# Pinwheel solutions to Schrödinger systems

Mónica Clapp111M. Clapp was supported by CONACYT (Mexico) through the research
grant A1-S-10457. and Angela Pistoia222A. Pistoia was partially supported by
INDAM-GNAMPA funds and Fondi di Ateneo “Sapienza” Università di Roma (Italy).

###### Abstract

We establish the existence of positive segregated solutions for competitive
nonlinear Schrödinger systems in the presence of an external trapping
potential, which have the property that each component is obtained from the
previous one by a rotation, and we study their behavior as the forces of
interaction become very small or very large.

As a consequence, we obtain optimal partitions for the Schrödinger equation by
sets that are linearly isometric to each other. 333 2020 Mathematics Subject
Classification: 35J50 (35J47, 35B06, 35B07, 35B40) 444 Keywords: Schrödinger
systems, segregated solutions, phase separation, optimal partition.

##  1 Introduction

Consider the nonlinear Schrödinger system

(1.1) $\begin{cases}-\Delta
u_{i}+V_{i}(x)u_{i}=|u_{i}|^{2p-2}u_{i}+\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{\ell}\beta_{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\\\
u_{i}\in H^{1}(\mathbb{R}^{N}),\quad u_{i}>0,\qquad
i=1,\ldots,\ell,\end{cases}$

where $N\geq 2$, $p>1$ and $p<\frac{N}{N-2}$ if $N\geq 3$,
$\beta_{ij}=\beta_{ji}\in\mathbb{R}$, and
$V_{i}\in\mathcal{C}^{0}(\mathbb{R}^{N}).$

For the cubic nonlinearity ($p=2$) in dimensions $N=2,3$ this system arises in
the study of Bose-Einstein condensation for a mixture of $\ell$ different
states which overlap in space. It has been widely studied in the last two
decades. Most work has been done in the autonomous case (i.e., for constant
$V_{i}$). We refer the reader to the recent paper [11] where the authors
provide an exhaustive list of references. The non-autonomous case turns out to
be much more difficult. Some results have been recently obtained by Peng and
Wang [13], Pistoia and Vaira [14], and Li, Wei and Wu [11].

The system (1.1) for a more general subcritical nonlinearity in higher
dimensions has been much less studied. Even if it does not have an immediate
physical motivation, finding a solution in this general setting is a quite
interesting and challenging problem from a mathematical point of view. To our
knowledge, the only result so far is that by Gao and Guo [10] who proved the
existence of infinitely many solutions for the system of only two equations
($\ell=2$) when the coupling parameter $\beta_{12}$ is negative, and both
equations have a common potential $V_{1}=V_{2}$, which does not enjoy any
symmetry properties, but satisfies suitable decay assumptions at infinity.
However, nothing is said about the sign of the solutions.

Here we study (1.1) in a fully symmetric setting, namely we consider the
nonlinear Schrödinger system

(1.2) $\begin{cases}-\Delta
u_{i}+V(x)u_{i}=|u_{i}|^{2p-2}u_{i}+\beta\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{\ell}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\\\ u_{i}\in
H^{1}(\mathbb{R}^{N}),\quad u_{i}>0,\qquad i=1,\ldots,\ell,\end{cases}$

where $N\geq 4$, $1<p<\frac{N}{N-2}$, $\beta<0$ and
$V\in\mathcal{C}^{0}(\mathbb{R}^{N})$ satisfies the following assumptions for
some $n\in\mathbb{N}$:

  * $(V_{1})$

$V$ is radial.

  * $(V_{2})$

$0<\inf_{x\in\mathbb{R}^{N}}V(x)$ and $V(x)\to V_{\infty}>0$ as
$|x|\to\infty$.

  * $(V_{3}^{n})$

There exist $C_{0},R_{0}>0$ and $\lambda\in(0,2\sin\frac{\pi}{\ell n})$ such
that

$V(x)\leq
V_{\infty}-C_{0}\mathrm{e}^{-\lambda\sqrt{V_{\infty}}|x|}\qquad\text{for every
\ }x\in\mathbb{R}^{N}\text{ \ with \ }|x|\geq R_{0}.$

We look for _fully nontrivial_ solutions to (1.2), i.e., solutions with all
components $u_{i}$ different from zero. Set

$\|u\|^{2}_{V}:=\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+V(x)u^{2}).$

We prove the following results.

######  Theorem 1.1.

Let $n\in\mathbb{N}$ and assume that $V$ satisfies $(V_{1})$, $(V_{2})$ and
$(V_{3}^{n})$. Then the system (1.2) has a fully nontrivial solution
$\boldsymbol{u}=(u_{1},\ldots,u_{\ell})$ satisfying

(1.3) $\begin{cases}u_{1}(\mathrm{e}^{2\pi\mathrm{i}/n}z,\theta
y)=u_{1}(z,y)&\text{for every \ }\theta\in O(N-2),\\\
u_{j+1}(z,y)=u_{1}(\mathrm{e}^{2\pi\mathrm{i}j/\ell n}z,y)&\text{for every \
}j=1,\ldots,\ell-1,\end{cases}$

and every $(z,y)\in\mathbb{C}\times\mathbb{R}^{N-2}\equiv\mathbb{R}^{N}$. This
solution has least energy among all nontrivial solutions satisfying (1.3).
Furthermore, the energy of each component satisfies

$\frac{p-1}{2p}\|u_{i}\|_{V}^{2}<n\,\mathfrak{c}_{\infty},$

where $\mathfrak{c}_{\infty}$ is the ground state energy of the Schrödinger
equation

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

As usual, $O(N-2)$ denotes the group of linear isometries of
$\mathbb{R}^{N-2}$. The symmetries (1.3) of the solutions given by Theorem 1.1
suggests calling them _pinwheel solutions_.

Since the potential $V$ is assumed to be radial, using the compactness of the
embedding of the subspace of radial functions in $H^{1}(\mathbb{R}^{N})$ into
$L^{2p}(\mathbb{R}^{N})$ and following the argument given in [6, Theorem 1.1],
it is easy to see that the system (1.2) has a solution all of whose components
are radial. Note however that, if $\boldsymbol{u}=(u_{1},\ldots,u_{\ell})$
satisfies (1.2) and (1.3) and some component $u_{i}$ is radial, then
$u_{1}=\cdots=u_{\ell}=:u$ and $u$ is a nontrivial solution of the equation

$-\Delta u+V(x)u=(1+(\ell-1)\beta)|u|^{2p-2}u,\qquad u\in
H^{1}(\mathbb{R}^{N}).$

Therefore, if $1+(\ell-1)\beta\leq 0$, a nontrivial solution to the system
(1.2) satisfying (1.3) cannot be radial. In fact, more can be said. The
following result, combined with Theorem (1.1), yields multiple positive
nonradial solutions when the assumption $(V_{3}^{n})$ is satisfied for large
enough $n$.

######  Proposition 1.2.

Let $\beta\leq-\frac{1}{\ell-1}$ and, for some $m,q\in\mathbb{N}$, let
$\boldsymbol{u}_{m},\boldsymbol{u}_{q}$ be solutions to (1.2) satisfying (1.3)
with $n=\ell^{m}$ and $n=\ell^{q}$ respectively. If $m\neq q$, then
$\boldsymbol{u}_{m}\neq\boldsymbol{u}_{q}$.

One may wonder if the solution given by Theorem 1.1 for
$\beta\in(-\frac{1}{\ell-1},0)$ is radial or not. The following result gives a
partial answer in terms of the nonautonomous Schrödinger equation (1.5).
Namely, if the least energy solutions to this equation that satisfy (1.6) are
nonradial, then the solutions to the system (1.2) satisfying (1.3) are
nonradial for $\beta$ close enough to $0$.

######  Theorem 1.3.

Let $n\in\mathbb{N}$ and assume that $V$ satisfies $(V_{1})$, $(V_{2})$ and
$(V_{3}^{n})$. Let $\boldsymbol{u}_{k}=(u_{k,1},\ldots,u_{k,\ell})$ be a least
energy fully nontrivial solution to (1.2) and (1.3) with $\beta=\beta_{k}$.
Assume that $\beta_{k}<0$ and $\beta_{k}\to 0$ as $k\to\infty$. Then, after
passing to a subsequence, $u_{k,j}\to u_{0,j}$ strongly in
$H^{1}(\mathbb{R}^{N})$, $u_{0,j}\geq 0$,
$\boldsymbol{u}_{0}=(u_{0,1},\ldots,u_{0,\ell})$ satisfies (1.3), $u_{0,j}$ is
a nontrivial solution to the equation

(1.5) $-\Delta u+V(x)u=|u|^{2p-2}u,\qquad u\in H^{1}(\mathbb{R}^{N}),$

and $u_{0,j}$ has least energy among all solutions to (1.5) satisfying

(1.6) $u(\mathrm{e}^{2\pi\mathrm{i}/n}z,\theta y)=u(z,y)\quad\text{for every \
}\theta\in O(N-2),\ (z,y)\in\mathbb{R}^{N}.$

Furthermore,

$\frac{p-1}{2p}\|u_{0,j}\|^{2}_{V}<n\,\mathfrak{c}_{\infty}.$

Next, we describe the behavior of the solutions given by Theorem 1.1 as
$\beta\to-\infty$. As shown by Conti, Terracini and Verzini [8, 9] and Chang,
Lin, Lin and Lin [2], there is a connection between variational elliptic
systems with strong competitive interaction and optimal partition problems.

We shall call an $\ell$-tuple $(\Omega_{1},\ldots,\Omega_{\ell})$ of nonempty
open subsets of $\mathbb{R}^{N}$ an _$(n,\ell)$ -pinwheel partition_ of
$\mathbb{R}^{N}$ if $\Omega_{i}\cap\Omega_{j}=\emptyset$ whenever $i\neq j$
and it satisfies following two symmetry conditions:

  * $(S_{1})$

$\Omega_{j+1}=\\{(z,y)\in\mathbb{C}\times\mathbb{R}^{N-2}:(\mathrm{e}^{2\pi\mathrm{i}j/\ell
n}z,y)\in\Omega_{1}\\}$ for each $j=1,\ldots,\ell-1$.

  * $(S_{2})$

If $(z,y)\in\Omega_{1}$ then $(\mathrm{e}^{2\pi\mathrm{i}/n}z,\theta
y)\in\Omega_{1}$ for every $\theta\in O(N-2)$.

We denote the set of all $(n,\ell)$-pinwheel partitions by
$\mathcal{P}^{n}_{\ell}$. If $\Omega$ is an open subset of $\mathbb{R}^{N}$
satisfying $(S_{2})$, a minimizer for

$\inf_{u\in\mathcal{M}_{\Omega}}\frac{p-1}{2p}\|u\|_{V}^{2}=:\mathfrak{c}_{\Omega}$

on the Nehari manifold

(1.7) $\displaystyle\mathcal{M}_{\Omega}:=$ $\displaystyle\\{u\in
H^{1}_{0}(\Omega):u\neq 0,\ \|u\|_{V}^{2}=\int_{\mathbb{R}^{N}}|u|^{2p},\text{
\ and \ }$ $\displaystyle u(\mathrm{e}^{2\pi\mathrm{i}/n}z,\theta
y)=u(z,y)\text{ \ for all \ }\theta\in O(N-2)\text{ \ and \
}(z,y)\in\Omega\\},$

is a _least energy solution_ to the problem

(1.8) $\begin{cases}-\Delta u+V(x)u=|u|^{2p-2}u,\qquad u\in
H_{0}^{1}(\Omega),\\\ u(\mathrm{e}^{2\pi\mathrm{i}/n}z,\theta
y)=u(z,y)\quad\text{for every \ }\theta\in O(N-2),\
(z,y)\in\Omega.\end{cases}$

We say that $(\Omega_{1},\ldots,\Omega_{\ell})$ is an _optimal
$(n,\ell)$-pinwheel partition_ for equation (1.5) if
$\mathfrak{c}_{\Omega_{j}}$ is attained on $\mathcal{M}_{\Omega_{j}}$ and

$\sum_{j=1}^{\ell}\mathfrak{c}_{\Omega_{j}}=\inf_{(\Theta_{1},\ldots,\Theta_{\ell})\in\mathcal{P}^{n}_{\ell}}\sum_{j=1}^{\ell}\mathfrak{c}_{\Theta_{j}}.$

######  Theorem 1.4.

Let $n\in\mathbb{N}$ and assume that $V$ satisfies $(V_{1})$, $(V_{2})$ and
$(V_{3}^{n})$. Let $\boldsymbol{u}_{k}=(u_{k,1},\ldots,u_{k,\ell})$ be a least
energy fully nontrivial solution to (1.2) and (1.3) with $\beta=\beta_{k}$.
Assume that $\beta_{k}\to-\infty$ as $k\to\infty$. Then, after passing to a
subsequence,

  * $(i)$

$u_{k,j}\to u_{\infty,j}$ strongly in $H^{1}(\mathbb{R}^{N})$,
$u_{\infty,j}\geq 0$, $u_{\infty,j}\neq 0$, $u_{\infty,i}u_{\infty,j}=0$ if
$i\neq j$, $\boldsymbol{u}_{\infty}=(u_{\infty,1},\ldots,u_{\infty,\ell})$
satisfies (1.3), and

$\int_{\mathbb{R}^{N}}\beta_{k}u_{k,j}^{p}u_{k,i}^{p}\to 0\text{ as
}k\to\infty\quad\text{whenever }i\neq j.$

  * $(ii)$

$u_{\infty,j}\in\mathcal{C}^{0}(\mathbb{R}^{N})$, the restriction of
$u_{\infty,j}$ to the open set
$\Omega_{j}:=\\{x\in\mathbb{R}^{N}:u_{\infty,j}(x)>0\\}$ is a least energy
solution to the problem (1.8) in $\Omega_{j}$, and
$(\Omega_{1},\ldots,\Omega_{\ell})$ is an optimal $(n,\ell)$-pinwheel
partition for equation (1.5).

  * $(iii)$

$\mathbb{R}^{N}\smallsetminus\bigcup_{j=1}^{\ell}\Omega_{j}=\mathscr{R}\cup\mathscr{S}$,
where $\mathscr{R}\cap\mathscr{S}=\emptyset$, $\mathscr{R}$ is an
$(m-1)$-dimensional $\mathcal{C}^{1,\alpha}$-submanifold of $\mathbb{R}^{N}$
and $\mathscr{S}$ is a closed subset of $\mathbb{R}^{N}$ with Hausdorff
measure $\leq m-2$. Furthermore, if $\xi\in\mathscr{R}$, there exist $i,j$
such that

$\lim_{x\to\xi^{+}}|\nabla u_{i}(x)|=\lim_{x\to\xi^{-}}|\nabla u_{j}(x)|\neq
0,$

where $x\to\xi^{\pm}$ are the limits taken from opposite sides of
$\mathscr{R}$ and, if $\xi\in\mathscr{S}$, then

$\lim_{x\to\xi}|\nabla u_{j}(x)|=0\qquad\text{for every \ }j=1,\ldots,\ell.$

  * $(iv)$

If $\ell=2$, then $u_{\infty,1}-u_{\infty,2}$ is a sign-changing solution to
equation (1.5) satisfying (1.6).

Note that $(iii)$ implies that the partition exhausts $\mathbb{R}^{N}$, i.e.,
$\mathbb{R}^{N}=\bigcup_{j=1}^{\ell}\overline{\Omega}_{j}$. Therefore, every
$\Omega_{j}$ is unbounded.

The regularity properties of optimal partitions have been established, in
different settings, for instance in [1, 4, 12, 15, 16] and some of the
references therein.

Theorem 1.4 establishes the existence of optimal partitions having an
additional property: each set of the partition is obtained from any other by
means of a linear isometry. Pinwheel partitions are an example of this type of
partitions, but others are conceivable. In Section 2 we present a general
symmetric variational setting for the system (1.2) that produces other
examples.

The existence of sign-changing solutions to equation (1.5) having the
additional property that their negative part is obtained from the positive one
by means of a linear isometry and a change of sign has been established in
[5]. This includes those given by Theorem 1.4$(iv)$. The tool for producing
this type of solutions is a homomorphism from some group of linear isometries
of $\mathbb{R}^{N}$ onto the group with two elements. As shown is Section 2
this tool also serves to get positive solutions of the system (1.2) for
$\ell=2$ with the property that each component is obtained from the other by
composition with a linear isometry. The general tool for obtaining a similar
result for the system (1.2) of $\ell$ equations is a homomorphism into the
group of permutations of a set of $\ell$ elements.

Rather than search for results in the general setting of Section 2, we
decided, for the sake of clarity, to look at pinwheel solutions only. The
solutions found in [13, 14] for $N=2,3$ and $p=2$ were of this type. Peng and
Wang [13] focused on the case where the potential $V$ is greater than its
limit at infinity and, for a system of two equations, they established the
existence of pinwheel solutions for $\beta$ sufficiently negative. Pistoia and
Vaira raised the question of whether solutions exist when $V$ is below its
limit at infinity and showed in [14] that the system (1.2) has a solution
satisfying (1.3) for $\beta$ close enough to $0$. The energy of each component
approaches $n\mathfrak{c}_{\infty}$ as $\beta\to 0$.

Our results can be easily extended to dimension $N=2$. In contrast, the
dimension $N=3$ requires a more delicate analysis because compactness can also
be lost by the presence of solutions to the autonomous system (with
$V=V_{\infty}$) that travel to infinity; see Remark 3.3.

The paper is organized as follows. In Section 2 we present the general
variational framework and in Section 3 we study the behavior of minimizing
sequences of pinwheel solutions for the system (1.2). In Section 4 we prove
Theorem 1.1 and Proposition 1.2. Section 5 is devoted to the proofs of
Theorems 1.3 and 1.4.

##  2 The symmetric variational setting

Let $G$ be a closed subgroup of the group $O(N)$ of linear isometries of
$\mathbb{R}^{N}$, and for $\ell\geq 2$ let $S_{\ell}$ be the group of
permutations of the set $\\{1,\ldots,\ell\\}$ acting on $\mathbb{R}^{\ell}$ in
the obvious way, i.e.,

$\sigma(u_{1},\ldots,u_{\ell})=(u_{\sigma(1)},\ldots,u_{\sigma(\ell)})\text{ \
for every \ }\sigma\in S_{\ell},\
(u_{1},\ldots,u_{\ell})\in\mathbb{R}^{\ell}.$

Let $\phi:G\to S_{\ell}$ be a continuous homomorphism of groups. A function
$\boldsymbol{u}:\mathbb{R}^{N}\to\mathbb{R}^{\ell}$ will be called _$\phi$
-equivariant_ if

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

Note that, if $\boldsymbol{u}:\mathbb{R}^{N}\to\mathbb{R}^{\ell}$ is
$\phi$-equivariant, then $\boldsymbol{u}$ is $K_{\phi}$-invariant, where
$K_{\phi}:=\ker(\phi)$.

These data define a $G$-action on
$\mathcal{H}:=(H^{1}(\mathbb{R}^{N}))^{\ell}$ as follows:

$(g\boldsymbol{u})(x):=\phi(g)\boldsymbol{u}(g^{-1}x)\text{ \ for every \
}g\in G,\ \boldsymbol{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{H}.$

For $u,v\in H_{0}^{1}(\mathbb{R}^{N})$ we set

$\langle u,v\rangle_{V}:=\int_{\mathbb{R}^{N}}(\nabla u\cdot\nabla
v+V(x)uv)\qquad\text{and}\qquad\|u\|_{V}:=\sqrt{\langle u,u\rangle_{V}}.$

The solutions to the system (1.2) are the positive critical points of the
functional $\mathcal{J}:\mathcal{H}\to\mathbb{R}$ given by

$\mathcal{J}(\boldsymbol{u}):=\frac{1}{2}\sum_{i=1}^{\ell}\|u_{i}\|_{V}^{2}-\frac{1}{2p}\sum_{i=1}^{\ell}\int_{\mathbb{R}^{N}}|u_{i}|^{2p}-\frac{\beta}{2p}\sum_{\begin{subarray}{c}i,j=1\\\
i\neq j\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|u_{i}|^{p}|u_{j}|^{p},$

which is of class $\mathcal{C}^{1}$. Its $i\text{-th}$ partial derivative is

$\partial_{i}\mathcal{J}(\boldsymbol{u})v=\langle
u_{i},v\rangle_{V}-\int_{\mathbb{R}^{N}}|u_{i}|^{2p-2}u_{i}v-\beta\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|u_{j}|^{p}|u_{i}|^{p-2}u_{i}v$

for any $\boldsymbol{u}\in\mathcal{H},\ v\in H^{1}(\mathbb{R}^{N})$. The
functional $\mathcal{J}$ is $G$-invariant, i.e.,

$\mathcal{J}(g\boldsymbol{u})=\mathcal{J}(\boldsymbol{u})\quad\text{ \ for
every \ }g\in G,\ \boldsymbol{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{H}.$

So, by the principle of symmetric criticality [17, Theorem 1.28], the critical
points of the restriction of $\mathcal{J}$ to the $G$-fixed point space of
$\mathcal{H}$,

$\mathcal{H}^{\phi}:=\\{\boldsymbol{u}\in\mathcal{H}:g\boldsymbol{u}=\boldsymbol{u}\
\forall g\in G\\}=\\{\boldsymbol{u}\in\mathcal{H}:\boldsymbol{u}\text{ is
}\phi\text{-equivariant}\\},$

are critical points of $\mathcal{J}$, i.e., they are the solutions to the
system (1.2) satisfying (2.1). We denote by $\mathcal{J}^{\phi}$ the
restriction of $\mathcal{J}$ to $\mathcal{H}^{\phi}$. Note that

$\displaystyle(\mathcal{J}^{\phi})^{\prime}(\boldsymbol{u})\boldsymbol{v}=\mathcal{J}^{\prime}(\boldsymbol{u})\boldsymbol{v}=\sum_{i=1}^{\ell}\partial_{i}\mathcal{J}(\boldsymbol{u})v_{i}\qquad\text{for
any \ }\boldsymbol{u},\boldsymbol{v}\in\mathcal{H}^{\phi}.$

The fully nontrivial critical points of $\mathcal{J}^{\phi}$ belong to the set

$\displaystyle\mathcal{N}^{\phi}:=\\{\boldsymbol{u}\in\mathcal{H}^{\phi}:u_{i}\neq
0,\ \partial_{i}\mathcal{J}(\boldsymbol{u})u_{i}=0\ \forall
i=1,\ldots,\ell\\}.$

Observe that

$\mathcal{J}^{\phi}(\boldsymbol{u})=\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{i}\|_{V}^{2}\qquad\text{if
\ }\boldsymbol{u}\in\mathcal{N}^{\phi}.$

Set

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

We consider also the single equation

(2.2) $-\Delta u+V(x)u=|u|^{2p-2}u,\qquad u\in H^{1}(\mathbb{R}^{N})^{G},$

where $H^{1}(\mathbb{R}^{N})^{G}:=\\{u\in H^{1}(\mathbb{R}^{N}):u\text{ is
}G\text{-invariant}\\}$, and we denote by
$J:H^{1}(\mathbb{R}^{N})^{G}\to\mathbb{R}$ and $\mathcal{M}^{G}$ energy
functional and the Nehari manifold associated to it, i.e.,

(2.3)
$J(u):=\frac{1}{2}\|u\|_{V}^{2}-\frac{1}{2p}\int_{\mathbb{R}^{N}}|u|^{2p}$

and

$\mathcal{M}^{G}:=\left\\{u\in H^{1}(\mathbb{R}^{N})^{G}:u\neq 0,\
\|u\|_{V}^{2}=\int_{\mathbb{R}^{N}}|u|^{2p}\right\\}.$

Similarly, we denote by $J_{\infty}:H^{1}(\mathbb{R}^{N})\to\mathbb{R}$ and
$\mathcal{M}_{\infty}$ be the energy functional and the Nehari manifold
associated to (1.4). Set

(2.4)
$\mathfrak{c}_{\infty}:=\inf_{u\in\mathcal{M}_{\infty}}J_{\infty}(u)\qquad\text{and}\qquad\mathfrak{c}^{G}:=\inf_{u\in\mathcal{M}^{G}}J(u).$

We shall focus our attention on the following example.

######  Example 2.1.

Let $\mathbb{Z}_{m}:=\\{\mathrm{e}^{2\pi\mathrm{i}j/m}:j=0,\ldots,m-1\\}$ act
on $\mathbb{C}$ by complex multiplication and $G_{m}:=\mathbb{Z}_{m}\times
O(N-2)$ act on $\mathbb{R}^{N}$ as

$\displaystyle\alpha x:=$ $\displaystyle(\alpha z,y)$
$\displaystyle\forall\alpha\in\mathbb{Z}_{m},$ $\displaystyle\theta x:=$
$\displaystyle(z,\theta y)$ $\displaystyle\forall\theta\in O(N-2),\quad
x=(z,y)\in\mathbb{C}\times\mathbb{R}^{N-2}\equiv\mathbb{R}^{N}.$

Let $\sigma_{1}\in S_{\ell}$ be the cyclic permutation
$\sigma_{1}(i):=i+1\mod\ell$, and $\phi_{n}:G_{\ell n}\to S_{\ell}$ be the
homomorphism given by $\phi_{n}(\mathrm{e}^{2\pi\mathrm{i}/\ell
n},\theta):=\sigma_{1}$ for any $\theta\in O(N-2)$. Then
$\boldsymbol{u}:\mathbb{R}^{N}\to\mathbb{R}^{\ell}$ is $\phi_{n}$-equivariant
iff

$(u_{1}(\mathrm{e}^{2\pi\mathrm{i}/\ell n}z,\theta
y),\ldots,u_{\ell}(\mathrm{e}^{2\pi\mathrm{i}/\ell n}z,\theta
y))=(u_{2}(z,y),\ldots,u_{\ell}(z,y),u_{1}(z,y))$

for every $(z,y)\in\mathbb{C}\times\mathbb{R}^{N-2}$ and $\theta\in O(N-2)$,
i.e., iff (1.3) hold true. Note that every $u_{j}$ is $G_{n}$-invariant.

##  3 The behavior of minimizing sequences

From now on we fix $n$ and we take $G_{n}$ and $\phi_{n}:G_{\ell n}\to
S_{\ell}$ as in Example 2.1. Then, for any
$\boldsymbol{u},\boldsymbol{v}\in\mathcal{H}^{\phi_{n}}$,

(3.1)
$\displaystyle(\mathcal{J}^{\phi_{n}})^{\prime}(\boldsymbol{u})\boldsymbol{v}=\sum_{i=1}^{\ell}\partial_{i}\mathcal{J}(\boldsymbol{u})v_{i}=\ell\,\partial_{j}\mathcal{J}(\boldsymbol{u})v_{j}\qquad\text{for
any \ }j=1,\ldots,\ell,$

and the set $\mathcal{N}^{\phi_{n}}$ is the usual Nehari manifold associated
to the functional
$\mathcal{J}^{\phi_{n}}:\mathcal{H}^{\phi_{n}}\to\mathbb{R}$, i.e.,

$\displaystyle\mathcal{N}^{\phi_{n}}=\\{\boldsymbol{u}\in\mathcal{H}^{\phi_{n}}:\boldsymbol{u}\neq
0,\ (\mathcal{J}^{\phi_{n}})^{\prime}(\boldsymbol{u})\boldsymbol{u}=0\\}.$

It has the following properties.

######  Proposition 3.1.

  * $(a)$

$\mathcal{N}^{\phi_{n}}\neq\emptyset$.

  * $(b)$

$c^{\phi_{n}}\geq\ell\mathfrak{c}^{G_{n}}>0$.

  * $(c)$

$\mathcal{N}^{\phi_{n}}$ is a closed $\mathcal{C}^{1}$-submanifold of
codimension $1$ of $\mathcal{H}^{\phi_{n}}$, and a natural constraint for
$\mathcal{J}^{\phi_{n}}$.

  * $(d)$

If $\boldsymbol{u}\in\mathcal{H}^{\phi_{n}}$ is such that, for each
$i=1,\ldots,\ell$,

$\int_{\mathbb{R}^{N}}|u_{i}|^{2p}+\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{\ell}\beta\int_{\mathbb{R}^{N}}|u_{i}|^{p}|u_{j}|^{p}>0,$

then there exists a unique $s_{\boldsymbol{u}}\in(0,\infty)$ such that
$s_{\boldsymbol{u}}\boldsymbol{u}\in\mathcal{N}^{\phi_{n}}$. Furthermore,

$\mathcal{J}^{\phi_{n}}(s_{\boldsymbol{u}}\boldsymbol{u})=\max_{s\in(0,\infty)}\mathcal{J}^{\phi_{n}}(s\boldsymbol{u}).$

  * $(e)$

$c^{\phi_{n}}\leq\ell n\mathfrak{c_{\infty}}$.

###### Proof.

The proof is easy. We give the details for the sake of completeness.

$(a):$ Let $\varphi\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{N})$ be a
nontrivial radial function such that
$\|\varphi\|_{V}^{2}=\int_{\mathbb{R}^{N}}|\varphi|^{2p}$. Set
$\xi_{i,j}:=(\mathrm{e}^{2\pi\mathrm{i}(i+\ell j)/\ell
n},0)\in\mathbb{C}\times\mathbb{R}^{N-2}\equiv\mathbb{R}^{N}$ and define

$\displaystyle u_{R,i+1}(x):=\sum_{j=0}^{n-1}\varphi(x-R\xi_{i,j}),\qquad
i=0,\ldots,n-1,$

where $R>0$ is taken large enough so that $u_{R,i}$ and $u_{R,j}$ have
disjoint supports for every $i\neq j$. Then,
$\boldsymbol{u}_{R}:=(u_{R,1},\ldots,u_{R,1})\in\mathcal{N}^{\phi_{n}}$.

$(b):$ Let $\boldsymbol{u}=(u_{1},\ldots,u_{\ell})\in\mathcal{N}^{\phi_{n}}$.
As $\beta<0$ we have

$0<\|u_{i}\|_{V}^{2}=\|u_{1}\|_{V}^{2}\leq\int_{\mathbb{R}^{N}}|u_{1}|^{2p}=\int_{\mathbb{R}^{N}}|u_{i}|^{2p}\qquad\forall
i=2,\ldots,\ell.$

Hence, there exists $s\in(0,1]$ such that $su_{i}\in\mathcal{M}^{G_{n}}$ for
every $i=1,\ldots,\ell$. Therefore,

$\ell\mathfrak{c}^{G_{n}}\leq\sum_{i=1}^{\ell}J(s_{i}u_{i})=\frac{p-1}{2p}\sum_{i=1}^{\ell}\|s_{i}u_{i}\|^{2}_{V}\leq\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{i}\|^{2}_{V}=\mathcal{J}^{\phi}(\boldsymbol{u}).$

It follows that $\ell\mathfrak{c}^{G_{n}}\leq c^{\phi_{n}}$.

$(c):$ The function
$\Psi:\mathcal{H}^{\phi_{n}}\smallsetminus\\{0\\}\to\mathbb{R}$ given by
$\Psi(\boldsymbol{u}):=(\mathcal{J}^{\phi_{n}})^{\prime}(\boldsymbol{u})\boldsymbol{u}$
is of class $\mathcal{C}^{1}$ and $\mathcal{N}^{\phi_{n}}=\Psi^{-1}(0)$. It
follows from $(b)$ that $\mathcal{N}^{\phi_{n}}$ is a closed subset of
$\mathcal{H}^{\phi_{n}}$. As

$\Psi^{\prime}(\boldsymbol{u})\boldsymbol{u}=(2-2p)\ell\,\|u_{1}\|_{V}^{2}\neq
0,$

we have that $0$ is a regular value of $\Psi$. This shows that
$\mathcal{N}^{\phi_{n}}$ is a $\mathcal{C}^{1}$-submanifold of codimension $1$
of $\mathcal{H}^{\phi_{n}}$. It also shows that
$\boldsymbol{u}\not\in\ker\Psi^{\prime}(\boldsymbol{u})=:T_{\boldsymbol{u}}\mathcal{N}^{\phi_{n}}$,
the tangent space of $\mathcal{N}^{\phi_{n}}$ at $\boldsymbol{u}$. Hence,

$\mathcal{H}^{\phi_{n}}=T_{\boldsymbol{u}}\mathcal{N}^{\phi_{n}}\oplus\mathbb{R}\boldsymbol{u}.$

Since, by definition,
$(\mathcal{J}^{\phi_{n}})^{\prime}(\boldsymbol{u})\boldsymbol{u}=0$ for every
$\boldsymbol{u}\in\mathcal{N}^{\phi_{n}}$, we infer that a critical point of
the restriction of $\mathcal{J}^{\phi_{n}}$ to $\mathcal{N}^{\phi_{n}}$ is a
critical point of $\mathcal{J}^{\phi_{n}}$.

$(d):$ The proof is straightforward. The number $s_{\boldsymbol{u}}$ is

$s_{\boldsymbol{u}}=\left(\frac{\|u_{1}\|_{V}^{2}}{\int_{\mathbb{R}^{N}}|u_{1}|^{2p}+\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
1\end{subarray}}^{\ell}\beta\int_{\mathbb{R}^{N}}|u_{1}|^{p}|u_{j}|^{p}}\right)^{1/(2p-2)}.$

$(e):$ Let $\omega$ be the least energy positive radial solution to (1.4) and
set $\xi_{i,j}=(\mathrm{e}^{2\pi\mathrm{i}(i+\ell j)/\ell
n},0)\in\mathbb{C}\times\mathbb{R}^{N-2}\equiv\mathbb{R}^{N}$. Define

$\displaystyle w_{R,i+1}(x)$
$\displaystyle:=\sum_{j=0}^{n-1}\omega(x-R\xi_{i,j}),\qquad i=0,\ldots,n-1.$

Then
$\boldsymbol{w}_{R}=(w_{R,1},\ldots,w_{R,\ell})\in\mathcal{H}^{\phi_{n}}$. If
$R$ is sufficiently large, statement $(d)$ yields $s_{R}\in(0,\infty)$ such
that $s_{R}\boldsymbol{w}_{R}\in\mathcal{N}^{\phi_{n}}$ and $s_{R}\to 1$ as
$R\to\infty$. Using assumption $(V_{2})$ we obtain

$\displaystyle c^{\phi_{n}}$
$\displaystyle\leq\mathcal{J}^{\phi_{n}}(s_{R}\boldsymbol{w}_{R})=\frac{p-1}{2p}\sum_{i=1}^{\ell}\|s_{R,i}w_{R,i}\|_{V}^{2}\to\ell
n\mathfrak{c}_{\infty}\quad\text{as \ }R\to\infty.$

This shows that $c^{\phi_{n}}\leq\ell n\mathfrak{c_{\infty}}$, as claimed. ∎

######  Lemma 3.2.

Let $(x_{k})$ be a sequence in $\mathbb{R}^{N}$, $N\geq 4$. After passing to a
subsequence, there exists a sequence $(\xi_{k})$ in $\mathbb{R}^{N}$ and a
constant $C_{0}>0$ such that

$|x_{k}-\xi_{k}|\leq C_{0}\qquad\text{for all \ }k\in\mathbb{N},$

and one of the following statements holds true:

  * •

either $\xi_{k}=0$ for all $k$,

  * •

or $\xi_{k}=(\zeta_{k},0)\in\mathbb{C}\times\mathbb{R}^{N-2}$ and
$|\zeta_{k}|\to\infty$,

  * •

or for each $m\in\mathbb{N}$ there exist $\gamma_{1},\ldots,\gamma_{m}\in
O(N-2)$ such that $|\gamma_{i}\xi_{k}-\gamma_{j}\xi_{k}|\to\infty$ if $i\neq
j$.

###### Proof.

See [3, Lemma 3.1]. ∎

######  Remark 3.3.

_Note that this lemma is not true in dimension $N=3$ because
$O(1)=\\{1,-1\\}$._

######  Theorem 3.4.

Let $\boldsymbol{u}_{k}=(u_{k,1},\ldots,u_{k,\ell})\in\mathcal{N}^{\phi_{n}}$
be such that $\mathcal{J}^{\phi_{n}}(\boldsymbol{u}_{k})\to c^{\phi_{n}}$ and
$u_{k,i}\geq 0$. Then, after passing to a subsequence, either
$\boldsymbol{u}_{k}\to\boldsymbol{u}$ strongly in $\mathcal{H}^{\phi_{n}}$
with $u_{i}\geq 0$, or there are points
$(z_{k},0)\in\mathbb{C}\times\mathbb{R}^{N-2}\equiv\mathbb{R}^{N}$ such that
$|z_{k}|\to\infty$,

$\lim_{k\to\infty}\Big{\|}u_{k,1}-\sum_{j=0}^{n}\omega\big{(}\ \cdot\
-\,(\mathrm{e}^{2\pi\mathrm{i}j/n}z_{k},0)\big{)}\Big{\|}=0,$

and $c^{\phi_{n}}=\ell n\mathfrak{c}_{\infty}$, where $\omega$ is the least
energy positive radial solution to (1.4).

###### Proof.

Invoking Ekeland’s variational principle [17, Theorem 8.5] we may assume that
$(\mathcal{J}^{\phi_{n}})^{\prime}(\boldsymbol{u}_{k})\to 0$ in
$(\mathcal{H}^{\phi_{n}})^{\prime}$.

Since $\beta<0$, Proposition 3.1$(b)$ yields $c_{0}>0$ such that

$\int_{\mathbb{R}^{N}}|u_{k,1}|^{2p}>c_{0}\qquad\forall k\in\mathbb{N}.$

By Lions’ lemma [17, Lemma 1.21] there exist $\delta>0$ and
$x_{k}\in\mathbb{R}^{N}$ such that, after passing to a subsequence,

$\int_{B_{1}(x_{k})}|u_{k,1}|^{2p}>\delta\qquad\forall k\in\mathbb{N}.$

For $(x_{k})$ we fix a sequence $(\xi_{k})$ and a constant $C_{0}>0$ such that
$|x_{k}-\xi_{k}|\leq C_{0}$ for all $k\in\mathbb{N}$, satisfying one of the
alternatives stated in Lemma 3.2. Then,

(3.2)
$\int_{B_{C_{0}+1}(\xi_{k})}|u_{k,1}|^{2p}\geq\int_{B_{1}(x_{k})}|u_{k,1}|^{2p}>\delta\qquad\forall
k\in\mathbb{N}.$

It follows that, either $\xi_{k}=0$, or
$\xi_{k}=(\zeta_{k},0)\in\mathbb{C}\times\mathbb{R}^{N-2}$ and
$\zeta_{k}\to\infty$. Otherwise, by Lemma 3.2, for each $m\in\mathbb{N}$ there
would exist $\gamma_{1},\ldots,\gamma_{m}\in O(N-2)$ such that
$|\gamma_{i}\xi_{k}-\gamma_{j}\xi_{k}|\geq 2(C_{0}+1)$ if $i\neq j$ for large
enough $k\in\mathbb{N}$ and, as $u_{k,1}$ is $G_{n}$-invariant, we would have
that

$\displaystyle\int_{\mathbb{R}^{N}}|u_{k,1}|^{2p}\geq\sum_{i=1}^{m}\int_{B_{C_{0}+1}(\gamma_{i}\xi_{k})}|u_{k,1}|^{2p}=m\int_{B_{C_{0}+1}(\xi_{k})}|u_{k,1}|^{2p}>m\delta,$

for all $m\in\mathbb{N}$. This is impossible because $(u_{k,1})$ is bounded in
$L^{2p}(\mathbb{R}^{N})$.

Next, we distinguish two cases.

Case 1. $\xi_{k}=0$ for all $k\in\mathbb{N}$.

Since the sequence $(u_{k,1})$ is bounded in $H^{1}(\mathbb{R}^{N})$, passing
to a subsequence, we have that $u_{k,1}\rightharpoonup u_{1}$ weakly in
$H^{1}(\mathbb{R}^{N})$, $u_{k,1}\to u_{1}$ in
$L^{2p}_{\mathrm{loc}}(\mathbb{R}^{N})$ and $u_{k,1}\to u_{1}$ a.e. in
$\mathbb{R}^{N}$. Hence, $u_{1}\geq 0$ and it follows from (3.2) that
$u_{1}\neq 0$. Note that, as $u_{k,1}\in H^{1}(\mathbb{R}^{N})^{G_{n}}$,
$u_{1}\in H^{1}(\mathbb{R}^{N})^{G_{n}}$. Set
$u_{j+1}(z,y):=u_{1}(\mathrm{e}^{2\pi\mathrm{i}j/\ell n}z,y)$ for
$(z,y)\in\mathbb{C}\times\mathbb{R}^{N-2}$, $j=1,\ldots,\ell-1$, and
$\boldsymbol{u}=(u_{1},\ldots,u_{\ell})$. Then, $u_{k,j+1}\rightharpoonup
u_{j+1}$ weakly in $H^{1}(\mathbb{R}^{N})$ and, as
$(\mathcal{J}^{\phi_{n}})^{\prime}(\boldsymbol{u}_{k})\to 0$ in
$(\mathcal{H}^{\phi_{n}})^{\prime}$, we derive from (3.1) that

$\displaystyle
0=\lim_{k\to\infty}\partial_{1}\mathcal{J}(\boldsymbol{u}_{k})\varphi=\partial_{1}\mathcal{J}(\boldsymbol{u})\varphi\qquad\text{for
every \ }\varphi\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{N})^{G_{n}}.$

Hence, $\boldsymbol{u}\in\mathcal{N}^{\phi_{n}}$ and

$\displaystyle c^{\phi_{n}}$
$\displaystyle\leq\mathcal{J}^{\phi_{n}}(\boldsymbol{u})=\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{i}\|_{V}^{2}\leq\liminf_{k\to\infty}\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{k,i}\|_{V}^{2}$
$\displaystyle=\lim_{k\to\infty}\mathcal{J}^{\phi_{n}}(\boldsymbol{u}_{k})=c^{\phi_{n}}.$

Therefore, $\boldsymbol{u}_{k}\to\boldsymbol{u}$ strongly in
$\mathcal{H}^{\phi_{n}}$. This shows that, in Case 1, the first alternative
stated in Theorem 3.4 holds true.

Case 2. $\xi_{k}=(\zeta_{k},0)\in\mathbb{C}\times\mathbb{R}^{N-2}$ and
$\zeta_{k}\to\infty$.

Set

$w_{k,i}(x):=u_{k,i}(x+\xi_{k}),\qquad i=1,\ldots,\ell.$

Note that $w_{k,i}$ is $O(N-2)$-invariant. Since the sequence $(w_{k,i})$ is
bounded in $H^{1}(\mathbb{R}^{N})$, a subsequence satisfies
$w_{k,i}\rightharpoonup w_{i}$ weakly in $H^{1}(\mathbb{R}^{N})^{O(N-2)}$,
$w_{k,i}\to w_{i}$ in $L^{2p}_{\mathrm{loc}}(\mathbb{R}^{N})$ and $w_{k,i}\to
w_{i}$ a.e. in $\mathbb{R}^{N}$. Hence, $w_{i}\geq 0$. To simplify notation,
set $\alpha:=\mathrm{e}^{2\pi\mathrm{i}/n}$. Note that, as
$|\alpha^{j}\xi_{k}-\alpha^{m}\xi_{k}|\to\infty$ if $j\neq m$, we have that

$w_{k,i}\circ\alpha^{-m}-\sum_{j=m+1}^{n-1}(w_{i}\circ\alpha^{-m})(\,\cdot\,-\alpha^{j}\xi_{k}+\alpha^{m}\xi_{k})\rightharpoonup
w_{i}\circ\alpha^{-m}$

weakly in $H^{1}(\mathbb{R}^{N})$. Hence, setting $V_{k}(x):=V(x+\xi_{k})$,
Lemma A.1 gives

$\displaystyle\|w_{i}\circ\alpha^{-m}\|_{V_{\infty}}^{2}$
$\displaystyle=\Big{\|}w_{k,i}\circ\alpha^{-m}-\sum_{j=m+1}^{n-1}(w_{i}\circ\alpha^{-j})(\,\cdot\,-\alpha^{j}\xi_{k}+\alpha^{m}\xi_{k})\Big{\|}_{V_{k}}^{2}$
$\displaystyle\quad-\Big{\|}w_{k,i}\circ\alpha^{-m}-\sum_{j=m}^{n-1}(w_{i}\circ\alpha^{-j})(\,\cdot\,-\alpha^{j}\xi_{k}+\alpha^{m}\xi_{k})\Big{\|}_{V_{k}}^{2}+o(1).$

Since $u_{k,i}$ is $G_{n}$-invariant, the change of variable
$y=z-\alpha^{m}\xi_{k}$ yields

$\displaystyle\Big{\|}u_{k,i}-\sum_{j=m+1}^{n-1}(w_{i}\circ\alpha^{-j})(\,\cdot\,-\alpha^{j}\xi_{k})\Big{\|}_{V}^{2}$
$\displaystyle\qquad=\Big{\|}u_{k,i}-\sum_{j=m}^{n-1}(w_{i}\circ\alpha^{-j})(\,\cdot\,-\alpha^{j}\xi_{k})\Big{\|}_{V}^{2}+\|w_{i}\|_{V_{\infty}}^{2}+o(1),$

and iterating this identity we obtain

(3.3)
$\|u_{k,i}\|_{V}^{2}=\Big{\|}u_{k,i}-\sum_{j=0}^{n-1}(w_{i}\circ\alpha^{-j})(\,\cdot\,-\alpha^{j}\xi_{k})\Big{\|}_{V}^{2}+n\|w_{i}\|_{V_{\infty}}^{2}+o(1).$

On the other hand, for any given $v\in H^{1}(\mathbb{R}^{N})^{O(N-2)}$ set
$v_{k}(y):=v(y-\xi_{k})$ and

$\widehat{v}_{k}(y):=\sum_{j=0}^{n-1}v_{k}(\alpha^{j}y).$

Recalling that $u_{k,i}$ is $G_{n}$-invariant and performing the translation
$y=x+\xi_{k}$, we obtain

$\displaystyle\partial_{i}\mathcal{J}(\boldsymbol{u}_{k})\widehat{v}_{k}$
$\displaystyle=\sum_{j=0}^{n-1}\partial_{i}\mathcal{J}(\boldsymbol{u}_{k})(v_{k}\circ\alpha^{j})=n\,\partial_{i}\mathcal{J}(\boldsymbol{u}_{k})v_{k}$
$\displaystyle=n\Big{(}\int_{\mathbb{R}^{N}}(\nabla w_{k,i}\cdot\nabla
v+V_{k}(x)w_{k,i}v)-\int_{\mathbb{R}^{N}}|w_{k,i}|^{2p-2}w_{k,i}v$
$\displaystyle\qquad-\beta\sum\limits_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|w_{k,j}|^{p}|w_{k,i}|^{p-2}w_{k,i}v\Big{)}.$

Note that $\widehat{v}_{k}$ is $G_{n}$-invariant. As
$(\mathcal{J}^{\phi_{n}})^{\prime}(\boldsymbol{u}_{k})\to 0$, invoking (3.1)
and assumption $(V_{2})$, and passing to the limit as $k\to\infty$ we get

(3.4) $0=\int_{\mathbb{R}^{N}}(\nabla w_{i}\cdot\nabla
v+V_{\infty}w_{i}v)-\int_{\mathbb{R}^{N}}|w_{i}|^{2p-2}w_{i}v-\beta\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|w_{j}|^{p}|w_{i}|^{p-2}w_{i}v$

for every $v\in H^{1}(\mathbb{R}^{N})^{O(N-2)}$ and $i=1,\ldots,\ell$. Since,
by (3.2),

$\int_{B_{C_{0}+1}(0)}|w_{k,1}|^{2p}\geq\int_{B_{C_{0}+1}(\xi_{k})}|u_{k,1}|^{2p}\geq\delta>0,$

we see that $w_{1}\neq 0$. Furthermore, equation (3.4) implies that

(3.5)
$\|w_{1}\|_{V_{\infty}}^{2}=\int_{\mathbb{R}^{N}}|w_{1}|^{2p}+\beta\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
1\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|w_{j}|^{p}|w_{1}|^{p}\leq\int_{\mathbb{R}^{N}}|w_{1}|^{2p},$

so there exists $t\in(0,1]$ such that
$\|tw_{1}\|_{V_{\infty}}^{2}=\int_{\mathbb{R}^{N}}|tw_{1}|^{2p}$. It follows
that $tw_{1}\in\mathcal{M}_{\infty}$, and from equation (3.3) and Proposition
3.1$(e)$ we derive

$\displaystyle
n\mathfrak{c}_{\infty}\leq\frac{p-1}{2p}n\|tw_{1}\|_{V_{\infty}}^{2}\leq\frac{p-1}{2p}n\|w_{1}\|_{V_{\infty}}^{2}\leq\lim_{k\to\infty}\frac{p-1}{2p}\|u_{k,1}\|_{V}^{2}=\frac{1}{\ell}c^{\phi_{n}}\leq
n\mathfrak{c}_{\infty}.$

Therefore, $t=1$, $w_{1}\in\mathcal{M}_{\infty}$ and
$J_{\infty}(w_{1})=\frac{p-1}{2p}\|w_{1}\|_{V_{\infty}}^{2}=\mathfrak{c}_{\infty}$,
i.e., $w_{1}$ is a least energy solution of (1.4). Moreover, from (3.3) we get
that

$\lim_{k\to\infty}\Big{\|}u_{k,1}-\sum_{j=0}^{n-1}(w_{1}\circ\alpha^{-j})(\,\cdot\,-\alpha^{j}\xi_{k})\Big{\|}_{V}^{2}=0.$

Since the positive least energy solution to (1.4) is unique up to translation
and $w_{1}$ is $O(N-2)$-invariant, there exists
$\xi=(\zeta,0)\in\mathbb{C}\times\mathbb{R}^{N-2}$ such that
$w_{1}(x)=\omega(x+\xi)$. Hence,
$(w_{1}\circ\alpha^{-j})(x-\alpha^{j}\xi_{k})=\omega(\alpha^{-j}x-\xi_{k}-\xi)=\omega(x-\alpha^{j}(\xi_{k}+\xi))$.
So, setting $z_{k}:=\zeta_{k}+\zeta$, we obtain

$\lim_{k\to\infty}\Big{\|}u_{k,1}-\sum_{j=0}^{n}\omega\big{(}\ \cdot\
-\,(\mathrm{e}^{2\pi\mathrm{i}j/n}z_{k},0)\big{)}\Big{\|}=0.$

This shows that, in Case 2, the second alternative stated in Theorem 3.4 holds
true. ∎

######  Corollary 3.5.

If $c^{\phi_{n}}<\ell n\mathfrak{c}_{\infty}$, the system (1.2) has a least
energy fully nontrivial solution satisfying (1.3).

##  4 Existence of a solution

We define the set of _weak $(n,\ell)$-pinwheel partitions_ as

$\displaystyle\mathcal{W}^{n}_{\ell}:=\\{(u_{1},\ldots,u_{\ell})\in\mathcal{H}^{\phi_{n}}:u_{i}\neq
0,\,\|u_{i}\|_{V}^{2}=|u_{i}|_{2p}^{2p},\,u_{i}u_{j}=0\text{ in
}\mathbb{R}^{N}\text{ if }i\neq j\\},$

and set

$\widehat{c}^{\phi_{n}}:=\inf_{(u_{1},\ldots,u_{\ell})\in\mathcal{W}^{n}_{\ell}}\,\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{i}\|_{V}^{2}.$

Our next goal is to give an upper estimate for $\widehat{c}^{\phi_{n}}$. To
this end, we choose $\varepsilon\in\big{(}0,\frac{d_{\ell n}-\lambda}{d_{\ell
n}+\lambda}\big{)}$ and a radial function
$\chi\in\mathcal{C}^{\infty}(\mathbb{R}^{N})$ satisfying $0\leq\chi\leq 1$,
$\chi(x)=1$ if $|x|\leq 1-\varepsilon$ and $\chi(x)=0$ if $|x|\geq 1$. Let
$\omega$ be the positive least energy radial solution to (1.4). For each $r>0$
define

$\omega_{r}(x):=\chi\left(\frac{x}{r}\right)\omega(x).$

######  Lemma 4.1.

As $r\to\infty$,

$\left|\|\omega\|^{2}-\|\omega_{r}\|^{2}\right|=O(\mathrm{e}^{-2(1-\varepsilon)\sqrt{V_{\infty}}r}),\qquad\left||\omega|^{2p}_{2p}-|\omega_{r}|^{2p}_{2p}\right|=O(\mathrm{e}^{-2p(1-\varepsilon)\sqrt{V_{\infty}}r}),$

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

###### Proof.

This statements follow easily from the well known estimates
$|\omega(x)|=O(|x|^{-\frac{N-1}{2}}\mathrm{e}^{-\sqrt{V_{\infty}}|x|})$ and
$|\nabla\omega(x)|=O(|x|^{-\frac{N-1}{2}}\mathrm{e}^{-\sqrt{V_{\infty}}|x|})$,
as in [7, Lemma 2]. ∎

Set $\varrho:=\frac{d_{\ell n}+\lambda}{4}$, and for $R>1$ define

$\widehat{w}_{1,R}(x):=\sum_{j=0}^{n-1}\omega_{\varrho
R}(x-R(\mathrm{e}^{2\pi\mathrm{i}j/n},0))\qquad\text{and}\qquad
w_{1,R}:=t_{R}\widehat{w}_{1,R},$

where $t_{R}\in(0,\infty)$ is such that
$\|w_{1,R}\|_{V}^{2}=|w_{1,R}|_{2p}^{2p}$. Note that $t_{R}\to 1$ as
$R\to\infty$, $w_{1,R}$ is $G_{n}$-invariant and

$\text{supp}\big{(}\omega_{\varrho
R}(\,\cdot\,-R(\mathrm{e}^{2\pi\mathrm{i}j/\ell
n},0))\big{)}\subset\overline{B_{\varrho R}(R(\mathrm{e}^{2\pi\mathrm{i}j/\ell
n},0))}.$

Set $w_{j+1,R}(\mathrm{e}^{2\pi\mathrm{i}j/\ell n}z,y):=w_{1,R}(z,y)$ for
$(z,y)\in\mathbb{C}\times\mathbb{R}^{N-2}$ and $j=1,\ldots,\ell-1$. Since
$\varrho<\frac{d_{\ell n}}{2}$ we have that
$\text{supp}(w_{i,R})\cap\text{supp}(w_{j,R})=\emptyset$ if $i\neq j$. Hence,
$\boldsymbol{w}_{R}=(w_{1,R},\ldots,w_{\ell,R})\in\mathcal{W}^{n}_{\ell}$.

######  Lemma 4.2.

There exist $C_{1},R_{1}>0$ such that

$\frac{p-1}{2p}\sum_{i=1}^{\ell}\|w_{i,R}\|_{V}^{2}=\mathcal{J}^{\phi_{n}}(\boldsymbol{w}_{R})\leq\ell
n\mathfrak{c}_{\infty}-C_{1}\mathrm{e}^{-\lambda\sqrt{V_{\infty}}R}\quad\text{for
all \ }R\geq R_{1}.$

###### Proof.

Since
$\boldsymbol{w}_{R}=(w_{1,R},\ldots,w_{\ell,R})\in\mathcal{W}^{n}_{\ell}$, the
equality holds true. To prove the inequality note that that
$t_{R}\in[\frac{1}{2},2]$ for $R$ large enough. Assumption $(V_{3}^{n})$
yields

$\displaystyle\int_{\mathbb{R}^{N}}(V(x)-V_{\infty})|t_{R}\omega_{\varrho
R}(x-R(1,0))|^{2}\,\mathrm{d}x$ $\displaystyle\qquad=\int_{|x|\leq\varrho
R}\big{(}V(x+R(1,0))-V_{\infty}\big{)}|t_{R}\omega_{\varrho
R}(x)|^{2}\,\mathrm{d}x$
$\displaystyle\qquad=-\frac{C_{0}}{4}\int_{|x|\leq\varrho
R}\mathrm{e}^{-\lambda\sqrt{V_{\infty}}\,|x+R(1,0)|}|\omega(x)|^{2}\,\mathrm{d}x$
$\displaystyle\qquad\leq-\frac{C_{0}}{4}\Big{(}\int_{\mathbb{R}^{N}}\mathrm{e}^{-\lambda\sqrt{V_{\infty}}|x|}|\omega(x)|^{2}\,\mathrm{d}x\Big{)}\mathrm{e}^{-\lambda\sqrt{V_{\infty}}R}=:-2C\mathrm{e}^{-\lambda\sqrt{V_{\infty}}R}.$

Using Lemma 4.1, for $R$ large enough we get

$\displaystyle\mathcal{J}^{\phi_{n}}(\boldsymbol{w}_{R})=\frac{1}{2}\sum_{i=1}^{\ell}\|w_{i,R}\|_{V}^{2}-\frac{1}{2p}\sum_{i=1}^{\ell}\int_{\mathbb{R}^{N}}|w_{i,R}|^{2p}-\frac{\beta}{2p}\sum_{\begin{subarray}{c}i,j=1\\\
i\neq j\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|w_{i,R}|^{p}|w_{j,R}|^{p}$
$\displaystyle=\ell n\left(\frac{1}{2}\|t_{R}\omega_{\varrho
R}(\,\cdot\,-R(1,0))\|_{V}^{2}-\frac{1}{2p}|t_{R}\omega_{\varrho
R}(\,\cdot\,-R(1,0))|^{2p}_{2p}\right)$ $\displaystyle=\ell
n\left(\frac{1}{2}\|t_{R}\omega_{\varrho
R}\|_{V_{\infty}}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}(V-V_{\infty})|t_{R}\omega_{\varrho
R}(\,\cdot\,-R(1,0))|^{2}-\frac{1}{2p}|t_{R}\omega_{\varrho
R}|^{2p}_{2p}\right)$ $\displaystyle=\ell
n\left(\frac{1}{2}\|t_{R}\omega\|_{V_{\infty}}^{2}-C\mathrm{e}^{-\lambda\sqrt{V_{\infty}}R}-\frac{1}{2p}|t_{R}\omega|^{2p}_{2p}+\,O\big{(}\mathrm{e}^{-2(1-\varepsilon)\sqrt{V_{\infty}}\varrho
R}\big{)}\right)$ $\displaystyle\leq\ell
n\mathfrak{c}_{\infty}-C_{1}\mathrm{e}^{-\lambda\sqrt{V_{\infty}}R},$

because $2(1-\varepsilon)\varrho>\frac{d_{\ell
n}+\lambda}{2}\big{(}1-\frac{d_{\ell n}-\lambda}{d_{\ell
n}+\lambda}\big{)}=\lambda$. ∎

###### Proof of Theorem 1.1.

Note that $\mathcal{W}^{n}_{\ell}\subset\mathcal{N}^{\phi_{n}}$. Hence, from
Lemma 4.2 we get

$c^{\phi_{n}}\leq\widehat{c}^{\phi_{n}}<\ell n\mathfrak{c}_{\infty},$

and Corollary 3.5 yields the result. ∎

###### Proof of Proposition 1.2.

Arguing by contradiction, assume that $\boldsymbol{u}$ is a solution to (1.2)
satisfying (1.3) with $n=\ell^{m}$ and with $n=\ell^{q}$ and that $1\leq m<q$.
Then, for $k=\ell^{q-m-1}j$ with $j=1,\ldots,\ell-1$ we have that

$u_{1}(x)=u_{1}(\mathrm{e}^{2\pi\mathrm{i}k/\ell^{q}}x)=u_{1}(\mathrm{e}^{2\pi\mathrm{i}j/\ell^{m}\ell}x)=u_{j+1}(x)$

and, as $1+\beta(\ell-1)\leq 0$, we obtain

$\displaystyle\|u_{1}\|_{V}^{2}=\int_{\mathbb{R}^{N}}|u_{1}|^{2p}+\beta\sum_{j=1}^{\ell-1}|u_{j+1}|^{p}|u_{1}|^{p}=(1+\beta(\ell-1))\int_{\mathbb{R}^{N}}|u_{1}|^{2p}\leq
0,$

a contradiction. ∎

##  5 The limit profiles of the solutions

We start with the case $\beta\to 0$.

###### Proof of Theorem 1.3.

We write $\mathcal{J}_{k}^{\phi_{n}}$ and $\mathcal{N}_{k}^{\phi_{n}}$ for the
functional and the Nehari set associated to the system (1.2) with
$\beta=\beta_{k}$, and we define

$c_{k}^{\phi_{n}}:=\inf_{\mathcal{N}_{k}^{\phi_{n}}}\mathcal{J}_{k}^{\phi_{n}}.$

As $\mathcal{W}^{n}_{\ell}\subset\mathcal{N}_{k}^{\phi_{n}}$ for every
$k\in\mathbb{N}$, invoking Lemma 4.2 we see that

(5.1)
$\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{k,i}\|_{V}^{2}=c_{k}^{\phi_{n}}\leq\widehat{c}^{\phi_{n}}<\ell
n\mathfrak{c}_{\infty}\qquad\forall k\in\mathbb{N}.$

After passing to a subsequence, we have that $u_{k,i}\rightharpoonup u_{0,i}$
weakly in $H^{1}(\mathbb{R}^{N})$, $u_{k,i}\to u_{0,i}$ strongly in
$L^{2}_{\mathrm{loc}}(\mathbb{R}^{N})$ and $u_{k,i}\to u_{0,i}$ a.e. in
$\mathbb{R}^{N}$, for each $i=1,\ldots,\ell$. Hence, $u_{0,i}\geq 0$ and
$\boldsymbol{u}_{0}=(u_{0,1},\ldots,u_{0,\ell})\in\mathcal{H}^{\phi_{n}}$.

We claim that

$u_{0,i}\neq 0\qquad\forall i=1,\ldots,\ell.$

To prove this claim assume, by contradiction, that $u_{0,i}=0$. Following the
argument in the proof of Theorem 3.4 we see that, after passing to a
subsequence, there exist $\xi_{k}\in\mathbb{R}^{N}$, $C_{0}>0$ and $\delta>0$
such that

(5.2) $\int_{B_{C_{0}+1}(\xi_{k})}|u_{k,i}|^{2p}>\delta>0\qquad\forall
k\in\mathbb{N},$

where, either $\xi_{k}=0$, or
$\xi_{k}=(\zeta_{k},0)\in\mathbb{C}\times\mathbb{R}^{N-2}$ and
$\zeta_{k}\to\infty$. Since $u_{k,i}\to 0$ strongly in
$L^{2}_{\mathrm{loc}}(\mathbb{R}^{N})$, equation (5.2) implies that
$\xi_{k}\neq 0$. Now, as in Case 2 of Theorem 3.4, we set

$w_{k,i}(x):=u_{k,i}(x+\xi_{k}),\qquad i=1,\ldots,\ell,$

and we take a subsequence satisfying $w_{k,i}\rightharpoonup w_{i}$ weakly in
$H^{1}(\mathbb{R}^{N})$, $w_{k,i}\to w_{i}$ in
$L^{2p}_{\mathrm{loc}}(\mathbb{R}^{N})$ and $w_{k,i}\to w_{i}$ a.e. in
$\mathbb{R}^{N}$. Hence, $w_{i}\in H^{1}(\mathbb{R}^{N})^{O(N-2)}$, $w_{i}\geq
0$ and following the proof of (3.3) we obtain

(5.3)
$\|u_{k,i}\|_{V}^{2}=\Big{\|}u_{k,i}-\sum_{j=0}^{n-1}(w_{i}\circ\alpha^{-j})(\,\cdot\,-\alpha^{j}\xi_{k})\Big{\|}_{V}^{2}+n\|w_{i}\|_{V_{\infty}}^{2}+o(1).$

Furthermore, following the proof of (3.4) we derive

$\int_{\mathbb{R}^{N}}(\nabla w_{i}\cdot\nabla
v+V_{\infty}w_{i}v)=\int_{\mathbb{R}^{N}}|w_{i}|^{2p-2}w_{i}v+\beta_{k}\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|w_{j}|^{p}|w_{i}|^{p-2}w_{i}v$

for every $v\in H^{1}(\mathbb{R}^{N})^{O(N-2)}$, and taking $v=w_{i}$ we get

$\|w_{i}\|_{V_{\infty}}^{2}=\int_{\mathbb{R}^{N}}|w_{i}|^{2p}+\beta_{k}\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|w_{j}|^{p}|w_{i}|^{p}\leq\int_{\mathbb{R}^{N}}|w_{i}|^{2p}.$

Since, by (5.2),

$\int_{B_{C_{0}+1}(0)}|w_{k,i}|^{2p}\geq\int_{B_{C_{0}+1}(\xi_{k})}|u_{k,i}|^{2p}\geq\delta>0,$

we see that $w_{i}\neq 0$. Hence, there exists $t\in(0,1]$ such that
$\|tw_{i}\|_{V_{\infty}}^{2}=\int_{\mathbb{R}^{N}}|tw_{i}|^{2p}$, and (5.3)
yields

$\displaystyle n\mathfrak{c}_{\infty}\leq
n\,\frac{p-1}{2p}\|tw_{i}\|_{V_{\infty}}^{2}\leq
n\,\frac{p-1}{2p}\|w_{i}\|_{V_{\infty}}^{2}\leq\frac{p-1}{2p}\|u_{k,i}\|_{V}^{2}.$

As a consequence,

$\displaystyle\ell
n\mathfrak{c}_{\infty}\leq\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{k,i}\|_{V}^{2},$

contradicting (5.1). This shows that $u_{0,i}\neq 0$, as claimed.

As $(\mathcal{J}_{k}^{\phi_{n}})^{\prime}(\boldsymbol{u}_{k})=0$, $u_{k,i}\geq
0$, $u_{0,i}\geq 0$ and $\beta_{k}<0$, we have that

$\displaystyle\langle u_{k,i},u_{0,i}\rangle_{V}$
$\displaystyle=\int_{\mathbb{R}^{N}}|u_{k,i}|^{2p-2}u_{k,i}u_{0,i}+\beta_{k}\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|u_{k,j}|^{p}|u_{k,i}|^{p-2}u_{k,i}u_{0,i}$
$\displaystyle\leq\int_{\mathbb{R}^{N}}|u_{k,i}|^{2p-2}u_{k,i}u_{0,i},$

and passing to the limit we obtain
$\|u_{0,i}\|_{V}^{2}\leq|u_{0,i}|_{2p}^{2p}$. Hence, there exists $s\in(0,1]$
such that $\|su_{0,i}\|_{V}^{2}=|su_{0,i}|_{2p}^{2p}$ and we have that

(5.4)
$\mathfrak{c}^{G_{n}}\leq\frac{p-1}{2p}\|su_{0,i}\|_{V}^{2}\leq\frac{p-1}{2p}\|u_{0,i}\|_{V}^{2}\leq\liminf_{k\to\infty}\frac{p-1}{2p}\|u_{k,i}\|_{V}^{2},$

with $\mathfrak{c}^{G_{n}}$ as in (2.4). We claim that these are equalities.

To prove this claim, let $v_{k}\in\mathcal{M}^{G_{n}}$ be such that
$J(v_{k})=\frac{p-1}{2p}\|v_{k}\|_{V}^{2}\to\mathfrak{c}^{G_{n}}$. Set
$u_{k,1}:=v_{k}$ and define $u_{k,j+1}$ as in (1.3) for $j=1,\ldots,\ell-1$.
Set $\boldsymbol{u}_{k}=(u_{k,1},\ldots,u_{k,\ell})$. Since $(v_{k})$ is
bounded in $H^{1}(\mathbb{R}^{N})$ and $\beta_{k}\to 0$, we have that

$\lim_{k\to\infty}\beta_{k}\int_{\mathbb{R}^{N}}|u_{k,j}|^{p}|u_{k,i}|^{p}=0\qquad\text{for
every \ }i,j,$

so, by Proposition 3.1$(d)$, for $k$ large enough there exists
$s_{k}\in(0,\infty)$ such that
$s_{k}\boldsymbol{u}_{k}\in\mathcal{N}_{k}^{\phi_{n}}$ and $s_{k}\to 1$ as
$k\to\infty$. Thus,

(5.5)
$c_{k}^{\phi_{n}}\leq\mathcal{J}^{\phi_{n}}(s_{k}\boldsymbol{u}_{k})=\frac{p-1}{2p}\sum_{i=1}^{\ell}\|s_{k}u_{k,i}\|_{V}^{2}=\frac{p-1}{2p}\ell
s_{k}^{2}\|v_{k}\|_{V}^{2}\longrightarrow\ell\mathfrak{c}^{G_{n}}.$

Combining (5.4) and (5.5) we see that $s=1$, thus
$u_{0,i}\in\mathcal{M}^{G_{n}}$, that $u_{k,i}\to u_{0,i}$ strongly in
$H^{1}(\mathbb{R}^{N})$ and that

$J(u_{0,i})=\mathfrak{c}^{G_{n}}=\frac{1}{\ell}\,c_{k}^{\phi_{n}}<n\,\mathfrak{c}_{\infty}.$

This completes the proof. ∎

Now we turn to the case $\beta\to-\infty$. For the proof of Theorem 1.4 we
need the following result.

######  Lemma 5.1.

Let $\beta_{k}<0$ and $(u_{k,1},\ldots,u_{k,\ell})$ be a solution to (1.2)
with $\beta=\beta_{k}$ such that $u_{k,i}\to u_{\infty,i}$ strongly in
$H^{1}(\mathbb{R}^{N})$ for every $i=1,\ldots,\ell$. Then $(u_{k,i})$ is
uniformly bounded in $L^{\infty}(\mathbb{R}^{N})$.

###### Proof.

Let $s\geq 0$ and assume that $u_{k,i}\in L^{2(s+1)}(\mathbb{R}^{N})$ for
every $k\in\mathbb{N}$. Fix $L>0$ and define
$w_{k,i}:=u_{k,i}\min\\{u^{s}_{k,i},L\\}$. Then,

(5.6) $\displaystyle\int_{\mathbb{R}^{N}}|\nabla
w_{k,i}|^{2}\leq(1+s)\int_{\mathbb{R}^{N}}\nabla
u_{k,i}\cdot\nabla(u_{k,i}\min\\{u^{2s}_{k,i},L^{2}\\})$
$\displaystyle=(1+s)\Big{(}\int_{\mathbb{R}^{N}}|u_{k,i}|^{2p-2}w_{k,i}^{2}+\beta\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|u_{k,j}|^{p}|u_{k,i}|^{p-2}w_{k,i}^{2}-\int_{\mathbb{R}^{N}}Vw_{k,i}^{2}\Big{)}$
$\displaystyle\leq(1+s)\int_{\mathbb{R}^{N}}|u_{k,i}|^{2p-2}w_{k,i}^{2}.$

On the other hand, for any $K>0$ we have that

$\displaystyle\int_{\mathbb{R}^{N}}|u_{k,i}|^{2p-2}w_{k,i}^{2}$
$\displaystyle\leq\int_{\mathbb{R}^{N}}(|u_{k,i}|^{2p-2}-|u_{\infty,i}|^{2p-2})w_{k,i}^{2}+\int_{|u_{\infty,i}|^{2p-2}\geq
K}|u_{\infty,i}|^{2p-2}w_{k,i}^{2}+K\int_{\mathbb{R}^{N}}w_{k,i}^{2}$
$\displaystyle\leq\Big{|}|u_{k,i}|^{2p-2}-|u_{\infty,i}|^{2p-2}\Big{|}_{2p}^{2p-2}|w_{k,i}|_{2p}^{2}+\Big{(}\int_{|u_{\infty,i}|^{2p-2}\geq
K}|u_{\infty,i}|^{2p}\Big{)}^{\frac{p-1}{p}}|w_{k,i}|_{2p}^{2}$
$\displaystyle\qquad+K|w_{k,i}|_{2}^{2}.$

As $u_{k,i}\to u_{\infty,i}$ strongly in $H^{1}(\mathbb{R}^{N})$, choosing
$k_{0}>0$ and $K$ sufficiently large, we get that

(5.7)
$\int_{\mathbb{R}^{N}}|u_{k,i}|^{2p-2}w_{k,i}^{2}\leq\frac{1}{2}|w_{k,i}|_{2p}^{2}+K|w_{k,i}|_{2}^{2}\qquad\text{for
every \ }k\geq k_{0}.$

Since $H^{1}(\mathbb{R}^{N})$ is continuously embedded into
$L^{2p}(\mathbb{R}^{N})$, we derive from (5.6) and (5.7) that, for every
$k\in\mathbb{N}$,

$|w_{k,i}|_{2p}^{2}\leq K_{s}|w_{k,i}|_{2}^{2},$

for some constant $K_{s}$ independent of $L$, and letting $L\to\infty$ we get

$|u_{k,i}|_{2p(s+1)}^{2(s+1)}=|u_{k,i}^{s+1}|_{2p}^{2}\leq
K_{s}|u_{k,i}^{s+1}|_{2}^{2}=K_{s}|u_{k,i}|_{2(s+1)}^{2(s+1)}.$

As $(u_{k,i})$ is uniformly bounded in $L^{2}(\mathbb{R}^{N})$, iterating this
inequality starting with $s=0$ and using interpolation, we conclude that
$(u_{k,i})$ is uniformly bounded in $L^{q}(\mathbb{R}^{N})$ for any
$q\in[2,\infty)$ and each $i=1,\ldots,\ell$. This implies that

$f_{k,i}:=|u_{k,i}|^{2p-2}u_{k,i}+\beta\sum_{j\neq
i}|u_{k,j}|^{p}|u_{k,i}|^{p-2}u_{k,i}$

is uniformly bounded in $L^{q}(\mathbb{R}^{N})$ for any $q\in[2,\infty)$.
Then, by the Calderón-Zygmund inequality, $(u_{k,i})$ is uniformly bounded in
$W^{2,q}(\mathbb{R}^{N})$ for every $q\in[2,\infty)$ and, choosing $q$ large
enough, we derive from the Sobolev embedding theorem that $(u_{k,i})$ is
uniformly bounded in $L^{\infty}(\mathbb{R}^{N})$, as claimed. ∎

###### Proof of Theorem 1.4.

$(i):$ As before, we write $\mathcal{J}_{k}^{\phi_{n}}$ and
$\mathcal{N}_{k}^{\phi_{n}}$ for the functional and the Nehari set associated
to the system (1.2) with $\beta=\beta_{k}$, and set

$c_{k}^{\phi_{n}}:=\inf_{\mathcal{N}_{k}^{\phi_{n}}}\mathcal{J}_{k}^{\phi_{n}}.$

Arguing as in the proof of Theorem 1.3, we see that, after passing to a
subsequence, $u_{k,i}\rightharpoonup u_{\infty,i}$ weakly in
$H^{1}(\mathbb{R}^{N})$, $u_{k,i}\to u_{\infty,i}$ strongly in
$L^{2}_{\mathrm{loc}}(\mathbb{R}^{N})$ and $u_{k,i}\to u_{\infty,i}$ a.e. in
$\mathbb{R}^{N}$, for each $i=1,\ldots,\ell$. Hence, $u_{\infty,i}\geq 0$ and
$\boldsymbol{u}_{\infty}=(u_{\infty,1},\ldots,u_{\infty,\ell})\in\mathcal{H}^{\phi_{n}}$,
so $\boldsymbol{u}_{\infty}$ satisfies (1.3). We also get that

$u_{\infty,i}\neq
0\quad\text{and}\quad\|u_{\infty,i}\|_{V}^{2}\leq|u_{\infty,i}|_{2p}^{2p}\qquad\forall
i=1,\ldots,\ell.$

Furthermore, as $(\mathcal{J}_{k}^{\phi_{n}})^{\prime}(\boldsymbol{u}_{k})=0$,
we have that, for every $j\neq i$,

$\displaystyle
0\leq\int_{\mathbb{R}^{N}}|u_{k,j}|^{p}|u_{k,i}|^{p}\leq\frac{|u_{k,i}|^{2p}_{2p}}{-\beta_{k}}\leq\frac{C}{-\beta_{k}}.$

As $\beta_{k}\to-\infty$, passing to the limit and using Fatou’s lemma we
obtain

$0\leq\int_{\mathbb{R}^{N}}|u_{\infty,j}|^{p}|u_{\infty,i}|^{p}\leq\liminf_{k\to\infty}\int_{\mathbb{R}^{N}}|u_{k,j}|^{p}|k_{n,i}|^{p}=0.$

This implies that $u_{\infty,j}u_{\infty,i}=0$ a.e. in $\mathbb{R}^{N}$
whenever $i\neq j$.

Let $s\in(0,1]$ be such that
$\|su_{\infty,i}\|_{V}^{2}=|su_{\infty,i}|_{2p}^{2p}$. Then,
$s\boldsymbol{u}_{\infty}\in\mathcal{W}^{n}_{\ell}$ and using (5.1) we get

$\displaystyle\widehat{c}^{\phi_{n}}$
$\displaystyle\leq\frac{p-1}{2p}\sum_{i=1}^{\ell}\|su_{\infty,i}\|_{V}^{2}\leq\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{\infty,i}\|_{V}^{2}$
$\displaystyle\leq\frac{p-1}{2p}\sum_{i=1}^{\ell}\liminf_{k\to\infty}\|u_{k,i}\|_{V}^{2}\leq\widehat{c}^{\phi_{n}}.$

This proves that $s=1$, $\boldsymbol{u}_{\infty}\in\mathcal{W}^{n}_{\ell}$,
$u_{k,i}\to u_{\infty,i}$ strongly in $H^{1}(\mathbb{R}^{N})$ and

(5.8)
$\widehat{c}^{\phi_{n}}=\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{\infty,i}\|_{V}^{2}.$

Finally, as
$\lim\limits_{k\to\infty}\|u_{k,i}\|_{V}^{2}=\|u_{\infty,i}\|_{V}^{2}=|u_{\infty,i}|_{2p}^{2p}=\lim\limits_{k\to\infty}|u_{k,i}|_{2p}^{2p}$,
from

$\lim_{k\to\infty}\|u_{k,i}\|_{V}^{2}=\lim_{k\to\infty}|u_{k,i}|_{2p}^{2p}+\lim_{k\to\infty}\beta_{k}\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{\ell}\int_{\mathbb{R}^{N}}|u_{k,j}|^{p}|u_{k,i}|^{p}$

we obtain

$\int_{\mathbb{R}^{N}}\beta_{k}u_{k,j}^{p}u_{k,i}^{p}\to 0\text{ as
}k\to\infty\quad\text{whenever }i\neq j.$

$(ii):$ It follows from Lemma 5.1 and [4, Theorem B.2] that $(u_{k,i})$ is
uniformly bounded in $\mathcal{C}^{0,\alpha}(K)$ for each compact subset $K$
of $\mathbb{R}^{N}$ and $\alpha\in(0,1)$. So from the Arzelà-Ascoli theorem we
get that $u_{\infty,i}\in\mathcal{C}^{0}(\mathbb{R}^{N})$. Therefore
$\Omega_{i}:=\\{x\in\mathbb{R}^{N}:u_{\infty,i}(x)>0\\}$ is open. Since
$u_{\infty,i}u_{\infty,j}=0$ if $i\neq j$ and $\boldsymbol{u}_{\infty}$
satisfies (1.3), we have that $\Omega_{i}\cap\Omega_{j}=\emptyset$ if $i\neq
j$ and the $\ell$-tuple $(\Omega_{1},\ldots,\Omega_{\ell})$ satisfies
$(S_{1})$ and $(S_{2})$. Thus, it is an $(n,\ell)$-pinwheel partition.

Since $\boldsymbol{u}_{\infty}\in\mathcal{W}^{n}_{\ell}$, we have that
$u_{\infty,i}$ belongs to the Nehari manifold $\mathcal{M}_{\Omega_{i}}$
defined in (1.7). Therefore,
$\frac{p-1}{2p}\|u_{\infty,i}\|_{V}^{2}\geq\mathfrak{c}_{\Omega_{i}}$.
Equality must hold true as, otherwise, there would exist
$v_{1}\in\mathcal{M}_{\Omega_{1}}$ such that
$\frac{p-1}{2p}\|u_{\infty,1}\|_{V}^{2}>\frac{p-1}{2p}\|v_{1}\|_{V}^{2}\geq\mathfrak{c}_{\Omega_{i}}$
and, defining $v_{j+1}$ as in (1.3), we would have that
$(v_{1},\ldots,v_{\ell})\in\mathcal{W}_{\ell}^{n}$ and

$\frac{p-1}{2p}\sum_{i=1}^{\ell}\|v_{i}\|_{V}^{2}<\frac{p-1}{2p}\sum_{i=1}^{\ell}\|u_{\infty,i}\|_{V}^{2}=\widehat{c}^{\phi_{n}}$

by (5.8), which is a contradiction. This shows that $u_{\infty,i}$ is a least
energy solution of (1.8) in $\Omega_{i}$. Now, since
$\frac{p-1}{2p}\|u_{\infty,i}\|_{V}^{2}=\mathfrak{c}_{\Omega_{i}}$, we get

$\displaystyle\inf_{(\Theta_{1},\ldots,\Theta_{\ell})\in\mathcal{P}^{n}_{\ell}}\sum_{j=1}^{\ell}\mathfrak{c}_{\Theta_{j}}\leq\sum_{j=1}^{\ell}\mathfrak{c}_{\Omega_{j}}=\widehat{c}^{\phi_{n}}\leq\inf_{(\Theta_{1},\ldots,\Theta_{\ell})\in\mathcal{P}^{n}_{\ell}}\sum_{j=1}^{\ell}\mathfrak{c}_{\Theta_{j}}.$

This shows that $(u_{\infty,1},\ldots,u_{\infty,\ell})$ is an optimal
$(n,\ell)$-pinwheel partition.

$(iii):$ This is a local statement. As mentioned above, $(u_{k,i})$ is
uniformly bounded in $\mathcal{C}^{0,\alpha}(\Omega)$ for each open subset
$\Omega$ compactly contained in $\mathbb{R}^{N}$ and $\alpha\in(0,1)$. So,
from the Arzelà-Ascoli theorem, we get that $u_{k,i}\to u_{\infty,i}$ in
$\mathcal{C}^{0,\alpha}(\Omega)$. Thus, all hypotheses of [4, Theorem C.1] are
satisfied and $(iii)$ follows from it.

$(iv):$ Let $G_{2n}$ be the group defined in Example 2.1 with $\ell=2$, and
let $\tau_{n}:G_{2n}\to\mathbb{Z}_{2}:=\\{1,-1\\}$ be the homomorphism given
by $\tau_{n}(\mathrm{e}^{2\pi\mathrm{i}/2n})=-1$ and $\tau_{n}(\theta)=1$ for
every $\theta\in O(N-2)$. A solution to the Schrödinger equation (1.5)
satisfying

(5.9) $u(gx)=\tau_{n}(g)u(x)\qquad\text{for all \ }g\in G_{2n},\
x\in\mathbb{R}^{N},$

is a critical point of the functional
$J:H^{1}(\mathbb{R}^{N})^{\tau_{n}}\to\mathbb{R}$ defined by (2.3) on the
space

$H^{1}(\mathbb{R}^{N})^{\tau_{n}}:=\\{u\in H^{1}(\mathbb{R}^{N}):u\text{
satisfies }\eqref{eq:nodal}\\}.$

The nontrivial ones belong to the Nehari manifold

$\mathcal{M}^{\tau_{n}}:=\\{u\in H^{1}(\mathbb{R}^{N})^{\tau_{n}}:u\neq 0,\
\|u\|_{V}^{2}=|u|_{2p}^{2p}\\},$

which is a natural constraint for $J$. Note that every nontrivial function
satisfying (5.9) is nonradial and changes sign.

There is a one-to-one correspondence

$\mathcal{W}^{n}_{2}\to\mathcal{M}^{\tau_{n}},\qquad(u_{1},u_{2})\mapsto
u_{1}-u_{2},$

whose inverse is $u\mapsto(u^{+},-u^{-})$, with $u^{+}:=\max\\{u,0\\}$ and
$u^{-}:=\min\\{u,0\\}$, satisfying

$\frac{p-1}{2p}\Big{(}\|u_{1}\|^{2}_{V}+\|u_{2}\|^{2}_{V}\Big{)}=J(u_{1}-u_{2}).$

Therefore,

$\displaystyle J(u_{\infty,1}-u_{\infty,2})$
$\displaystyle=\frac{p-1}{2p}\Big{(}\|u_{\infty,1}\|^{2}_{V}+\|u_{\infty,2}\|^{2}_{V}\Big{)}$
$\displaystyle=\inf_{(u_{1},u_{2})\in\mathcal{W}^{n}_{2}}\frac{p-1}{2p}\Big{(}\|u_{1}\|^{2}_{V}+\|u_{2}\|^{2}_{V}\Big{)}=\inf_{u\in\mathcal{M}^{\tau_{n}}}J(u).$

This shows that $u_{\infty,1}-u_{\infty,2}$ is a least energy solution to
(1.5) and (5.9). ∎

##  Appendix A An auxiliary result

######  Lemma A.1.

Assume $v_{k}\rightharpoonup v$ weakly in $H^{1}(\mathbb{R}^{N})$,
$\xi_{k}\in\mathbb{R}^{N}$ satisfies $|\xi_{k}|\to\infty$ and
$V\in\mathcal{C}^{0}(\mathbb{R}^{N})$ satisfies $(V_{2})$. Set
$V_{k}(x):=V(x+\xi_{k})$. Then,

$\lim_{k\to\infty}\|v_{k}\|_{V_{k}}^{2}-\lim_{k\to\infty}\|v_{k}-v\|_{V_{k}}^{2}=\|v\|_{V_{\infty}}^{2}.$

###### Proof.

As $v_{k}\rightharpoonup v$ weakly in $H^{1}(\mathbb{R}^{N})$ one has

$\displaystyle\|v\|_{V_{\infty}}^{2}+o(1)$
$\displaystyle=\|v_{k}\|_{V_{\infty}}^{2}-\|v_{k}-v\|_{V_{\infty}}^{2}$
$\displaystyle=\|v_{k}\|_{V_{k}}^{2}-\|v_{k}-v\|_{V_{k}}^{2}+2\int_{\mathbb{R}^{N}}(V_{\infty}-V_{k})v_{k}v-\int_{\mathbb{R}^{N}}(V_{\infty}-V_{k})v^{2}.$

Given $\varepsilon>0$, choose $R>0$ large enough so that

$\int_{\mathbb{R}^{N}\smallsetminus B_{R}}|V_{\infty}-V_{k}||v|^{2}\leq
2\sup_{x\in\mathbb{R}^{N}}V(x)\int_{\mathbb{R}^{N}\smallsetminus
B_{R}}|v|^{2}<\frac{\varepsilon}{2}.$

Now, take $k_{0}$ such that

$|V_{\infty}-V(x+\xi_{k})|<\frac{\varepsilon}{2|v|_{2}^{2}}=:\delta\quad\text{for
every \ }x\in B_{R}\text{ \ and \ }k\geq k_{0}.$

Then, for $k\geq k_{0}$ we have that

$\int_{\mathbb{R}^{N}}|V_{\infty}-V_{k}||v|^{2}\leq\int_{B_{R}}|V_{\infty}-V_{k}||v|^{2}+\int_{\mathbb{R}^{N}\smallsetminus
B_{R}}|V_{\infty}-V_{k}||v|^{2}<\varepsilon$

and

$\displaystyle\int_{\mathbb{R}^{N}}|(V_{\infty}-V_{k})v_{k}v|$
$\displaystyle\leq\left(\int_{\mathbb{R}^{N}}|V_{\infty}-V_{k}||v_{k}|^{2}\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{N}}|V_{\infty}-V_{k}||v|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq C\sqrt{\varepsilon}.$

This completes the proof. ∎

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Mónica Clapp
Instituto de Matemáticas

Universidad Nacional Autónoma de México

Campus Juriquilla

Boulevard Juriquilla 3001

76230 Querétaro, Qro., Mexico

<EMAIL_ADDRESS>
Angela Pistoia
Dipartimento di Metodi e Modelli Matematici

La Sapienza Università di Roma

Via Antonio Scarpa 16

00161 Roma, Italy

<EMAIL_ADDRESS>