}\|\omega\|^{2}$ if $h\in Q^{+}$, and
$\|\overline{v}_{h}\|^{2}<10\,\mu_{h}\|\omega\|^{2}$ if $h\in Q^{-}$.

###### Proof.

Let $\widetilde{\boldsymbol{u}}_{n}$ be a least energy block-wise nontrivial
solution to $(\mathscr{S}^{\boldsymbol{\phi}}_{\Omega_{\varepsilon_{n}}})$.
Note that $\mathrm{Fix}(G^{\prime})\neq 0$ because $N\geq 5$. So, after
passing to a subsequence, $\widetilde{\boldsymbol{u}}_{n}$ is fully nontrivial
by Proposition 6.1 and, invoking Proposition 2.5, we derive from Theorem 2.7
that, for each $h=1,\ldots,q$ there exist a sequence $(\xi_{hn})$ in
$\mathbb{R}^{N}$, a closed subgroup $G^{\prime}_{h}$ of $G^{\prime}$, and a
least energy fully nontrivial solution $\overline{v}_{h}$ to the cooperative
system

($\mathscr{S}_{h}^{\boldsymbol{\phi}|G_{h}}$) $\qquad\begin{cases}-\Delta
v_{i}+v_{i}=\displaystyle\sum\limits_{j\in
I_{h}}\beta_{ij}|v_{j}|^{p}|v_{i}|^{p-2}v_{i},\\\ v_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{h}|G^{\prime}_{h}},\qquad i\in I_{h},\end{cases}$

satisfying properties $(ii)$, $(iv)$ and $(v)$. Property $(v)$ combined with
Proposition 2.5 yields

$c^{\boldsymbol{\phi}}=\sum_{h=1}^{q}|G^{\prime}/G^{\prime}_{h}|\,c^{\boldsymbol{\phi}|G^{\prime}_{h}}_{h}=\sum_{h=1}^{q}c^{\boldsymbol{\phi}}_{h}.$

If $h\in Q^{+}$, then $\phi_{h}\equiv 1$ and Corollary 4.2 gives
$c^{\boldsymbol{\phi}|G^{\prime}_{h}}_{h}=c^{\boldsymbol{\phi}}_{h}$. On the
other hand, if $h\in Q^{-}$, then $\phi_{h}=\theta$ and arguing as in
Proposition 4.5 we see that

$\mathfrak{c}^{\theta}<|G^{\prime}\xi|\,\mathfrak{c}^{\theta|G^{\prime}_{\xi}}\quad\forall\
\xi=(z_{1},z_{2},y)\in\mathbb{C}\times\mathbb{C}\times\mathbb{R}^{N-4}\text{ \
with \ }(z_{1},z_{2})\neq(0,0).$

Then, Proposition 4.1 yields
$c_{h}^{\boldsymbol{\phi}}<|G^{\prime}/G^{\prime}_{h}|\,c^{\boldsymbol{\phi}|G^{\prime}_{h}}_{h}$
if $G^{\prime}_{h}\neq G^{\prime}$. As a consequence,
$G^{\prime}_{h}=G^{\prime}$ for every $h$. So $v_{i}$ changes sign for every
$i\in I_{Q^{-}}$. Furthermore, $\xi_{hn}\in\mathrm{Fix}(G^{\prime})$, so from
property $(ii)$ in Theorem 2.7 we get that $|\xi_{hn}-\xi_{kn}|\to\infty$ if
$h\neq k$, and property $(iv)$ becomes

$\lim_{n\to\infty}\left\|\widetilde{u}_{in}-v_{i}(\ \cdot\
-\xi_{hn})\right\|=0\qquad\text{for every \ }i\in I_{h}.$

Since $\dfrac{p-1}{2p}\|\overline{v}_{h}\|^{2}=c_{h}^{\boldsymbol{\phi}}$ and
$\mathfrak{c}=\dfrac{p-1}{2p}\|\omega\|^{2}$, the statement $(d)$ follows from
the estimates in Corollary 4.2 and Proposition 4.5. ∎

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

This result is easily derived from Theorem 6.2, performing the change of
variable described at the beginning of this section. ∎

The following two theorems establish the existence of solutions whose limit
profile remains coupled.

######  Theorem 6.3.

Assume $(B_{1})-(B_{3})$. Let $N=4$ or $N\geq 6$, $q=2$, $G:=G_{6}$ and
$\theta$ be as in Example 4.3 (with $m=6$) and $\Omega$ be a $G$-invariant
bounded domain in $\mathbb{R}^{N}$ such that $0\in\Omega$. Let
$\\{1,2\\}=Q^{+}\cup Q^{-}$ with $Q^{+}\cap Q^{-}=\emptyset$ and set
$\phi_{h}\equiv 1$ if $h\in Q^{+}$ and $\phi_{h}=\theta$ if $h\in Q^{-}$.
Then, for any $\varepsilon_{n}>0$ with $\varepsilon_{n}\to 0$, after passing
to a subsequence, there exist a least energy fully nontrivial solution
$\widetilde{\boldsymbol{u}}_{n}=(\widetilde{u}_{1n},\ldots,\widetilde{u}_{\ell
n})$ to the system
$(\mathscr{S}_{\Omega_{\varepsilon_{n}}}^{\boldsymbol{\phi}})$ with
$\widetilde{u}_{in}$ positive if $i\in I_{Q^{+}}$ and $\widetilde{u}_{in}$
sign-changing if $i\in I_{Q^{-}}$, and a least energy fully nontrivial
solution $\boldsymbol{w}=(\overline{w}_{1},\overline{w}_{2})$ to the system

($\mathscr{S}^{\boldsymbol{\phi}}$) $\qquad\begin{cases}-\Delta
v_{i}+v_{i}=\displaystyle\sum\limits_{j\in
I_{h}}\beta_{ij}|v_{j}|^{p}|v_{i}|^{p-2}v_{i},\\\ v_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{h}},\qquad i\in I_{h},\qquad h=1,2,\end{cases}$

such that

  * $(a)$

the components of $\overline{w}_{h}$ are positive if $h\in Q^{+}$ and they
change sign if $h\in Q^{-}$,

  * $(b)$

$\displaystyle\lim_{n\to\infty}\left\|\widetilde{u}_{in}-w_{i}\right\|=0$ for
every $i\in I_{\\{1,2\\}}$,

  * $(c)$

$\boldsymbol{w}$ satisfies the energy estimates (5.2).

###### Proof.

Let $\widetilde{\boldsymbol{u}}_{n}$ be a least energy block-wise nontrivial
solution to $(\mathscr{S}^{\boldsymbol{\phi}}_{\Omega_{\varepsilon_{n}}})$. By
Theorem 3.3 it is fully nontrivial. Comparing Lemma 5.1 with property $(v)$ of
Theorem 2.7 we see that the set $Q$ given by that theorem must be all of
$\\{1,2\\}$. Hence, there exists a solution $\boldsymbol{w}$, as claimed, to
the system $(\mathscr{S}^{\boldsymbol{\phi}})$ satisfying $(a)$ and $(b)$. The
energy estimates $(c)$ are given by Theorem 5.3. ∎

######  Theorem 6.4.

Assume $(B_{1})-(B_{3})$. Let $N=4$ or $N\geq 6$, $G:=G_{\infty}$ and $\Theta$
be as in _Example_ 2.11 and $\Omega$ be a $G$-invariant bounded domain in
$\mathbb{R}^{N}$ such that $0\in\Omega$. Let $\\{1,\ldots,q\\}=Q^{+}\cup
Q^{-}$ with $Q^{+}\cap Q^{-}=\emptyset$ and set $\phi_{h}\equiv 1$ if $h\in
Q^{+}$ and $\phi_{h}=\Theta$ if $h\in Q^{-}$. Then, for any
$\varepsilon_{n}>0$ with $\varepsilon_{n}\to 0$, after passing to a
subsequence, there exist a least energy fully nontrivial solution
$\widetilde{\boldsymbol{u}}_{n}=(\widetilde{u}_{1n},\ldots,\widetilde{u}_{\ell
n})$ to the system
$(\mathscr{S}_{\Omega_{\varepsilon_{n}}}^{\boldsymbol{\phi}})$ with
$\widetilde{u}_{in}$ positive if $i\in I_{Q^{+}}$ and $\widetilde{u}_{in}$
sign-changing if $i\in I_{Q^{-}}$, and a least energy fully nontrivial
solution $\boldsymbol{w}=(\overline{w}_{1},\ldots,\overline{w}_{q})$ to the
system

($\mathscr{S}^{\boldsymbol{\phi}}$) $\qquad\begin{cases}-\Delta
v_{i}+v_{i}=\displaystyle\sum\limits_{j\in
I_{h}}\beta_{ij}|v_{j}|^{p}|v_{i}|^{p-2}v_{i},\\\ v_{i}\in
H^{1}(\mathbb{R}^{N})^{\phi_{h}},\qquad i\in I_{h},\qquad
h=1,\ldots,q,\end{cases}$

satisfying

  * $(a)$

the components of $\overline{w}_{h}$ are positive if $h\in Q^{+}$ and they
change sign if $h\in Q^{-}$,

  * $(b)$

$\displaystyle\lim_{n\to\infty}\left\|\widetilde{u}_{in}-w_{i}\right\|=0$ for
every $i\in I_{h}$.

###### Proof.

Let $\widetilde{\boldsymbol{u}}_{n}$ be a least energy block-wise nontrivial
solution to $(\mathscr{S}^{\boldsymbol{\phi}}_{\Omega_{\varepsilon_{n}}})$. By
Theorem 3.3 it is fully nontrivial. Since the only finite $G$-orbit is the
origin, property $(ii)$ of Theorem 2.7 says that splitting is impossible. So
the set $Q$ given by that theorem must be the whole $\\{1,\ldots,q\\}$. Hence,
there exists a solution $\boldsymbol{w}$ to the system
$(\mathscr{S}^{\boldsymbol{\phi}})$ with the stated properties. ∎

###### Proof of Theorem 1.4..

This result follows immediately from Theorems 6.3 and 6.4, performing the
change of variable described at the beginning of this section. ∎

## References

  * [1] Ackermann, Nils; Dancer, Norman: Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity. Differential Integral Equations 29 (2016), no. 7-8, 757–774.
  * [2] Bracho, Javier; Clapp, Mónica; Marzantowicz, Wacław: Symmetry breaking solutions of nonlinear elliptic systems. Topol. Methods Nonlinear Anal. 26 (2005), no. 1, 189–201.
  * [3] Byeon, Jaeyoung; Sato, Yohei; Wang, Zhi-Qiang: Pattern formation via mixed attractive and repulsive interactions for nonlinear Schrödinger systems. J. Math. Pures Appl. (9) 106 (2016), no. 3, 477–511.
  * [4] Cerami, Giovanna; Clapp, Mónica: Sign changing solutions of semilinear elliptic problems in exterior domains. Calc. Var. Partial Differential Equations 30 (2007), no. 3, 353–367.
  * [5] Chen, Zhijie; Lin, Chang-Shou; Zou, Wenming: Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrödinger system. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 859–897.
  * [6] Cingolani, Silvia; Clapp, Mónica; Secchi, Simone: Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 63 (2012), no. 2, 233–248.
  * [7] Clapp, Mónica; Pistoia, Angela: Fully nontrivial solutions to elliptic systems with mixed couplings. Nonlinear Analysis 216 (2022), 112694.
  * [8] Clapp, Mónica; Soares, Mayra: Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems. Preprint, arXiv:2108.00299.
  * [9] Clapp, Mónica; Srikanth, P. N.: Entire nodal solutions of a semilinear elliptic equation and their effect on concentration phenomena. J. Math. Anal. Appl. 437 (2016), no. 1, 485–497.
  * [10] Dovetta, Simone; Pistoia Angela: Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime. Math. Eng. 4 (2022), no. 4, Paper No. 027, 21 pp.
  * [11] Esry, B. D.; Greene, Chris H.; Burke, Jr., James P.; Bohn, John L.: Hartree-Fock theory for double condensates, Phys. Rev. Lett. 78 (1997), 3594-3597.
  * [12] Lin, Tai-Chia; Wei, Juncheng: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 4, 403–439.
  * [13] Lin, Tai-Chia; Wei, Juncheng: Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbb{R}^{n}$, $n\leq 3$. Comm. Math. Phys. 255 (2005), no. 3, 629–653.
  * [14] Liu, Jiaquan; Liu, Xiangqing; Wang, Zhi-qiang: Multiple mixed states of nodal solutions for nonlinear Schrödinger systems. Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 565–586.
  * [15] Sato, Yohei; Wang, Zhi-Qiang: On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system. Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), no. 1, 1–22.
  * [16] Sato, Yohei; Wang, Zhi-Qiang: On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete Contin. Dyn. Syst. 35 (2015), no. 5, 2151–2164.
  * [17] Sato, Yohei; Wang, Zhi-Qiang: Least energy solutions for nonlinear Schrödinger systems with mixed attractive and repulsive couplings. Adv. Nonlinear Stud. 15 (2015), no. 1, 1–22.
  * [18] Sato, Yohei; Wang, Zhi-Qiang: Multiple positive solutions for Schrödinger systems with mixed couplings. Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1373–1392.
  * [19] Sirakov, Boyan: Least energy solitary waves for a system of nonlinear Schrödinger equations in Rn. Comm. Math. Phys. 271 (2007), no. 1, 199–221.
  * [20] Soave, Nicola: On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition. Calc. Var. Partial Differential Equations 53 (2015), no. 3-4, 689–718.
  * [21] Soave, Nicola; Tavares, Hugo: New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms. J. Differential Equations 261 (2016), no. 1, 505–537.
  * [22] Tavares, Hugo; You, Song: Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case. Calc. Var. Partial Differential Equations 59 (2020), no. 1, Paper No. 26, 35 pp.
  * [23] Tavares, Hugo; You, Song; Zou, Wenming: Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: the higher dimensional case. J. Funct. Anal. 283 (2022), no. 2, Paper No. 109497, 50 pp.
  * [24] Tavares, Hugo; Terracini, Susanna: Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems. Ann. Inst. H. Poincaré C Anal. Non Linéaire 29 (2012), no. 2, 279–300.
  * [25] Wei, Juncheng; Wu, Yuanze: Ground states of nonlinear Schrödinger systems with mixed couplings. J. Math. Pures Appl. (9) 141 (2020), 50–88.
  * [26] Willem, Michel: Minimax theorems. Progress in Nonlinear Differential Equations and their Applications 24. Birkhäuser Boston, Inc., Boston, MA (1996).

Mónica Clapp and Mayra Soares
Instituto de Matemáticas

Universidad Nacional Autónoma de México

Circuito Exterior, Ciudad Universitaria

04510 Coyoacán, Ciudad de México, Mexico

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>