# Saddle solutions for the Choquard equation with a general nonlinearity

Jiankang Xia E-mail address<EMAIL_ADDRESS>
Abstract: In the spirit of Berestycki and Lions, we prove the existence of
saddle type nodal solutions for the Choquard equation

$-\Delta u+u=\big{(}I_{\alpha}\ast F(u)\big{)}F^{\prime}(u)\qquad\text{ in
}\;\mathbb{R}^{N}$

where $N\geq 2$ and $I_{\alpha}$ is the Riesz potential of order
$\alpha\in(0,N)$. Without any compact setting, we construct saddle solutions
in a unified way for the Choquard equation whose nodal domains are of cone
shapes demonstrating Coxeter’s symmetric configurations in $\mathbb{R}^{N}$.
Moreover, if $F^{\prime}$ is odd and has constant sign on $(0,+\infty)$, then
the saddle solution maintains signed on the fundamental domain of the
corresponding Coxeter group and receives opposite signs on any two adjacent
domains. These results further complete the variational framework in
constructing sign-changing solutions for the Choquard equation but still
require a quadratic or super-quadratic growth on $F$ near the origin.

Key words: Choquard equation; saddle solutions; Coxeter group; nodal domains.

2020 Mathematics Subject Classification: 35A01 $\cdot$ 35B08 $\cdot$ 35J20
$\cdot$ 35J60

##  1 Introduction and main results

We consider nodal solutions for the Choquard type nonlinear equation

$-\Delta u+u=\big{(}I_{\alpha}\ast F(u)\big{)}f(u),\qquad u\in
H^{1}(\mathbb{R}^{N}),$ ($\mathcal{C}$)

which has been widely studied in recent years for its diverse applications in
physics. Here the dimension $N\geq 2$ and $F\in C^{1}(\mathbb{R},\mathbb{R})$
is the primitive function of $f$. The Riesz potential $I_{\alpha}$ with order
$\alpha\in(0,N)$ is defined for $x\in\mathbb{R}^{N}\setminus\\{0\\}$ by

$I_{\alpha}(x)=\frac{A_{\alpha}}{\lvert x\rvert^{N-\alpha}}\quad\text{ with
}\quad
A_{\alpha}=\frac{\Gamma(\frac{N-\alpha}{2})}{2^{\alpha}\pi^{\frac{N}{2}}\Gamma(\frac{\alpha}{2})}$

where $\Gamma$ denotes the classical Gamma function and $\ast$ stands for the
convolution on $\mathbb{R}^{N}$.

The Choquard equation ($\mathcal{C}$) is a canonically nonlocal equation due
to the appearance of the convolution. When $N=3$, $\alpha=2$ and $F(u)=\lvert
u\rvert^{2}$, as an important model, it arises in various physics contexts,
such as the theory of one-component plasma in describing the electron trapped
in its own hole [20] and the model of a polaron at rest in the quantum physics
[29]. This equation also has applications in Einstein–Klein–Gordon system and
Einstein–Dirac system [11]. It is usually referred to as the
Schrödinger–Newton equation in the description of self-gravitating matter
[25]. We refer to [6, 18, 28] and the references therein for more physical
interpretations.

The $p$-homogeneous Choquard equation, e.g., $F(u)=\lvert u\rvert^{p}$, has
already been extensively investigated by a plenty of literatures during last
decades. We refer to [20, 22, 23] for the existence and multiplicity results
for the Choquard equation ($\mathcal{C}$) in $\mathbb{R}^{3}$ with
$F(u)=\lvert u\rvert^{2}$ and $\alpha=2$. Moroz and Van Schaftingen [26]
obtained the ground state solution for the $p$-homogeneous Choquard equation.
Here and in the sequel, the ground state solution is understood in the sense
that it minimizes the energy functional among all nontrivial solutions. It is
shown that nontrivial solutions with finite energies to the $p$-homogeneous
Choquard equation can exist if and only if the exponent $p$ satisfies
$\frac{N-2}{N+\alpha}<\frac{1}{p}<\frac{N}{N+\alpha}$, see e.g., [26, 27, 32].
Several papers have also studied the existence of sign-changing solutions to
the $p$-homogeneous Choquard equation in recent years. In a compact setting,
we mention [32] for the least energy sign-changing solution and refer to [12,
15] for the multiple radially symmetric solutions with prescribed number of
nodal domains. We refer readers to the survey paper [28] and the references
therein for more recent results in this field.

For the Choquard equation ($\mathcal{C}$), Moroz and Van Schaftingen [27]
investigated the ground state solution under the so-called Berestycki–Lions
conditions as $N\geq 3$,

  * ($F_{0}$)

$F\in C^{1}(\mathbb{R},\mathbb{R})$ and there exists $s_{0}\in\mathbb{R}$ such
that $F(s_{0})\neq 0$.

  * ($f_{1}$)

There exists $C>0$ such that for every $s\in\mathbb{R}$, $\lvert
sF^{\prime}(s)\rvert\leq C(\lvert s\rvert^{\frac{N+\alpha}{N}}+\lvert
s\rvert^{\frac{N+\alpha}{N-2}}).$

  * ($f_{2}$)

$\lim\limits_{s\to 0}\frac{F(s)}{\lvert s\rvert^{\frac{N+\alpha}{N}}}=0$ and
$\lim\limits_{s\to\infty}\frac{F(s)}{\lvert
s\rvert^{\frac{N+\alpha}{N-2}}}=0$.

Besides the nontrivial assumption $(F_{0})$, Battaglia and Van Schaftingen [1]
obtained the ground state solution for the planar case by assuming the
following hypotheses.

  * ($f^{\prime}_{1}$)

For every $\theta>0$, there exists $C_{\theta}>0$ such that for every
$s\in\mathbb{R}$,

$\lvert F^{\prime}(s)\rvert\leq C_{\theta}\min\\{1,\lvert
s\rvert^{\frac{\alpha}{2}}\\}e^{\theta\lvert s\rvert^{2}}.$

  * ($f^{\prime}_{2}$)

$\lim\limits_{s\to 0}\frac{F(s)}{\lvert s\rvert^{\frac{2+\alpha}{2}}}=0$.

We emphasis that the above assumptions on the nonlinearity are quit mild and
reasonable. It turns out that these assumptions are actually "almost
necessary" for the existence of nontrivial finite energy solutions [1, 27].

On the other hand, some progresses have been made on the saddle type nodal
solutions. It was Ghimenti and Van Schaftingen who constructed a surprisingly
odd solution for the $p$-homogeneous Choquard equation in their seminal work
[8], this kind of odd solution has exactly two half-space nodal domains and
makes the issue of constructing sign-changing solutions become more
interesting because the compact assumption there is no longer needed. Actually
they took advantage of the reflection group with order $2$ to grantee the
nodal property. Motivated by the highlight results in [8], Wang and the author
of current paper recently developed an effective variational approach in
constructing saddle type nodal solutions for the $p$-homogeneous Choquard
equation in [33, 36] by making use of the dihedral group and the full
polyhedral groups of the Platonic solids, respectively. As a result, finding
saddle solutions is reduced to solving a minimization problem constrained on
the saddle type Nehari manifold. In a very recent joint work with Zhang [37],
we developed a unified variational framework for the construction of saddle
solutions to the upper critical Choquard equation. It turns out that the
Coxeter symmetry was introduced and these groups contain more reflections than
before. All of the symmetric groups used in [8, 33, 36, 37] consist of
distinct number of reflections giving different tiling and configurations in
$\mathbb{R}^{N}$. Note that finite Coxeter groups can be generated by finite
reflections in $\mathbb{R}^{N}$. Therefore, nodal sets of these saddle
solutions are all made up by hyperplanes [8, 33, 36, 37], each of saddle
solutions keeps a constant sign on every conic (sectorial) region and receives
opposite signs on any two adjacent regions. This reveals a distinguished
nonlocal feature of the Choquard equation. As a contrast, it is impossible to
admit such type of nodal solutions either for its local counterpart of the
nonlinear Schrödinger equation with the form $-\Delta u+u=|u|^{q-2}u$ [7] or
for the Yamabe problem $-\Delta u=\lvert u\rvert^{\frac{4}{N-2}}u$ on
$\mathbb{R}^{N}$ [30].

Whereas all of the saddle solutions constructed in [8, 33, 36, 37] up until
now were only available for the homogeneous Choquard equation. It seems that
little is known on the existence of nodal solutions to the Choquard equation
($\mathcal{C}$) except a very recent paper [39] in which the compact setting
is assumed and a sequence of radial or non-radial sign-changing solutions are
obtained. The condition of Ambrosetti–Rabinowtiz type is also required in [39]
on the nonlinearity.

In the present paper, we shall deal with the existence of saddle solutions for
the Choquard equation ($\mathcal{C}$) with a general nonlinearity by
introducing the Coxeter symmetry. The approach for seeking saddle solutions in
the previous work heavily depends upon the Nehari manifold method, in turn,
the homogeneous or the monotonic property of the nonlinearity plays a
significant role. As a consequence, the Nehari manifold method can not work
well in tacking with the general Choquard equation ($\mathcal{C}$) if we do
not have any additional monotonic assumptions on $F$. This has been observed
in the process of finding ground state solutions in [1, 27].

To state our results, we shall first recall the Coxeter group and some
notations. Let $G$ be a finite Coxeter group with rank $k\leq N$ and $\lvert
G\rvert$ denote its cardinality. It follows that $G=\langle\mathcal{S}\rangle$
with $\mathcal{S}$ being the generating set and having $k$ distinct
reflections in $\mathcal{S}$. More precisely, there exists a unique
epimorphism $\psi:G\to\\{\pm 1\\}$ induced by $\psi(s)=-1$ for all
$s\in\mathcal{S}$. A function $u:\mathbb{R}^{N}\to\mathbb{R}$ is said to be
$G$-symmetry if it satisfies

$g\diamond u(x)=\psi(g)u(x),\quad\text{ for }g\in G,\quad x\in\mathbb{R}^{N}.$

Here $\diamond$ denotes the group action which will be explained in Section 2.
With this kind of symmetric groups, the symmetric critical principle is
applicable and we shall consider our problem in a closed $G$-subspace defined
by

$\mathcal{H}_{G}=\\{u\in H^{1}(\mathbb{R}^{N})\,|\,g\diamond
u=\psi(g)u,\forall g\in G\\}.$

It turns out that all of the symmetric groups used in [8, 33, 36] are of
Coxeter with their ranks less than $4$ yielding different symmetric nodal
structures. In what follows, we say $u\in\mathcal{H}_{G}\setminus\\{0\\}$ is a
$G$-saddle solution if $u$ solves the Choquard equation ($\mathcal{C}$) and
$u$ is called $G$-groundstate if in addition $u$ minimizes the energy
functional amongst all the $G$-saddle solutions.

The first result is about the existence of $G$-groundstate. Besides the
nontrivial assumption $(F_{0})$, in the present paper, we assume a little
stronger assumptions than that in [1, 27] on the nonlinearity $F$, the reason
will be explained later.

  * ($F_{1}$)

$\lim\limits_{s\to 0}\frac{f(s)}{s}=\mu\in[0,+\infty)$.

  * ($F_{2}$)

$\lim\limits_{s\to\infty}\frac{F(s)}{\lvert s\rvert^{\frac{N+\alpha}{N-2}}}=0$
for $N\geq 3$; $\lim\limits_{s\to\infty}\frac{F(s)}{\lvert
s\rvert^{\bar{p}}}=0$ for some $\bar{p}\in[2,+\infty)$ as $N=2$.

Our results on the saddle solutions can be stated as follows.

######  Theorem 1.1.

Let $N\geq 2$ and $G$ be a finite Coxeter group of rank $k$ with $1\leq k\leq
N$. If $F$ is even and satisfies the assumptions $(F_{0})$, $(F_{1})$ and
$(F_{2})$, then the Choquard equation ($\mathcal{C}$) admits a $G$-saddle
solution $u\in H^{1}(\mathbb{R}^{N})\cap C^{2}(\mathbb{R}^{N})$ which turns to
be a $G$-groundstate.

We remark that in the $p$-homogeneous case that $F(u)=\lvert u\rvert^{p}$,
Theorem 1.1 holds provided that $N\geq 2$ and $p\in[2,\frac{N+\alpha}{N-2})$,
which are exactly the same with that in [8, 33, 36]. In fact, the odd solution
constructed in [8] makes use of the 1-rank Coxeter groups ${\bf{A}_{1}}$,
generated by one reflection. The 2-rank Coxeter group, dihedral group
${\bf{I}}_{2}(m)$, was used in [36, Theorem 1.1] to construct the dihedral
type saddle solution that has exactly $2m$ nodal domains. The irreducible
Coxeter groups with rank 3 including ${\bf{A}}_{3}$, ${\bf{B}}_{3}$,
${\bf{H}}_{3}$ in [33] yield the saddle solutions with polyhedral symmetries,
where ${\bf{A}}_{3}$, ${\bf{B}}_{3}$, ${\bf{H}}_{3}$ are exactly the three
kinds of full polyhedral groups for the Plato’s polyhedrons in
$\mathbb{R}^{3}$. Different saddle solutions for $p$-homogeneous Choquard
equation were also obtained separately by taking advantage of the symmetric
groups ${\bf{I}}_{2}(m)\times{\rm\bf{A}}_{1}$,
${\bf{I}}_{2}(m_{1})\times{\bf{I}}_{2}(m_{2})$ and
${\rm\bf{A}}_{1}\times\cdots\times{\rm\bf{A}}_{1}$ in the previous work [33,
36]. In summary, all the symmetric groups used in [8, 33, 36] are types of
Coxeter so that each of the saddle solution demonstrates a unique Coxeter’s
symmetric configurations in $\mathbb{R}^{N}$. The notations and the
classification of Coxeter groups will be explained in the final appendix. In
this sense, Theorem 1.1 will further complete the variational framework
initiated in [8] and further developed in [33, 36, 37]. Here we also would
like to mention the existence results on sign-changing solutions with higher
energies under higher dimensional symmetries [2, 3]. Nevertheless, the key
assumption $(Z_{0})$ imposed on the group seems necessary and the group action
they used can not be realized in $\mathbb{R}^{3}$. Our results manifest that
the compactness can still fail below the threshold value in [2, 3] which
equals the ground state energy multiplying by the order of the group $G$, see
Proposition 4.1.

As we shall see that for the existence of saddle solutions, the higher rank of
the Coxeter group we use, the more existing saddle solutions we should
consider when restoring the compactness. In fact, as is clarified in [33, 36],
given a finite Coxeter group $G$, there exists a threshold value $c_{G}^{*}$
below which the compactness can be restored, and the threshold $c_{G}^{*}$
actually depends upon the energy levels of the preceding saddle solutions that
correspond to subgroups of $G$. As a result, the existing saddle solutions
come into play like a hierarchy when we build one type of saddle solutions
with higher dimensional symmetries.

In the process of constructing saddle solutions, our ultimate task is to
restore the compactness in $\mathbb{R}^{N}$ under the group action. To this
end, a suitable local compactness condition is needed which leads to a prior
estimate on the mountain pass type energy level and in turn it attributes to
the decay behavior of the sign-changing solutions. For the $p$-homogeneous
Choquard equation, as indicated in [8], finding the odd solution relies on the
decay behavior of the ground state solution. For the sign-changing solutions
of $p$-homogeneous Choquard equation, the decay behavior is only clear when
$p\geq 2$, see [36, Proposition 2.1]. For the Choquard equation
($\mathcal{C}$), up until now, only the exponential decay of the ground state
solution was known [38, Proposition 2.2], which requires the nonlinearity $f$
having superlinear and subcritical growth. However, there has no decay results
for sign-changing solutions of ($\mathcal{C}$).

In this paper, under our assumptions on the nonlinearity, we shall prove that
any weak solutions of ($\mathcal{C}$) has an exponential decay, which plays a
big part in determining the threshold for the compactness, see Proposition 4.1
below. We remark that this decay estimate particularly holds for sign-changing
solutions. This partly complements the results of [38, Proposition 2.2] where
the decay behavior is only considered for positive solutions.

######  Theorem 1.2.

Let $N\geq 2$, $\alpha\in(0,N)$. If $F\in C^{1}(\mathbb{R},\mathbb{R})$
satisfies $(F_{1})$–$(F_{2})$ and $u\in H^{1}(\mathbb{R}^{N})$ is a weak
solution of ($\mathcal{C}$). Then, $u\in W^{2,s}_{\rm{loc}}(\mathbb{R}^{N})$
for any $s\geq 1$ and $u\in C^{2}(\mathbb{R}^{N})\cap
L^{\infty}(\mathbb{R}^{N})$. Moreover, there exist positive constants $C,c>0$
such that $\lvert u(x)\rvert\leq Ce^{-c\lvert x\rvert}.$

The regularity results have been obtained in [27] and the decay estimate
depends on a $L^{\infty}$ estimates on the convolution term and the comparison
principle. In particular, as is revealed in [36], the additional limit
behavior of the convolution term enable us to deal with the quadratic case,
which also extends the decay result in [38] for the positive solutions.
However, the decay behavior is still open for the sign-changing solutions to
the sub-quadratic Choquard equation ($\mathcal{C}$) and only the groundstate
solution to the $p$-homogeneous Choquard equation is proved to have a
polynomial decay [8]. This is the reason why we limit the nonlinearity $f$
admitting linear or superlinear growth near the origin in all previous work
[33, 36]. This also in contrast with the critical Choquard equation [37],
because it has been shown that all the solutions have the polynomial decay
behavior [14]. We also bring attention that the assumptions $(F_{2})$ is
somewhat stronger than the hypothesis $(f^{\prime}_{1})$ as $N=2$, where only
a weaker Moser–Trudinger type growth assumption is assumed. This is because
the asymptotic decay behavior of the convolution term $I_{\alpha}\ast F(u)$ is
still unclear for sign-changing solutions.

In a similar manner as in that [8, Proposition 2.6], we can also prove the
constant signed property of the $G$-groundstate on the strict fundamental
domain of the Coxeter group $G$ under some stronger assumptions.

######  Theorem 1.3.

Let $N\geq 2$ and $G$ be a finite Coxeter group of rank $k$ with $1\leq k\leq
N$. If $F\in C^{1}(\mathbb{R},\mathbb{R})$ satisfies $(F_{1})$ and $(F_{2})$,
in addition, if $F^{\prime}$ is odd and has a constant sign on $(0,+\infty)$,
then any $G$-groundstate of the Choquard equation ($\mathcal{C}$) maintains a
constant sign on the strict fundamental domain and has exactly $\lvert
G\rvert$ nodal domains.

When compared with the local counterpart of the nonlinear Schrödinger
equations [34], one significant difference is that the energy of the
${\bf{A}_{1}}$-groundstate is strictly less than twice the groundstate energy,
i.e., the energy doubling property does not hold any more for the nonlocal
Choquard equation. This fact makes it possible to find the global minimal
nodal solution which minimizes the functional amongst all the sign-changing
solutions. This have been done in [8] for the $p$-homogeneous Choquard
equation. It turns out that the minimal nodal solution exist once the Choquard
equation ($\mathcal{C}$) has the Nehari structure and the
${\bf{A}_{1}}$-groundstate. To make it clear, we assume that

  * ($F^{\prime}_{1}$)

$\lim\limits_{s\to 0}\frac{f(s)}{s^{r-1}}=\mu\in[0,+\infty)$ for some $r>2$
and the function $\frac{f(s)}{\lvert s\rvert}$ is increasing in
$s\in\mathbb{R}\setminus\\{0\\}$.

  * ($F^{\prime}_{2}$)

there exists some $p\in(2,\frac{N+\alpha}{N-2})$ such that
$\lim\limits_{\lvert s\rvert\to+\infty}\frac{F(s)}{\lvert s\rvert^{p}}=0$.

######  Theorem 1.4.

Let $N\geq 2$ and $\alpha\in(0,N)$. If $F\in C^{1}(\mathbb{R},\mathbb{R})$ is
even and satisfies $(F^{\prime}_{1})$ and $(F^{\prime}_{2})$, then the
Choquard equation ($\mathcal{C}$) has a global minimal nodal solution.

We sketch our main idea and organize the paper as follows. By introducing the
Coxeter group $G$ with rank $k\leq N$, we shall furnish a suitable framework
space for searching the $G$-saddle solutions, see Section 2. We then prove
Theorem 1.2 in Section 3 that all the solutions of the Choquard equation
($\mathcal{C}$) decay exponentially at infinity. In addition, the regularities
for the solutions enable us to establish the Pohozaev identity for this
equation. As is mentioned above, the lack of monotonic property on the
nonlinearity makes the Nehari manifold method can not work well any more. We
shall consider the mountain pass level as in [1, 27] rather than the
minimization on the corresponding Nehari manifold, this is possible since the
nontrivial assumption $(F_{0})$ yields the mountain pass geometry, see the
details in Section 2. To avoid the condition of Ambrosetti–Rabinowtiz type on
the nonlinearity, we shall construct the Pohozaev–Palais–Smale sequence at the
mountain pass level $c_{G}$ by using a scaling technique, see Proposition 2.3.
To prove the key energy estimate on this level, we shall construct test
functions by using the preceding saddle solutions, which can be done by an
induction argument, see Section 4. More precisely, we first show that
$c_{{\bf{A}}_{1}}<2c_{0}$ by using the exponential decay of the ground state
solution for the Choquard equation ($\mathcal{C}$). And then, we conclude the
existence of a nontrivial saddle solution with ${\bf{A}_{1}}$-symmetry by
employing the concentration compactness principle. As in [27], a crucial
minimax characterization on this mountain pass level $c_{G}$ will be given,
which reveals that $c_{\bf{A}_{1}}$ actually equals the infimum of the energy
functional constrained on the corresponding Pohozaev manifold with
${\bf{A}_{1}}$-symmetry, this also involves the scaling technique, see
Proposition 2.4 below. Therefore, $c_{\bf{A}_{1}}$ is indeed the least energy
amongst ${\bf{A}}_{1}$-saddle solutions. The existence of
${\bf{A}}_{1}$-groundstate will be the start point to construct the saddle
solution with the dihedral nodal structures. By repeating this procedure, we
then conclude step by step the existence of $G$-saddle solutions with the rank
$k\leq N$. Every saddle solutions that we obtained turns out to be the
$G$-groundstate, see Proposition 2.4. Theorem 1.1 will be inductively
completed in Section 5 by employing the concentration compactness principle.
We also prove Theorem 1.3 in Section 5 under the additional assumptions on the
nonlinearity. We finally complete the proof of Theorem 1.4. This will be done
by considering the minimization problem on the set of nodal solutions,

$c_{\rm{nod}}=\inf_{\mathcal{M}_{\rm{nod}}}\mathcal{E}$

where the constraint $\mathcal{M}_{\rm{nod}}=\\{u\in
H^{1}(\mathbb{R}^{N})\,|\,u^{\pm}\neq 0,\mathcal{E}^{\prime}(u)=0\\}$ with
$u^{\pm}=\max\\{\pm u,0\\}$. Up to translations, this problem can be achieved
along any minimizing sequence. We shall see that this is a direct consequence
of the fact that $c_{\rm{nod}}\leq c_{{\bf{A}}_{1}}<2c_{0}$. Since the
background knowledge on the theory of Coxeter group has been summarized in
[37], we give an appendix at the end of this paper for the convenience of
readers.

##  2 Variational framework and minimax characterizations

The Choquard equation ($\mathcal{C}$) has a variational structure and
corresponds to an energy functional $\mathcal{E}$, defined for each function
$u\in H^{1}(\mathbb{R}^{N})$ by

$\mathcal{E}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}\lvert\nabla
u\rvert^{2}+\lvert
u\rvert^{2}-\frac{1}{2}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u))F(u).$

According to the arguments in [1, 27], this functional is well defined and is
of $C^{1}$ in $H^{1}(\mathbb{R}^{N})$. In the sprite of Berestycki–Lions
assumptions, the authors in [1, 27] investigated the existence of ground state
solutions for the Choquard equation ($\mathcal{C}$) by employing a scaling
technique introduced by Jeanjean [16] to construct the so called
Pohozaev–Palais–Smale sequence. Precisely, they proved that the mountain pass
level defined below actually yields the ground state solution, it takes the
form of

$c_{0}=\inf_{\gamma\in\Gamma_{0}}\max_{t\in[0,1]}\mathcal{E}(\gamma(t))$

where the paths set is defined by

$\Gamma_{0}=\\{\gamma\in C([0,1];H^{1}(\mathbb{R}^{N}))\,|\,\gamma(0)=0\text{
and }\mathcal{E}(\gamma(1))<0\\}.$

To obtain the symmetric nodal structures, we shall introduce the Coxeter
symmetry, see the details in the Appendix. For $k\geq 1$, we denote
$\mathcal{G}_{k}$ and $\mathcal{W}_{k}$ be the collections of finite Coxeter
groups and the irreducible Coxeter groups with rank $k$, respectively. Given
$G\in\mathcal{G}_{k}$, by Proposition A.4, we see that
$G=W_{k_{1}}\times\cdots\times W_{k_{\ell}}$ where
$W_{k_{i}}\in\mathcal{W}_{k_{i}}$ for each $k_{i}\geq 1$ and
$\sum_{i}k_{i}=k$. Recalling the unique epimorphism $\psi:G\to\\{\pm 1\\}$
defined in Lemma A.6, we now set up the framework space

$\mathcal{H}_{G}:=\\{u\in H^{1}(\mathbb{R}^{N})\,|\,\texttt{g}\diamond
u=\psi(g)u\text{ for all }\texttt{g}\in G\\}.$

Here the group action on a function $u:\mathbb{R}^{N}\to\mathbb{R}$ is defined
for each $g\in G$ by

$g\diamond u(x)=u(g^{-1}x)\quad\text{ with }\quad gx:=(g\oplus 1_{N-k})x.$

The energy functional permits the mountain pass geometry, so that the mountain
pass level is a natural candidate for seeking weak solutions. Namely, we
consider

$c_{G}=\inf_{\gamma\in\Gamma_{G}}\max_{t\in[0,1]}\mathcal{E}(\gamma(t))$ (2.1)

where paths set belongs to $\mathcal{H}_{G}$, i.e.,
$\Gamma_{G}=\\{\gamma\in\Gamma_{0}\,|\,\gamma(t)\in\mathcal{H}_{G}\text{ for
all }t\in[0,1]\\}$. Comparing with the saddle type nodal solutions obtained in
[26, 36, 33], we have neither monotonic assumption nor Ambrosetti–Rabinowitz
type growth condition on the nonlinearity. To overcome this, we recall the
Pohozaev identity, which is allowed by the regularities exhibited in Theorem
1.2. For the detailed proofs, we refer to [27, Theorem 3] and [1, Proposition
5.2].

######  Proposition 2.1.

Let $N\geq 2$. If $F$ satisfies $(F_{1})$–$(F_{2})$ and $u\in
H^{1}(\mathbb{R}^{N})\cap W^{2,2}_{\rm{loc}}(\mathbb{R}^{N})$ solves the
equation ($\mathcal{C}$), then

$\mathcal{P}(u)=\frac{N-2}{2}\int_{\mathbb{R}^{N}}\lvert\nabla
u\rvert^{2}+\frac{N}{2}\int_{\mathbb{R}^{N}}\lvert
u\rvert^{2}-\frac{N+\alpha}{2}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u))F(u)=0.$
(2.2)

We define the Pohozaev manifold that

$\mathcal{P}=\\{u\in
H^{1}(\mathbb{R}^{N})\setminus\\{0\\}\,|\,\mathcal{P}(u)=0\\}$

and set $\mathcal{P}_{G}=\mathcal{P}\cap\mathcal{H}_{G}$. To simplify our
notations, we define

$A(u)=\int_{\mathbb{R}^{N}}\lvert\nabla u\rvert^{2},\quad
B(u)=\int_{\mathbb{R}^{N}}\lvert u\rvert^{2},\quad\text{ and }\quad
Q(u)=\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u))F(u).$

######  Lemma 2.2.

Let $N\geq 2$ and $F\in C^{1}(\mathbb{R};\mathbb{R})$ satisfy the assumptions
$(F_{1})$–$(F_{2})$. Then for each $u\in\mathcal{H}_{G}\setminus\\{0\\}$ that
satisfies $Q(u)>0$, there exist a path $\gamma_{u}\in
C([0,+\infty);\mathcal{H}_{G})$ and a unique $t_{u}\in(0,+\infty)$ such that
$\gamma_{u}(0)=0$, $\gamma_{u}(t_{u})\in\mathcal{P}_{G}$ and
$\mathcal{E}(\gamma_{u}(t_{u}))=\max_{t\geq 0}\mathcal{E}(\gamma_{u}(t))$.
Particularly, if $\mathcal{P}(u)\leq 0$, then $t_{u}\leq 1$.

This lemma essentially is the results of [27, Propositons 4.1] (see [1,
Propositons 5.1] for $N=2$) and the proof relies on the scaling technique
developed by Jeanjean and Tanaka [17].

###### Proof.

Let $u\in\mathcal{H}_{G}\setminus\\{0\\}$ that satisfies $Q(u)>0$. We define a
path $\gamma_{u}:[0,+\infty)\to\mathcal{H}_{G}$ such that for $N\geq 3$

$\gamma_{u}(t)(x)=\begin{cases}u(x/t),&\text{if }t>0,\\\ 0,&\text{if
}t=0.\end{cases}$

Then the path $\gamma_{u}(t)\in C([0,+\infty);\mathcal{H}_{G})$ and satisfies
$\gamma_{u}(0)=0$. Direct computations show us two functions that

$\alpha(t):=\mathcal{E}(\gamma_{u}(t))=\frac{t^{N-2}}{2}A(u)+\frac{t^{N}}{2}B(u)-\frac{t^{N+\alpha}}{2}Q(u)$

and

$\beta(t):=\mathcal{P}(\gamma_{u}(t))=\frac{N-2}{2}t^{N-2}A(u)+\frac{N}{2}t^{N}B(u)-\frac{N+\alpha}{2}t^{N+\alpha}Q(u).$

It is easy to see that $\beta(t)=t\alpha^{\prime}(t)$ and $\alpha(t)$ admits a
global maxima because $\alpha(t)>0$ for small $t$ and $\alpha(t)<0$ when $t$
is sufficient large. It then follows that there exists $t_{u}\in(0,+\infty)$
such that $\alpha(t_{u})=\max_{t\geq 0}\alpha(t)$. Moreover, $t_{u}$ is unique
since the function $\beta(t)$ only has one zero point so that $\beta(t_{u})=0$
and $\gamma_{u}(t_{u})\in\mathcal{P}_{G}$. In fact, suppose in contrary that
$t_{u}^{1}<t_{u}^{2}$ such that $\beta(t_{u}^{1})=\beta(t_{u}^{2})=0$, we then
deduce a contradiction that

$\Big{(}\frac{1}{(t_{u}^{1})^{2}}-\frac{1}{(t_{u}^{2})^{2}}\Big{)}\frac{N-2}{2}A(u)=\Big{(}(t_{u}^{1})^{\alpha}-(t_{u}^{2})^{\alpha}\Big{)}\frac{N+\alpha}{2}Q(u).$

Note that $\beta(t)>0$ on the interval $(0,t_{u})$. $\mathcal{P}(u)\leq 0$
implies that $\beta(1)\leq 0$, this brings us that $(0,t_{u})\subset(0,1)$,
which amounts to $t_{u}\leq 1$.

For the case $N=2$, the above path $\gamma_{u}$ is not anymore continuous at
$0$ in $\mathcal{H}_{G}$, we shall connect the function $\gamma_{u}(t_{u})$
with $0$ by the multiplication. Since $Q(u)>0$, we can find a $t_{u}>0$ such
that $\beta(t_{u})=0$, so that $\gamma_{u}(t_{u})\in\mathcal{P}_{G}$. We then
define that

$\tilde{\gamma}_{u}(t)(x)=\begin{cases}\frac{t}{\tau_{0}}u(\frac{x}{\tau_{0}t_{u}}),&\text{if
}t\leq\tau_{0},\\\ u(\frac{x}{tt_{u}}),&\text{if }t\geq\tau_{0}.\end{cases}$

The parameter $\tau_{0}\in(0,1)$ will be determined later. Clearly, we see
that $\gamma_{u}$ is continuous in $\mathcal{H}_{G}$. For $t\geq\tau_{0}$, the
Pohozaev identity gives that

$\displaystyle\tilde{\alpha}(t)=\mathcal{E}(\tilde{\gamma}_{u}(t))$
$\displaystyle=\frac{1}{2}A(u)+\frac{(t_{u}t)^{2}}{2}B(u)-\frac{(tt_{u})^{2+\alpha}}{2}Q(u)$
$\displaystyle=\frac{1}{2}A(u)+\Big{(}\frac{t^{2}}{2}-\frac{t^{2+\alpha}}{2+\alpha}\Big{)}t_{u}^{2}B(u).$

It is not difficult to check that $\tilde{\alpha}(t)$ achieves its strict
maximum at $t=1$ and is negative for sufficient large $t$. Without loss of
generality, we assume that $t_{u}=1$. For $t\leq\tau_{0}$, by the assumptions
$(F_{1})$–$(F_{2})$ and the Hardy–Littlewood–Sobolev inequality, we deduce
that

$\displaystyle\frac{\tau_{0}}{t}\int_{\mathbb{R}^{2}}(I_{\alpha}*$
$\displaystyle
F(tu/\tau_{0})f(tu/\tau_{0})u=\frac{\tau^{2}_{0}}{t^{2}}\int_{\mathbb{R}^{2}}(I_{\alpha}*F(tu/\tau_{0})f(tu/\tau_{0})tu/\tau_{0}$
$\displaystyle\leq
C\frac{\tau^{2}_{0}}{t^{2}}\Big{(}\int_{\mathbb{R}^{2}}\lvert
F(tu/\tau_{0})\rvert^{\frac{4}{2+\alpha}}\Big{)}^{\frac{2+\alpha}{4}}\Big{(}\int_{\mathbb{R}^{2}}\lvert
f(tu/\tau_{0})tu/\tau_{0}\rvert^{\frac{4}{2+\alpha}}\Big{)}^{\frac{2+\alpha}{4}}$
$\displaystyle\leq
C\frac{\tau^{2}_{0}}{t^{2}}\Big{(}\int_{\mathbb{R}^{2}}\lvert
tu/\tau_{0}\rvert^{\frac{8}{2+\alpha}}+\lvert
tu/\tau_{0}\rvert^{\frac{4\bar{p}}{2+\alpha}}\Big{)}^{\frac{2+\alpha}{2}}\leq
C\big{(}\|u\|^{4}+\|u\|^{2\bar{p}}\big{)}.$

We conclude our proof by fixing a proper $\tau_{0}<1$ such that
$\frac{\,\mathrm{d}}{\,\mathrm{d}\tau}\mathcal{E}(\tilde{\gamma}_{u}(t))>0$
for $t\leq\tau_{0}$. Actually, we have for sufficiently small $\tau_{0}$,

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\tilde{\alpha}(t)$
$\displaystyle=\frac{\,\mathrm{d}}{\,\mathrm{d}t}\Big{(}\frac{t^{2}}{2\tau_{0}^{2}}A(u)+\frac{t^{2}}{2}B(u)-\frac{\tau_{0}^{2+\alpha}}{2}\int_{\mathbb{R}^{2}}(I_{\alpha}*F(tu/\tau_{0}))F(tu/\tau_{0})\Big{)}$
$\displaystyle=t\Big{(}\frac{1}{\tau_{0}^{2}}A(u)+B(u)-\tau_{0}^{\alpha}\frac{\tau_{0}}{t}\int_{\mathbb{R}^{2}}(I_{\alpha}*F(tu/\tau_{0})f(tu/\tau_{0})u\Big{)}$
$\displaystyle\geq
t\Big{(}\frac{1}{\tau_{0}^{2}}A(u)+B(u)-C\tau_{0}^{\alpha}(\|u\|^{4}+\|u\|^{2p})\Big{)}>0.$

Hence, $\tilde{\alpha}(t)$ is strictly increasing when $t\leq\tau_{0}$, which
allows $t=t_{u}$ being the unique global maximum point. ∎

A standard argument shows that the functional $\mathcal{E}$ has the mountain
pass geometry, which furnishes a Pohozaev–Palais–Smale sequence at the energy
level $c_{G}$ defined by (2.1). This sequence has an additional property that
the equation (2.2) asymptotically holds.

######  Proposition 2.3.

Let $N\geq 2$, $\alpha\in(0,N)$. Assume the growth assumptions
$(F_{0})$–$(F_{2})$ hold. Then there exists a sequence
$(u_{n})_{n\in\mathbb{N}}$ in $\mathcal{H}_{G}$ such that, as $n\to\infty$,

$\displaystyle\mathcal{E}(u_{n})\to
c_{G}\in(0,+\infty),\;\mathcal{P}(u_{n})\to 0,\;\text{\rm{ and}
}\mathcal{E}^{\prime}(u_{n})\to 0\;\text{\rm{ strongly in the dual space
}}\mathcal{H}_{G}^{*}.$

###### Proof.

To show $c_{G}<+\infty$, we need construct a path in $\mathcal{H}_{G}$. By
Lemma 2.2, this turns out to find a function $w_{0}\in\mathcal{H}_{G}$ such
that $Q(w_{0})>0$. In order to derive this, by recalling the nontrivial
assumption $(F_{0})$ and the unit ball $B_{1}\subset\mathbb{R}^{N}$ centered
at the origin, we take $w_{0}=s_{0}\sum_{g\in
G}\psi(g)g\diamond\chi_{B_{1}\cap\mathcal{F}}(x)$ so that
$w_{0}\in\mathcal{H}_{G}$ and $Q(w_{0})=F(s_{0})^{2}Q(\chi_{B_{1}})>0$. Here
$\mathcal{F}$ is the strict fundamental domain of the Coxeter group $G$ and
$\chi_{B_{1}\cap\mathcal{F}}$ denotes the characteristic function on
$B_{1}\cap\mathcal{F}$. In fact, for any $\bar{g}\in G$, we have

$\bar{g}\diamond w_{0}=s_{0}\sum_{g\in
G}\psi(g)(\bar{g}g)\diamond\chi_{B_{1}\cap\mathcal{F}}(x)=s_{0}\psi(\bar{g})\sum_{g\in
G}\psi(\bar{g}g)(\bar{g}g)\diamond\chi_{B_{1}\cap\mathcal{F}}(x)=\psi(\bar{g})w_{0}.$

The remanent proof can be verbatim copied from [27, Proposition 2.1] and [1,
Proposition 3.1], respectively. The only difference is the Sobolev space
$H^{1}(\mathbb{R}^{N})$ should be replaced by the framework subspace
$\mathcal{H}_{G}$, so we omit the details but emphasis that the scaling
technique of Jeanjean [16] can be used to get the additional asymptotic
property for the Palais–Smale sequence that $\mathcal{P}(u_{n})\to 0$ as
$n\to\infty$. ∎

We remark that the nontrivial assumption $(F_{0})$ indicates both the paths
set $\Gamma_{G}$ and the Pohozaev manifold $\mathcal{P}$ are nonempty. We
shall prove the following characterizations.

######  Proposition 2.4.

Let $N\geq 2$, $\alpha\in(0,N)$. Assume that the assumptions
$(F_{0})$–$(F_{2})$ hold. Then

$c_{G}=p_{G}=m_{G}.$

Here $p_{G}=\inf_{\mathcal{P}_{G}}\mathcal{E}$ and
$m_{G}=\inf_{\mathcal{M}_{G}}\mathcal{E}$ with
$\mathcal{M}_{G}=\\{u\in\mathcal{H}_{G}\setminus\\{0\\}\,|\,\mathcal{E}^{\prime}(u)=0\\}.$

###### Proof.

For any $u\in\mathcal{P}_{G}$, we have $\mathcal{P}(u)=0$ so that $Q(u)>0$. By
Lemma 2.2 and a suitable change of variable, we conclude that
$\gamma_{u}(t)\in\Gamma_{G}$. It then follows that $c_{G}\leq\max_{t\geq
0}\mathcal{E}(\gamma_{u}(t))=\mathcal{E}(u)$. The arbitrariness of
$u\in\mathcal{P}_{G}$ implies that $c_{G}\leq p_{G}$. We easily deduce that
$p_{G}\leq m_{G}$ since $\mathcal{M}_{G}\subset\mathcal{P}_{G}$. The last
inequality $m_{G}\leq c_{G}$ will be proved by the existence of a $G$-saddle
solution $u\in\mathcal{H}_{G}$ that satisfies $\mathcal{E}(u)\leq c_{G}$. It
turns out that this proposition will be complete by an induction argument. ∎

##  3 Exponential decays for the solutions

We shall prove that solutions of Choquard equation ($\mathcal{C}$) decay
exponentially at infinity, which shall play a big part in determining the
threshold values for the compactness, see Proposition 4.1 below.

Proof of Theorem 1.2. The local regularity that $u\in
W^{2,s}_{\rm{loc}}(\mathbb{R}^{N})$ for $s\geq 1$ follows from [27, Theorem 2]
for $N\geq 3$ and from [1, Proposition 5.1] for $N=2$, respectively. Next, we
show that $I_{\alpha}*F(u)\in L^{\infty}(\mathbb{R}^{N})$. For $N\geq 3$, let
$H(u)=F(u)/u$ and $K(u)=f(u)$. In view of the assumptions $(F_{1})$–$(F_{2})$,
we deduce that there exists constant $C>0$ such that for $x\in\mathbb{R}^{N}$,

$\lvert H(u)\rvert,\lvert K(u)\rvert\leq C\big{(}\lvert
u(x)\rvert^{\frac{\alpha}{N}}+\lvert
u(x)\rvert^{\frac{2+\alpha}{N-2}}\big{)}.$

It then follows that $H,K\in
L^{\frac{2N}{\alpha}}(\mathbb{R}^{N})+L^{\frac{2N}{2+\alpha}}(\mathbb{R}^{N})$.
Here a function $g\in L^{p_{1}}(\mathbb{R}^{N})+L^{p_{2}}(\mathbb{R}^{N})$
means that $g\leq\lvert g_{1}\rvert+\lvert g_{2}\rvert$ with $g_{i}\in
L^{p_{i}}(\mathbb{R}^{N})$ for $i=1,2$. We then conclude by [27, Proposition
3.1] that $u\in L^{p}(\mathbb{R}^{N})$ for any
$p\in[2,\frac{N}{\alpha}\frac{2N}{N-2})$. For $N=2$, we conclude directly by
the Rellich–Kondrachov embedding theorem that $u\in L^{p}(\mathbb{R}^{N})$ for
any $p\in[2,\infty)$.

We denote for convenience $\frac{2N}{N-2}=\infty$ in case that $N=2$. By the
growth assumptions $(F_{1})$–$(F_{2})$, there exists a constant $C>0$ such
that for all $t\in\mathbb{R}$

$\lvert F(t)\rvert\leq C(\lvert t\rvert^{2}+\lvert
t\rvert^{\frac{N+\alpha}{N-2}}).$

For the planar case, the exponent $\frac{N+\alpha}{N-2}$ will be replaced by
fixed $\bar{p}\in[2,\infty)$. We then conclude that $I_{\alpha}*F(u)\in
L^{\infty}(\mathbb{R}^{N})$ by proving that $I_{\alpha}*\lvert u\rvert^{s}\in
L^{\infty}(\mathbb{R}^{N})$ for any $s\in(\frac{2\alpha}{N},\frac{2N}{N-2})$.
In fact, we have by direct computations that

$\begin{split}\int_{\mathbb{R}^{N}}\frac{\lvert u(y)\rvert^{s}}{\lvert
x-y\rvert^{N-\alpha}}\,\mathrm{d}y&=\int_{\mathbb{R}^{N}}\frac{\lvert
u(x-y)\rvert^{s}}{\lvert y\rvert^{N-\alpha}}\,\mathrm{d}y\\\
&=\int_{B_{1}}\frac{\lvert u(x-y)\rvert^{s}}{\lvert
y\rvert^{N-\alpha}}\,\mathrm{d}y+\int_{B_{1}^{c}}\frac{\lvert
u(x-y)\rvert^{s}}{\lvert y\rvert^{N-\alpha}}\,\mathrm{d}y.\end{split}$ (3.1)

By choosing $t\in(\frac{N}{\alpha},\frac{N}{s\alpha}\frac{2N}{N-2})$ and
$r\in(\frac{2}{s},\frac{N}{\alpha})$ such that
$\frac{t}{t-1}<\frac{N}{N-\alpha}<\frac{r}{r-1}$, we deduce that $\lvert
x\rvert^{\alpha-N}\in L^{\frac{t}{t-1}}(B_{1})$ and $\lvert
x\rvert^{\alpha-N}\in L^{\frac{r}{r-1}}(B^{c}_{1})$. Here and in the sequel,
the notation $B_{r}$ denotes the ball in $\mathbb{R}^{N}$ with radius $r$ and
centered at the origin. By the Hölder inequality, we have

$\begin{split}\int_{\mathbb{R}^{N}}\frac{\lvert u(y)\rvert^{s}}{\lvert
x-y\rvert^{N-\alpha}}\,\mathrm{d}y&\leq\Big{(}\int_{B_{1}}\lvert
x\rvert^{\frac{t(\alpha-N)}{t-1}}\,\mathrm{d}x\Big{)}^{\frac{t-1}{t}}\Big{(}\int_{\mathbb{R}^{N}}\lvert
u\rvert^{st}\Big{)}^{\frac{1}{t}}\\\ &\quad+\Big{(}\int_{B^{c}_{1}}\lvert
x\rvert^{\frac{r(\alpha-N)}{r-1}}\,\mathrm{d}x\Big{)}^{\frac{r-1}{r}}\Big{(}\int_{\mathbb{R}^{N}}\lvert
u\rvert^{sr}\Big{)}^{\frac{1}{r}}.\end{split}$ (3.2)

Hence, (3.1) and (3.2) conclude the claim. Note that $u\in
W^{2,s}_{\rm{loc}}(\mathbb{R}^{N})$ for $s\geq 1$. We then derive by combining
the Rellich–Kondrachov embedding theorem and a standard Schauder estimate that
$u\in C^{2,\lambda}(B_{1}(x))$ for some $\lambda\in(0,1)$ and there exists a
constant $C$ independent of $x$ such that

$\lvert u(x)\rvert\leq\|u\|_{C^{1,\lambda}(B_{1}(x))}\leq
C\|u\|_{W^{2,s}(B_{1}(x))}.$

This leads to $\lim_{\lvert x\rvert\to+\infty}\lvert u(x)\rvert=0$. We thus
have $u\in L^{\infty}(\mathbb{R}^{N})$ since $u\in C^{1}(\mathbb{R}^{N})$.
Moreover, for $s\in[2,\frac{N+\alpha}{N-2}]$, we can take
$\varepsilon\in(0,s-\frac{2\alpha}{N})$ so that

$\begin{split}\int_{\mathbb{R}^{N}}&\frac{\lvert u(y)\rvert^{s}}{\lvert
x-y\rvert^{N-\alpha}}\,\mathrm{d}y=\int_{B_{\lvert x\rvert/2}}\frac{\lvert
u(y)\rvert^{s}}{\lvert x-y\rvert^{N-\alpha}}\,\mathrm{d}y+\int_{B^{c}_{\lvert
x\rvert/2}}\frac{\lvert u(y)\rvert^{s}}{\lvert
x-y\rvert^{N-\alpha}}\,\mathrm{d}y\\\ &\leq\C\lvert
x\rvert^{\alpha-N}\int_{\mathbb{R}^{N}}\lvert
u\rvert^{s}+\|u\|^{\varepsilon}_{L^{\infty}(B^{c}_{\lvert
x\rvert/2})}(I_{\alpha}*\lvert u\rvert^{s-\varepsilon})(x).\end{split}$

This gives that $\lim_{\lvert x\rvert\to+\infty}(I_{\alpha}*F(u))(x)=0$,
which, together with the growth assumption $(F_{1})$, amounts to

$\lim_{\lvert x\rvert\to+\infty}\big{(}I_{\alpha}*F(u)\big{)}\lvert
f(u)/u\rvert=0.$

Therefore, there exists $\gamma\in(0,1)$ such that $(I_{\alpha}*F(u))\lvert
f(u)\rvert\leq\gamma\lvert u(x)\rvert$ for sufficiently large $x$. Then the
desired decay estimate can be derived by using the Kato’s inequality [19] and
by virtue of the comparison principle, we omit the details and refer to [26,
36].

##  4 Energy estimates

######  Proposition 4.1.

Let $N\geq 2$ and $\alpha\in(0,N)$. Assume that $G\in\mathcal{G}_{k}$ with
$1\leq k\leq N$. Then there exists $c_{G}^{*}$ such that

$\displaystyle 0<c_{G}<c_{G}^{*}$ $\displaystyle:=\min\big{\\{}\lvert Gq\rvert
c_{S_{q}}\big{|}q\in\partial^{k-1}\mathcal{F}\big{\\}}<\min\big{\\{}\lvert
Gq\rvert
c_{S_{q}}\big{|}q\in\partial^{i}\mathcal{F},i=0,1,\cdots,k-2\big{\\}}.$

According to the classification results of the finite Coxeter group,
$\mathcal{W}_{1}$ contains only ${\rm\bf{A}}_{1}$ and its fundamental domain
is $\mathbb{R}^{N}_{+}$. In this case, the threshold value $c_{G}^{*}=2c_{0}$
[8, Proposition 2.4]. For $k=2$, $\mathcal{W}_{2}$ is consist of dihedral
groups ${\rm\bf{I}}_{2}(m)$ for $m\geq 2$ and the associated fundamental
domain is a sectorial region with dihedral angle ${\pi}/{m}$. The last two
inequalities in this case turn out to be
$c_{G}<c_{G}^{*}:=mc_{{\bf{A}_{1}}}<2mc_{0}$ [36, Proposition 3.1].

###### Proof.

We shall construct test functions for each rank by induction. We remark that
Proposition 2.4 holds for $G=\\{1\\}$, see [1, 27]. This will be the start
point. We first take a cut-off function $\zeta\in
C_{c}^{2}(\mathbb{R}^{N};[0,1])$ such that $\zeta=1$ on $B_{1}$ and
$\operatorname{supp}\zeta\subset B_{2}$. We begin our construction from $k=1$.
We then have $G=\rm\bf{A}_{1}$ and $\mathcal{F}=\mathbb{R}^{N}_{+}$. Let
$U_{0}$ be the ground state solution for the Choquard equation
($\mathcal{C}$). We then define for $x=(x^{\prime},x^{N})\in\mathbb{R}^{N}$
that

$U_{R}(x):=\big{(}\zeta_{R}U_{0}\big{)}(x^{\prime},x^{N}-3R)-\big{(}\zeta_{R}U_{0}\big{)}(x^{\prime},-x^{N}-3R).$

It is easy to check that $\tau_{N}\diamond U_{R}=-U_{R}$ where $\tau_{N}$ is
the reflection such that $\tau_{N}(x^{\prime},x^{N})=(x^{\prime},-x^{N})$.
Repeating the subsequent arguments, we conclude that
$c_{{\rm\bf{A}}_{1}}<2c_{0}$. With this estimate, we shall prove successively
that the Choquard equation ($\mathcal{C}$) admits a saddle solution
$U_{{\rm\bf{A}_{1}}}$ such that $\mathcal{E}(U_{{\rm\bf{A}_{1}}})\leq
c_{{\rm\bf{A}_{1}}}$, which implies that Proposition 2.4 holds for $k=1$.

Recall that $G=W_{k_{1}}\times\cdots\times W_{k_{\ell}}:=W_{k_{1}}\times W$.
We now suppose this proposition holds for $k-1$. By Lemma A.8, for each
$q\in\partial^{k-1}\mathcal{F}\cap\partial B_{1}$, we see that
$S_{q}\in\mathcal{G}_{k-1}$. Without loss of generality, we may assume
$\partial^{k-1}\mathcal{F}\subset\cap_{s\in\mathcal{S}_{k-1}\subset\mathcal{S}}H_{s}$
with $\mathcal{S}\setminus\mathcal{S}_{k-1}=\\{s_{1}\\}$ and $s_{1}\in
W_{k_{1}}$. We carry out the proof into two subcases. The first case is
$k_{1}=1$. We thus have $S_{q}=W$ and $q$ is perpendicular to $H_{s_{1}}$.
Hence, we may assume that $H_{s_{1}}=\\{(x_{1},x^{\prime})\,|\,x_{1}=0\\}$. By
the inductive assumption, the Choquard equation ($\mathcal{C}$) admits saddle
solutions $U_{S_{q}}\in H^{1}(\mathbb{R}^{N})$ that achieves $c_{S_{q}}$ and
satisfies that

$w\diamond U_{S_{q}}=\psi(w)U_{S_{q}},\qquad\forall w\in W,$

where $\psi:G\to\\{\pm 1\\}$ is the epimorphism. Particularly, Proposition 2.4
holds for any proper subgroup of $G$, that is,
$c_{S_{q}}=p_{S_{q}}=m_{S_{q}}$. We define

$U^{q}_{R}(x):=\big{(}\zeta_{R}U_{S_{q}}\big{)}(x^{1}-3R,x^{\prime})-\big{(}\zeta_{R}U_{S_{q}}\big{)}(-x^{1}-3R,x^{\prime}).$

We then conclude $s_{1}\diamond U^{q}_{R}=-U_{R}^{q}$ and $w\diamond
U_{R}^{q}=\psi(w)U^{q}_{R}$ for any $w\in W$, so that
$U^{q}_{R}\in\mathcal{H}_{G}$ since $s_{1}w=ws_{1}$ for $w\in W$. Here the
group action $S_{q}$ is realized in the space
$\\{0\\}\times\mathbb{R}^{k-1}=\mathbb{R}^{k-1}$.

We now deal with the remaining case that $k_{1}\geq 2$. We realize the group
action $S_{q}$ in the whole space $\mathbb{R}^{k}$. In other words, we assume
the saddle solution $U_{S_{q}}\in H^{1}(\mathbb{R}^{N})$ that achieves
$c_{S_{q}}$ satisfies

$w\diamond U_{S_{q}}=\psi(w)U_{S_{q}},\qquad\forall\;w\in S_{q}.$

Note that $W_{k_{1}}$ is irreducible and $s_{1}\not\in S_{q}$. By the
Coxeter’s classification theorem A.5 (or by the connectedness of the Coxeter
diagrams), we see that there exists at most three subgroups of $W_{k_{1}}$
such that $S_{q}=W_{t_{1}}\times W_{t_{2}}\times W_{t_{3}}\times W$, here
$W_{t_{i}}\in\mathcal{W}_{t_{i}}$, $t_{1}\geq 1$ and $t_{2},t_{3}\geq 0$
satisfy that $t_{1}+t_{2}+t_{3}=k_{1}-1$. Recalling Lemma A.7, we then define
a test function

$U^{q}_{R}(x):=\frac{1}{\lvert S_{q}\rvert}\sum_{n\in
N_{k}}\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}s_{1}x-l_{G}Rq)-\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}x-l_{G}Rq)$

where $N_{k}=N_{k_{1}}\times W$ and the constant $l_{G}$ is determined so that
the balls $B_{2R}(l_{G}RGq)$ are separated far away from each other. Here we
recall that $Gq$ is the orbit of $q\in\partial^{k-1}\mathcal{F}\cap\partial
B_{1}$ under the action of $G$. This can be done provided that $l_{G}$ is
sufficiently large, since there exist constants $K_{1},K_{2}>0$ that depend
only on the group $G$ such that for $x\neq y\in Gq$,

$K_{1}:=\min_{x\neq y\in Gq}\lvert x-y\rvert\leq\lvert
x-y\rvert\leq\max_{x\neq y\in Gq}\lvert x-y\rvert:=K_{2}.$

We claim that $U^{q}_{R}\in\mathcal{H}_{G}$. By the facts that
$wW_{k_{1}}=W_{k_{1}}w$ and $wq=q$ for $w\in W$, we deduce

$w^{-1}\diamond U^{q}_{R}=\frac{1}{\lvert S_{q}\rvert}\sum_{n\in
N_{k}}\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}s_{1}wx-
l_{G}Rq)-\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}wx-l_{G}Rq)=\psi(w)U_{R}^{q}.$

On the other hand, for every $s\in\mathcal{S}\cap(W_{t_{1}}\times
W_{t_{2}}\times W_{t_{3}})$, it follows that $s\in
W_{k_{1}}=N_{k_{1}}\rtimes\langle s_{1}\rangle$, so that there exist unique
$n_{s},n_{s}^{\prime}\in N_{k_{1}}$ such that
$s=s_{1}n_{s}=n_{s}^{\prime}s_{1}$. We then have

$\begin{split}&s\diamond U^{q}_{R}=\frac{1}{\lvert S_{q}\rvert}\sum_{n\in
N_{k}}\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}s_{1}sx-
l_{G}Rq)-\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}sx-l_{G}Rq)\\\
&=\frac{1}{\lvert S_{q}\rvert}\sum_{n\in
N_{k}}\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}n_{s}x-l_{G}Rq)-\frac{1}{\lvert
S_{q}\rvert}\sum_{n\in
N_{k}}\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}n_{s}^{\prime}s_{1}x-l_{G}Rq)\\\
&=-\frac{1}{\lvert S_{q}\rvert}\sum_{n\in
N_{k}}\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}s_{1}x-l_{G}Rq)+\frac{1}{\lvert
S_{q}\rvert}\sum_{n\in
N_{k}}\big{(}\zeta_{R}U_{S_{q}}\big{)}(n^{-1}x-l_{G}Rq)=-U_{R}^{q}.\end{split}$

By the construction, we easily see $s_{1}\diamond U_{R}^{q}=-U_{R}^{q}$. We
thus deduce $s\diamond U_{R}^{q}=\psi(s)U_{R}^{q}$ for any $s\in\mathcal{S}$.
This concludes the claim that $U^{q}_{R}\in\mathcal{H}_{G}$.

We now prove that $Q(U_{R}^{q})>0$ for sufficiently large $R$. To this end, we
denote $B^{j}_{2R}=B_{2R}(l_{G}R{z}_{j}+x^{0})$ for given
$x^{0}\in\\{0\\}\times\mathbb{R}^{N-k}$ and ${z}_{j}\in Gq$. Note that all of
the balls $B^{j}_{2R}$ for $1\leq j\leq\lvert Gq\rvert$ are separated from
each other. We then deduce that

$\begin{split}\int_{\mathbb{R}^{N}}&(I_{\alpha}*F(U_{R}^{q}))F(U_{R}^{q})=A_{\alpha}\sum_{z_{j}\in
Gq}\sum_{z_{m}\in
Gq}\int_{B^{j}_{2R}}\int_{B^{m}_{2R}}K_{jm}(x,y)\,\mathrm{d}x\,\mathrm{d}y\\\
&=A_{\alpha}\sum_{z_{j}\in
Gq}\int_{B^{j}_{2R}}\int_{B^{j}_{2R}}K_{jj}(x,y)\,\mathrm{d}x\,\mathrm{d}y+A_{\alpha}\sum_{z_{j}\neq
z_{m}\in
Gq}\int_{B^{j}_{2R}}\int_{B^{m}_{2R}}K_{jm}(x,y)\,\mathrm{d}x\,\mathrm{d}y\end{split}$

where the function $K_{jm}:\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}$ is
defined by

$K_{jm}(x,y):=\frac{F\big{(}(\zeta_{R}U_{R}^{q})(x-l_{G}Rz_{j})\big{)}F\big{(}(\zeta_{R}U_{R}^{q})(y-l_{G}Rz_{m})\big{)}}{\lvert
x-y\rvert^{N-\alpha}}.$

Moreover, we observe that $(K_{1}l_{G}-4)R\leq\lvert
x-y\rvert\leq(K_{2}l_{G}+4)R$ for any $x\in B^{j}_{2R}$ and $y\in B^{m}_{2R}$
with $z_{j}\neq z_{m}$. It then follows that

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*F(U^{q}_{R}))F(U^{q}_{R})$
$\displaystyle\geq{\lvert
Gq\rvert}\int_{B_{2R}}\int_{B_{2R}}A_{\alpha}\frac{F\big{(}(\zeta_{R}U_{S_{q}})(x)\big{)}F\big{(}(\zeta_{R}U_{S_{q}})(y)\big{)}}{\lvert
x-y\rvert^{N-\alpha}}$ (4.1) $\displaystyle\quad+\sum_{z_{j}\neq z_{m}\in
Gq}\frac{\C}{R^{N-\alpha}}\int_{B^{j}_{2R}}F\big{(}(\zeta_{R}U_{S_{q}})(x-l_{G}Rz_{j})\big{)}\,\mathrm{d}x\int_{B^{m}_{2R}}F\big{(}(\zeta_{R}U_{S_{q}})(y-l_{G}Rz_{m})\big{)}\,\mathrm{d}y$
$\displaystyle={\lvert
Gq\rvert}\int_{B_{2R}}\int_{B_{2R}}A_{\alpha}\frac{F\big{(}(\zeta_{R}U_{S_{q}})(x)\big{)}F\big{(}\zeta_{R}U_{S_{q}})(y)\big{)}}{\lvert
x-y\rvert^{N-\alpha}}\,\mathrm{d}x\,\mathrm{d}y+\frac{\C}{R^{N-\alpha}}\Big{(}\int_{B_{2R}}F\big{(}\zeta_{R}U_{S_{q}}\big{)}\Big{)}^{2}.$

We now estimate the first term that

$\begin{split}&\int_{B_{2R}}\int_{B_{2R}}A_{\alpha}\frac{F\big{(}(\zeta_{R}U_{S_{q}})(x)\big{)}F\big{(}\zeta_{R}U_{S_{q}})(y)\big{)}}{\lvert
x-y\rvert^{N-\alpha}}\,\mathrm{d}x\,\mathrm{d}y=\int_{\mathbb{R}^{N}}(I_{\alpha}*F(\zeta_{R}U_{S_{q}})F(\zeta_{R}U_{S_{q}})\\\
&\geq\int_{\mathbb{R}^{N}}(I_{\alpha}*F(U_{S_{q}})F(U_{S_{q}})-2\int_{\mathbb{R}^{N}}\big{(}I_{\alpha}*F(U_{S_{q}})\big{)}\big{(}F(U_{S_{q}})-F(\zeta_{R}U_{S_{q}})\big{)}.\end{split}$
(4.2)

In view of the Hardy–Littlewood–Sobolev inequality [21, Theorem 4.3], we
deduce that

$\displaystyle\Big{|}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(U_{S_{q}})$
$\displaystyle\big{(}F(U_{S_{q}})-F(\zeta_{R}U_{S_{q}})\big{)}\Big{|}$
$\displaystyle\leq\C\Big{(}\int_{\mathbb{R}^{N}}\lvert
F(U_{S_{q}})\rvert^{\frac{2N}{N+\alpha}}\int_{B_{R}^{c}}\big{(}\lvert
F(U_{S_{q}})\rvert+\lvert
F(\zeta_{R}U_{S_{q}})\rvert\big{)}^{\frac{2N}{N+\alpha}}\Big{)}^{\frac{N+\alpha}{2N}}.$

By the assumptions $(F_{1})$–$(F_{2})$, we see that there exists $C>0$ such
that for all $s\in\mathbb{R}$

$\lvert F(s)\rvert\leq C(\lvert s\rvert^{\frac{N+\alpha}{N}}+\lvert
s\rvert^{\frac{N+\alpha}{N-2}}),$

this implies that $\int_{\mathbb{R}^{N}}\lvert
F(U_{S_{q}})\rvert^{\frac{2N}{N+\alpha}}$ is bounded and

$\int_{B_{R}^{c}}\lvert
F(\zeta_{R}U_{S_{q}})\rvert^{\frac{2N}{N+\alpha}},\;\int_{B_{R}^{c}}\lvert
F(U_{S_{q}})\rvert^{\frac{2N}{N+\alpha}}\leq\C\int_{B_{R}^{c}}\lvert
U_{S_{q}}\rvert^{2}+\C\int_{B_{R}^{c}}\lvert
U_{S_{q}}\rvert^{\frac{2N}{N-2}}.$

Noticing that $U_{S_{q}}$ decays exponential at infinity, we conclude that

$2\int_{\mathbb{R}^{N}}(I_{\alpha}*F(U_{S_{q}}))\big{(}F(U_{S_{q}})-F(\zeta_{R}U_{S_{q}})\big{)}=o(\frac{1}{R^{N-\alpha}}).$
(4.3)

Similarly, by the assumptions $(F_{1})$ and $(F_{2})$ again we derive that
$\lvert F(t)\rvert\leq C(\lvert t\rvert^{2}+\lvert
t\rvert^{\frac{N+\alpha}{N-2}})$ for some positive constant $C$, so that

$\frac{1}{R^{N-\alpha}}\Big{(}\int_{\mathbb{R}^{N}}F(U_{S_{q}})\Big{)}^{2}-\frac{1}{R^{N-\alpha}}\Big{(}\int_{\mathbb{R}^{N}}F(\zeta_{R}U_{S_{q}})\Big{)}^{2}=o(\frac{1}{R^{N-\alpha}}).$
(4.4)

By combining the above (4)–(4.4), we arrive at an asymptotic lower bound for
$Q(U_{R}^{q})$ that

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*F(U^{q}_{R}))F(U_{R}^{q})$
$\displaystyle\geq\frac{\lvert G\rvert}{\lvert
S_{q}\rvert}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(U_{S_{q}}))F(U_{S_{q}})+\frac{\C}{R^{N-\alpha}}\Big{(}\int_{B_{R}}F(\zeta_{R}U_{S_{q}})\Big{)}^{2}+o(\frac{1}{R^{N-\alpha}})$
$\displaystyle=\frac{\lvert G\rvert}{\lvert
S_{q}\rvert}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(U_{S_{q}}))F(U_{S_{q}})+\frac{\C}{R^{N-\alpha}}\Big{(}\int_{\mathbb{R}^{N}}F(U_{S_{q}})\Big{)}^{2}+o\big{(}\frac{1}{R^{N-\alpha}}\big{)}.$
(4.5)

This completes the claim. By Lemma 2.2, there exist a path $\gamma_{R}\in
C([0,+\infty);\mathcal{H}_{G})$ and a unique $t_{R}\in(0,+\infty)$ such that
$\mathcal{P}(\gamma_{R}(t_{R}))=0$ so that
$\gamma_{R}(t_{R})\in\mathcal{P}_{G}$. Direct calculations give us that

$\frac{N-2}{2}t_{R}^{N-2}\int_{\mathbb{R}^{N}}\lvert\nabla
U^{q}_{R}\rvert^{2}+\frac{N}{2}t_{R}^{N}\int_{\mathbb{R}^{N}}\lvert
U_{R}^{q}\rvert^{2}=\frac{N+\alpha}{2}t_{R}^{N+\alpha}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(U^{q}_{R}))F(U^{q}_{R}).$
(4.6)

On the other hand, we deduce by integration by parts that

$\begin{split}\int_{\mathbb{R}^{N}}\lvert\nabla U^{q}_{R}\rvert^{2}+\lvert
U^{q}_{R}\rvert^{2}&=\sum_{z_{j}\in Gq}\int_{B_{2R}^{j}}\lvert\nabla
U^{q}_{R}\rvert^{2}+\lvert U^{q}_{R}\rvert^{2}=\frac{\lvert G\rvert}{\lvert
S_{q}\rvert}\int_{B_{2R}}\lvert\nabla\big{(}\zeta_{R}U_{S_{q}}\big{)}\rvert^{2}+\lvert\zeta_{R}U_{S_{q}}\rvert^{2}\\\
&=\frac{\lvert G\rvert}{\lvert
S_{q}\rvert}\int_{\mathbb{R}^{N}}\zeta_{R}^{2}(\lvert\nabla
U_{S_{q}}\rvert^{2}+\lvert U_{S_{q}}\rvert^{2})-\frac{\lvert G\rvert}{\lvert
S_{q}\rvert}\int_{\mathbb{R}^{N}}\zeta_{R}(\Delta\zeta_{R})\lvert
U_{S_{q}}\rvert^{2}\\\ &\leq\frac{\lvert G\rvert}{\lvert
S_{q}\rvert}\int_{\mathbb{R}^{N}}\lvert\nabla U_{S_{q}}\rvert^{2}+\lvert
U_{S_{q}}\rvert^{2}+\frac{\C}{R^{2}}\int_{B_{2R}\setminus B_{R}}\lvert
U_{S_{q}}\rvert^{2}.\end{split}$

Again, the exponential decay of the solution $U_{S_{q}}$ implies that

$\int_{\mathbb{R}^{N}}\big{(}\frac{N-2}{2}\lvert\nabla
U^{q}_{R}\rvert^{2}+\frac{N}{2}\lvert U^{q}_{R}\rvert^{2}\big{)}\leq\lvert
Gq\rvert\int_{\mathbb{R}^{N}}\big{(}\frac{N-2}{2}\lvert\nabla
U_{S_{q}}\rvert^{2}+\frac{N}{2}\lvert
U_{S_{q}}\rvert^{2}\big{)}+o\big{(}\frac{1}{R^{N-\alpha}}\big{)}.$ (4.7)

Taking into account of these estimates (4) and (4.7), we then derive from
(4.6) that $t_{R}$ is uniformly bounded in $R$. We thus deduce an asymptotic
bound that

$\begin{split}\mathcal{E}(\gamma_{R}(t_{R}))&=\frac{t_{R}^{N-2}}{2}\int_{\mathbb{R}^{N}}\lvert\nabla
U_{R}^{q}\rvert+\frac{t_{R}^{N}}{2}\int_{\mathbb{R}^{N}}\lvert
U_{R}^{q}\rvert^{2}-\frac{t_{R}^{N+\alpha}}{2}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(U_{R}^{q}))F(U_{R}^{q})\\\
&\leq\lvert
Gq\rvert\mathcal{E}(\gamma_{U_{S_{q}}}(t_{R}))-\frac{C}{R^{N-\alpha}}+o(\frac{1}{R^{N-\alpha}})\\\
&\leq\lvert Gq\rvert\max_{t\geq
0}\mathcal{E}(\gamma_{U_{S_{q}}}(t))-\frac{C}{R^{N-\alpha}}+o(\frac{1}{R^{N-\alpha}})\\\
&=\lvert
Gq\rvert\mathcal{E}(U_{S_{q}})-\frac{C}{R^{N-\alpha}}+o(\frac{1}{R^{N-\alpha}}).\end{split}$

The desired conclusion $c_{G}<{\lvert Gq\rvert}c_{S_{q}}$ follows provided
that $R$ is sufficiently large. We note that the Proposition 2.4 is used in
the final equality that $U_{S_{q}}$ achieves $p_{G}$. We also mention the
Lagrange’s theorem on the finite group that $\lvert G\rvert=\lvert
Gq\rvert\lvert S_{q}\rvert.$ ∎

##  5 Proof of Theorems

We now prove that, up to translations, the Pohozaev–Palais–Smale sequence
constructed in Proposition 2.3 has a nontrivial weak limit below the energy
level $c_{G}$.

######  Lemma 5.1.

Let $N\geq 2$, $\alpha\in(0,N)$ and $G\in\mathcal{G}_{k}$ with $1\leq k\leq
N$. Assume that the sequence $(u_{n})_{n\in\mathbb{N}}\subset\mathcal{H}_{G}$
satisfies as $n\to\infty$,

$\mathcal{E}(u_{n})\to c_{G},\quad\mathcal{E}^{\prime}(u_{n})\to
0\quad\text{\rm{ strongly in } }\mathcal{H}_{G}^{*}\quad\text{ \rm{and}
}\mathcal{P}(u_{n})\to 0.$

Then, there exists a sequence of points
$(a_{n})_{n\in\mathbb{N}}\subset\\{0\\}\times\mathbb{R}^{N-k}\subset\mathbb{R}^{N}$
such that, up to a subsequence, $(u_{n}(\cdot-a_{n}))_{n\in\mathbb{N}}$
converges weakly to $u\in\mathcal{H}_{G}$. Moreover,

$0<\mathcal{E}(u)\leq
c_{G}\qquad\text{\rm{and}}\qquad\mathcal{E}^{\prime}(u)=0\quad\text{\rm{in}}\quad\mathcal{H}_{G}^{*}.$

###### Proof.

The sequence $(u_{n})_{n\in\mathbb{N}}$ is bounded in $H^{1}(\mathbb{R}^{N})$
because we have

$(\alpha+2)A(u_{n})+\alpha
B(u_{n})=2(N+\alpha)\mathcal{E}(u_{n})-2\mathcal{P}(u_{n})+o(1).$

We claim that there exists $R>0$ such that for some
$p\in(\frac{N+\alpha}{N},\frac{N+\alpha}{N-2})$

$\liminf_{n\to\infty}\int_{O_{R}}\lvert u_{n}\rvert^{\frac{2Np}{N+\alpha}}>0.$
(5.1)

Here $O_{R}:=[-R,R]^{k}\times\mathbb{R}^{N-k}$. We acknowledge this claim
(5.1) temporarily and first finish the proof of the lemma. Taking a function
$\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})$ such that $\varphi=1$ on $O_{R}$,
$\operatorname{supp}\varphi\subset O_{3R/2}$, $0\leq\varphi\leq 1$ on
$\mathbb{R}^{N}$ and $\nabla\varphi\in L^{\infty}(\mathbb{R}^{N})$, we then
deduce from [23, Lemma 1.1] (see also [26, Lemma 2.3] or [35, Lemma 1.21])
that there exists a constant $K_{G}>0$ such that

$\begin{split}\int_{O_{R}}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}&\leq\int_{\mathbb{R}^{N}}\lvert\varphi
u_{n}\rvert^{\frac{2Np}{N+\alpha}}\\\
&\leq\C\Big{(}\sup_{a\in\mathbb{R}^{N}}\int_{B_{R/2}(a)}\lvert\varphi
u_{n}\rvert^{\frac{2Np}{N+\alpha}}\Big{)}^{1-\frac{N+\alpha}{Np}}\int_{\mathbb{R}^{N}}\lvert\nabla(\varphi
u_{n})\rvert^{2}+\lvert\varphi u_{n}\rvert^{2}\\\
&\leq\C\Big{(}\sup_{a\in\\{0\\}\times\mathbb{R}^{N-k}}\int_{B_{K_{G}R}(a)}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}\Big{)}^{1-\frac{N+\alpha}{Np}}\int_{\mathbb{R}^{N}}\lvert\nabla
u_{n}\rvert^{2}+\lvert u_{n}\rvert^{2}.\end{split}$

Hence, there exists a sequence of points
$(a_{n})_{n\in\mathbb{N}}\subset\\{0\\}\times\mathbb{R}^{N-k}$ such that

$\liminf_{n\to\infty}\int_{B_{K_{G}R}(a_{n})}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}>0.$

We then conclude by a similar argument as in [27, Proposition 2.2] that, up to
translations and a subsequence, $(u_{n})_{n\in\mathbb{N}}$ converges weakly to
some function $u\in\mathcal{H}_{G}\setminus\\{0\\}$ and
$\mathcal{E}^{\prime}(u)=0$ in $\mathcal{H}_{G}^{*}$. We thus conclude that
$\mathcal{P}(u)=0$. By the weakly lower semicontinuity of the norm, we are
left that $\mathcal{E}(u)\leq c_{G}$. In fact, we have

$\begin{split}c_{G}&=\lim_{n\to\infty}\Big{(}\mathcal{E}(u_{n})-\frac{1}{N+\alpha}\mathcal{P}(u_{n})\Big{)}\\\
&=\lim_{n\to\infty}\frac{\alpha}{2(N+\alpha)}\int_{\mathbb{R}^{N}}\lvert\nabla
u_{n}\rvert^{2}+\frac{\alpha}{2(N+\alpha)}\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{2}\\\
&\geq\frac{\alpha}{2(N+\alpha)}\int_{\mathbb{R}^{N}}\lvert\nabla
u\rvert^{2}+\lvert
u\rvert^{2}=\mathcal{E}(u)-\frac{1}{N+\alpha}\mathcal{P}(u)=\mathcal{E}(u).\end{split}$

We now return to finish the proof of the claim (5.1). Up to a subsequence,
there exists $\Lambda\in(0,+\infty)$ such that

$\lim_{n\to\infty}\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}=\Lambda.$

Note that the sequence $(u_{n})_{n\in\mathbb{N}}$ is bounded in
$H^{1}(\mathbb{R}^{N})$. The Rellich–Kondrachov embedding theorem guarantees
the finiteness of $\Lambda$. We suppose that $\Lambda=0$. By $(F_{1})$ and
$(F_{2})$, we deduce that for any $\varepsilon>0$, there exists
$C_{\varepsilon}$ such that

$\lvert F(u)\rvert\leq\varepsilon(\lvert u\rvert^{\frac{N+\alpha}{N}}+\lvert
u\rvert^{\frac{N+\alpha}{N-2}})+C_{\varepsilon}\lvert u\rvert^{p}.$

By the Hardy–Littlewood–Sobolev inequality and by the Rellich–Kondrachov
embedding theorem, we derive that

$\lim_{n\to\infty}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u_{n}))F(u_{n})=0.$

Inserting this into $\lim_{n\to\infty}\mathcal{P}(u_{n})=0$, we deduce
$\lim_{n\to\infty}\|u_{n}\|=0$, then a contradiction follows that
$c_{G}=\lim_{n\to\infty}\mathcal{E}(u_{n})=0.$

Since $G=\langle\mathcal{S}\rangle$, in the subsequent parts of this paper, we
assume each of the hyperplane fixed by the reflection $s\in\mathcal{S}$
passing through the origin and $\mathcal{F}$ always refer to the strict
fundamental domain under the group action $G\times 1_{N-k}$, see Appendix.
Without loss of generality, we can fix $\mathcal{F}$ to be the infinite cone
formed by $H_{s}$ for all $s\in\mathcal{S}$. Let $\partial^{i}\mathcal{F}$
denote its $k-i$ dimensional facets of the polyhedral cone $\mathcal{F}$ for
$i=0,1,\cdots,k-1$, here $\partial^{0}\mathcal{F}$ refers to the interior of
$\mathcal{F}$ for convenience. We claim that for any $r>0$,

$\liminf_{n\to\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B_{r}(y)}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}>0.$

Otherwise, by Lions’ Lemma (e.g., [35]) the sequence
$(u_{n})_{n\in\mathbb{N}}\subset H^{1}(\mathbb{R}^{N})$ converges strongly to
zero in $L^{\frac{2Np}{N+\alpha}}(\mathbb{R}^{N})$ as $n\to\infty$. Then a
similar argument as above implies $c_{G}=0$ again. In other words, the
sequence $(u_{n})_{n\in\mathbb{N}}$ is non-vanishing. According to the P. -L.
Lions’ concentration-compactness principle [23], up to a subsequence, there
are still two possibilities: compactness and dichotomy.
Compactness: there exists $(x_{n})_{n\in\mathbb{N}}\subset\mathbb{R}^{N}$ such
that

  * $\forall\varepsilon>0,\exists R>0\text{ such that }\liminf_{n\to\infty}\int_{B_{R}(x_{n})}\lvert u_{n}\rvert^{\frac{2Np}{N+\alpha}}\geq\Lambda-\varepsilon.$

Without loss generality, we assume that
$(x_{n})_{n\in\mathbb{N}}\subset\mathcal{F}$. We then easily deduce that there
exists $M>0$ such that $\sum_{j=1}^{k}\lvert x_{n}^{j}\rvert\leq M$ for all
$x_{n}=(x_{n}^{1},x_{n}^{2},\cdots,x_{n}^{N})$. Otherwise, for largely $n$,
there exists some $\mathcal{R}\in G$ such that
$B_{R}(x_{n})\cap\mathcal{R}(B_{R}(x_{n}))=\emptyset.$ In fact, if
$x_{n}\in\partial^{0}\mathcal{F}$, we can choose any
$\mathcal{R}\in\mathcal{S}$; if $x_{n}\in\partial^{i}\mathcal{F}\subset\\{\cap
H_{s}\,|\,s\in\mathcal{S}_{i}\subset\mathcal{S}\\}\times\mathbb{R}^{N-k}$,
then we take a reflection $\mathcal{R}\in\mathcal{S}\setminus\mathcal{S}_{i}$.
It then follows that

$\liminf_{n\to\infty}\int_{(B_{R}(\mathcal{R}x_{n}))}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}=\liminf_{n\to\infty}\int_{B_{R}(x_{n})}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}\geq\Lambda-\varepsilon.$

This would lead to a contradiction once $\varepsilon$ is sufficiently small
since

$\liminf_{n\to\infty}\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}\geq\liminf_{n\to\infty}\int_{B_{R}(x_{n})}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}+\liminf_{n\to\infty}\int_{\mathcal{R}(B_{R}(x_{n}))}\lvert
u_{n}\rvert^{\frac{4N}{N+\alpha}}\geq 2\Lambda-2\varepsilon.$

Dichotomy: there exists $(y_{n})_{n\in\mathbb{N}}\subset\mathbb{R}^{N}$ and
$\delta\in(0,1)$ such that $\forall\varepsilon>0$, there exists $R_{0}>0$ such
that for any $r,r^{\prime}\geq R_{0}$, there holds

$\limsup_{n\to\infty}\bigg{|}\int_{B_{r}(y_{n})}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}-\delta\Lambda\bigg{|}+\limsup_{n\to\infty}\bigg{|}\int_{B^{c}_{r^{\prime}}(y_{n})}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}-(1-\delta)\Lambda\bigg{|}\leq\varepsilon.$

For this situation, we also assume by the symmetry that
$(y_{n})_{n\in\mathbb{N}}\subset\mathcal{F}$.

Case 1: If there exists $R_{1}>0$ such that $(y_{n})_{n\in\mathbb{N}}\subset
B_{R_{1}}\cap\mathcal{F}$, we then conclude the claim (5.1) straight away
since $\delta\Lambda>0$.

For the subsequent proofs, we always assume the cut-off function $\xi\in[0,1]$
satisfies $\xi=0$ for $s\leq 1$ or $s\geq 4$, $\xi(s)=1$ for $2\leq s\leq 3$
and $\lvert\xi^{\prime}(s)\rvert\leq 2$; the cut-off function $\eta\in[0,1]$
satisfies that $\eta(t)=1$ for $t\in[0,2]$, $\eta(t)=0$ whenever $t\geq 3$ and
$\lvert\eta^{\prime}(t)\rvert\leq 2$. Suppose that the claim (5.1) is not
true, by a diagonal process and up to a subsequence we may choose
$\varepsilon_{n}\to 0$, $r_{n}\to+\infty$ as $n\to\infty$ and
$r_{n}^{\prime}=4r_{n}$ such that

$\bigg{|}\int_{B_{r_{n}}(y_{n})}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}-\delta\Lambda\bigg{|}+\bigg{|}\int_{B^{c}_{r^{\prime}_{n}}(y_{n})}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}-(1-\delta)\Lambda\bigg{|}\leq\varepsilon_{n}.$

The subsequent proofs will be completed by splitting into several cases
according to the asymptotic distance between the sequence
$(y_{n})_{n\in\mathbb{N}}$ and the boundary $\partial\mathcal{F}$.
Case 2: up to a subsequence, there exists some $i\in\\{1,\cdots,k-1\\}$ such
that

$\lim_{n\to\infty}\operatorname{dist}(y_{n},\partial^{i}\mathcal{F})=0\text{
and
}\lim_{n\to\infty}\operatorname{dist}(y_{n},\partial^{\ell}\mathcal{F})=0\text{
for all }\ell>i.$

In this situation, we fix a unit vector
$q\in\partial^{i}\mathcal{F}\cap\partial B_{1}$ and, up to a subsequence, we
may assume that $y_{n}=\nu_{n}q$ with $\nu_{n}\to+\infty$ for each subcases
such that $\lvert x-y\rvert\geq 4r_{n}$ for $x\in B_{3r_{n}}(z_{1})$ and $y\in
B_{3r_{n}}(z_{2})$ where $z_{1}\neq z_{2}\in Gy_{n}$. We define a test
function such that

$\phi_{n}^{q}(x)=\sum_{z\in Gy_{n}}\xi(\lvert
x-z\rvert/r_{n})u_{n}(x)=\frac{1}{\lvert S_{q}\rvert}\sum_{g\in G}\xi(\lvert
x-gy_{n}\rvert/r_{n})u_{n}(x).$

It then follows that, $g\diamond\phi_{n}^{q}=\psi(g)\phi_{n}^{q}$ for any
$g\in G$ so that $\phi_{n}^{q}\in\mathcal{H}_{G}$. Testing the limiting
equality that $\mathcal{E}^{\prime}(u_{n})\to 0$ strongly in
$\mathcal{H}_{G}^{*}$ as $n\to\infty$ against the function $\phi_{n}^{q}$, we
reach that,

$\int_{\mathbb{R}^{N}}\nabla
u_{n}\nabla\phi_{n}^{q}+u_{n}\phi_{n}^{q}=\int_{\mathbb{R}^{N}}(I_{\alpha}*F(u_{n}))f(u_{n})\phi_{n}^{q}+o(1).$

Taking into account the growth assumption $(F_{1})$ and $(F_{2})$ and
combining the boundedness of the sequence
$(u_{n})_{n\in\mathbb{N}}\subset\mathcal{H}_{G}$, the Hardy–Littlewood–Sobolev
inequality and the fact that

$\int_{B_{4r_{n}}(y_{n})\setminus B_{r_{n}}(y_{n})}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}\leq 2\varepsilon_{n},$ (5.2)

we then conclude that

$\int_{B_{3r_{n}}(y_{n})\setminus B_{2r_{n}}(y_{n})}\lvert\nabla
u_{n}\rvert^{2}+\lvert u_{n}\rvert^{2}=o(1).$ (5.3)

Define for each $z_{j}\in G{y_{n}}$ that

$v_{n}^{j}(x)=\eta(\lvert x-z_{j}\rvert/r_{n})u_{n}(x)\;\text{ and
}\;w_{n}=u_{n}-\sum v_{n}^{j}.$

We clearly see that $\operatorname{supp}v_{n}^{j}\subset B_{3r_{n}}(z_{j})$
for every $1\leq j\leq\lvert G{q}\rvert$. On the other hand, Lemma A.8 states
that the isotropic group $S_{q}\in\mathcal{G}_{i}$ and
$S_{q}=\langle\mathcal{S}_{i}\rangle$ for some
$\mathcal{S}_{i}\subset\mathcal{S}$ with $\lvert\mathcal{S}_{i}\rvert=i$,
which leads to $sq=q$ for $s\in\mathcal{S}_{i}$. Let $z_{j}=g_{j}y_{n}$ for
some $g_{j}\in G$, we then deduce for every $s\in\mathcal{S}_{i}$,

$g_{j}sg_{j}^{-1}\diamond v_{n}^{j}=\eta(\lvert
g_{j}sg_{j}^{-1}x-z_{j}\rvert/r_{n})u_{n}(g_{j}sg_{j}^{-1}x)=\psi(g_{j}sg_{j}^{-1})v_{n}^{j}=\psi(s)v_{n}^{j}.$

This means that $v_{n}^{j}\in\mathcal{H}_{S_{q}}$ for all $1\leq j\leq\lvert
Gq\rvert$ since the group $g_{j}S_{q}g_{j}^{-1}$ is isomorphic to $S_{q}$ for
every $j$. For simplicity, we denote $v_{n}=\sum_{j}v_{n}^{j}$, so that

$v_{n}=\sum_{z_{j}\in G{\nu_{n}q}}\eta(\lvert
x-z_{j}\rvert/r_{n})u_{n}=\frac{1}{\lvert S_{q}\rvert}\sum_{g\in G}\eta(\lvert
x-g\nu_{n}q\rvert/r_{n})u_{n}.$

We thus have $v_{n}\in\mathcal{H}_{G}$ since $g\diamond v_{n}=\psi(g)v_{n}$
for any $g\in G$. It follows that $w_{n}=u_{n}-v_{n}\in\mathcal{H}_{G}$. We
denote $B_{r}^{j}=B_{r}(z_{j})$ with $r>0$ for $z_{j}\in G{\nu_{n}q}$, and
take $\mathcal{B}_{r}=\cup_{z_{j}}B_{r}^{j}$ for simplicity. By symmetry and
by the choice of the cut-off function, we see from (5.3) that

$\displaystyle\int_{\mathbb{R}^{N}}$ $\displaystyle\nabla v_{n}\nabla
w_{n}+v_{n}w_{n}=\sum_{z_{j}\in G{\nu_{n}q}}\int_{B_{3r_{n}}(z_{j})\setminus
B_{2r_{n}}(z_{j})}\nabla{v_{n}^{j}}\nabla w_{n}+v_{n}^{j}w_{n}$
$\displaystyle\leq\frac{\C}{r_{n}}\int_{\mathbb{R}^{N}}\lvert\nabla
u_{n}\rvert^{2}+\lvert u_{n}\rvert^{2}+\C\int_{B_{3r_{n}}(y_{n})\setminus
B_{2r_{n}}(y_{n})}\lvert\nabla u_{n}\rvert^{2}+\lvert u_{n}\rvert^{2}=o(1).$
(5.4)

Note that $\lvert x-y\rvert\geq 4r_{n}$ for any
$x\in\operatorname{supp}v_{n}^{j}$, $y\in\operatorname{supp}v_{n}^{m}$
provided that $j\neq m$. Taking $\beta\in(\alpha,N)$ such that
$\beta<N\frac{2+\alpha}{N-2}$, we then deduce that for $1\leq j\neq
m\leq{\lvert Gq\rvert}$,

$\displaystyle\int_{\mathbb{R}^{N}}(I_{\alpha}*F(v^{j}_{n}))F(v^{m}_{n})$
$\displaystyle\leq\frac{\C}{r_{n}^{\beta-\alpha}}\int_{\mathbb{R}^{N}}(I_{\beta}*F(u_{n}))F(u_{n})$
$\displaystyle\leq\frac{\C}{r_{n}^{\beta-\alpha}}\Big{(}\int_{\mathbb{R}^{N}}\lvert
F(u_{n})\rvert^{\frac{2N}{N+\beta}}\Big{)}^{\frac{N+\beta}{N}}=o(1).$ (5.5)

This is a consequence of Rellich–Kondrachov embedding theorem. In fact, our
choice of $\beta$ ensures the exponent
$\frac{2N}{N+\beta}\frac{N+\alpha}{N-2}\in(2,\frac{2N}{N-2})$, we then deduce
by $(F_{1})$ and $(F_{2})$ that

$\int_{\mathbb{R}^{N}}\lvert
F(u_{n})\rvert^{\frac{2N}{N+\beta}}\leq\C\int_{\mathbb{R}^{N}}\lvert
u_{n}\rvert^{2}+\lvert
u_{n}\rvert^{\frac{2N}{N+\beta}\frac{N+\alpha}{N-2}}\leq\C\big{(}\|u_{n}\|^{2}+\|u_{n}\|^{\frac{2N}{N+\beta}\frac{N+\alpha}{N-2}}\big{)}.$

Here we recall that $\frac{N+\alpha}{N-2}=\bar{p}$ and $\frac{2N}{N-2}=\infty$
for $N=2$.

We shall prove the following two decompositions on the functionals:

$\mathcal{E}(u_{n})=\sum_{1\leq j\leq\lvert
Gq\rvert}\mathcal{E}(v_{n}^{j})+\mathcal{E}(w_{n})+o(1),$ (5.6)
$\mathcal{P}(u_{n})=\sum_{1\leq j\leq\lvert
Gq\rvert}\mathcal{P}(v_{n}^{j})+\mathcal{P}(w_{n})+o(1).$ (5.7)

For brevity, we set

$J_{n}(x,y)=A_{\alpha}\bigg{(}\frac{F(u_{n}(x))F(u_{n}(y))}{\lvert
x-y\rvert^{N-\alpha}}-\frac{F(v_{n}(x))F(v_{n}(y))}{\lvert
x-y\rvert^{N-\alpha}}-\frac{F(w_{n}(x))F(w_{n}(y))}{\lvert
x-y\rvert^{N-\alpha}}\bigg{)}.$

By direct calculations, we expand $\mathcal{E}(u_{n})$ in terms of
$\mathcal{E}(v_{n}^{j})$ and $\mathcal{E}(w_{n})$ that

$\begin{split}&\mathcal{E}(u_{n})=\mathcal{E}(v_{n})+\mathcal{E}(w_{n})+\int_{\mathbb{R}^{N}}\nabla
v_{n}\nabla
w_{n}+v_{n}w_{n}-\frac{1}{2}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y\\\
&\quad\;\quad\;=\sum_{1\leq j\leq\lvert
Gq\rvert}\mathcal{E}(v^{j}_{n})+\mathcal{E}(w_{n})-\frac{1}{2}\sum_{1\leq
j\neq m\leq\lvert
Gq\rvert}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(v^{j}_{n}))F(v^{m}_{n})\\\
&\quad\;\quad\qquad+\int_{\mathbb{R}^{N}}\nabla v_{n}\nabla
w_{n}+v_{n}w_{n}-\frac{1}{2}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y\end{split}$

Similarly, we can also obtain the following expansion,

$\displaystyle\mathcal{P}(u_{n})=\sum_{1\leq j\leq\lvert
Gq\rvert}\mathcal{P}(v^{j}_{n})+\mathcal{P}(w_{n})-\frac{N+\alpha}{2}\sum_{j\neq
m}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(v^{j}_{n}))F(v^{m}_{n})$
$\displaystyle\qquad\;\qquad\;+(N-2)\int_{\mathbb{R}^{N}}\nabla v_{n}\nabla
w_{n}+N\int_{\mathbb{R}^{N}}v_{n}w_{n}-\frac{N+\alpha}{2}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y.$

In view of the estimates (5.4) and (5.5), the conclusions (5.6) and (5.7)
follows once we prove

$\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y=o(1).$
(5.8)

On one hand, similar manipulations as in (5.5) bring us that

$\begin{split}\int_{\mathcal{B}_{2r_{n}}}\int_{\mathcal{B}^{c}_{3r_{n}}}\frac{F(\varphi_{n}(x))F(\varphi_{n}(y))}{\lvert
x-y\rvert^{N-\alpha}}\,\mathrm{d}x\,\mathrm{d}y&\leq\frac{\C}{r_{n}^{\beta-\alpha}}\Big{(}\int_{\mathbb{R}^{N}}\lvert
F(\varphi_{n})\rvert^{\frac{2N}{N+\beta}}\Big{)}^{\frac{N+\beta}{N}}\\\
&\leq\frac{\C}{r_{n}^{\beta-\alpha}}\big{(}\|u_{n}\|^{2}+\|u_{n}\|^{\frac{2N}{N+\beta}\frac{N+\alpha}{N-2}}\big{)}=o(1).\end{split}$

Here $\varphi_{n}$ can be any sequence among $u_{n}$, $v_{n}$ and $w_{n}$. We
thus conclude that

$\int_{\mathcal{B}_{2r_{n}}}\int_{\mathcal{B}^{c}_{3r_{n}}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y=o(1).$

On the other hand, by the Hardy–Littlewood–Sobolev inequality and by the
Rellich–Kondrachov embedding theorem, we infer from (5.2) and (5.3) that for
any $\varphi_{n}$ amongst $u_{n}$, $v_{n}$ and $w_{n}$

$\begin{split}&\int_{\mathcal{B}_{3r_{n}}\setminus\mathcal{B}_{2r_{n}}}\int_{\mathbb{R}^{N}}\frac{F(\varphi_{n}(x))F(\varphi_{n}(y))}{\lvert
x-y\rvert^{N-\alpha}}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\C\bigg{(}\int_{\mathbb{R}^{N}}\lvert
F(\varphi_{n})\rvert^{\frac{2N}{N+\alpha}}\bigg{)}^{\frac{N+\alpha}{2N}}\bigg{(}\int_{B_{3r_{n}}(y_{n})\setminus
B_{2r_{n}}(y_{n})}\lvert
F(\varphi_{n})\rvert^{\frac{2N}{N+\alpha}}\bigg{)}^{\frac{N+\alpha}{2N}}\\\
&\leq\C\Big{(}\|u_{n}\|^{2}+\|u_{n}\|^{\frac{N+\alpha}{N-2}\frac{2N}{N+\alpha}}\Big{)}^{\frac{N+\alpha}{2N}}\\\
&\qquad\times\bigg{(}\int_{B_{3r_{n}}(y_{n})\setminus B_{2r_{n}}(y_{n})}\lvert
u_{n}\rvert^{\frac{N+\alpha}{N-2}\frac{2N}{N+\alpha}}+\lvert
u_{n}\rvert^{2}+C_{\varepsilon}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}\bigg{)}^{\frac{N+\alpha}{2N}}=o(1).\end{split}$

We then obtain

$\int_{\mathcal{B}_{3r_{n}}\setminus\mathcal{B}_{2r_{n}}}\int_{\mathbb{R}^{N}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y=o(1).$

The very similar analyses as above also indicate that

$\int_{\mathcal{B}^{c}_{3r_{n}}}\int_{\mathcal{B}_{2r_{n}}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y=o(1)\;\;\text{
and
}\;\;\int_{\mathbb{R}^{N}}\int_{\mathcal{B}_{3r_{n}}\setminus\mathcal{B}_{2r_{n}}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y=o(1).$

We are now ready to complete the final estimate on the convolution term (5.8)
that,

$\begin{split}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}&J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y=\int_{\mathcal{B}_{2r_{n}}\cup(\mathcal{B}_{3r_{n}}\setminus\mathcal{B}_{2r_{n}})\cup\mathcal{B}^{c}_{3r_{n}}}\int_{\mathcal{B}_{2r_{n}}\cup(\mathcal{B}_{3r_{n}}\setminus\mathcal{B}_{2r_{n}})\cup\mathcal{B}^{c}_{3r_{n}}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\int_{\mathbb{R}^{N}}\int_{\mathcal{B}_{3r_{n}}\setminus\mathcal{B}_{2r_{n}}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y+\int_{\mathcal{B}_{2r_{n}}}\int_{\mathcal{B}^{c}_{3r_{n}}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y\\\
&\quad+\int_{\mathcal{B}_{3r_{n}}\setminus\mathcal{B}_{2r_{n}}}\int_{\mathbb{R}^{N}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y+\int_{\mathcal{B}^{c}_{3r_{n}}}\int_{\mathcal{B}_{2r_{n}}}J_{n}(x,y)\,\mathrm{d}x\,\mathrm{d}y=o(1).\end{split}$

By the Rellich–Kondrachov embedding theorem and the fact that

$\liminf_{n\to\infty}\int_{\mathbb{R}^{N}}\lvert
v^{j}_{n}\rvert^{\frac{2Np}{N+\alpha}}\geq\liminf_{n\to\infty}\int_{B^{j}_{r_{n}}}\lvert
u_{n}\rvert^{\frac{2Np}{N+\alpha}}\geq\delta\Lambda>0,$

for each $1\leq j\leq\lvert Gq\rvert$, we then deduce that
$\liminf_{n\to\infty}\|v_{n}^{j}\|>0$. Note that

$\mathcal{P}(u_{n})=\sum\mathcal{P}(v_{n}^{j})+\mathcal{P}(w_{n})=o(1).$

We claim that $\mathcal{P}(w_{n})=o(1)$, so that $\mathcal{P}(v_{n}^{j})=o(1)$
for every $1\leq j\leq\lvert Gq\rvert$, which bring us that
$\lim_{n\to\infty}Q(v^{j}_{n})>0$ for each $j$. By Lemma 2.2, there exists
$s_{n}^{j}\in(0,+\infty)$ such that
$v_{n}^{j}(\frac{\cdot}{s_{n}^{j}})\in\mathcal{P}_{S_{q}}$. In particular,
$\mathcal{P}(v_{n}^{j})=o(1)$ implies $\lim_{n\to\infty}s_{n}^{j}=1$. On the
other hand,

$\liminf_{n\to\infty}\mathcal{E}(w_{n})=\liminf_{n\to\infty}\mathcal{E}(w_{n})-\frac{1}{N+\alpha}\lim_{n\to\infty}\mathcal{P}(w_{n})\geq
0.$

We thus deduce

$\displaystyle
c_{G}=\lim_{n\to\infty}\sum\mathcal{E}(v_{n}^{j})+\lim_{n\to\infty}\mathcal{E}(w_{n})\geq\lim_{n\to\infty}\sum\mathcal{E}(v_{n}^{j}(\frac{\cdot}{s_{n}^{j}}))\geq\lvert
Gq\rvert c_{S_{q}}\geq c_{G}^{*}.$

This contradicts with the proposition 4.1 that $c_{G}<c_{G}^{*}$.

We now set out to complete the claim that $\mathcal{P}(w_{n})=o(1).$ In fact,
since the sequence $(u_{n})_{n\in\mathbb{N}}$ is bounded, we then easily
derive the boundedness of $\mathcal{P}(v^{j}_{n})$ and $\mathcal{P}(w_{n})$.
Up to a subsequence, we assume both $\mathcal{P}(w_{n})$ and
$\mathcal{P}(w_{n})$ take finite limit values. If
$\lim_{n\to\infty}\mathcal{P}(w_{n})>0$, then
$\lim_{n\to\infty}\mathcal{P}(v_{n}^{j})<0$. By Lemma 2.2, there exist
$t_{n}\in(0,1]$ such that
$v_{n}^{j}(\frac{\cdot}{t_{n}})\in\mathcal{P}_{S_{q}}$. Namely,

$\frac{t_{n}^{N-2}}{2}\int_{\mathbb{R}^{N}}\lvert\nabla
v^{j}_{n}\rvert^{2}+\frac{t_{n}^{N}}{2}\int_{\mathbb{R}^{N}}\lvert
v^{j}_{n}\rvert^{2}=\frac{t_{n}^{N+\alpha}}{2}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(v^{j}_{n}))F(v^{j}_{n}).$

In view of the fact that $\lim_{n\to\infty}\mathcal{P}(v_{n}^{j})<0$ and the
fact that $\mathcal{E}(w_{n})-\frac{1}{N+\alpha}\mathcal{P}(w_{n})\geq 0$, we
thus conclude that

$\displaystyle c_{G}$
$\displaystyle=\mathcal{E}(u_{n})-\frac{1}{N+\alpha}\mathcal{P}(u_{n})+o(1)\geq\sum\mathcal{E}(v_{n}^{j})-\frac{1}{N+\alpha}\mathcal{P}(v_{n}^{j})+o(1)$
$\displaystyle=\sum\Big{(}\frac{1}{2}-\frac{N-2}{2(N+\alpha)}\Big{)}\int_{\mathbb{R}^{N}}\lvert\nabla
v_{n}^{j}\rvert^{2}+\Big{(}\frac{1}{2}-\frac{N}{2(N+\alpha)}\Big{)}\int_{\mathbb{R}^{N}}\lvert
v_{n}^{j}\rvert^{2}+o(1)$ $\displaystyle\geq\sum
t_{n}^{N-2}\Big{(}\frac{1}{2}-\frac{N-2}{2(N+\alpha)}\Big{)}\int_{\mathbb{R}^{N}}\lvert\nabla
v_{n}^{j}\rvert^{2}+t_{n}^{N}\Big{(}\frac{1}{2}-\frac{N}{2(N+\alpha)}\Big{)}\int_{\mathbb{R}^{N}}\lvert
v_{n}^{j}\rvert^{2}+o(1)$
$\displaystyle=\sum\mathcal{E}\big{(}v_{n}^{j}(\frac{\cdot}{t_{n}})\big{)}-\mathcal{P}\big{(}v_{n}^{j}(\frac{\cdot}{t_{n}})\big{)}+o(1)\geq\lvert
Gq\rvert c_{S_{q}}+o(1).$

This is a contradiction. For the other case that
$\lim_{n\to\infty}\mathcal{P}(w_{n})<0$, we conclude that there exists
$t_{n}\leq 1$ such that $w_{n}(\frac{\cdot}{t_{n}})\in\mathcal{P}_{G}$. We
therefore deduce

$\displaystyle c_{G}$
$\displaystyle=\mathcal{E}(u_{n})-\frac{1}{N+\alpha}\mathcal{P}(u_{n})+o(1)\geq\mathcal{E}(w_{n})-\frac{1}{N+\alpha}\mathcal{P}(w_{n})+o(1)$
$\displaystyle\geq\mathcal{E}(w_{n}(\frac{\cdot}{t_{n}}))-\mathcal{P}(w_{n}(\frac{\cdot}{t_{n}}))+o(1)\geq
c_{G}+o(1).$

This leads to $\lim_{n\to\infty}t_{n}=1$ so that
$\lim_{n\to\infty}\mathcal{E}(w_{n})=c_{G}$ and
$\lim_{n\to\infty}\mathcal{E}(v^{j}_{n})=0$. We thus deduce that for each $j$

$\lim_{n\to\infty}\mathcal{P}(v^{j}_{n})=\lim_{n\to\infty}\mathcal{P}(v^{j}_{n})-(N+\alpha)\lim_{n\to\infty}\mathcal{E}(v^{j}_{n})\leq-\frac{\alpha}{2}\lim_{n\to\infty}\|v_{n}^{j}\|^{2}<0.$

A contradiction again.

Case 3: up to a subsequence, for any $\ell\geq 1$,
$\lim_{n\to\infty}\operatorname{dist}(y_{n},\partial^{\ell}\mathcal{F})=+\infty.$

In such case, up to a subsequence, we may assume for $z_{i}\neq z_{j}\in
Gy_{n}$, there holds $\lvert x-y\rvert\geq 4r_{n}$ for any $x\in
B_{3r_{n}}(z_{i})$ and $y\in B_{3r_{n}}(z_{j})$. We take the test function
that

$\phi_{n}^{G}=\sum_{g\in G}\xi(\lvert x-gy_{n}\rvert/r_{n})u_{n}.$

We conclude that $\phi_{n}^{G}\in\mathcal{H}_{G}$ since for every $\bar{g}\in
G$,

$\bar{g}\diamond\phi_{n}^{G}=\sum_{g\in G}\xi(\lvert
x-g\bar{g}y_{n}\rvert/r_{n})u_{n}(\bar{g}^{-1}x)=\psi(\bar{g})\phi_{n}^{G}.$

By repeating the arguments as above, we conclude that

$\int_{B_{3r_{n}}(y_{n})\setminus B_{2r_{n}}(y_{n})}\lvert\nabla
u_{n}\rvert^{2}+\lvert u_{n}\rvert^{2}=o(1).$

Similarly, we define for each $z_{j}\in G{y_{n}}$ that

$v_{n}^{i}(x)=\eta(\lvert x-z_{j}\rvert/r_{n})u_{n}(x)\;\text{ and
}\;w_{n}=u_{n}-\sum v_{n}^{j}.$

Since $\operatorname{supp}v_{n}^{j}\subset B_{3r_{n}}(z_{j})$ for each $j$, we
deduce that $v_{n}^{j}\in H^{1}(\mathbb{R}^{N})$. It can also be checked that
$w_{n}\in\mathcal{H}_{G}$. Similarly, we have the decompositions that

$\mathcal{E}(u_{n})=\sum\mathcal{E}(v_{n}^{j})+\mathcal{E}(w_{n})+o(1),\text{
and }\,\mathcal{P}(u_{n})=\sum\mathcal{P}(v_{n}^{j})+\mathcal{P}(w_{n})+o(1).$
(5.9)

Again by the Rellich–Kondrachov embedding theorem, we observe that for all
$z_{j}\in G{y_{n}}$,

$\lim_{n\to\infty}\int_{\mathbb{R}^{N}}\lvert\nabla v^{j}_{n}\rvert^{2}+\lvert
v_{n}^{j}\rvert^{2}>0.$

We conclude that $\mathcal{P}(v_{n}^{j})=o(1)$ and $\mathcal{P}(w_{n})=o(1)$
so that there exists $s_{n}^{j}\in(0,+\infty)$ such that
$v_{n}^{j}(\frac{\cdot}{s_{n}^{j}})\in\mathcal{P}$ with
$\lim_{n\to\infty}s_{n}^{j}=1$. We therefrom deduce by the above (5.9) that

$\displaystyle c_{G}=\lim_{n\to\infty}\mathcal{E}(u_{n})$
$\displaystyle=\sum\lim_{n\to\infty}\mathcal{E}(v_{n}^{j})+\lim_{n\to\infty}\mathcal{E}(w_{n})\geq\sum\lim_{n\to\infty}\mathcal{E}(v_{n}^{j}(\frac{\cdot}{s_{n}^{j}}))\geq\lvert
G\rvert c_{0}.$

This is in contradiction to the proposition 4.1 that $c_{G}<c_{G}^{*}<\lvert
G\rvert c_{0}$. ∎

Proof of Theorem 1.1. By induction, we suppose Proposition 2.4 holds provided
that the rank of Coxeter group is no more than $k-1$. We consider the mountain
pass problem (2.1) in which the Coxeter group $G$ is of rank $k$. By
Proposition 2.3, there exists a Pohozaev–Palais–Smale sequence
$(u_{n})_{n\in\mathbb{N}}$ in $\mathcal{H}_{G}$ at the energy level $c_{G}$.
We then obtain a nontrivial weak solution $u\in\mathcal{H}_{G}$ with
$\mathcal{E}(u)\leq c_{G}$ by applying Proposition 4.1 and Lemma 5.1. This
completes Lemma 2.4 that $c_{G}=p_{G}=m_{G}=\mathcal{E}(u)$, which in turn
yields a strong convergence of $(u_{n})_{n\in\mathbb{N}}$ in
$H^{1}(\mathbb{R}^{N})$. By the symmetric criticality principle [35, Theorem
1.28]), we see that $u$ is a critical point of $\mathcal{E}$ in
$H^{1}(\mathbb{R}^{N})$, so that $u$ is a $G$-groundstate. The regularity
$u\in C^{2}(\mathbb{R}^{N})$ follows from Theorem 1.2.

We next show that $u$ keeps a constant sign on the interior of the strict
fundamental domain $\mathcal{F}$. Our proofs essentially follow the arguments
of [8, Proposition 2.6], see also [33, 36, 37].

Proof of Theorem 1.3. Taking the subspace

$H^{1}_{0}(\mathcal{F})=\\{v\in H^{1}(\mathcal{F})\,|\,v=0\text{ almost
everywhere in }\mathbb{R}^{N}\setminus{\rm{int}}(\mathcal{F})\\},$

we define two functionals $\mathcal{E}_{0},\mathcal{P}_{0}\in
C^{1}(H^{1}_{0}(\mathcal{F}),\mathbb{R})$ such that for $v\in
H^{1}_{0}(\mathcal{F})$,

$\mathcal{E}_{0}(v)=\frac{\lvert G\rvert}{2}\int_{\mathcal{F}}\lvert\nabla
v\rvert^{2}+\lvert
v\rvert^{2}-\frac{1}{2}\int_{\mathcal{F}}\int_{\mathcal{F}}K_{\alpha}(x,y)F(v(x))F(v(y))\,\mathrm{d}x\,\mathrm{d}y$

and

$\mathcal{P}_{0}(v)=\frac{(N-2)\lvert
G\rvert}{2}\int_{\mathcal{F}}\lvert\nabla v\rvert^{2}+\frac{N\lvert
G\rvert}{2}\int_{\mathcal{F}}\lvert
v\rvert^{2}-\frac{N+\alpha}{2}\int_{\mathcal{F}}\int_{\mathcal{F}}K_{\alpha}(x,y)F(v(x))F(v(y))\,\mathrm{d}x\,\mathrm{d}y$

with the kernel $K_{\alpha}:\mathcal{F}\times\mathcal{F}\to\mathbb{R}$ being
defined by $K_{\alpha}(x,y)=\sum_{g_{i},g_{j}\in G}A_{\alpha}\lvert
g_{i}x-g_{j}y\rvert^{\alpha-N}$. We observe that $\chi_{\mathcal{F}}u\in
H^{1}_{0}(\mathcal{F})$ for $u\in\mathcal{H}_{G}$ where $\chi_{\mathcal{F}}$
is the characteristic function of the conic region $\mathcal{F}$. Conversely,
for each $v\in H^{1}_{0}(\mathcal{F})$, we construct a $G$-symmetry function
$\mathcal{U}(v):\mathbb{R}^{N}\to\mathbb{R}$ such that

$\mathcal{U}(v)(x):=\sum_{g\in
G}\psi(g)\big{(}\chi_{\mathcal{F}}v\big{)}(gx).$

Direct calculations show us that for any $u\in\mathcal{H}_{G}$, $v\in
H^{1}_{0}(\mathcal{F})$,

$u=\mathcal{U}(\chi_{\mathcal{F}}u),\;\;\mathcal{E}(\mathcal{U}(v))=\mathcal{E}_{0}(v),\;\;\text{
and }\;\;\mathcal{P}(\mathcal{U}(v))=\mathcal{P}_{0}(v).$

These facts suggest that
$\inf_{\mathcal{P}_{0}}\mathcal{E}_{0}=\inf_{\mathcal{P}_{G}}\mathcal{E}=c_{G}$,
where $\mathcal{P}_{0}=\\{v\in
H^{1}_{0}(\mathcal{F})\setminus\\{0\\}\,|\,\mathcal{P}_{0}(v)=0\\}$.

Let $u$ be the nodal solution that achieves $c_{G}$, then
$w:=\chi_{\mathcal{F}}u\in H^{1}_{0}(\mathcal{F})$ and
$\mathcal{E}_{0}(w)=\inf_{\mathcal{P}_{0}}\mathcal{E}_{0}=c_{G}$. Note that
$\lvert w\rvert\in H^{1}_{0}(\mathcal{F})$. Since $f$ is odd, $F$ is even and
we thus deduce that

$\mathcal{E}_{0}(\lvert w\rvert)=\mathcal{E}_{0}(w)=c_{G}\;\;\text{ and
}\;\;\mathcal{P}_{0}(\lvert w\rvert)=\mathcal{P}_{0}(w)=0.$

We thus assume that $w\geq 0$ in $\mathcal{F}$. On the other hand,
$\mathcal{P}(\mathcal{U}(w))=\mathcal{P}_{0}(w)=0$ and
$\mathcal{E}(\mathcal{U}(w))=\mathcal{E}_{0}(w)=c_{G}$ imply that
$\mathcal{U}(w)\in\mathcal{P}_{G}$ also achieves $c_{G}=p_{G}$, thus
$\mathcal{U}(w)$ is a weak solution to the Choquard equation ($\mathcal{C}$),
see [27, Lemma 5.1]. By Theorem 1.2, we infer that $\mathcal{U}(w)\in C^{2}$.
Therefore, $\mathcal{U}(w)$ satisfies in the classical sense that
$-\Delta\mathcal{U}(w)+\mathcal{U}(w)\geq 0$ in $\mathcal{F}$, which, together
with the strong maximum principle for classical supersolutions (see [10,
Theorem 3.5]), yields $\mathcal{U}(w)>0$ everywhere in
${\rm{int}}(\mathcal{F})$. From the definition of $\mathcal{U}$, we see that
$u>0$ everywhere in ${\rm{int}}(\mathcal{F})$. Thus $u$ has exactly $\lvert
G\rvert$ number of nodal domains in the whole space $\mathbb{R}^{N}$.

We finally complete the existence of the global minimal nodal solutions.

Proof of Theorem 1.4. We consider the minimization problem on the nodal
solution set,

$c_{\rm{nod}}=\inf_{u\in\mathcal{M}_{\rm{nod}}}\mathcal{E}(u),$

with the constraint being defined by $\mathcal{M}_{\rm{nod}}=\\{u\in
H^{1}(\mathbb{R}^{N})\,|\,u^{\pm}\neq 0,\mathcal{E}^{\prime}(u)=0\\}$. Clearly
we have by Theorem 1.1 that the ${\bf{A}_{1}}$-groundstates do exist, which
leads us $c_{0}\leq c_{\rm{nod}}\leq c_{\bf{A}_{1}}<2c_{0}$, here $c_{0}$
refers to the ground state energy of the Choquard equation ($\mathcal{C}$).
Moreover, $c_{0}=\inf_{\mathcal{N}}\mathcal{E}$ where $\mathcal{N}$ is the
Neari manifold. This can be done due to our monotonic assumption
$(F^{\prime}_{1})$.

Taking a minimizing sequence
$(v_{n})_{n\in\mathbb{N}}\subset\mathcal{M}_{\rm{nod}}$ so that
$\mathcal{E}^{\prime}(v_{n})=0$ and $\mathcal{E}(v_{n})\to c_{\rm{nod}}$ as
$n\to\infty$. By combining the Pohozaev identity, we obtain the boundedness of
$(v_{n})_{n\in\mathbb{N}}$ in $H^{1}(\mathbb{R}^{N})$. By the assumptions
$(F^{\prime}_{1})$ and $(F^{\prime}_{2})$, there exists some constant $C>0$
such that

$\lvert F(t)\rvert,\lvert f(t)t\rvert\leq C(\lvert t\rvert^{r}+\lvert
t\rvert^{p}).$

This, together with the facts that
$\langle\mathcal{E}^{\prime}(v_{n}),v_{n}^{\pm}\rangle=0$ and the
Hardy–Littlewood–Sobolev inequality, implies that
$\liminf_{n\to\infty}\|v_{n}^{\pm}\|>0$. We now claim that

$\sum_{p_{1},p_{2}\in\\{r,p\\}}\liminf_{n\to\infty}\int_{\mathbb{R}^{N}}(I_{\alpha}*\lvert
v_{n}^{+}\rvert^{p_{1}})\lvert v_{n}^{-}\rvert^{p_{2}}>0.$

Otherwise, we deduce by the assumptions $(F^{\prime}_{1})$ and
$(F^{\prime}_{2})$ again that

$\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(v_{n}^{\pm}))F(v_{n}^{\mp})\leq
C\sum_{p_{1},p_{2}\in\\{r,p\\}}\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}(I_{\alpha}*\lvert
v_{n}^{+}\rvert^{p_{1}})\lvert v_{n}^{-}\rvert^{p_{2}}=0$

and

$\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}(I_{\alpha}*F(v_{n}^{\pm}))f(v_{n}^{\mp})v_{n}^{\mp}\leq
C\sum_{p_{1},p_{2}\in\\{r,p\\}}\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}(I_{\alpha}*\lvert
v_{n}^{+}\rvert^{p_{1}})\lvert v_{n}^{-}\rvert^{p_{2}}=0$

We therefrom obtain that

$\mathcal{E}(v_{n})=\mathcal{E}(v_{n}^{+})+\mathcal{E}(v_{n}^{-})+o(1),\;\text{
and
}\;\langle\mathcal{E}^{\prime}(v_{n}^{\pm}),v_{n}^{\pm}\rangle=\langle\mathcal{E}^{\prime}(v_{n}),v_{n}^{\pm}\rangle+o(1)=o(1).$

It then follows that there exist $s_{n}^{\pm}\in(0,+\infty)$ with
$\lim_{n\to\infty}s_{n}^{\pm}=1$ such that
$s^{\pm}_{n}v_{n}^{\pm}\in\mathcal{N}$. This leads a contradiction that

$\displaystyle 2c_{0}>c_{\rm{nod}}+o(1)$
$\displaystyle=\mathcal{E}(v_{n})=\mathcal{E}(v_{n}^{+})+\mathcal{E}(v_{n}^{-})+o(1)$
$\displaystyle=\mathcal{E}(s_{n}^{+}v_{n}^{+})+\mathcal{E}(s_{n}^{-}v_{n}^{-})+o(1)\geq
2c_{0}+o(1).$

By adopting the proof of [8, Lemma 3.6], we conclude that there exist $R>0$
and one pair $(p_{1},p_{2})\in\\{(r,r),(r,p),(p,r),(p,p)\\}$ such that

$\liminf_{n\to\infty}\sup_{a\in\mathbb{R}^{N}}\int_{B_{R}(a)}\lvert
v^{+}_{n}\rvert^{\frac{2Np_{1}}{N+\alpha}}\int_{B_{R}(a)}\lvert
v_{n}^{-}\rvert^{\frac{2Np_{2}}{N+\alpha}}>0.$

Up to a translation and a subsequence, we can assume that

$\liminf_{n\to\infty}\int_{B_{R}(0)}\lvert
v_{n}^{+}\rvert^{\frac{2Np_{1}}{N+\alpha}}>0,\text{ and
}\liminf_{n\to\infty}\int_{B_{R}(0)}\lvert
v_{n}^{-}\rvert^{\frac{2Np_{2}}{N+\alpha}}>0.$

This means that $v_{n}\rightharpoonup v\in H^{1}(\mathbb{R}^{N})$ and
$v^{\pm}\neq 0$. In particular, $\mathcal{E}^{\prime}(v)=0$ and
$\mathcal{P}(v)=0$. We then deduce by the weakly lower semicontinuity of the
norm that

$\displaystyle c_{\rm{nod}}$
$\displaystyle\leq\mathcal{E}(v)=\mathcal{E}(v)-\frac{1}{N+\alpha}\mathcal{P}(v)$
$\displaystyle=\frac{\alpha}{2(N+\alpha)}\int_{\mathbb{R}^{N}}\lvert\nabla
v\rvert^{2}+\frac{2+\alpha}{2(N+\alpha)}\int_{\mathbb{R}^{N}}\lvert
v\rvert^{2}$
$\displaystyle\leq\frac{\alpha}{2(N+\alpha)}\int_{\mathbb{R}^{N}}\lvert\nabla
v_{n}\rvert^{2}+\frac{2+\alpha}{2(N+\alpha)}\int_{\mathbb{R}^{N}}\lvert
v_{n}\rvert^{2}+o(1)$
$\displaystyle=\mathcal{E}(v_{n})-\frac{1}{N+\alpha}\mathcal{P}(v_{n})+o(1)=\mathcal{E}(v_{n})+o(1).$

Hence, the infimum value $c_{\rm{nod}}$ is achieved by the sign-changing
solution $v$ and then the strongly convergence follows.

## Appendix

##  Appendix A Finite Coxeter groups

In this section, we shall recall some fundamental theory on Coxeter groups
including the classification result and the Coxeter diagram. Most of these
background knowledge are contained in the references [5, 31]. We only list
some definitions and results because the related proofs can be found either in
[5, 31] or in the previous work [37]. The first concept is the reflection.

######  Definition A.1.

[5, Definition 6.1.1] A linear reflection on a vector space $V$ is a linear
automorphism $r:V\to V$ such that $r^{2}=1$ and such the fixed subspace
$H_{r}$ is a hyperplane.

With the help of reflections, we introduce the formal definition of Coxeter
groups.

######  Definition A.2.

[31, Definition 1.18] Let $I$ be a finite indexing set with cardinal number
$\lvert I\rvert=k$ and $\mathcal{S}=\\{s_{i}\\}_{i\in I}$. A Coxeter matrix
$M=(m_{ij})_{i,j\in I}$ on the set $\mathcal{S}$ is a $k\times k$ symmetric
matrix with entries in $\mathbb{N}\cup\\{\infty\\}$ such that $m_{ii}=1$ for
all $i\in I$ and $m_{ij}\geq 2$ for all distinct $i,j\in I$. The associated
Coxeter group $G=G_{\mathcal{S}}$ is defined by the presentation

$G=\langle\mathcal{S}\,|\,s_{i}^{2}=1,\,\forall i\in I\;\text{ and
}\;(s_{i}s_{j})^{m_{ij}}=1,\;\text{ for }i\neq j\in I\rangle.$

The set $\mathcal{S}$ is a Coxeter generating set for $G$ and the cardinality
$\lvert I\rvert$ is called the rank of $G$.

The following Coxeter diagram is a useful tool in classifying the finite
Coxeter groups.

######  Definition A.3.

[5, Definition 3.5.1] Suppose that $M=(m_{ij})$ is a Coxeter matrix on a set
$I$. We associate to $M$ a graph $\Upsilon$ called its Coxeter graph. The
vertex set of $\Upsilon$ is $I$. Distinct vertices $i$ and $j$ are connected
by an edge if and only if $m_{ij}\geq 3$. The edge ${i,j}$ is labeled by
$m_{ij}$ if $m_{ij}\geq 4$. If $m_{ij}=3$, the edge is left unlabeled. The
graph together with the labeling of its edges is called the Coxeter diagram
associated to $M$.

The following proposition (see e.g., [5, Theorem 4.1.6 and Proposition 4.1.7])
shows that the study of finite Coxeter groups can be largely reduced to the
case when the Coxeter is irreducible, which means its diagram is connected.

######  Proposition A.4.

For each $\mathcal{T}\subset\mathcal{S}$, $G_{\mathcal{T}}$ is a Coxeter
group. Moreover, suppose $\mathcal{S}$ can be partitioned into two nonempty
disjoint subsets $\mathcal{T}$ and $\mathcal{T}^{\prime}$ such that
$(st)^{2}=1$ for all $s\in\mathcal{T}$ and $t\in\mathcal{T}^{\prime}$. Then
$G=G_{\mathcal{T}}\times G_{\mathcal{T}^{\prime}}$.

As a result, any finite Coxeter group $G$ is the direct products of some
irreducible Coxeter groups. We now recall the famous classification of the
finite irreducible Coxeter groups, which was first completely enumerated by
Coxeter [4].

######  Theorem A.5.

[5, Theorem 6.9.1] The finite irreducible Coxeter groups and the associated
Coxeter diagrams are listed in Table 1.

${\bf{A}}_{n}\;(n\geq 1)$ |  |  ${\bf{E}}_{8}$ |
---|---|---|---
${\bf{B}}_{n}\;(n\geq 2)$ |  4 |  ${\bf{H}}_{3}$ |  5
${\centering\@add@centering}{\bf{D}}_{n}\;(n\geq 4)$ |  |  ${\bf{H}}_{4}$ |  5
${\bf{E}}_{6}$ |  |  ${\bf{F}}_{4}$ |  4
${\bf{E}}_{7}$ |  |  ${\bf{I}}_{2}(m)$ |  $m$
Table 1:  Classification of the finite irreducible Coxeter groups

The finite irreducible Coxeter groups consist of three one-parameter families
of increasing rank ${\bf{A}}_{n}$, ${\bf{B}}_{n}$, ${\bf{D}}_{n}$, one one-
parameter family of dimension two ${\bf{I}}_{2}(m)$, and six exceptional
groups: ${\bf{E}}_{6}$, ${\bf{E}}_{7}$, ${\bf{E}}_{8}$ and ${\bf{H}}_{3}$,
${\bf{H}}_{4}$, ${\bf{F}}_{4}$. Each of these groups corresponds to a symmetry
group of certain regular polytope (polygons) or semiregular polytope in the
corresponding dimensional space. Here we note for the diagram of
${\bf{I}}_{2}(m)$ that if $m=2$, there is no edge, and if $m=3$, the edge is
not labeled.

The following lemmas is a direct consequence of the finite Coxeter groups, we
refer to [31, Lemma 2.4] and [37, Lemma 2.7], respectively.

######  Lemma A.6.

There exists a unique epimorphism $\psi:G\to\\{\pm 1\\}$ induced by
$\psi(s)=-1$ for all $s\in\mathcal{S}$, so that for any $g\in G$,
$\psi(g^{-1})=\psi(g)^{-1}$.

######  Lemma A.7.

For every finite irreducible Coxeter group $W$ of rank $k$, there exists a
normal subgroup $N_{k}\lhd W$ such that $W=N_{k}\rtimes{\bf{A}}_{1}$.

Here we recall the semidirect product $N_{k}\rtimes{\bf{A}_{1}}$ means that
$W=N_{k}{\bf{A}_{1}}$ and $N_{k}\cap{\bf{A}_{1}}=\\{1\\}$.

We finally recall the fundamental domain for the Coxeter group $G$, it is a
closed, connected subset $\mathcal{F}\subset\mathbb{R}^{N}$ such that
$Gx\cap\mathcal{F}\neq\emptyset$ for every $x\in\mathbb{R}^{N}$ and
$Gx\cap\mathcal{F}=\\{x\\}$ for every $x$ in the interior of $\mathcal{F}$. It
is called a strict fundamental domain if it intersects each $G$-orbit in
exactly one point. Here $Gx$ refers to the $G$-orbit of the point
$x\in\mathbb{R}^{k}$. It turns out that every Coxeter group admits a strict
fundamental domain $\mathcal{F}$ [5, Corollary 6.6.9]. In fact, under the
group action $G$, for any $g\in G$, $g\mathcal{F}$ is also a fundamental
domain. Without loss of generality, we can fix $\mathcal{F}$ to be the
infinite cone formed by $H_{s}$ for all $s\in\mathcal{S}$. Let
$\partial^{i}\mathcal{F}$ denote its $k-i$ dimensional facets of the
polyhedral cone $\mathcal{F}$ for $i=0,1,\cdots,k-1$, here
$\partial^{0}\mathcal{F}$ refers to the interior of $\mathcal{F}$ for
convenience. Let $\mathcal{G}_{k}$ be the collection of finite Coxeter groups
of rank $k$. Also, $\mathcal{G}_{0}$ contains only the trivial group. We now
state an elementary results on the isotropic groups [37, Lemma 2.8].

######  Lemma A.8.

Let $k\geq 1$ and $\mathcal{F}$ be the strict fundamental domain of
$G\in\mathcal{G}_{k}$. Then for any $q\in\partial^{i}\mathcal{F}$ with
$i=0,\cdots,k-1$, $S_{q}\in\mathcal{G}_{i}$.

We recall that the isotropic group $S_{q}$ for $q\in\mathcal{F}$ is the
subgroup of $G$ that satisfies $S_{q}=\\{g\in G\,|\,gq=q\\}$. We close this
appendix by citing the representation result for the finite Coxeter groups,
which is owe to Tits [31, Theorem 3.1].

######  Lemma A.9.

Let $G$ be a finite Coxeter group with rank $k$. There exists a faithful
representation

$\rho:G\to GL(\mathbb{R}^{k}).$

A faithful representation means that group homomorphism $\rho$ is injective.
Namely, the realized space for the group $G\in\mathcal{G}_{k}$ at least is
$\mathbb{R}^{k}$, so that the restriction $k\leq N$ in Theorem 1.1 is to
ensure the well realization of the action of Coxeter group $G$.

Acknowledgement. This paper is supported by National Natural Science
Foundation of China (Grant No. 11771324, 11811540026 and 11901456) and by the
Natural Science Foundation of Shaanxi Province (Grant no. 2020JQ–120).

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