11institutetext:  Dipartimento di Fisica e Geologia, Università degli studi di
Perugia, and INFN, sezione di Perugia, via A. Pascoli snc, 06123, Perugia,
Italy 22institutetext: Old Dominion University, Norfolk, Virginia 23529, USA.

# Deeply virtual Compton scattering off Helium nuclei with positron beams

Sara Fucini 11 Mohammad Hattawy 22 Matteo Rinaldi 11 Sergio Scopetta 11

(Received: date / Revised version: date)

###### Abstract

Positron initiated deeply virtual Compton scattering (DVCS) off 4He and 3He
nuclei is described. The way the so-called $d-$term could be obtained from the
real part of the relevant Compton form factor is summarized, and the
importance and novelty of this measurement is discussed. The measurements
addressed for 3He targets could be very useful even in a standard unpolarized
target setup, measuring beam spin and beam charge asymmetries only. The
unpolarized beam charge asymmetries for DVCS off 3He and 4He are also
estimated, at JLab kinematics and, for 4He, also at a configuration typical at
the future Electron-Ion Collider. Incoherent DVCS processes, in particular the
ones with tagging the internal target by measuring slow recoiling nuclei, and
the unique possibility offered by positron beams for the investigation of
Compton form factors of higher twist, are also briefly addressed.

###### pacs:

13.60.Hb,14.20.Dh,27.10.+h

## Introduction

In recent years, a growing interest on nuclear deeply virtual Compton
scattering (DVCS), i.e., hard photon electroproduction from nuclear targets,
has arisen. This is mainly motivated by the possibility to shed light on the
European Muon Collaboration (EMC) effect, i.e., the elusive nuclear
modifications of the nucleon parton structure (see, e.g., Refs. Dupre:2015jha
; Cloet:2019mql  for recent reports), as well as the possibility to
distinguish the so-called coherent and incoherent channels of the DVCS
process. This latter feature has been experimentally recently achieved by the
CLAS collaboration at JLab using a 4He gaseous target Hattawy:2017woc ;
Hattawy:2018liu ; Dupre:2021uco , paving the way to a new class of precise
measurements at high luminosity facilities. Coherent DVCS takes place when the
nucleus recoils elastically, while in the incoherent process the struck proton
is detected in the final state. Recently, the measurement of positron
initiated DVCS has been experimentally proposed, in particular at JLab with
the 12 GeV electron beam Accardi:2020swt . In the present paper, we analyze
the impact that these measurements may have using 4He and 3He targets. This is
done separately for the coherent and incoherent channels in the next two
sections. Some additional remarks on higher twist effects are reported in the
fourth section, followed by the conclusions of our investigation.

Figure 1: The leading-order, twist-2, handbag approximation to the coherent
DVCS process off 4He, where the four-vectors of the initial/final electron,
photon, and 4He nucleus are denoted by $k/k^{\prime}$, $q/q^{\prime}$, and
$p/p^{\prime}$, respectively. The quantity $x+\xi$ is the hadron’s
longitudinal momentum fraction carried by the struck quark, -2$\xi$ is the
longitudinal momentum fraction of the momentum transfer $\Delta$
($=q-q^{\prime}=p^{\prime}-p$), and $t$ ($=\Delta^{2}$) is the squared
momentum transfer.

## Coherent DVCS

For the sake of concreteness, to explore the insights on how positron beams
could help in this field, let us think first to coherent DVCS off 4He, the
only process which has been clearly accessed so far, illustrated, in the
handbag approximation, in Fig. 1. In DVCS, the relevant non-perturbative
information is encoded in the so called generalized parton distribution
functions (GPDs), giving access, in specific kinematic configurations, to the
location, in the transverse plane of the target, of one parton of given
longitudinal momentum (the so-called hadron tomography). We recall that 4He
has only one chiral-even GPD at leading-twist.

In the coherent DVCS data analysis from the EG6 experiment by the CLAS
collaboration Hattawy:2017woc , the crucial measured observable was the single
beam-spin asymmetry, $A_{LU}$, which can be extracted from the reaction yields
with the two electron helicities ($N^{\pm}$):

$A_{LU}=\frac{1}{P_{B}}\frac{N^{+}-N^{-}}{N^{+}+N^{-}},$ (1)

where $P_{B}$ is the degree of longitudinal polarization of the incident
electron beam.

Within the accessed kinematical phase-space of the EG6 experiment, the cross
section of real photon electroproduction is dominated by the so-called Bethe-
Heitler(BH) process, where the real photon is emitted by the incoming or the
outgoing lepton, while the pure DVCS contribution is very small. However, the
DVCS contribution is enhanced in the observables sensitive to the interference
term, i.e. the quantity $A_{LU}$ given above, which depends on the azimuthal
angle $\phi$ between the $(e,e^{\prime})$ and
$(\gamma^{*},^{4}$He${}^{\prime})$ planes. The asymmetry $A_{LU}$ for a spin-
zero target can be approximated at leading-twist as

$A_{LU}(\phi)=\frac{\alpha_{0}(\phi)\,\Im m(\mathcal{H}_{A})}{den(\phi)}\,,$
(2)

with:

$\displaystyle den(\phi)$ $\displaystyle=$
$\displaystyle\alpha_{1}(\phi)+\alpha_{2}(\phi)\,\Re e(\mathcal{H}_{A})$ (3)
$\displaystyle+$ $\displaystyle\alpha_{3}(\phi)\,\big{(}\Re
e(\mathcal{H}_{A})^{2}+\Im m(\mathcal{H}_{A})^{2}\big{)}\,.$

The kinematic factors $\alpha_{i}$ are known (see, e.g., Ref. Belitsky:2001ns
; Belitsky:2008bz ). In the experimental analysis, using the different
contributions proportional to $\sin(\phi)$ and $\cos(\phi)$ in Eq. (3), both
the real and imaginary parts of the so-called Compton Form Factor (CFF)
$\mathcal{H}_{A}$, $\Re e(\mathcal{H}_{A})$ and $\Im m(\mathcal{H}_{A})$,
respectively, have been extracted by fitting the $A_{LU}(\phi)$ distribution.
Results of the impulse approximation calculation of Ref. Fucini:2018gso , are
shown together with the data of Ref. Hattawy:2017woc  in Figs. 2 and 3. In the
theoretical calculation, use is made of state-of-the-art ingredients for the
description of the nuclear and nucleon structure (older calculations are found
in Liuti:2005gi ; Guzey:2003jh ). In particular, a convolution formula is
obtained where the nuclear effects are governed by a model of a one-body non-
diagonal spectral function based on the Av18 nucleon nucleon interaction
Wiringa:1994wb  and the Urbana IX three body forces Pudliner:1997ck  (see Ref.
Fucini:2018gso  for details), while the nucleon GPDs are modelled following
the Goloskokv-Kroll model Kroll:2012sm ; Goloskokov:2008ib ; Diehl:2013xca .
Big statistical errors are seen everywhere in the data but, in particular,
$\Re e(\mathcal{H}_{A})$ is less precisely extracted than $\Im
m(\mathcal{H}_{A})$, due to the small coefficient $\alpha_{2}$ in Eq. (3).
This fact is easily understood looking at Fig. 3 where it is apparent that the
predicted contribution of the real part of the CFF to the symmetry is really
small.

Figure 2:  The imaginary part of the CFF measured in coherent DVCS off 4He.
Data from Ref. Hattawy:2017woc ; calculations (red crosses) from Ref.
Fucini:2018gso

.

Figure 3:  The real part of the CFF measured in coherent DVCS off 4He. Data
from Ref. Hattawy:2017woc ; calculations (red crosses) from Ref.
Fucini:2018gso .

Forth-coming data from JLab with 12 GeV electron beams at high luminosity,
using also the detector system developed by the ALERT run-group
Armstrong:2017wfw , will be affected by much smaller statistical errors.
Together with refined realistic theoretical calculations, in progress for
light nuclei Fucini:2020vpr , the new data will help to unveil a possible
exotic behavior of the real and imaginary part of $\mathcal{H}_{A}$, beyond
that predicted in a conventional realistic scenario using the Impulse
approximation. Nonetheless, the extracted $\Re e(\mathcal{H}_{A})$ will be
always less precise than $\Im m(\mathcal{H}_{A})$, intrinsically, due to the
small coefficient $\alpha_{2}$ in (3) previously introduced. A precise
knowledge of $\Re e(\mathcal{H}_{A})$ for light nuclei would be instead
crucial, as it is briefly analyzed in what follows, specified initially for
the 4He target. Formally one can write, for the quantities $\Re
e(\mathcal{H}_{A})$ and $\Im m\mathcal{H}_{A}$ shown in Figs. 3 and 2
respectively Guidal:2013rya :

$\Re e\mathcal{H}_{A}(\xi,t)\equiv{\cal
P}\int_{0}^{1}dxH_{+}(x,\xi,t)C_{+}(x,\xi)\,,$ (4)

and

$\Im m\mathcal{H}_{A}=-\pi H_{+}(\xi,\xi,t)\,,$ (5)

with:

$H_{+}=H(x,\xi,t)-H(-x,\xi,t)\,,$ (6)

amd

$C_{+}(x,\xi)=\frac{1}{x+\xi}+\frac{1}{x-\xi}\,,$ (7)

with $H(x,\xi,t)$ the chiral-even, leading twist generalized parton
distribution (GPD).

Besides, it is also known that $\Re e(\mathcal{H}_{A})$ satisfies a once
subtracted dispersion relation at fixed $t$ and can be therefore related to
$\Im m\mathcal{H}_{A}$, leading to Anikin:2007yh ; Diehl:2007jb ;
Radyushkin:2011dh ; Pasquini:2014vua

$\Re e\mathcal{H}_{A}(\xi,t)\equiv{\cal
P}\int_{0}^{1}dxH_{+}(x,x,t)C_{+}(x,\xi)-\delta(t)\,.$ (8)

t

Figure 4: The beam spin asymmetry measured in coherent DVCS off 4He, for
$\phi=90^{o}$. Data from Ref. Hattawy:2017woc ; calculations (red dots) from
Ref. Fucini:2018gso . The result obtained neglecting the real part of the CFF
$\mathcal{H}_{A}$ is also shown (blue triangles)

One notices that, in contrast to the convolution integral defining the real
part of the CFF in Eq. (4), where the GPD enters for unequal values of its
first and second argument, the integrated in the dispersion relation
corresponds to the GPD where its first and second arguments are equal. The
subtraction term $\delta(t)$ can be related to the so-called $d-$term and
accurate measurements, supplemented by precise calculations, would allow
therefore to study this quantity for nuclei, for the first time. This
$d-$term, introduced initially to recover the so-called polinomiality property
in double distribution (DD) approaches to GPDs modelling Polyakov:1999gs , has
been related to the form factor of the QCD energy momentum tensor (see e.g.
Ref. Polyakov:2018zvc ; Dutrieux:2021nlz ). It encodes information on the
distribution of forces and pressure between elementary QCD degrees of freedom
in the target. For nuclei, it has been predicted to behave as $A^{7/3}$ in a
mean field scheme, either in the liquid drop model of nuclear structure
Polyakov:2002yz  or in the Walecka model Jung:2014jja . None of these
approaches makes much sense for light nuclei, for which accurate realistic
calculations are possible. Using light nuclei one would therefore explore, at
the parton level, the onset and evolution of the mean field behavior across
the periodic table, from deuteron to 4He, whose density and binding are not
far from those of finite nuclei. In this sense, coherent DVCS off 3He targets
acquire an important role: an intermediate behavior is expected between that
of the almost unbound deuteron system and that of the deeply bound alpha
particle. The formal description of coherent DVCS off 3He follows that for the
proton Belitsky:2001ns ; Belitsky:2008bz , a spin one-half target, in terms of
CFFs and related GPDs, accessed properly defining spin dependent asymmetries.
Realistic theoretical calculations are available for GPDs Scopetta:2004kj ;
Scopetta:2009sn ; Rinaldi:2012ft ; Rinaldi:2012pj ; Rinaldi:2014bba  and are
in progress for the relevant CFFs, cross sections and asymmetries,
representing an important support to the planning of measurements
Fucini:2020vpr . We remark that in Refs. Rinaldi:2012ft ; Rinaldi:2012pj  it
has been shown that the 3He chiral-even GPDs are strongly dominated by those
of the neutron, which can be safely extracted by properly taking into account
the nuclear effects described in the Impulse Approximation. One could object
that the use of 3He, either longitudinally or transversely polarized,
represents at the moment a challenge, either with electron or positron beams.
Actually beam-charge asymmetries, built using electron and positron data,
would represent, even with unpolarized 3He targets and unpolarized beams, a
possible access to $\Re e\mathcal{H}_{A}(\xi,t)$, as previously described for
4He, with the same potential to explore the ”d-”term for this relevant light
nucleus.

Positron beams would guarantee this achievement: as a matter of fact,
combining data for properly defined asymmetries measured using electrons and
positrons, the role of $\Re e\mathcal{H}_{A}$ would be directly accessed. Let
us recall how it is possible.

One should notice that, between the quantities appearing in the above
equations and the cross sections defining the generic photo-$e^{\pm}$
production, in the following schematic general expression Accardi:2020swt :

$\displaystyle\sigma^{e}_{\lambda 0}$ $\displaystyle=$
$\displaystyle\sigma_{BH}+\sigma_{DVCS}+\lambda\tilde{\sigma}_{DVCS}$ (9)
$\displaystyle+$ $\displaystyle e\sigma_{INT}+e\lambda\tilde{\sigma}_{INT}\,,$

the following relations hold:

$\displaystyle\sigma_{BH}$ $\displaystyle\propto$
$\displaystyle\alpha_{1}(\phi)\,,$ $\displaystyle\sigma_{DVCS}$
$\displaystyle\propto$ $\displaystyle\alpha_{3}(\phi)\big{(}\Re
e(\mathcal{H}_{A})^{2}+\Im m(\mathcal{H}_{A})^{2}\big{)}\,,$
$\displaystyle\sigma_{INT}$ $\displaystyle\propto$
$\displaystyle\alpha_{2}(\phi)\,\Re e(\mathcal{H}_{A})\,,$
$\displaystyle\tilde{\sigma}_{INT}$ $\displaystyle\propto$
$\displaystyle\alpha_{0}(\phi)\,\Im m(\mathcal{H}_{A})\,,$ (10)

while $\tilde{\sigma}_{DVCS}$ is proportional to a term kinematically
suppressed at JLab kinematics, dependent on higher twist CFFs.

From a combined analysis of data taken with polarized electrons and/or
positrons, one could access all the five cross sections in Eq. (9). We stress
in particular that, using just unpolarized electrons and positrons, $\Re
e(\mathcal{H}_{A})$ would be directly accessed, building charge beam
asymmetries. To expose this fact, in order to extract the real part of CFF,
the beam charge asymmetry (BCAA), for a nucleus with $A$ nucleons, is
introduced. By following the same formalism adopted in Ref. Belitsky:2001ns ,
one can define the BCAA as follows:

$\displaystyle\mbox{BCA}_{A}=\dfrac{d\sigma^{+}-d\sigma^{-}}{d\sigma^{+}-d\sigma^{-}}\leavevmode\nobreak\
,$ (11)

where here $d\sigma^{\pm}$ represents the five times differential unpolarized
cross section for an electron ($-$) or a positron ($+$) beam, respectively.
Let us recall that such a cross section for, e.g., the process
$e^{\pm}\leavevmode\nobreak\ {}^{3}He\rightarrow e^{\pm}\leavevmode\nobreak\
{}^{3}He\gamma$ Belitsky:2001ns ; Belitsky:2008bz , reads:

$\displaystyle d\sigma^{C}\equiv\dfrac{d\sigma^{C}}{dx_{B}\leavevmode\nobreak\
dy\leavevmode\nobreak\ dt\leavevmode\nobreak\ d\phi\leavevmode\nobreak\
d\varphi}=\dfrac{\alpha^{3}x_{B}y}{16\pi^{2}Q^{2}\sqrt{1+\varepsilon^{2}}}\left|\dfrac{T_{C}}{e^{3}}\right|^{2}$
(12)

where $\varepsilon=2x_{B}M_{3}/Q$, $C=\pm 1$ is the lepton charge, $M_{3}$ is
the 3He mass and $y$ is the lepton energy fraction. Moreover $\phi$ is the
azimuthal angle between the lepton plane and the recoiled nucleus. Finally,
$T^{2}_{C}=|T_{BH}|^{2}+|T_{DVCS}|^{2}+I_{C}$. Let us remark that only the
interference term depends on the beam charge Belitsky:2001ns . Therefore:

$\displaystyle
d\sigma^{+}-d\sigma^{-}=\dfrac{2\alpha^{3}x_{B}y}{16\pi^{2}Q^{2}\sqrt{1+\varepsilon^{2}}}\dfrac{I}{e^{6}}$
(13)

where $I=I_{C}/C$. Within this formalism:

$\displaystyle I$
$\displaystyle=\dfrac{e^{6}}{x_{B}y^{3}tP_{1}(\phi)P_{2}(\phi)}$ (14)
$\displaystyle\times\Big{\\{}c_{0}^{I}+\sum_{1}^{3}\big{[}c_{n}^{I}\cos(n\phi)+s_{n}^{I}\sin(n\phi)\big{]}\Big{\\}}.$

These harmonic coefficients depend on the 3He CFFs. In particular, if
unpolarized targets and beams are considered, $s_{n}^{I}$ do not contribute to
the BCA3. The functions $P_{1}$ and $P_{2}$ Belitsky:2001ns  cancel out in the
evaluation of BCA3. On the other hand:

$\displaystyle
d\sigma^{+}+d\sigma^{-}=\dfrac{2\alpha^{3}x_{B}y}{16\pi^{2}Q^{2}\sqrt{1+\varepsilon^{2}}}\dfrac{|T_{DVCS}|^{2}+|T_{BH}|^{2}}{e^{3}}\leavevmode\nobreak\
.$ (15)

We remind that, when terms of the order $t/Q^{2}$ can be neglected, BCA3 is
directly related to the real part of CFFs. As a matter of facts, in this
kinematic conditions,

$\displaystyle\mbox{BCA}_{3}(\phi)\sim\dfrac{e^{6}}{x_{B}y^{3}\Delta^{2}P_{1}(\phi)P_{2}(\phi)}\dfrac{c_{1}^{I}\cos(\phi)}{|T_{BH}|^{2}}\leavevmode\nobreak\
.$ (16)

We remind that, since $c_{3}^{I}$ is related to the gluon distribution, in
this initial analysis such a contribution is also neglected, together with
$c_{0}^{I}$ and $c_{2}^{I}$. Finally $c_{1}^{I}\propto Re\leavevmode\nobreak\
c_{unp}^{I}$ where
$c_{unp}^{I}=F_{1}\mathcal{H}+x_{B}/(2-x_{B})(F_{1}+F_{2})\mathcal{\tilde{H}}-\Delta^{2}/(4M^{2})F_{2}\mathcal{E}$
(for details see Ref. Belitsky:2001ns ). We recall that $F_{1}$ and $F_{2}$
are the Dirac and Pauli form factors respectively and
$\mathcal{F}=\mathcal{H,E,\tilde{H}}$ is the CFF. Finally:

$\displaystyle\mbox{BCA}_{3}(\phi)=\dfrac{x_{B}(1+\varepsilon^{2})^{2}}{y}\dfrac{c_{1}^{I}\cos(\phi)}{c_{0}^{BH}+c_{1}^{BH}\cos(\phi)}\leavevmode\nobreak\
.$ (17)

Numerical predictions for the 3He BCA, evaluated in a kinenatical range
typical at JLab, are shown for the first time in the upper panel of Fig. 5.
Here the 3He GPDs have been calculated by following the line of Refs.
Rinaldi:2012ft ; Rinaldi:2012pj , i.e. by means of the off-forward spin
dependent spectral function calculated from the 3He wave function
corresponding to the AV18 nuclear potential Pace:2001cm ; Kievsky:1996gz ;
Kievsky:1997bg . The nucleonic GPDs, necessary as an input to the
calculations, have been chosen among the phenomenological parametrizations of
Refs. Kroll:2012sm ; Goloskokov:2008ib ; Diehl:2013xca . In addition one can
introduce the nuclear beam spin asymmetry (BSA):

$\displaystyle\mbox{BSA}_{A}=\dfrac{d\sigma^{\uparrow}-d\sigma^{\downarrow}}{d\sigma^{\uparrow}+d\sigma^{\downarrow}}\leavevmode\nobreak\
.$ (18)

By using the previously discussed strategy and approximations, one can relate
the above quantity to the imaginary part of CFF. For a 1/2 spin target like
the 3He one gets:

$\displaystyle\mbox{BSA}_{3}\sim\pm\dfrac{x_{Bj}}{y}\dfrac{s_{1}^{I}}{c_{0}^{BH}}\sin(\phi)\leavevmode\nobreak\
,$ (19)

where now $s^{I}_{1}\propto\leavevmode\nobreak\ Im\leavevmode\nobreak\
c^{I}_{unp}$.

Figure 5: Upper panel: The 3He beam charge asymmetry Eq. (17) for a specific
kinematic configuration, typical at JLab. Lower panel: the same but for the
3He beam spin asymmetry Eq. (19).

The formalism above discussed for the beam charge asymmetry of the 3He nucleus
can be easily extended to the 4He nucleus. While the beam spin asymmetry for a
longitudinally polarized electron beam off an unpolarized target has already
demonstrated to be a key observable to access the the imaginary part of the
CFF, the BCA is mostly useful to extract information for the real part Eq.
(8). Looking at Eq. (17), the numerator cleanly encapsulates the real part of
CFF linked to the chiral even GPD of the $H_{q}^{A}$. Conversely to the 3He
case, the 4He spinless nature allows one the access to only this GPDs without
the contamination of other GPDs. We show here our first estimates of the BCA
for 4He. We make use, for the expression of the GPD $H_{q}^{A}$ of the impulse
approximation model presented in Ref. Fucini:2018gso  where, as already
reported, a semi-realistic model for the nuclear part, accounting for a
realistic momentum dependence obtained with realistic NN potential (Av18) and
three-body forces, was adopted. While the non diagonal momentum and the energy
dependencies of the spectral function is just modeled, a confirmation of the
capability of our model to describe conventional nuclear effects is at hand.
As an example of this fact, one can check Figs. 2-3. In passing by, we remind
that the twist-two CFF of the 4He nucleus receives the contribution of the GPD
$H$ and $E$ of the bound nucleons in the form
$H_{q}^{N/A}=\sqrt{(1-\xi^{2})}\bigg{[}H_{q}^{N}-\frac{\xi^{2}}{1-\xi^{2}}E_{q}^{N}\bigg{]}$.
As noted in Ref. Fucini:2020vpr , the contribution of the GPD $E_{q}^{N}$ is
practically negligible at the kinematics probed at JLab due to the smallness
of the skweness (an even greater kinematical suppression for such a factor
will be observed in the typical kinematic ranges expected at the Electron Ion
Collider (EIC) AbdulKhalek:2021gbh ). For the GPDs of the nucleons, we made
use of the same GK models used for 3He, i.e. the ones described in Refs.
Goloskokov:2006hr ; Goloskokov:2008ib . In Fig. 7, a sizable asymmetry is
observed for 4He plotted ad various kinematics. Two of these are the same
already probed in Ref. Hattawy:2017woc , while another one is the same
foreseen for the free proton using positron beams (see. Ref. Burkert:2021rxz
). As a reference, we show in Fig. 6 the same quantity for one of the
kinematic points envisioned at the EIC for a positron beam at 18 GeV colliding
with an helium-4 beam with an energy of 110 GeV. The kinematic points plotted
is the present paper have been obtained as pseudo-data generated with the
software TOPEG.
While in Ref. AbdulKhalek:2021gbh  the possibility to reach the first
tomographic view of the nucleus is shown, here the big asymmetry found in some
of the kinematical ranges investigated, allowing one an easier extraction of
information from the observable, confirms the prominent role that the EIC
together with the upgraded JLab will play with the aim to reach a deeper
comprehension of the innermost structure of hadronic targets and hadronic
matter.

Figure 6: The 4He beam charge asymmetry shown at different kinematic points,
typical at JLab. Figure 7: The 4He beam charge asymmetry shown at different
EIC kinematics. The energy configuration corresponding to these plots is
$E_{electron}\times E_{nucleus}=(18x110)$ GeV. Figure 8: The leading-order,
twist-2, handbag approximation to incoherent DVCS off 4He. See Fig. 1 for the
definition of the variables.

## Incoherent DVCS

A subject aside is represented by incoherent DVCS off He nuclei, i.e., the
process where, in the final state, the struck proton is detected, its CFFs
accessed, its GPDs, in principle, extracted and, ultimately, its tomography
obtained. Fig. 8 illustrates the leading-twist handbag diagram for the
incoherent DVCS process off 4He. Such measurements would provide a pictorial
representation of the realization of the EMC effect and a great progress
towards the understanding of its dynamical origin. As already stressed, this
channel has been successfully isolated by the EG6 experiment of the CLAS
collaboration at JLab Hattawy:2018liu  and a first glimpse at the parton
structure of the bound proton in the transverse coordinate space is therefore
at hand. Recent impulse approximation calculations, using again a description
based on the Av18 interaction for the nuclear effects and, for the nucleon
part, the GK model and the one proposed in Ref Mezrag:2013mya , evaluated
using the PARTON tool Berthou:2015oaw , have been presented in Refs
Fucini:2019xlc ; Fucini:2020lxi . The theoretical description of the recent
data with conventional realistic ingredients is rather encouraging. The
program at JLab 12 includes an improvement of the accuracy of these
measurements, in particular, for the first time in DVCS, tagging the struck
nucleon detecting slow recoiling nuclei, using the detector developed by the
ALERT run group Armstrong:2017zcm . This would allow to keep possible final
state interactions, relevant in principle in this channel, under control.
Measurements performed with electron and positron beams would allow for
example the measurement of the $d-$term for the bound nucleon, either proton
in 3He (tagging 2H from DVCS on 3He) or in 4He (tagging 3H from DVCS on 4He)
or neutron in 4He (tagging 3He from DVCS on 4He). Modifications of the
$d-$term of the nucleon in the nuclear medium, studied e.g. in Ref.
Jung:2014jja , would be at hand, as well as a glimpse at the structure of the
neutron in the transverse plane, complementary to that obtained with deuteron
targets.

## Beyond a chiral even GPDs description of DVCS on 4He

As a last argument, we quickly note that, from the measurement of beam spin
asymmetries built using cross sections measured with polarized electrons and
positrons in coherent DVCS off 4He, the terms $\tilde{\sigma}_{DVCS}$ and
$\tilde{\sigma}_{INT}$, appearing in Eq. 9, could be independently accessed.
This would allow, for example for 4He, to study the other leading twist CFF of
a spinless target, the so called gluon transversity GPD $H_{T}$, giving a
corresponding name to the CFF ${\cal{H}}_{T}$, appearing in
$\tilde{\sigma}_{DVCS}$. In Ref. Belitsky:2008bz , it is shown how the
contribution of ${\cal{H}}_{T}$ to the cross section occurs through an
interference between twist-two and effective twist-three CFFs. A first glimpse
at this complicated interrelation would be obtained for a spin-less target, in
particular for a nuclear target, for the first time. As for any other gluon-
sensitive observable, data for the cross section $\tilde{\sigma}_{DVCS}$ would
be a perfect tool to study gluon dynamics in nuclei. For example, a comparison
between the above observable and calculations performed in an Impulse
Approximation scheme, where the relevant nuclear degrees of freedom are
colorless nucleons and mesons, would have the potential to expose possible
exotic gluon dynamics in nuclei. This would be a pretty new possibility,
complementary to that planned at JLab with 12 GeV, using exclusive vector
meson electroproduction off 4He Armstrong:2017zcm . Such an interersting
behavior would be very hardly seen using electrons only, due to the strong
kinematical suppression of $\tilde{\sigma}_{DVCS}$ with respect to the other
contributions in Eq. (9).

## Conclusions

The unique possibilities offered by the use of positron beams in DVCS off
three- and four-body nuclear systems have been reviewed. Summarizing, we can
conclude that the main advantages will be:

  * •

in coherent DVCS off 3He and 4He, using polarized electrons and unpolarized
positrons, the real part of the chiral even unpolarized CFFs would be measured
with a precision comparable to that of their imaginary part, providing a tool
for the study of the so called $d$-term. In turn, the distribution of pressure
and forces between the partons in nuclei, a new way to look at the nuclear
medium modifications of nucleon structure, could be investigated;

  * •

in incoherent DVCS off 3He and 4He, possibly tagging slow recoiling nuclear
systems, the same programme could be run for the bound proton and neutron;

  * •

using polarized 3He targets, a more complicated setup for the moment, spin
dependent and parton helicity flip CFFs would be accessed for the first time
for a nucleus, in both their real and imaginary parts. Moreover, in this case
these quantities would be dominated by the neutron contributions and, in turn,
an extraction of the neutron CFFs would be feasible;

  * •

coherent DVCS off 4He, initiated with polarized positrons, would allow a first
analysis of nuclear chiral odd CFFs and GPDs, with higher twist contamination
suitable to tentatively explore gluon dynamics in nuclei.

A programme of nuclear measurements with positron beams would represent
therefore an exciting complement to the experiments planned with nucleon
targets, and to those planned with nuclear targets and electron beams. In this
paper, we have shown for the first time realistic Impulse Approximation
predictions for spin-independent charge asymmetries, at JLab kinematics for
3He and 4He and, in this last case, also in one of the typical kinematic
setups foreseen at the EIC. In an extended forthcoming investigation these
calculations will be extended to the incoherent channels and to other spin
dependent asymmetries.

## References

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#  Multiple normalized solutions to a system of nonlinear Schrödinger
equations

Jarosław Mederski  and  Andrzej Szulkin
Institute of Mathematics,
Polish Academy of Sciences,
ul. Śniadeckich 8, 00-656 Warsaw, Poland<EMAIL_ADDRESS>
Department of Mathematics,
Stockholm University,
106 91 Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract.

We find a normalized solution $u=(u_{1},\ldots,u_{K})$ to the system of $K$
coupled nonlinear Schrödinger equations

$\left\\{\begin{array}[]{l}-\Delta
u_{i}+\lambda_{i}u_{i}=\sum_{j=1}^{K}\beta_{i,j}u_{i}|u_{i}|^{p/2-2}|u_{j}|^{p/2}\quad\mathrm{in}\
\mathbb{R}^{3},\\\ u_{i}\in H^{1}_{rad}(\mathbb{R}^{3}),\\\
\int_{\mathbb{R}^{3}}|u_{i}|^{2}\,dx=\rho_{i}^{2}\quad\text{for
}i=1,\ldots,K,\end{array}\right.$

where $\rho=(\rho_{1},\ldots,\rho_{K})\in(0,\infty)^{K}$ is prescribed,
$(\lambda,u)\in\mathbb{R}^{K}\times H^{1}(\mathbb{R}^{3})^{K}$ are the unknown
and $4\leq p<6$. In the case of two equations we show the existence of
multiple solutions provided that the coupling is sufficiently large. We also
show that for negative coupling there are no ground state solutions. The main
novelty in our approach is that we use the Cwikel-Lieb-Rozenblum theorem in
order to estimate the Morse index of a solution as well as a Liouville-type
result in an exterior domain.

###### Key words and phrases:

System of nonlinear Schrödinger equations, normalized solution, ground state,
Nehari manifold, Pohozaev manifold

###### 2010 Mathematics Subject Classification:

Primary: 35J50, 35J60, 35Q55; Secondary: 35J20, 78A25

##  1\. Introduction

Consider the system of time-dependent nonlinear Schrödinger equations of the
form

$\mathrm{i}\frac{\partial\Phi_{i}}{\partial
t}-\Delta\Phi_{i}=g_{i}(\Phi),\quad i=1,\ldots,K.$

In this paper we look for solutions which are normalized by prescribing the
$L^{2}$-bounds for $\Phi=(\Phi_{1},\ldots,\Phi_{K})$:

$\int_{\mathbb{R}^{3}}\left|\Phi_{i}(t,x)\right|^{2}\,dx=\rho_{i}^{2}\quad\hbox{for
}i=1,\ldots,K.$

Here $\rho=(\rho_{1},\ldots,\rho_{K})\in(0,\infty)^{K}$ represents the total
number of atoms in Bose–Einstein condensation [29, 16] or the power supply in
nonlinear optics [11]. For a particular power-type nonlinearity in three
dimensions, seeking for solitary wave solutions of the form
$\Phi_{i}(x,t)=u_{i}(x)e^{-\mathrm{i}\lambda_{i}t}$ leads to the system of
time-independent Schrödinger equations

(1.1) $\left\\{\begin{array}[]{l}-\Delta
u_{i}+\lambda_{i}u_{i}=\sum_{j=1}^{K}\beta_{i,j}u_{i}|u_{i}|^{p/2-2}|u_{j}|^{p/2}\quad\mathrm{in}\
\mathbb{R}^{3},\\\ u_{i}\in H^{1}_{rad}(\mathbb{R}^{3}),\\\
\int_{\mathbb{R}^{3}}|u_{i}|^{2}\,dx=\rho_{i}^{2}\quad\text{for
}i=1,\ldots,K,\end{array}\right.,$

where we assume $\beta_{i,i}>0$ and $\beta_{i,j}=\beta_{j,i}\in\mathbb{R}$ for
$i,j=1,\ldots,K$. Here

$(\lambda,u)=(\lambda_{1},\ldots,\lambda_{K},u_{1},\ldots,u_{K})\in\mathbb{R}^{K}\times
H^{1}_{rad}(\mathbb{R}^{3})^{K}$

are the unknown and $H^{1}_{rad}(\mathbb{R}^{3})^{K}$ is the subspace of
$H^{1}(\mathbb{R}^{3})^{K}$ whose all components are radial. Note that this
covers the most important example from the physical point of view which is
$p=4$; see [11, 29, 14, 26] and the references therein. We look for solutions
in $H^{1}_{rad}(\mathbb{R}^{3})^{K}$ and not in $H^{1}(\mathbb{R}^{3})^{K}$ in
order to get some additional compactness for the functional $J$ defined below.

Recall that Kwong [23] showed that there exists a unique positive radial
solution to the equation

$-\Delta u+u=u^{p-1}.$

After the rescaling

$u_{\beta_{1,1}}:=\alpha u(\gamma\cdot),$

for suitable $\alpha,\gamma>0$ this gives a unique positive radial solution to
(1.1) with $K=1$. In general, if $K\geq 2$, a scaling-type argument fails. It
is well known that weak solutions to (1.1) are critical points of the energy
functional $J|_{\mathcal{S}}$ given by

$J(u):=\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\,dx-\frac{1}{p}\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx,\quad u\in
H^{1}_{rad}(\mathbb{R}^{3})^{K}$

where

(1.2) ${\mathcal{S}}:=\left\\{u=(u_{1},\ldots,u_{K})\in
H^{1}_{rad}(\mathbb{R}^{3})^{K}\ :\
\int_{\mathbb{R}^{3}}|u_{i}|^{2}\,dx=\rho_{i}^{2}\quad\text{for
}i=1,\ldots,K\right\\}.$

The constants $\lambda_{i}\in\mathbb{R}$ in (1.1) appear as the Lagrange
multipliers. Similarly as for one equation, a global minimizer for $J$
restricted to ${\mathcal{S}}$ can only exist in the $L^{2}$-subcritical case
($2<p<\frac{10}{3}$), and in the $L^{2}$-critical case ($p=\frac{10}{3}$)
provided that $\rho_{i}$ are sufficiently small [37, 27, 33, 9]. If
$p>\frac{10}{3}$, then $\inf_{\mathcal{S}}J=-\infty$ and the purpose of this
work is to construct a minimax theory for $J$ on ${\mathcal{S}}$. Therefore,
from now on we assume that $\frac{10}{3}<p<6$.

Let us recall that every critical point of the functional above belongs to
$W^{2,q}_{\textup{loc}}(\mathbb{R}^{N})^{K}$ for all $q<\infty$ and satisfies
the Pohožaev [10] and the Nehari identities

$\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,dx=\frac{6}{p}\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx-3\sum_{i=1}^{K}\lambda_{i}\int_{\mathbb{R}^{3}}|u_{i}|^{2}\,dx$

and

$J^{\prime}(u)[u]+\sum_{i=1}^{K}\lambda_{i}\int_{\mathbb{R}^{3}}|u_{i}|^{2}\,dx=0.$

By a linear combination of the two equalities above it is easily checked that
every solution $u$ satisfies the Nehari-Pohožaev identity

(1.3) $M(u):=\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\,dx-\frac{3(p-2)}{2p}\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx=0.$

In a usual way (cf. e.g. [3, 4, 30]) we introduce the constraint

(1.4) ${\mathcal{M}}:=\left\\{u\in
H^{1}_{rad}(\mathbb{R}^{3})^{K}\setminus\\{0\\}:M(u)=0\right\\}$

which contains all nontrivial solutions to (1.1) and does not depend on
$\lambda=(\lambda_{1},\ldots,\lambda_{K})$.

In the case of two equations Bartsch, Jeanjean and Soave proved in [3] that
for $p=4$ and every sufficiently small or sufficiently large
$\beta:=\beta_{1,2}=\beta_{2,1}>0$ there exists a solution
$(u,\lambda)\in{\mathcal{S}}\times(0,\infty)^{2}$ to (1.1). See also [7] for
an extension and [2, 24] for more general exponents, or [4, 5] for $\beta<0$.
Moreover, each component of $u$ is positive, radial, radially nonincreasing,
of class ${\mathcal{C}}^{2}$ and

$J(u)=\inf_{{\mathcal{S}}\cap{\mathcal{M}}}J$

provided that $\beta>0$ is sufficiently large – see [30] where also a more
general nonlinearity has been considered.

While there are several multiplicity results concerning normalized solutions
for a single equation (under different assumptions on the right-hand side),
the problem of multiple normalized solutions to (1.1) with $K\geq 2$ has been
little investigated. To our knowledge there are only a few results, even in
the case $p=4$ which is most interesting from the point of view of physics.
Here we can mention that a second positive solution to (1.1) has been found by
Gou and Jeanjean [18] for $\beta>0$ small and $K=2$. Next, Bartsch and Soave
[6] observed that the system (1.1) with $K=2$ is invariant with respect to the
involution $(u_{1},u_{2})\mapsto(u_{2},u_{1})$ provided that
$\rho_{1}=\rho_{2}$, $\beta_{1,1}=\beta_{2,2}$ and $p=4$. Inspired by [13]
they showed that for any $\beta<-\beta_{1,1}=-\beta_{2,2}$ the problem (1.1)
has infinitely many pairs $(u_{1},u_{2})$, $(u_{2},u_{1})$ ($u_{1}\neq u_{2}$)
of positive radial solutions in this particular situation.

The main purpose of this work is to show that if $K=2$ and $m\geq 1$, then for
$\beta_{1,2}=\beta_{2,1}$ large enough there exist at least $m$ distinct
solutions to (1.1). We impose no restrictions on $\rho_{i}$ and
$\beta_{i,i}>0$.

We call a solution to (1.1) a _ground state_ if it minimizes $J$ on
${\mathcal{S}}\cap{\mathcal{M}}$.

Now we state the main result of this paper.

######  Theorem 1.1.

Let $4\leq p<6$ and suppose that $K=2$. For any $m\geq 1$ there is a constant
$\beta_{m}>0$ such that if $\beta:=\beta_{1,2}=\beta_{2,1}>\beta_{m}$, then
(1.1) either has infinitely many solutions or at least $m$ solutions $u^{j}\in
H^{1}_{rad}(\mathbb{R}^{3})^{2}$ such that
$J(u^{1})<J(u^{2})<\cdots<J(u^{m})$. One of these solutions has all components
positive and is a ground state.

Note that the system (1.1) is $\mathbb{Z}_{2}^{K}$-invariant where
$\mathbb{Z}_{2}:=\\{\pm 1\\}$, i.e. if $u=(u_{1},\ldots,u_{K})$ is a solution
to (1.1), then so are $(\pm u_{1},\ldots,\pm u_{K})$ (any combination of
signs). These solutions will be called the $\mathbb{Z}_{2}^{K}$-_orbit_ (or
simply the _orbit_) of $u$. In particular, there are at least $m$ different
$\mathbb{Z}^{2}_{2}$-orbits of solutions in Theorem 1.1.

For general systems with $K\geq 2$ and $\beta_{i,j}>0$ we are only able to
prove the existence of one solution under rather restrictive assumptions.

######  Theorem 1.2.

Let $4\leq p\leq\frac{14}{3}$ and $K\geq 2$. If

(1.5) $\max_{\begin{subarray}{c}{\mathcal{I}}\subset\\{1,\ldots,K\\},\\\
1\leq|{\mathcal{I}}|\leq
K-1\end{subarray}}\Big{(}\max_{i\in{\mathcal{I}}}\\{\beta_{i,i}\rho_{i}^{\frac{6-p}{2}}\\}+\frac{|{\mathcal{I}}|-1}{|{\mathcal{I}}|^{\frac{3p-10}{4}}}\max_{\begin{subarray}{c}i,j\in{\mathcal{I}}\\\
i\neq
j\end{subarray}}\big{\\{}\beta_{i,j}(\rho_{i}\rho_{j})^{\frac{6-p}{4}}\big{\\}}\Big{)}<\frac{\sum_{1\leq
i,j\leq
K}\beta_{i,j}(\rho_{i}\rho_{j})^{p/2}}{\big{(}\sum_{i=1}^{K}\rho_{i}^{2}\big{)}^{\frac{3}{4}(p-2)}}$

and $\beta_{i,j}>0$ for all $i,j$, then (1.1) has a solution $u\in
H^{1}_{rad}(\mathbb{R}^{3})^{K}$ such that
$J(u)=\inf_{{\mathcal{S}}\cap{\mathcal{M}}}J$. So $u$ is a ground state and we
may assume all its components are positive.

Here $|{\mathcal{I}}|$ is the cardinality of the set ${\mathcal{I}}$. Note
that if $K=2$, then $|{\mathcal{I}}|-1=0$ and (1.5) holds for
$\beta:=\beta_{1,2}=\beta_{2,1}$ sufficiently large. For general $K\geq 2$ the
following is true.

######  Proposition 1.3.

A sufficient condition for (1.5) to hold is that $4\leq p\leq\frac{14}{3}$,
$\rho_{i}=\rho$ for all $i$, $\beta_{i,j}=\beta$ for all $i\neq j$ and $\beta$
is sufficiently large.

###### Proof.

If $\rho_{i}=\rho$ and $\beta_{i,j}=\beta$, then (1.5) can be re-written as

$\max_{\begin{subarray}{c}{\mathcal{I}}\subset\\{1,\ldots,K\\},\\\
1\leq|{\mathcal{I}}|\leq
K-1\end{subarray}}\Big{(}\frac{1}{\beta}\max_{i\in{\mathcal{I}}}\\{\beta_{i,i}\\}+\frac{|{\mathcal{I}}|-1}{|{\mathcal{I}}|^{\frac{3p-10}{4}}}\Big{)}<\frac{\frac{1}{\beta}K+K(K-1)}{K^{\frac{3}{4}(p-2)}}.$

So the conclusion will follow if we can show that
$(K-2)K^{\frac{3}{4}(p-2)}(K-1)^{-\frac{3p-10}{4}}<K(K-1)$, i.e.

(1.6) $(K-2)K^{\frac{3p-10}{4}}<(K-1)^{\frac{3p-6}{4}}$

By the inequality of arithmetic and geometric means we get

$(K-2)^{\frac{4}{3p-6}}K^{\frac{3p-10}{3p-6}}<\frac{4}{3p-6}(K-2)+\frac{3p-10}{3p-6}K\leq
K-1$

where the second inequality is a consequence of the assumption
$p\leq\frac{14}{3}$. Clearly, this implies (1.6). ∎

######  Remark 1.4.

_We would like to mention that a variant of the condition ( 1.5) has appeared
in [3] for a system of $K$ equations in the particular case $p=4$. However,
[3, Lemma 5.3] is not true as stated. It will become true after replacing the
condition [3, (1.14)] appearing in [3, Theorem 1.5] with_

$\frac{\Big{(}\sum_{i=1}^{k}a_{i}^{2}\Big{)}^{3}}{\Big{(}\sum_{1\leq i,j\leq
k}\beta_{ij}a_{i}^{2}a_{j}^{2}\Big{)}^{2}}<\min_{\begin{subarray}{c}{\mathcal{I}}\subset\\{1,\ldots,k\\}\\\
1\leq|{\mathcal{I}}|\leq
k-1\end{subarray}}\frac{1}{\Big{(}\max_{i\in{\mathcal{I}}}\\{\beta_{ii}a_{i}\\}+\frac{k-2}{\sqrt{k-1}}\max_{i,j\in{\mathcal{I}},i\neq
j}\\{\beta_{ij}a_{i}^{1/2}a_{j}^{1/2}\\}\Big{)}^{2}}.$

_Under this condition[3, Theorem 1.5] is a special case of Theorem 1.2 in the
present paper. _

As will be seen later, we shall give a variant of condition (1.5) which
implies the existence of at least $m$ solutions $u^{j}\in
H^{1}_{rad}(\mathbb{R}^{3})^{K}$ for any $K\geq 2$ (see Lemma 4.3 below).
However, if $m\geq 2$, we are able to check this condition only for $K=2$ and
it is an open question whether Theorem 1.1 can be generalized for systems with
at least three equations.

Even for the system with $K=2$ one encounters a considerable difficulty in the
variational approach because not all Palais-Smale sequences need to converge
strongly. In $H^{1}_{rad}(\mathbb{R}^{3})^{2}$, if $(u^{n})$ is a bounded
Palais-Smale sequence, then $u^{n}\rightharpoonup u=(u_{1},u_{2})$ after
passing to a subsequence and one can show $J(u^{n})\to J(u)$. However, since
the embedding of $H^{1}_{rad}(\mathbb{R}^{3})^{2}$ into
$L^{2}(\mathbb{R}^{3})^{2}$ is not compact, $u_{1},u_{2}$ may not have the
$L^{2}$-norms as required. In many cases the strong convergence can be shown
at the ground state level but it is unclear what happens at higher levels. The
only paper we know of where this issue could be dealt with is the already
mentioned [6]. Unfortunately, the approach used there cannot be carried over
to more general situations.

In order to get multiple normalized solutions to (1.1) we study the energy
functional $J|_{\mathcal{S}}$ and the crucial step is to show that each
Palais-Smale sequence at $c$ which approaches ${\mathcal{M}}$ either has a
convergent subsequence or converges weakly to a semitrivial solution of the
equations in (1.1), see Theorem 1.5 below. The proof is based on the Morse
index, Cwikel-Lieb-Rozenblum theorem [12, 25, 32] and a Liouville-type result
in an exterior domain (see Lemma A.1).

######  Theorem 1.5.

Let $c>0$ and $4\leq p<6$. If $p=4$ and $\min\\{\beta_{i,j}:\;i\neq j\\}<0$,
assume in addition that $K=2$. Let $(u^{n})\subset{\mathcal{S}}$ be a sequence
such that $J(u^{n})\to c$, $J|_{{\mathcal{S}}}^{\prime}(u^{n})\to 0$ and
$M(u^{n})\to 0$ as $n\to\infty$. Then $(u^{n})$ is bounded, so
$u^{n}\rightharpoonup u=(u_{1},\ldots,u_{K})$ after passing to a subsequence
and for each $i=1,\ldots,K$ either $u_{i}=0$ or $u_{i}^{n}\to u_{i}$ in
$H_{rad}^{1}(\mathbb{R}^{3})$.

We emphasize that although in this paper we mainly focus on the case
$\beta_{i,j}>0$, the result above holds true regardless of the signs of
$\beta_{i,j}$ for $i\neq j$.

In [4] it has been shown that for $K=2$, $p=4$ and $\beta<0$ the system (1.1)
has a solution which is of mountain pass type. A natural question to ask is
whether there also exists a ground state. We show in the next theorem that
this is not the case. Let $u_{\beta_{i,i}}$ be the unique positive solution to

$\left\\{\begin{array}[]{l}-\Delta u+\lambda
u=\beta_{i,i}u^{p-1}\quad\mathrm{in}\ \mathbb{R}^{3},\\\ u\in
H^{1}_{rad}(\mathbb{R}^{3}),\\\
\int_{\mathbb{R}^{3}}|u|^{2}\,dx=\rho_{i}^{2}.\end{array}\right.$

As before, $\lambda$ appears as a Lagrange multiplier.

######  Theorem 1.6.

Let $\frac{10}{3}<p<6$ and suppose that $K\geq 2$ and $\beta_{i,j}<0$ for all
$i\neq j$. Then

$\inf_{{\mathcal{S}}\cap{\mathcal{M}}}J=\min\big{\\{}J(0,\ldots,u_{\beta_{i,i}},\dots,0):\;i=1,\ldots,K\big{\\}}$

and (1.1) has no radial ground state solution, i.e.
$\inf_{{\mathcal{S}}\cap{\mathcal{M}}}J$ is not attained. The same holds true
if $H^{1}_{rad}(\mathbb{R}^{3})$ is replaced by $H^{1}(\mathbb{R}^{3})$ in
(1.1) _(_ then ${\mathcal{S}},{\mathcal{M}}$ are subsets of
$H^{1}(\mathbb{R}^{3})^{K}$ instead of $H^{1}_{rad}(\mathbb{R}^{3})^{K}$_)_.

Note that if $\lambda_{i}$ are prescribed and positive while $\rho_{i}$ are
free, then there are no ground state solutions in $H^{1}(\mathbb{R}^{3})^{K}$
if $p=4$ and all $\beta_{i,j}<0$, $i\neq j$ [28]. On the other hand, in this
case ground states do exist in $H^{1}_{rad}(\mathbb{R}^{3})^{K}$ [34].

The paper is organized as follows. In Section 2 we prove Theorem 1.5. Theorem
1.1 is proved in Section 3 and Theorem 1.2 in Section 4. The latter result
will follow from a more general condition specified in Lemma 4.3. In fact also
Theorem 1.1 follows from this lemma but for technical simplicity we prefer to
prove it separately. Nonexistence of ground states in the repulsive case
$\beta_{i,j}<0$ (Theorem 1.6) is proved in Section 5. A Liouville-type lemma
which is used in Theorem 1.5 is proved in Appendix A in a more general setting
than needed since we think it can be interesting in itself.

Notation. $|\cdot|_{k}$ is the norm in $L^{k}(\mathbb{R}^{3})$, $B(0,\rho)$ is
the ball of radius $\rho$ and center at the origin, $\rightharpoonup$ denotes
the weak convergence, $V^{+}:=\max\\{V,0\\}$, $V^{-}:=\min\\{V,0\\}$.

##  2\. Proof of Theorem 1.5

In order to study the compactness of Palais-Smale sequences of $J$ on
${\mathcal{S}}$ we apply the Cwikel-Lieb-Rozenblum theorem [12, 25, 32]. We
also need a Liouville-type lemma outside a large ball. More precisely, suppose
that the potential $V$ in the Schrödinger operator $-\Delta+V$ on
$L^{2}(\mathbb{R}^{3})$ satisfies $\lim_{|x|\to\infty}V(x)|x|^{2}=0$. If $u\in
H^{1}(\mathbb{R}^{3})$ and there is $\rho>0$ such that $u\geq 0$ and $-\Delta
u+V(x)u\geq 0$ for $|x|\geq\rho$, then $\inf\\{u(x):|x|=M\\}=0$ for any
$M\geq\rho$ (so in particular, $u(x)=0$ for $|x|\geq\rho$ if $u\in
H^{1}_{rad}(\mathbb{R}^{3})$). A more general version of this result is proved
in Appendix A since we believe it can be interesting in itself. It extends the
results in [4, 20].

The main novelty here is that we show that if $\lambda_{i}=0$, then $u_{i}=0$
also if we a priori do not know whether $u_{i}\geq 0$.

Proof of Theorem 1.5. First of all, note that $M(u^{n})\to 0$ implies

(2.1) $\frac{3p-10}{6(p-2)}\int_{\mathbb{R}^{3}}|\nabla
u^{n}|^{2}\,dx=J(u^{n})-\frac{2}{3(p-2)}M(u^{n})\to c>0$

and hence $(u^{n})$ is bounded in $H^{1}_{rad}(\mathbb{R}^{3})^{K}$. Therefore
we may assume $u^{n}\rightharpoonup u=(u_{1},\ldots,u_{K})$ in
$H^{1}_{rad}(\mathbb{R}^{3})^{K}$ and $u^{n}\to u$ in
$L^{p}(\mathbb{R}^{3})^{K}$. Note also that

$J^{\prime}(u^{n})+\lambda^{n}u^{n}=o(1)$

for some Lagrange multipliers $\lambda^{n}\in\mathbb{R}^{K}$ (here
$\lambda^{n}u^{n}$ should be understood as an element of the dual space of
$H^{1}_{rad}(\mathbb{R}^{3})$). It follows that

(2.2) $-\lambda^{n}_{i}\rho_{i}^{2}=\int_{\mathbb{R}^{3}}|\nabla
u^{n}_{i}|^{2}\,dx-\sum_{1\leq j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u^{n}_{i}|^{p/2}|u^{n}_{j}|^{p/2}\,dx+o(1),$

so $(\lambda^{n})$ is bounded and passing to a subsequence,
$\lambda^{n}\to\lambda$ for some $\lambda\in\mathbb{R}^{K}$. Observe that

$J^{\prime}(u)+\lambda u=0$

and as a solution, $u$ satisfies the Nehari-Pohožaev identity (1.3). Thus
$M(u)=0$ and therefore

(2.3) $\int_{\mathbb{R}^{3}}(|\nabla u^{n}|^{2}-|\nabla
u|^{2})\,dx=\frac{3(p-2)}{2p}\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}(|u^{n}_{i}|^{p/2}|u_{j}^{n}|^{p/2}-|u_{i}|^{p/2}|u_{j}|^{p/2})\,dx+o(1)\to
0$

as $n\to\infty$. So $J(u)=c$.

We shall show that if $u_{i}\neq 0$, then $\lambda_{i}\neq 0$. Arguing by
contradiction, suppose $\lambda_{1}=0$ and $u_{1}\neq 0$. Then
$\partial_{u_{1}}J(u)=0$. Let $(\Omega_{k})_{k=1}^{m}$ be the sequence of
nodal domains of $u_{1}$ and let $u_{1,k}:=u_{1}$ on $\Omega_{k}$ and
$u_{1,k}=0$ on $\mathbb{R}^{3}\setminus\Omega_{k}$. Clearly, $u_{1,k}\in
H^{1}_{rad}(\mathbb{R}^{3})$. We claim $m$ must be finite. Indeed, we have

(2.4)
$\displaystyle\partial^{2}_{u_{1}}J(u)[u_{1,k},u_{1,k}]=\int_{\mathbb{R}^{3}}|\nabla
u_{1,k}|^{2}\,dx-(p-1)\beta_{1,1}\int_{\mathbb{R}^{3}}|u_{1,k}|^{p}\,dx$
$\displaystyle\hskip
227.62204pt-\frac{p-2}{2}\sum_{j=2}^{K}\beta_{1,j}\int_{\mathbb{R}^{3}}|u_{1,k}|^{p/2}|u_{j}|^{p/2}\,dx$
$\displaystyle<\int_{\mathbb{R}^{3}}|\nabla
u_{1,k}|^{2}\,dx-\beta_{1,1}\int_{\mathbb{R}^{3}}|u_{1,k}|^{p}\,dx-\sum_{j=2}^{K}\beta_{1,j}\int_{\mathbb{R}^{3}}|u_{1,k}|^{p/2}|u_{j}|^{p/2}\,dx=\partial_{u_{1}}J(u)[u_{1,k}]=0$

for $k=1,\ldots,m$ where the inequality above is a consequence of the identity

$\displaystyle(p-2)\beta_{1,1}\int_{\mathbb{R}^{3}}|u_{1,k}|^{p}\,dx+\frac{p-4}{2}\sum_{j=2}^{K}\beta_{1,j}\int_{\mathbb{R}^{3}}|u_{1,k}|^{p/2}|u_{j}|^{p/2}\,dx$
$\displaystyle\quad=(p-2)\beta_{1,1}\int_{\mathbb{R}^{3}}|u_{1,k}|^{p}\,dx+\frac{p-4}{2}\sum_{j=2}^{K}\beta_{1,j}\int_{\mathbb{R}^{3}}|u_{1,k}|^{p/2}|u_{j}|^{p/2}\,dx+\frac{p-4}{2}\partial_{u_{1}}J(u)[u_{1,k}]$
$\displaystyle\quad=\frac{p-4}{2}\int_{\mathbb{R}^{3}}|\nabla
u_{1,k}|^{2}\,dx+\frac{p}{2}\int_{\mathbb{R}^{3}}\beta_{1,1}|u_{1,k}|^{p}>0.$

It follows from (2.4) that $m\leq m(u_{1})$ where $m(u_{1})$ stands for the
Morse index of

$H^{1}_{rad}(\mathbb{R}^{3})\ni v\mapsto J(v,u_{2},\ldots,u_{K})$

at $v=u_{1}$. The idea of estimating the number of nodal domains by the Morse
index goes back to Benci and Fortunato [8], see also Bahri and Lions [1].
According to the Cwikel-Lieb-Rozenblum theorem [12, 25, 32], the operator

$Av:=-\Delta v+W(x)v$

has discrete spectrum on the negative real line and the number of negative
eigenvalues is bounded by a constant times
$\int_{\mathbb{R}^{3}}|W^{-}|^{3/2}\,dx$ provided this integral is finite and
$W\in L^{1}_{loc}(\mathbb{R}^{N})$. A convenient reference to this result is
[15, Theorem 4.31]. If we put

$W(x):=-(p-1)\beta_{1,1}|u_{1}|^{p-2}-\frac{p-2}{2}\sum_{j=2}^{K}\beta_{1,j}|u_{1}|^{p/2-2}|u_{j}|^{p/2},$

then $W\in L^{3/2}(\mathbb{R}^{3})$ because $2\leq\frac{3}{2}(p-2)<6$.
Therefore $m(u_{1})$ (which equals the number of negative eigenvalues of
$-\Delta+W)$ is finite and hence so is $m$ which proves our claim. So there is
$\rho>0$ such that $u_{1}$ has constant sign on $\mathbb{R}^{3}\setminus
B(0,\rho)$ or $u_{1}=0$ on $\mathbb{R}^{3}\setminus B(0,\rho)$. If the latter
case occurs, then by the unique continuation property [17, Theorem 1.4] we
obtain $u_{1}=0$ which is a contradiction. So we may assume $u_{1}>0$ on
$\mathbb{R}^{3}\setminus B(0,\rho)$. In view of [35], there is a constant
$C>0$ such that $|u_{i}(x)|\leq C/|x|$ for $i=1,\ldots,K$ and $|x|\geq 1$.
Hence setting

$V(x):=-\sum_{j=1}^{K}\beta_{1,j}|u_{1}|^{p/2-2}|u_{j}|^{p/2}$

we get

$\lim_{|x|\to\infty}V(x)|x|^{2}=0$

provided $p>4$. Since $\partial_{u_{1}}J(u)=0$, it follows that

$-\Delta u_{1}+V(x)u_{1}=0,$

and by Lemma A.1 we get $u_{1}=0$, a contradiction again. Let us assume now
that $p=4$. If $\beta_{i,j}\geq 0$ for all $i,j$, then $-\Delta u_{1}\geq 0$
for $|x|\geq\rho$ and once more by Lemma A.1 we get a contradiction. We have
shown that $\lambda_{1}\neq 0$ and in a similar way we show $\lambda_{i}\neq
0$ for $i=2,\ldots,K$.

Now we consider the remaining case, i.e. $p=4$ and $K=2$. Since
$u\in{\mathcal{M}}$ and $\lambda_{1}=0$,

(2.5)
$-\lambda_{2}\int_{\mathbb{R}^{3}}|u_{2}|^{2}\,dx=J^{\prime}(u)[u]-\frac{2p}{3(p-2)}M(u)=\frac{p-6}{3(p-2)}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\,dx<0,$

hence $\lambda_{2}>0$. Observe that

$-\Delta
u_{2}+\lambda_{2}u_{2}=\beta_{2,1}|u_{1}|^{p/2}|u_{2}|^{p/2-2}u_{2}+\beta_{2,2}|u_{2}|^{p-2}u_{2}=:f(x,u_{2}).$

As $0$ is not in the essential spectrum of $-\Delta+\lambda_{2}$ and
$\lim_{u_{2}\to 0}\text{ess}\sup_{x\in\mathbb{R}^{N}}f(x,u_{2})/u_{2}=0$, it
follows from [31, Theorem 2] that $u_{2}$ decays exponentially. Hence
$\lim_{|x|\to\infty}|u_{1}|^{p/2-2}|u_{2}|^{p/2}|x|^{2}=0$ and

$-\Delta
u_{1}-\beta_{1,2}|u_{1}|^{p/2-2}|u_{2}|^{p/2}u_{1}=\beta_{1,1}|u_{1}|^{p-2}u_{1}\geq
0$

for $|x|\geq\rho$. Again, by Lemma A.1 we get a contradiction. So
$\lambda_{1}\neq 0$ also in this case.

It remains to show that $u_{i}^{n}\to u_{i}$ in $L^{2}(\mathbb{R}^{3})$ for
$i=1,\ldots,K$. But as $\lambda_{i}\neq 0$, using (2.3) we obtain

$\displaystyle o(1)$ $\displaystyle=$
$\displaystyle(\partial_{u_{i}}J(u_{n})-\partial_{u_{i}}J(u))[u^{n}_{i}-u_{i}]+\int_{\mathbb{R}^{3}}(\lambda^{n}_{i}u^{n}_{i}-\lambda_{i}u_{i})(u^{n}_{i}-u_{i})\,dx$
$\displaystyle=$
$\displaystyle-\lambda^{i}\int_{\mathbb{R}^{3}}|u^{n}_{i}-u_{i}|^{2}\,dx+o(1)$

and the conclusion follows.  $\Box$

######  Remark 2.1.

_In fact we have shown a little more: For each $i=1,\ldots,K$ either
$\lambda_{i}=0$ and $u_{i}=0$, or $\lambda_{i}\neq 0$ and $u_{i}^{n}\to u_{i}$
in $H^{1}_{rad}(\mathbb{R}^{3})$. In particular, in the latter case
$|u_{i}|_{2}=\rho_{i}$. _

##  3\. Proof of Theorem 1.1

In order to prove Theorem 1.1 we need some auxiliary results. As they will
also be needed in Theorem 1.2, we formulate them for a general $K$ and not
separately for $K=2$.

For $u\in H^{1}(\mathbb{R}^{3})^{K}\setminus\\{0\\}$ and $s>0$, define

$s\star u(x):=s^{3/2}u(sx)\quad\text{and}\quad\varphi(s):=J(s\star u)$

(this definition goes back to [21]). Then

(3.1) $\varphi(s)=J(s\star u)=\frac{s^{2}}{2}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\,dx-\frac{s^{\frac{3(p-2)}{2}}}{p}\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx.$

Recall the following lemma.

######  Lemma 3.1 ([30, 22]).

If $u\in H^{1}_{rad}(\mathbb{R}^{3})^{K}\setminus\\{0\\}$, then there exists a
global maximizer $s_{u}>0$ for $\varphi$ and $\varphi$ is strictly increasing
on $(0,s_{u})$ and strictly decreasing on $(s_{u},+\infty)$. Moreover, $s\star
u\in{\mathcal{M}}$ if and only if $s=s_{u}$, $M(s\star u)>0$ if and only if
$s\in(0,s_{u})$ and $M(s\star u)<0$ if and only if $s>s_{u}$.

The proof is by a simple computation using (3.1); in [30, 22] this result has
been proved in a more general situation. It follows from Lemma 3.1 that
${\mathcal{S}}\cap{\mathcal{M}}\neq\emptyset$. We show that ${\mathcal{S}}$
and ${\mathcal{M}}$ are ${\mathcal{C}}^{2}$-manifolds which intersect
transversally. That ${\mathcal{S}}$ is of class ${\mathcal{C}}^{2}$ is
obvious. It is easily seen from (1.3) that $M$ is a
${\mathcal{C}}^{2}$-functional and $M^{\prime}(u)[u]\neq 0$ for
$u\in{\mathcal{M}}$. Hence ${\mathcal{M}}$ is a ${\mathcal{C}}^{2}$-manifold.
We need to show that $\partial_{u_{i}}M(u)$ and $u_{i}$ are linearly
independent for $i=1,2,\ldots,K$. Assuming the contrary, we have

$\partial_{u_{i}}M(u)+\lambda_{i}u_{i}=0\hbox{ for some
}\lambda_{i}\in\mathbb{R},\ i=1,\ldots,K$

and

$\displaystyle\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,dx$
$\displaystyle=\frac{3(p-2)}{2p}\,\frac{3(p-2)}{2}\Big{(}\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx\Big{)}$
$\displaystyle>\frac{3(p-2)}{2p}\Big{(}\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx\Big{)}$
$\displaystyle=\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\,dx-M(u)=\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,dx$

where the first equality above is the Nehari-Pohožaev identity for $M$. So we
obtain a contradiction which shows that ${\mathcal{S}}$ and ${\mathcal{M}}$
intersect transversally and ${\mathcal{S}}\cap{\mathcal{M}}$ is a
${\mathcal{C}}^{2}$-manifold of codimension $K+1$ in
$H^{1}_{rad}(\mathbb{R}^{3})^{K}$.

For a closed and symmetric set $A$ in a Banach space we define the genus
$\gamma(A)$ as the smallest integer $n$ such that there exists a continuous
and odd map $h:A\to\mathbb{R}^{n}\setminus\\{0\\}$. If no such $h$ exists, we
set $\gamma(A):=\infty$, and we also define $\gamma(\emptyset)=0$. Basic
properties of genus may be found e.g. in [36]. In the proposition below we
compute the genus of a product of spheres. The result may be known but we
could not find any convenient reference to it. Let $S^{k-1}$ denote the unit
sphere in $\mathbb{R}^{k}$.

######  Proposition 3.2.

Let $S:=S^{m_{1}-1}\times\cdots\times S^{m_{k}-1}$. Then
$\gamma(S)=\min\\{m_{1},\ldots,m_{k}\\}$.

###### Proof.

Assume without loss of generality that $m_{1}\leq m_{j}$ for $2\leq j\leq k$.
The map

$h:S\to S^{m_{1}-1},\qquad h(x_{1},\ldots,x_{k})=x_{1}$

is odd, hence $\gamma(S)\leq\gamma(S^{m_{1}-1})=m_{1}$. Since also

$g:S^{m_{1}-1}\to S,\qquad g(x_{1})=(x_{1},\ldots,x_{1})$

is odd, $\gamma(S^{m_{1}-1})\leq\gamma(S)$. ∎

From now on and to the end of this section we assume $K=2$. Let

$\Sigma:=\\{A\subset{\mathcal{S}}\cap{\mathcal{M}}:A=-A\text{ and $A$ is
compact}\\}$

and

(3.2) $c_{m}(\beta):=\inf_{\begin{subarray}{c}A\in\Sigma\\\ \gamma(A)\geq
m\end{subarray}}\max_{u\in A}J(u)$

where $\beta:=\beta_{1,2}=\beta_{2,1}$.

Consider the equation

(3.3) $\left\\{\begin{array}[]{l}-\Delta
u_{i}+\lambda_{i}u_{i}=\beta_{i,i}|u_{i}|^{p-2}u_{i}\quad\mathrm{in}\
\mathbb{R}^{3},\\\ u_{i}\in H^{1}_{rad}(\mathbb{R}^{3}),\\\
\int_{\mathbb{R}^{3}}|u_{i}|^{2}\,dx=\rho_{i}^{2}\end{array}\right.$

for $i=1,2$. As we have mentioned in the introduction, by a result of Kwong
[23] this equation has a unique positive solution and it is known to be a
ground state. Denote the smaller of the ground state energies for $i=1,2$ by
$c_{0}$. Next we show that there exists $\beta_{m}>0$ such that if
$\beta>\beta_{m}$, then $c_{m}(\beta)<c_{0}$. This is an immediate consequence
of the following result.

######  Lemma 3.3.

$\lim_{\beta\to\infty}c_{m}(\beta)=0$.

###### Proof.

Let $A_{1}\subset H^{1}_{rad}(\mathbb{R}^{3})$ be an $(m-1)$-dimensional
sphere such that $|u_{1}|_{2}=\rho_{1}$ for all $u_{1}\in A_{1}$ and let
$\rho:=\rho_{2}/\rho_{1}$. Then $A:=A_{1}\times\rho A_{1}\subset{\mathcal{S}}$
and $\gamma(A)=m$ according to Proposition 3.2. For $u=(u_{1},\rho u_{1})\in
A$ we have

$J(s\star u)\leq\frac{(1+\rho^{2})s^{2}}{2}\int_{\mathbb{R}^{3}}|\nabla
u_{1}|^{2}\,dx-\beta\rho^{p/2}s^{3(p-2)/2}\int_{\mathbb{R}^{3}}|u_{1}|^{p}\,dx$

(cf. (3.1)). Since the first integral on the right-hand side above is bounded
and the second integral is bounded away from 0 on $A_{1}$, a simple
computation shows that $J(s\star u)\leq C/\beta^{\alpha}$ for some $C>0$,
$\alpha=4/(3p-10)$ and all $u\in A$, $s>0$. Note that $\alpha>0$ because
$p>10/3$. Let $\overline{A}:=\\{s_{u}\star u:u\in A\\}$ where $s_{u}$ is as in
Lemma 3.1. Then $\overline{A}\in\Sigma$ and
$\gamma(\overline{A})=\gamma(A)=m$, so
$c_{m}(\beta)\leq\max_{v\in\overline{A}}J(v)\to 0$ as $\beta\to\infty$ and the
conclusion follows. ∎

Proof of Theorem 1.1. Choose $\beta_{m}$ such that $c_{m}(\beta)<c_{0}$ for
$\beta>\beta_{m}$. Introduce a map $\widetilde{J}:{\mathcal{S}}\to\mathbb{R}$
by setting $\widetilde{J}(u):=J(s_{u}\ast u)$. Then

${\mathcal{S}}\ni u\mapsto s_{u}\ast u\in{\mathcal{S}}\cap{\mathcal{M}}$

is a homeomorphism, and in a similar way as in [6, Lemma 3.8], $\widetilde{J}$
is of class ${\mathcal{C}}^{1}$ with

(3.4) $\widetilde{J}^{\prime}(u)[v]=J^{\prime}(s_{u}\ast u)[s_{u}\ast v]$

for every $u\in{\mathcal{S}}$ and $v\in T_{u}{\mathcal{S}}$ where
$T_{u}{\mathcal{S}}$ is the tangent space of ${\mathcal{S}}$ at $u$. Now
observe that

$c_{m}(\beta)=\inf_{\begin{subarray}{c}A\in\widetilde{\Sigma}\\\ \gamma(A)\geq
m\end{subarray}}\max_{u\in A}\widetilde{J}(u)$

where

$\widetilde{\Sigma}:=\\{A\subset{\mathcal{S}}:A=-A\text{ and $A$ is
compact}\\}$

and critical points of $\widetilde{J}$ are in one-to-one correspondence with
critical points of $J|_{\mathcal{S}}$.

Set $c_{j}:=c_{j}(\beta)$. The conclusion will now follow from standard
arguments, see e.g. [36, Theorem II.5.7], if we can show that $\widetilde{J}$
satisfies the Palais-Smale condition at each level $c\in(0,c_{m}]$. Assuming
this, we have $0<c_{1}\leq\cdots\leq c_{m}$ and each $c_{j}$ ($1\leq j\leq m$)
is a critical value for $\widetilde{J}$. So if all $c_{j}$ are different, then
to each $c_{j}$ there corresponds a solution $u^{j}$ (or an orbit of solutions
to be more precise). If $c_{j}=c_{j+1}$ for some $j$, then
$\gamma(K_{c_{j}})\geq 2$ where $K_{c}$ denotes the set of critical points of
$\widetilde{J}$ such that $\widetilde{J}(u)=c$. But then $K_{c_{j}}$ is
infinite and so is the number of (orbits of) solutions.

To complete the proof, let $(u^{n})$ be a Palais-Smale sequence for
$\widetilde{J}$ such that $\widetilde{J}(u^{n})\to c\in(0,c_{m}]$. Denote
$w^{n}:=s_{u^{n}}\star u^{n}$. Then $w_{n}\in{\mathcal{S}}\cap{\mathcal{M}}$
(so $M(w^{n})=0$), $J(w^{n})\to c$ and by (3.4),
$J|_{\mathcal{S}}^{\prime}(w^{n})\to 0$. According to Theorem 1.5 and since
$c>0$, $(w^{n})$ is bounded (cf. (2.1)), so passing to a subsequence,
$w^{n}\rightharpoonup w\neq 0$ and either $w^{n}\to w$ or one of the
components of $w$, say $w_{1}$, equals zero. But then $J(0,w_{2})=c$ and
$w_{2}$ is a solution to (3.3). This is impossible because $c<c_{0}$. Hence
the Palais-Smale condition is satisfied.  $\Box$

##  4\. Proof of Theorem 1.2

Let $C_{p}>0$ denote the optimal constant in the Gagliardo-Nirenberg
inequality

(4.1) $|u|_{p}\leq C_{p}|\nabla
u|_{2}^{\delta_{p}}|u|_{2}^{1-\delta_{p}}\quad\hbox{for }u\in
H^{1}(\mathbb{R}^{3}),$

$\delta_{p}=3\big{(}\frac{1}{2}-\frac{1}{p}\big{)}$ and $\delta_{p}p>2$.

Define

(4.2)
$c_{0}:=\min\big{\\{}\inf_{{\mathcal{M}}_{\mathcal{I}}\cap{\mathcal{S}}_{\mathcal{I}}}J:\;{\mathcal{I}}\subset\\{1,\ldots,K\\},\;1\leq|{\mathcal{I}}|\leq
K-1\big{\\}},$

where $|{\mathcal{I}}|$ is the cardinality of ${\mathcal{I}}$ and

$\displaystyle{\mathcal{S}}_{\mathcal{I}}$ $\displaystyle:=$
$\displaystyle\left\\{u\in
H^{1}_{rad}(\mathbb{R}^{3})^{K}:\int_{\mathbb{R}^{3}}|u_{i}|^{2}\,dx=\rho_{i}^{2}\text{
if }i\in{\mathcal{I}}\hbox{ and }u_{i}=0\text{ if
}i\notin{\mathcal{I}}\right\\},$ $\displaystyle{\mathcal{M}}_{\mathcal{I}}$
$\displaystyle:=$ $\displaystyle\left\\{u\in{\mathcal{M}}:u_{i}=0\text{ if
}i\notin{\mathcal{I}}\right\\}.$

Note that

$\inf_{{\mathcal{M}}_{\mathcal{I}}\cap{\mathcal{S}}_{\mathcal{I}}}J>0$

for each ${\mathcal{I}}\subset\\{1,\ldots,K\\},\;1\leq|{\mathcal{I}}|\leq K-1$
and $c_{0}>0$, see e.g. [30, Lemma 2.3, Lemma 2.5].

Next we want to obtain a lower bound of
$\inf_{{\mathcal{M}}_{\mathcal{I}}\cap{\mathcal{S}}_{\mathcal{I}}}J$. In order
to do this, we need the following inequalities.

######  Lemma 4.1.

For any $\alpha\geq 2$, $q\geq 1$, $m\geq 2$ and
$a_{1},a_{2},\ldots,a_{m}\in\mathbb{R}^{+}$ there holds

(4.3) $\displaystyle\Big{(}\sum_{i=1}^{m}a_{i}^{\alpha}\Big{)}^{1/\alpha}$
$\displaystyle\leq$
$\displaystyle\Big{(}\sum_{i=1}^{m}a_{i}^{2}\Big{)}^{1/2},$ (4.4)
$\displaystyle\Big{(}\frac{1}{m(m-1)}\sum_{1\leq i\neq j\leq
m}a_{i}a_{j}\Big{)}^{1/2}$ $\displaystyle\leq$
$\displaystyle\Big{(}\frac{1}{m}\sum_{i=1}^{m}a_{i}^{q}\Big{)}^{1/q}.$

###### Proof.

Inequality (4.3) is a special case of [19, (2.10.3)] (it holds for any
$\alpha\geq\beta>0$ where $\beta$ replaces 2 on the right-hand side).

In order to prove (4.4), note that

$\sum_{1\leq i\neq j\leq
m}a_{i}a_{j}\leq(m-1)\sum_{i=1}^{m}a_{i}^{2}=(m-1)\Big{(}\sum_{i=1}^{m}a_{i}\Big{)}^{2}-(m-1)\sum_{1\leq
i\neq j\leq m}a_{i}a_{j}.$

Hence

$\sum_{1\leq i\neq j\leq
m}a_{i}a_{j}\leq\frac{m-1}{m}\Big{(}\sum_{i=1}^{m}a_{i}\Big{)}^{2}$

and

$\sum_{1\leq i\neq j\leq m}a_{i}a_{j}\leq
m(m-1)\Big{(}\frac{1}{m}\sum_{i=1}^{m}a_{i}\Big{)}^{2}\leq
m(m-1)\Big{(}\frac{1}{m}\sum_{i=1}^{m}a_{i}^{q}\Big{)}^{2/q}$

where the last inequality is a special case of [19, (2.10.4)]. ∎

######  Lemma 4.2.

Let ${\mathcal{I}}\subset\\{1,\ldots,K\\},\;1\leq|{\mathcal{I}}|\leq K-1$.
There holds

$\inf_{{\mathcal{M}}_{\mathcal{I}}\cap{\mathcal{S}}_{\mathcal{I}}}J\geq\frac{3p-10}{6(p-2)}\Big{[}\frac{3(p-2)}{2p}C_{p}^{p}\Big{(}\max_{i\in{\mathcal{I}}}\\{\beta_{i,i}\rho_{i}^{\frac{6-p}{2}}\\}+\frac{|{\mathcal{I}}|-1}{|{\mathcal{I}}|^{\frac{3p-10}{4}}}\max_{i,j\in{\mathcal{I}},i\neq
j}\\{\beta_{i,j}(\rho_{i}\rho_{j})^{\frac{6-p}{4}}\\}\Big{)}\Big{]}^{-\frac{4}{3p-10}}.$

###### Proof.

By the Hölder inequality,

$\sum_{i,j\in{\mathcal{I}}}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx\leq\sum_{i\in{\mathcal{I}}}\beta_{i,i}|u_{i}|_{p}^{p}+\sum_{i,j\in{\mathcal{I}},i\neq
j}\beta_{i,j}|u_{i}|_{p}^{p/2}|u_{j}|_{p}^{p/2}.$

Taking into account (4.1) and using (4.3) with $a_{i}=|\nabla u_{i}|_{2}$,
$\alpha=\frac{3}{2}(p-2)$ in the first term on the right-hand side above and
(4.4) with $a_{i}=|\nabla u_{i}|_{2}^{\frac{3}{4}(p-2)}$, $m=|{\mathcal{I}}|$,
$q=\frac{8}{3(p-2)}$ in the second term we get the inequality

$\displaystyle\sum_{i,j\in{\mathcal{I}}}\beta_{ij}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx$
$\displaystyle\leq
C_{p}^{p}\Big{(}\max_{i\in{\mathcal{I}}}\\{\beta_{i,i}\rho_{i}^{\frac{6-p}{2}}\\}|\nabla
u|_{2}^{\frac{3}{2}(p-2)}+\frac{|{\mathcal{I}}|-1}{|{\mathcal{I}}|^{\frac{3p-10}{4}}}\max_{i,j\in{\mathcal{I}},i\neq
j}\\{\beta_{i,j}(\rho_{i}\rho_{j})^{\frac{6-p}{4}}\\}|\nabla
u|_{2}^{\frac{3}{2}(p-2)}\Big{)}$

for $u\in{\mathcal{S}}_{\mathcal{I}}$. Note that $p\leq\frac{14}{3}$ is needed
here in order to have $q\geq 1$. Using (1.3) we obtain

$\displaystyle|\nabla u|_{2}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{3(p-2)}{2p}C_{p}^{p}\Big{(}\max_{i\in{\mathcal{I}}}\\{\beta_{i,i}\rho_{i}^{\frac{6-p}{2}}\\}+\frac{|{\mathcal{I}}|-1}{|{\mathcal{I}}|^{\frac{3p-10}{4}}}\max_{i,j\in{\mathcal{I}},i\neq
j}\\{\beta_{i,j}(\rho_{i}\rho_{j})^{\frac{6-p}{4}}\\}\Big{)}|\nabla
u|_{2}^{\frac{3}{2}(p-2)}$

for $u\in{\mathcal{M}}_{\mathcal{I}}\cap{\mathcal{S}}_{\mathcal{I}}$. Since
$M(u)=0$, it follows using (2.1) that

$\displaystyle J(u)$ $\displaystyle=$
$\displaystyle\frac{3p-10}{6(p-2)}|\nabla u|_{2}^{2}$ $\displaystyle\geq$
$\displaystyle\frac{3p-10}{6(p-2)}\Big{[}\frac{3(p-2)}{2p}C_{p}^{p}\Big{(}\max_{i\in{\mathcal{I}}}\\{\beta_{i,i}\rho_{i}^{\frac{6-p}{2}}\\}+\frac{|{\mathcal{I}}|-1}{|{\mathcal{I}}|^{\frac{3p-10}{4}}}\max_{i,j\in{\mathcal{I}},i\neq
j}\\{\beta_{i,j}(\rho_{i}\rho_{j})^{\frac{6-p}{4}}\\}\Big{)}\Big{]}^{-\frac{4}{3p-10}}$

as claimed. ∎

As in the preceding section, let

$\Sigma:=\\{A\subset{\mathcal{M}}\cap{\mathcal{S}}:A=-A\text{ and $A$ is
compact}\\}.$

We shall show that

$c_{m}(\boldsymbol{\beta}):=\inf_{\begin{subarray}{c}A\in\Sigma\\\
\gamma(A)\geq m\end{subarray}}\max_{u\in A}J(u)<c_{0}$

for suitable $\boldsymbol{\beta}:=\\{\beta_{i,j}:1\leq i,j\leq K\\}$.

Let ${\mathcal{S}}_{m}$ denote the collection of all $(m-1)$-dimensional unit
spheres in $L^{2}_{rad}(\mathbb{R}^{3})$. We set

$\Theta_{m}:=\inf_{A\in{\mathcal{S}}_{m}}\sup_{u\in
A}\frac{\big{(}\int_{\mathbb{R}^{3}}|\nabla
u|^{2}\,dx\big{)}^{\frac{3(p-2)}{3p-10}}}{\big{(}\int_{\mathbb{R}^{3}}|u|^{p}\,dx\big{)}^{\frac{4}{3p-10}}}.$

######  Lemma 4.3.

For any $m\geq 1$, $c_{m}(\boldsymbol{\beta})<c_{0}$ provided that

(4.5)
$\displaystyle\max_{\begin{subarray}{c}{\mathcal{I}}\subset\\{1,\ldots,K\\}\\\
1\leq|{\mathcal{I}}|\leq
K-1\end{subarray}}\Big{(}\max_{i\in{\mathcal{I}}}\\{\beta_{i,i}\rho_{i}^{\frac{6-p}{2}}\\}+\frac{|{\mathcal{I}}|-1}{|{\mathcal{I}}|^{\frac{3p-10}{4}}}\max_{i,j\in{\mathcal{I}},i\neq
j}\\{\beta_{i,j}(\rho_{i}\rho_{j})^{\frac{6-p}{4}}\\}\Big{)}\frac{\big{(}\sum_{i=1}^{K}\rho_{i}^{2}\big{)}^{\frac{3}{4}(p-2)}}{\sum_{1\leq
i,j\leq K}\beta_{i,j}(\rho_{i}\rho_{j})^{p/2}}$
$\displaystyle\qquad<\Theta_{m}^{-\frac{3p-10}{4}}C_{p}^{-p}.$

Moreover, $\Theta_{m}^{-\frac{3p-10}{4}}C_{p}^{-p}\leq 1$ for all $m\geq 1$
and $\Theta_{1}^{-\frac{3p-10}{4}}C_{p}^{-p}=1$.

###### Proof.

Let $A_{1}\in{\mathcal{S}}_{m}$. Note that
$\widehat{A}:=\rho_{1}A_{1}\times\rho_{2}A_{1}\times\ldots\times\rho_{K}A_{1}\subset{\mathcal{S}}$
and by Proposition 3.2, $\gamma(\widehat{A})=m$. In view of Lemma 3.1 we may
define

$\widetilde{A}:=\\{s_{u}\ast u:\;u\in\widehat{A}\\}.$

Observe that $\widetilde{A}\in\Sigma$ and $\gamma(\widetilde{A})=m$. For $v\in
A_{1}$ we take $u:=(\rho_{1}v,\ldots,\rho_{K}v)\in\widehat{A}$ and $s_{u}\star
u\in\widetilde{A}$. By (3.1),

$s_{u}^{\frac{3p-10}{2}}=\frac{|\nabla
u|_{2}^{2}}{\frac{3(p-2)}{2p}\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx}$

Hence using (2.1) and $M(u)=0$,

$\displaystyle J(s_{u}\star u)$ $\displaystyle=$
$\displaystyle\frac{3p-10}{6(p-2)}s_{u}^{2}|\nabla
u|_{2}^{2}=\frac{3p-10}{6(p-2)}\,\frac{|\nabla
u|_{2}^{\frac{6(p-2)}{3p-10}}}{\big{(}\frac{3(p-2)}{2p}\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}|^{p/2}|u_{j}|^{p/2}\,dx\big{)}^{\frac{4}{3p-10}}}$
$\displaystyle=$
$\displaystyle\frac{3p-10}{6(p-2)}\,\frac{\big{(}\sum_{i=1}^{K}\rho_{i}^{2}\big{)}^{\frac{3(p-2)}{3p-10}}|\nabla
v|_{2}^{\frac{6(p-2)}{3p-10}}}{\big{(}\frac{3(p-2)}{2p}\big{)}^{\frac{4}{3p-10}}\big{(}\sum_{1\leq
i,j\leq
K}\beta_{i,j}(\rho_{i}\rho_{j})^{p/2}\big{)}^{\frac{4}{3p-10}}\big{(}\int_{\mathbb{R}^{3}}|v|^{p}\,dx\big{)}^{\frac{4}{3p-10}}}$

and

$\displaystyle c_{m}(\boldsymbol{\beta})$ $\displaystyle\leq$ $\displaystyle
J(s_{u}\star u)$ $\displaystyle\leq$ $\displaystyle\sup_{v\in
A_{1}}\frac{\big{(}\int_{\mathbb{R}^{3}}|\nabla
v|^{2}\,dx\big{)}^{\frac{3(p-2)}{3p-10}}}{\big{(}\int_{\mathbb{R}^{3}}|v|^{p}\,dx\big{)}^{\frac{4}{3p-10}}}\,\frac{3p-10}{6(p-2)}\,\frac{\big{(}\sum_{i=1}^{K}\rho_{i}^{2}\big{)}^{\frac{3(p-2)}{3p-10}}}{\big{(}\frac{3(p-2)}{2p}\big{)}^{\frac{4}{3p-10}}\big{(}\sum_{1\leq
i,j\leq K}\beta_{i,j}(\rho_{i}\rho_{j})^{p/2}\big{)}^{\frac{4}{3p-10}}}.$

Since this holds for all $A_{1}\in{\mathcal{S}}_{m}$, it follows that

$c_{m}(\boldsymbol{\beta})\leq\Theta_{m}\,\frac{3p-10}{6(p-2)}\,\frac{\big{(}\sum_{i=1}^{K}\rho_{i}^{2}\big{)}^{\frac{3(p-2)}{3p-10}}}{\big{(}\frac{3(p-2)}{2p}\big{)}^{\frac{4}{3p-10}}\big{(}\sum_{1\leq
i,j\leq K}\beta_{i,j}(\rho_{i}\rho_{j})^{p/2}\big{)}^{\frac{4}{3p-10}}}.$

By (4.5),

$\Big{[}C_{p}^{p}\Big{(}\max_{i\in{\mathcal{I}}}\\{\beta_{i,i}\rho_{i}^{\frac{6-p}{2}}\\}+\frac{|{\mathcal{I}}|-1}{|{\mathcal{I}}|^{\frac{3p-10}{4}}}\max_{i,j\in{\mathcal{I}},i\neq
j}\\{\beta_{i,j}(\rho_{i}\rho_{j})^{\frac{6-p}{4}}\\}\Big{)}\Big{]}^{-\frac{4}{3p-10}}>\Theta_{m}\frac{\big{(}\sum_{i=1}^{K}\rho_{i}^{2}\big{)}^{\frac{3(p-2)}{3p-10}}}{\big{(}\sum_{1\leq
i,j\leq K}\beta_{i,j}(\rho_{i}\rho_{j})^{p/2}\big{)}^{\frac{4}{3p-10}}},$

and in view of Lemma 4.2,

$\inf_{{\mathcal{M}}_{\mathcal{I}}\cap{\mathcal{S}}_{\mathcal{I}}}J>\Theta_{m}\frac{3p-10}{6(p-2)}\frac{\big{(}\sum_{i=1}^{K}\rho_{i}^{2}\big{)}^{\frac{3(p-2)}{3p-10}}}{\big{(}\frac{3(p-2)}{2p}\big{)}^{\frac{4}{3p-10}}\big{(}\sum_{1\leq
i,j\leq K}\beta_{i,j}(\rho_{i}\rho_{j})^{p/2}\big{)}^{\frac{4}{3p-10}}}\geq
c_{m}(\boldsymbol{\beta}).$

for each ${\mathcal{I}}\subset\\{1,\ldots,K\\},\;1\leq|{\mathcal{I}}|\leq
K-1$. Hence, recalling the definition (4.2),
$c_{m}(\boldsymbol{\beta})<c_{0}$.

Observe that $\Theta_{m}^{-\frac{3p-10}{4}}C_{p}^{-p}\leq 1$ follows from the
Gagliardo-Nirenberg inequality (4.1). If $m=1$, then we may take

$A_{1}:=\\{u,-u\\},$

where $u\in H^{1}_{rad}(\mathbb{R}^{3})$ is an optimizer for (4.1) and
$|u|_{2}=1$. That such $u$ exists is a consequence of [38, Corollaries 2.1 and
2.2]. This shows that $\Theta_{1}=C_{p}^{-\frac{4p}{3p-10}}$. ∎

Proof of Theorem 1.2. Since $\Theta_{1}^{-\frac{3p-10}{4}}C_{p}^{-p}=1$, (4.5)
is equivalent to (1.5). Hence for $\boldsymbol{\beta}$ such that (1.5) holds,
we have $c_{1}(\boldsymbol{\beta})<c_{0}$. Now the argument is exactly the
same as in the proof of Theorem 1.1 (in fact a little simpler because only
$c_{1}(\boldsymbol{\beta})$ is of interest here).  $\Box$

######  Remark 4.4.

_As we have already mentioned in the introduction, if $K=2$ and $m\geq 1$,
then (4.5) holds for all $\beta$ large enough ($\beta>\beta_{m}$) because
$|{\mathcal{I}}|-1=0$. Looking into the proof of Lemma 4.2 we also see that
the restriction $p\leq\frac{14}{3}$ is not necessary in this case. Hence we
have reproved Theorem 1.1. However, since our argument in Section 3 is
considerably simpler, we preferred to treat the case $K=2$ separately. If
$K>2$ and (4.5) holds for some $m$, then the system (1.1) either has
infinitely many solutions or at least $m$ solutions at different energy
levels. However, we do not know whether (4.5) can hold for $K>2$ and $m>1$. _

##  5\. Nonexistence of ground states

Denote $J_{i}(v):=J(0,\ldots,v,\ldots,0)$ and

$\displaystyle{\mathcal{S}}_{i}$ $\displaystyle:=$ $\displaystyle\Big{\\{}u\in
H^{1}_{rad}(\mathbb{R}^{3}):\;\int_{\mathbb{R}^{3}}|u|^{2}\,dx=\rho_{i}^{2}\Big{\\}},$
$\displaystyle{\mathcal{M}}_{i}$ $\displaystyle:=$ $\displaystyle\big{\\{}u\in
H^{1}_{rad}(\mathbb{R}^{3}):M_{i}(u):=M_{i}(0,\dots,u,\ldots,0)=0\big{\\}}$

for $i=1,\ldots,K$.

Proof of Theorem 1.6. We prove the nonexistence of radial ground states. The
proof in the non-radial case is exactly the same except that in the
definitions of
${\mathcal{S}},{\mathcal{M}},{\mathcal{S}}_{i},{\mathcal{M}}_{i}$ one needs to
replace $H^{1}_{rad}(\mathbb{R}^{3})$ with $H^{1}(\mathbb{R}^{3})$.

As we have mentioned earlier, by the uniqueness result of Kwong [23] the
equation

$\left\\{\begin{array}[]{l}-\Delta u+\lambda u=\beta_{i,i}|u|^{p-2}u,\\\
\int_{\mathbb{R}^{3}}u^{2}\,dx=\rho_{i}^{2}\end{array}\right.$

has a unique positive ground state solution $u_{i}:=u_{\beta_{i,i}}$ which is
in $H^{1}_{rad}(\mathbb{R}^{3})$. Thus

$J(u_{i})=\inf_{{\mathcal{M}}_{i}\cap{\mathcal{S}}_{i}}J_{i}.$

Denote $c_{i}=J_{i}(u_{i})$ and assume without loss of generality that
$c_{1}\leq c_{i}$ for all $i$. We claim that $c_{1}$ is the ground state level
for (1.1) and that it is not attained. To prove this we first show that no
level $c\leq c_{1}$ is attained by $J$ on ${\mathcal{M}}\cap{\mathcal{S}}$ and
then exhibit a sequence $(v^{n})\subset{\mathcal{M}}\cap{\mathcal{S}}$ such
that $J(v^{n})\to c_{1}$.

Choose $w=(w_{1},\ldots,w_{K})\in{\mathcal{M}}\cap{\mathcal{S}}$. Then (cf.
(1.3))

$0=M(w)\geq M_{1}(w_{1})+\cdots+M_{K}(w_{K})$

because $\beta_{i,j}<0$ for all $i\neq j$. Hence at least one of
$M_{i}(w_{i})$ is $\leq 0$. Using (2.1), $M(w)=0$ and the equality
$\int_{\mathbb{R}^{3}}|\nabla(s\star
w)|^{2}\,dx=s^{2}\int_{\mathbb{R}^{3}}|\nabla w|^{2}\,dx$, we obtain

$J(s_{w}\star w)=\frac{3p-10}{6(p-2)}s_{w}^{2}\int_{\mathbb{R}^{3}}|\nabla
w|^{2}\,dx=\frac{3p-10}{6(p-2)}\int_{\mathbb{R}^{3}}(|\nabla
w_{1}|^{2}+\cdots+|\nabla w_{K}|^{2})\,dx$

($s_{w}=1$ because $w\in{\mathcal{M}}\cap{\mathcal{S}}$). Likewise,

$J_{i}(s_{w_{i}}\star
w_{i})=\frac{3p-10}{6(p-2)}s_{w_{i}}^{2}\int_{\mathbb{R}^{3}}|\nabla
w_{i}|^{2}\,dx,\quad i=1,\ldots,K.$

Note that in order that $s_{w}$ and $s_{w_{i}}$ exist it is necessary to have
$p>\frac{10}{3}$. Choose $i$ with $M_{i}(w_{i})\leq 0$. Since $M_{i}(s\star
w_{i})>0$ for $s<s_{w_{i}}$ and $<0$ for $s>s_{w_{i}}$, we have $s_{w_{i}}\leq
1$ and since $s_{w_{i}}\star w_{i}\in{\mathcal{M}}_{i}\cap{\mathcal{S}}_{i}$,

$\displaystyle J(w)$
$\displaystyle=\frac{3p-10}{6(p-2)}\int_{\mathbb{R}^{3}}|\nabla
w|^{2}\,dx>\frac{3p-10}{6(p-2)}\int_{\mathbb{R}^{3}}|\nabla w_{i}|^{2}\,dx$
$\displaystyle\geq\frac{3p-10}{6(p-2)}s_{w_{i}}^{2}\int_{\mathbb{R}^{3}}|\nabla
w_{i}|^{2}\,dx\geq c_{i}\geq c_{1}.$

So no level $c\leq c_{1}$ is attained by $J$ on
${\mathcal{M}}\cap{\mathcal{S}}$.

Let $u^{n}:=(u_{1},s_{n}\star u_{2},\ldots,s_{n}\star u_{K})$ where $s_{n}\to
0^{+}$. Then $u^{n}\in{\mathcal{S}}$, $|\nabla(s_{n}\star u_{i})|_{2}\to 0$
and $|s_{n}\star u_{i}|_{p}\to 0$ for $2\leq i\leq K$. Set $u_{1}^{n}:=u_{1}$
and $u_{i}^{n}:=s_{n}\star u_{i}$ for $2\leq i\leq K$. Since
$|u_{i}^{n}|_{p}\to 0$ for $2\leq i\leq K$,

$\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}^{n}|^{p/2}|u_{j}^{n}|^{p/2}\,dx>0$

for $n$ large enough. For such $n$, $s_{u^{n}}$ exists, $s_{u^{n}}\star
u^{n}\in{\mathcal{M}}\cap{\mathcal{S}}$ and using (3.1) we obtain

$s_{u^{n}}^{\frac{3p-10}{2}}=\frac{2p}{3(p-2)}\,\frac{\int_{\mathbb{R}^{3}}|\nabla
u^{n}|^{2}\,dx}{\sum_{1\leq i,j\leq
K}\beta_{i,j}\int_{\mathbb{R}^{3}}|u_{i}^{n}|^{p/2}|u_{j}^{n}|^{p/2}\,dx}\to\frac{2p}{3(p-2)}\,\frac{\int_{\mathbb{R}^{3}}|\nabla
u_{1}|^{2}\,dx}{\beta_{1,1}\int_{\mathbb{R}^{3}}|u_{1}|^{p}\,dx}=s_{u_{1}}^{\frac{3p-10}{2}}=1$

because $u_{1}\in{\mathcal{M}}_{1}\cap{\mathcal{S}}_{1}$. It follows that

$J(s_{u^{n}}\star
u^{n})=\frac{3p-10}{6(p-2)}s_{u^{n}}^{2}\int_{\mathbb{R}^{3}}|\nabla
u^{n}|^{2}\,dx\to\frac{3p-10}{6(p-2)}\int_{\mathbb{R}^{3}}|\nabla
u_{1}|^{2}\,dx=c_{1}.$

This completes the proof.  $\Box$

##  Appendix A A Liouville-type lemma

Let $V\in L^{\infty}_{loc}(\mathbb{R}^{N}\setminus B(0,\rho))$ and let
$A:=-\Delta+V$ be a Schrödinger operator on
$L^{2}\big{(}\mathbb{R}^{N}\setminus\overline{B(0,\rho)}\big{)}$. We assume
$V$ satisfies the condition

  * (V)

$V(x)\leq\frac{N(4-N)}{4}|x|^{-2}$ for $|x|\geq\rho$.

######  Lemma A.1.

Let $u\in H^{1}(\mathbb{R}^{N})$ and $N\in\\{3,4\\}$. Suppose that $Au\geq 0$,
$u\geq 0$ and $(V)$ hold for $x\in\mathbb{R}^{N}\setminus B(0,\rho)$. Then
$\inf\\{u(x):|x|=M\\}=0$ for any $M\geq\rho$. If, in addition,
$u\in{\mathcal{C}}^{1}(\mathbb{R}^{N}\setminus\overline{B(0,\rho)})$, then
$u=0$ .

Note that if $N=4$, then $V^{+}=0$ in $\mathbb{R}^{N}\setminus B(0,\rho)$.

###### Proof.

Suppose that $c:=\inf\\{u(x):|x|=M\\}>0$ for some $M\geq\rho$. We may assume
$M=\rho$. Let $v(x):=c\rho^{\alpha}|x|^{-\alpha}$, and note that
$\int_{|x|\geq\rho}|v|^{2}\,dx=\infty$, $\lim_{R\to\infty}\int_{R\leq|x|\leq
R+1}|v|^{2}\,dx=0$ and $\lim_{R\to\infty}\int_{R\leq|x|\leq R+1}|\nabla
v|^{2}\,dx=0$ provided $\frac{N-1}{2}<\alpha\leq\frac{N}{2}$. Choosing
$\alpha=\frac{N}{2}$, we have

$\displaystyle(-\Delta+V^{+})v$ $\displaystyle=$ $\displaystyle
c\rho^{\alpha}\big{(}-\alpha(\alpha+2-N)|x|^{-\alpha-2}+V^{+}(x)|x|^{-\alpha}\big{)}$
$\displaystyle\leq$ $\displaystyle
c\rho^{\alpha}\Big{(}-\alpha(\alpha+2-N)+\frac{N(4-N)}{4}\Big{)}|x|^{-\alpha-2}=0.$

Therefore $(-\Delta+V^{+})(u-v)\geq Au-(-\Delta+V^{+})v\geq 0$ in
$\mathbb{R}^{N}\setminus B(0,\rho)$ and $u-v\geq 0$ for $|x|=\rho$. Let
$R>\rho$ and $\eta_{R}\in{\mathcal{C}}_{0}^{\infty}(\mathbb{R}^{N})$ be a
nonnegative function such that $|\nabla\eta_{R}|\leq C$, $\eta_{R}(x)=1$ for
$x\in B(0,R)$ and $\eta_{R}(x)=0$ for $x\in\mathbb{R}^{N}\setminus B(0,R+1)$.
Testing the inequality $(-\Delta+V^{+})(u-v)\geq 0$ with
$(u-v)^{-}\eta_{R}\leq 0$, we obtain

$(-\Delta+V^{+})(u-v)(u-v)^{-}\eta_{R}\leq 0$

for $x\in\mathbb{R}^{N}\setminus B(0,\rho)$. Hence

(A.1) $\displaystyle\int_{\mathbb{R}^{N}\setminus
B(0,\rho)}\big{(}|\nabla(u-v)^{-}|^{2}+V^{+}(x)|(u-v)^{-}|^{2}\big{)}\eta_{R}\,dx\leq-\int_{\mathbb{R}^{N}\setminus
B(0,\rho)}(\nabla(u-v)^{-}\nabla\eta_{R})(u-v)^{-}\,dx$
$\displaystyle\qquad\leq\frac{1}{2}\int_{\mathbb{R}^{N}\setminus
B(0,\rho)}\big{(}|\nabla(u-v)^{-}|^{2}+|(u-v)^{-}|^{2}\big{)}|\nabla\eta_{R}|\,dx.$

Since $\lim_{R\to\infty}\int_{R\leq|x|\leq
R+1}|v|^{2}\,dx=\lim_{R\to\infty}\int_{R\leq|x|\leq R+1}|\nabla v|^{2}\,dx=0$,
we obtain

(A.2) $\displaystyle\int_{\mathbb{R}^{N}\setminus
B(0,\rho)}\big{(}|\nabla(u-v)^{-}|^{2}+|(u-v)^{-}|^{2}\big{)}|\nabla\eta_{R}|\,dx$
$\displaystyle\qquad\leq C\int_{B(0,R+1)\setminus
B(0,R)}|\nabla(u-v)|^{2}+|u-v|^{2}\,dx\to 0\text{ as }R\to\infty.$

It follows from (A.1), (A.2) and the Lebesgue dominated convergence theorem
that

$\int_{\mathbb{R}^{N}\setminus B(0,\rho)}|\nabla(u-v)^{-}|^{2}\,dx=0,$

so $c_{1}:=(u-v)^{-}$ is constant for $|x|>\rho$. Note that
$u-v=(u-v)^{+}+c_{1}\geq c_{1}$, thus $u\geq v+c_{1}$ and

$\int_{|x|\geq\rho}|u|^{2}\,dx\geq\int_{|x|\geq\rho}|v+c_{1}|^{2}\,dx=\infty.$

We get a contradiction since $u\in L^{2}(\mathbb{R}^{N})$. Therefore
$\inf\\{u(x):|x|=M\\}=0$ for any $M\geq\rho$, hence $u\geq 0$. If
$u\in{\mathcal{C}}^{1}(\mathbb{R}^{N}\setminus\overline{B(0,\rho)})$, then
$u=0$ on $\mathbb{R}^{N}\setminus\overline{B(0,\rho)}$ by the maximum
principle. ∎

Acknowledgements. The authors would like to thank Louis Jeanjean for helpful
comments.

J. Mederski was partly supported by the National Science Centre, Poland (Grant
No. 2020/37/B/ST1/02742).

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