# Study of the 7Be($d$,3He)6Li* reaction at 5 MeV/u

Sk M. Ali1<EMAIL_ADDRESS>Present Address: FRIB, Michigan State
University, East Lansing, MI 48824, USA  D. Gupta1<EMAIL_ADDRESS>K.
Kundalia1 S. Maity1 Swapan K Saha1 Visiting faculty, School of Astrophysics,
Presidency University, 86/1 College Street, Kolkata 700073, India  O.
Tengblad2 J.D. Ovejas2 A. Perea2 I. Martel3 J. Cederkall4 J. Park4 Present
Address: Center for Exotic Nuclear Studies, Institute for Basic Science, 34126
Daejeon, South Korea  A. M. Moro5 1Department of Physical Sciences, Bose
Institute, EN 80, Sector V, Bidhannagar, Kolkata 700091, India  2Instituto de
Estructura de la Materia $-$ CSIC, Serrano 113 bis, ES-28006 Madrid, Spain
3University of Huelva, Avenida Fuerzas Armadas sin numero Campus “El Carmen”,
21007, Huelva, Spain  4Department of Physics, Lund University, Box 118, SE-221
00 Lund, Sweden  5Departamento de Física Atómica, Molecular y Nuclear,
Facultad de Física, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla,
Spain

###### Abstract

The measurement of the 7Be($d$,3He)6Li* transfer cross section at 5 MeV/u is
carried out. The population of the 2.186 MeV excited state of 6Li in this
reaction channel is observed for the first time. The experimental angular
distributions have been analyzed in the finite range DWBA and coupled-channel
frameworks. The effect of the 7Be($d$,3He)6Li reaction on both the 6Li and 7Li
abundances are investigated at the relevant big-bang nucleosynthesis energies.
The excitation function is calculated by TALYS and normalized to the
experimental data. The $S$ factor of the ($d$,3He) channel from the present
work is about 50$\%$ lower than existing data at nearby energies. At big-bang
energies, the $S$ factor is about three orders of magnitude smaller than that
of the ($d,p$) channel. The ($d$,3He) reaction rate is found to have a less
than 0.1$\%$ effect on the 6,7Li abundances.

Big-bang nucleosynthesis, primordial lithium abundance, lithium anomalies,
reaction rates

###### pacs:

24.10.Eq, 25.40.Hs, 26.35.+c, 25.60.Je, 29.30.Ep

##  I Introduction

In nuclear astrophysics, the cosmological lithium problem is widely known and
the solution does not exist at present. The problem involves an anomaly in the
observed 7Li abundance in metal-poor stars as compared to the big-bang
nucleosynthesis (BBN) theory CO04 ; SP82 ; BO10 ; FI11 ; CY16 . The BBN theory
overestimates the 7Li abundance by a factor of $\sim$ 3 and substantial
efforts have been devoted to solve this problem by studying the relevant
nuclear reactions and searching for new resonances inside the Gamow window
AN05 ; HA13 ; BA16 ; DA18 ; LA19 ; RI19 ; AL22 . At present, it is believed
that nuclear physics solutions to this problem are highly improbable KI11 ;
AL22 , and therefore solutions beyond standard model are contemplated FI11 ;
CY16 .

A number of studies also point towards a second cosmological lithium problem
involving 6Li. Here the BBN calculations give the 6Li/7Li ratio $\sim$ 10-5
SE04 ; MU16 , as compared to the observed values of $0.01-0.10$ MO17 ; GO19 ;
WA22 . The 6Li abundance in metal-poor stars exhibit a plateau as a function
of metallicity NO97 ; AS06 ; AS08 ; AN14  similar to 7Li, suggesting a
primordial origin. In contrast to 7Li, the BBN predictions here underestimate
the observed abundance by a factor of $\sim$ 103 BO10 ; AN14 . However, more
recent studies MO17 ; GO19 ; WA22  have contradicted the 6Li detections in
Spite plateau stars. These new analyses employ 3D non-local thermodynamic
equilibrium (NLTE) modelling of the stellar atmosphere. The analyses provide
upper limits to the isotopic ratio of 6Li/7Li $<10\%$ at very low metallicity.
The results are compatible with no detection of 6Li as well, and thus
supporting the negligible amount of 6Li predicted by BBN models. The results
neither solve the cosmological 7Li problem nor worsen it and do not support
exotic scenarios for significant 6Li production in the early Universe KU08 ;
JE09 ; LU21 . However, there are also a number of peculiar stars where the 6Li
excess has been verified TA17 . It is to be noted that the determination of
primordial 6Li abundance inferred from observations, requires high sensitivity
spectrometers and improved stellar modelling. The 3D-NLTE analyses from
different authors TR17  do not yield the same 6Li/7Li ratios and in some
cases, there is also good agreement between 1D and 3D models.

The disagreement in case of the lithium isotopes is of considerable
importance, especially when one finds remarkable agreement between BBN
predictions and primordial abundances of 2H and 4He FI11 . It may be noted
that, simultaneous solutions to both the 6Li and 7Li problems have also been
proposed through the existence of massive, negatively charged particles
binding to light nuclei in the early universe. The bound exotic particles
would reduce the reaction Coulomb barriers, thereby extending BBN to lower
temperatures KU07 . Thus 6,7Li problems may provide insight to important post-
primordial scenarios.

In this context, a 7Be destruction reaction that may impact both the lithium
problems simultaneously, is 7Be($d,^{3}$He)6Li. This reaction not only
produces 6Li but also destroys 7Be thereby decreasing 7Li abundance
indirectly, as primordial 7Li mostly originated from 7Be. Sensitivity studies
by Boyd et al. BO10  concluded that the 7Be($d$,3He)6Li reaction could not
change the BBN abundances of 6,7Li by more than 0.1% assuming a factor of 1000
enhancement in the reaction rate. However, if the reaction were to proceed
through a thermally populated first excited state, 7Be∗ with excitation energy
of 429 keV, then it produced a 30% decrease in the mass-7 abundance when the
rates were artificially increased by thousand times. Broggini et al. BR12
also found that the effect of the 7Be($d$,3He)6Li reaction is too small to
explain the observations of 6Li AS06 . The study of the
7Be($d$,3He)6Li∗($\alpha$,d) reaction, with breakup of 6Li from its excited
states, can also shed light on the 7Be($d$,t)6Be reaction and relevant
resonances in 6Be GU03 ; CH12 . At present, the only experimental data for
7Be($d$,3He)6Li reaction, are by Li et al. LI18 , at center of mass energies
of $E_{cm}$= 4.0 and 6.7 MeV.

The authors in Ref. LI18  initially considered the ($d$,3He) reaction rate to
be the same as the ($d,p$) rate. They also artificially multiplied the rate by
a factor of 100. As a result, their BBN calculations show a 45% decrease in
abundance of 7Li and 47% increase in abundance of 6Li. The experiment by Li et
al. LI18  used low 7Be beam intensities of 5000 pps. The reaction products 3He
and 6Li were selected from $\Delta E-E$ spectrum using gates from Monte Carlo
simulations of the 7Be($d$,3He)6Li reaction. Although background subtraction
was done using a carbon target, no kinematical signatures were shown to verify
that the selected 3He and 6Li are indeed from the above reaction. Thus,
experimental investigation of the 7Be($d$,3He)6Li reaction in further details
is required. This would also help to ascertain the relative importance of the
($d,p$) and ($d,^{3}$He) channels affecting the lithium abundances. The
($d,p$) reaction cross sections at $E_{cm}=7.8$ MeV have been reported in Ref.
AL22 , while the present paper involves the ($d,^{3}$He) reaction.

Figure 1:  $E$ vs $\theta$ plot of events from 7Be($d$,3He)6Li* reaction at 5
MeV/u. The kinematic lines corresponding to the 3He and 6Li bands at S3
covering 8∘-25∘ and W1 detectors covering above 40∘ are also shown (see text
for details). The inset shows the excitation energy spectrum of 6Li.

##  II Experiment

The experiment was carried out at the HIE-ISOLDE radioactive ion beam facility
of CERN. A 5 MeV/u 7Be beam of intensity $\sim$ 5 $\times$ 105 pps and energy
resolution $\sim$ 168 keV was incident on a CD2 target of thickness 15 $\mu$m.
The details of the beam production and the experiment are described in AL22 .
The charged particles emitted from the reaction were detected by a compact
array of silicon strip detectors. The forward angles from 8∘ $-$ 25∘ were
covered by a 1000 $\mu$m thick annular detector (S3) with 24 rings and 32
spokes AL22 ; MI . A set of five detector telescopes assembled in a pentagon
geometry covered the angular range 40∘ \- 80∘. The thickness of the thin
$\Delta E$ (W1) and thick $E$(MSX25) layers were 60 $\mu$m and 1500 $\mu$m
respectively. The W1 detector is a 16x16 silicon strip detector while the
MSX25 is an unsegmented silicon pad detector AL22 ; MI . Two more telescopes
covered the back angles from 127∘ \- 165∘ AL22 , but are not relevant for the
present work. The size of the detector strips and the detector to target
distance imposed an angular resolution of $\sim 1^{\circ}$ in S3 and
$2^{\circ}$ in W1. The energy calibrations of the detectors were carried out
by a mixed $\alpha$-source as well as by elastic peaks from the Rutherford
scattering of ${5}$ MeV/u 7Be and ${5.15}$ MeV/u 12C beams on a 208Pb target.
A 15 $\mu$m thick CH2 target and a 1 mg/cm2 208Pb target were used for
background measurements and normalization respectively. To normalize the
present data, one of the forward rings of S3 was used as a monitor detector,
and the 7Be + 208Pb elastic scattering data from this ring was assumed to be
the Rutherford cross section. In addition to statistical uncertainty, the
present data also include a 10$\%$ error due to target thickness and a 10$\%$
error due to beam intensity KU22 .

##  III Results

At the forward angles covered by S3, the 7Be($d$,3He)6Li reaction can give
rise to events where both 3He and 6Li are incident on S3. These events are
identified by imposing the condition of two hits at S3 and no hit at the W1
pentagon detectors. In addition to CH2 background subtraction, the
experimental data were further filtered by applying a gate on the total energy
of the two hits. The total energy was obtained from Monte Carlo simulations of
the reaction. The simulations, using the package NPTool MA16 , took into
account the beam energy spread, geometry and resolution of the detectors as
well as energy and angular straggling in the target and detectors. The
resultant energy ($E$) vs scattering angle ($\theta$) plot in Fig. 1 shows a
clear band of 3He from 7Be($d$,3He)6Li reaction. The 6Li detected in
coincidence with 3He can also be seen in the figure. The kinematics for 3He
and 6Li are shown in Fig. 1 by the red and black lines respectively. Another
3He band is also seen corresponding to the 2.186 MeV excitation of 6Li. The
population of this state is observed for the first time in the
7Be($d$,3He)6Li∗ reaction channel. The 2.186 MeV state being above the 6Li
breakup threshold of 1.474 MeV, breaks up into $\alpha$ and $d$. This 3He band
is seen with the red-dotted kinematic line in Fig. 1. It becomes more distinct
when we consider the triple-coincidence (3He-$\alpha$-$d$), identifying the
3He in coincidence with two hits at S3.

The 7Be($d$,3He)6Li reaction can also give rise to events where the 3He is
incident on W1 while the corresponding 6Li falls on S3. For such events,
multiplicity condition of one hit at S3 and one hit at W1 were considered. The
3He nuclei corresponding to the ground state (g.s) of 6Li are completely
stopped at $\Delta E$ (W1) and are detected up to an angle $\sim$
$50^{\circ}$, as can be seen from Fig. 1. The 6Li corresponding to these 3He
were also detected and analyzed. However, for the sake of clarity, these 6Li
are not shown in Fig. 1. For the 2.186 MeV state, the 3He are barely detected
at W1, as can be seen by the red-dotted line around $40^{\circ}$ in Fig. 1.
These events are therefore not considered in the analysis. Fig. 1 thus shows
the superimposed contribution of three types of events that are analysed
separately. These are events for (1) two S3 hits with no W1 hit, (2) three S3
hits and (3) one S3 and one W1 hits. The relevant efficiency corrections for
cross section calculations are also separately done as explained later. The
3He may also originate from other channels like breakup of 7Be. These 3He
traverse the $\Delta E$ detectors, stopping at the $E$ detectors. It may be
noted that, these events were vetoed in the present analysis. The inset in
Fig. 1 shows the excitation energy spectrum of 6Li. If we further gate on the
3He bands, the background in the excitation energy spectrum goes away.

Figure 2: Angular distribution for the ground state of 6Li from
7Be($d$,3He)6Li reaction at 5 MeV/u. The CCBA calculations with (without) the
coupling of the 7Be∗(1/2-) excitation at 0.429 keV are shown by the black
solid (dotted) lines. The separate $p$ and $\alpha$-transfer contributions
with the coupling are also shown by the red dotted and blue dash-dotted lines
respectively.

The 7Be($d,^{3}$He)6Li* angular distributions for the g.s ($1^{+}$) and 2.186
MeV($3^{+}$) state of 6Li are obtained from 3He counts in S3 and W1 detectors.
For the g.s, there is no loss of 3He counts as all the 3He and 6Li are
detected in coincidence at S3. However, for 6Li($3^{+}$), the 3He counts are
corrected by taking into account the coincidence efficiency of simultaneous
detection of 3He with $\alpha$ and $d$ at S3 from NPTool simulations. The
absolute normalization for the cross sections were obtained from Rutherford
scattering with a 208Pb target AL22 . The angular distributions for the g.s
and 2.186 MeV state of 6Li are shown in Figs. 2 and 3 respectively. Initially,
finite range distorted-wave Born approximation (DWBA) calculations were
carried out using FRESCO TH88 , assuming one-step $p-$transfer ($d,^{3}$He)
and $\alpha$-transfer ($d,^{6}$Li). For the $p-$transfer, a $p$ \+ 6Li
configuration of 7Be is considered, with the proton occupying a $1p_{3/2}$ or
$1p_{1/2}$ configuration. The proton transferred to $d$ is in 1$s_{1/2}$,
forming 3He. The optical model potential (OMP) parameters for $d$ \+ 7Be
entrance channel are obtained in the same experiment by fitting the elastic
scattering data AL22 . The 3He + 6Li exit channel potential LU68 , $d$ \+ 6Li
core-core potential BI65 , and $p$ \+ 6Li BA80  and $d+p$ SA20  binding
potentials, used in the present work are listed in Table 1. To investigate the
possible influence of the 7Be first excited state on the calculated transfer
cross sections, we have carried out coupled-channel calculations (CCBA) SA83
including the 7Be∗(1/2-) excitation at 0.429 keV. These calculations are
explained later and shown in Figs. 2 and 3.

Figure 3: Angular distribution for the 2.186 MeV excited state of 6Li from
7Be($d$,3He)6Li∗ reaction at 5 MeV/u. The CCBA calculation is shown by the
black solid line. The separate $p$ and $\alpha$-transfer contributions are
shown by the red dotted and blue dash-dotted lines respectively.

The proton spectroscopic factors used in the calculations are 0.438 and 0.225
for $p_{3/2}$ and $p_{1/2}$ components in 7Be = $p$ \+ 6Li($1^{+})$
respectively BR11  as given in Table 2. For the 7Be = $p$ \+ 6Li($3^{+})$
configuration, the proton spectroscopic factor is 0.457 BR11 . The
spectroscopic factor for 3He in $d+p$ configuration is taken as 1.16 KI20 .
For $\alpha-$transfer, the 7Be is assumed to have the $\alpha$ \+ 3He
configuration in a $2P$ state of relative motion. The transferred $\alpha$
forms 6Li with $\alpha+d$ configuration in a $2S$ state. The spectroscopic
factor for 7Be in $\alpha$ \+ 3He configuration is taken as 1.19 RU96 .
According to the three-body calculations of Ref. WA15 , the probability for
the 6Li(gs) = $\alpha+d$ and 6Li($3^{+}$) = $\alpha+d$ configurations are 0.70
and 0.72 respectively. In the CCBA calculations presented here, we adopted
these values for the corresponding spectroscopic factors. This should be a
good approximation as long as antisymmetrization effects, not included in
three-body calculations, are small. The core-core $d$ \+ 3He potential PE76 ,
and binding potentials for $\alpha$ \+ 3He BU88  and $\alpha$ \+ $d$ systems
FE15  are also given in Table 1. It may be noted that the 6Li($3^{+}$) state
is populated by both $p$-transfer and $\alpha$-transfer. However, 6Li($3^{+}$)
is unbound, being above the breakup threshold. To include its contribution in
the $\alpha$-transfer calculations, a continuum bin is defined to be centred
at the resonance energy of 0.71 MeV above the continuum. The width of the bin
is taken as 0.7 MeV, as the experimental energy resolution is $\sim$ 660 keV.
The $p$ and $\alpha$-transfer contributions are added coherently to give the
total cross section.

Figure 4: Astrophysical $S$ factor for the 7Be($d$,3He)6Li reaction. The red
solid circle and square correspond to the present work while the blue open
squares correspond to the data from Li et al. LI18 . The open black circle
corresponds to ($d$,$p$) data AL22 . The red and gray bands are TALYS
calculations for ($d$,$p$) and ($d$,3He) channels respectively, normalized to
the present data at 7.8 MeV (green vertical line). The red dotted line is the
($d$,$p$) estimate by Parker PA72 .

In the CCBA calculations SA83 , a collective model has been used to couple the
3/2- ground state and 1/2- excited state of 7Be. The Coulomb and nuclear
potentials are deformed assuming the rotor model for 7Be. For the Coulomb
part, the intrinsic reduced matrix element is taken as 11.4 $e$fm2, as deduced
from the $B(E2;3/2^{-}\rightarrow 1/2^{-})$ value quoted in Ref. HE19 .
Assuming a pure rotor model, and equal mass and charge deformations, this
leads to the nuclear deformation length $\delta_{2}$ = 2 fm. The proton
spectroscopic factors for the 7Be∗(1/2-) state are taken as 0.854 and 0.038
for 1$p_{3/2}$ and 1$p_{1/2}$ configurations respectively. They are obtained
from the Shell model calculations with Cohen-Kurath interaction CO65 . The
bare OMPs for the $d$ \+ 7Be system AL22  are re-adjusted due to the inclusion
of deformation couplings in the entrance channel. The code FRESCO is used to
adjust the depths and radii of the volume real and surface imaginary parts to
best fit the elastic angular distribution. The modified OMPs are also included
in Table 1. The CCBA calculation for the ground state with (without)
7Be∗(1/2-) coupling is shown by the black solid (dotted) line in Fig. 2. It is
observed that the addition of the 7Be∗(1/2-) state does not alter much the
CCBA result, reducing only the dip of the first minimum. For the
7Be($d$,3He)6Li∗ calculation, the direct coupling between the 7Be excited
state and the 6Li(3+) final state is not possible within the restricted
$p$-shell orbital considered in the Cohen-Kurath modelspace. Thus, the CCBA
calculation for 7Be($d$,3He)6Li∗ reaction does not include the 7Be∗(1/2-)
coupling, and is shown by the black solid line in Fig. 3. The contributions
from $p$ and $\alpha$-transfer are also shown separately by the red dotted and
blue dash-dotted lines respectively. The apparent disagreement between the
calculated and measured angular distributions for the population of the
6Li($3^{+})$ state might be due to the ambiguities of the optical potentials
involved in the CCBA calculations or to the effect of channels not included in
the calculations, such as the coupling to higher states in 7Be or between the
6Li states.

The total cross section $\sigma_{\textrm{exp}}$ leading to the g.s and 2.186
MeV state of 6Li are given in Table 3. They are obtained by integrating the
angular distributions in Figs. 2 and 3 by using LISE++ TA16 . The error in the
total cross section ($\Delta\sigma$) is obtained from the variation in total
cross section considering the upper error limit
$(d\sigma/d\Omega)_{\textrm{high}}$ and lower error limit
$(d\sigma/d\Omega)_{\textrm{low}}$ of the angular distribution data. The error
is given as $\Delta\sigma=(\sigma_{\textrm{high}}-\sigma_{\textrm{low}})/2$,
where
$\sigma_{\textrm{high/low}}=2\pi\int_{0}^{\pi}(d\sigma/d\Omega)_{\textrm{high/low}}(\sin\theta
d\theta)$. The integrated total cross sections from CCBA calculations
$\sigma_{\textrm{CCBA}}$ are also included in the table for comparison. The
reason for the variation in the data and calculations for the 2.186 MeV state,
is probably the absence of data at $\theta_{cm}>$ 70∘. To estimate the
reaction cross section at the Gamow window ($T=0.5-1$ GK, $E_{cm}=0.11-0.56$
MeV), the excitation function was calculated with TALYS 1.95 KO19 . The
calculations are then normalized to the present data at $E_{cm}=7.8$ MeV with
the experimental uncertainty limits, shown by the coloured bands (Fig. 4). The
$Q$ value of this reaction is $-0.11$ MeV as compared to the high $Q$ value of
+16.67 MeV in case of 7Be($d,p$)8Be(2$\alpha$). For the 7Be($d$,3He)6Li∗
reaction to the 2.186 MeV state, the TALYS calculations give zero cross
section below $E_{cm}=0.71$ MeV. The energies in the Gamow window are below
the reaction threshold to populate the 2.186 MeV state of 6Li. So this state
is not considered in the $S$ factor normalization. The gray band in Fig. 4
shows the $S$ factor for the ($d$,3He) channel from TALYS calculations with
experimental uncertainties. The earlier work by Li et al. LI18  did not give
any estimate of systematic uncertainties associated with the extrapolation of
the cross sections to lower energies. The input parameters included in the
TALYS calculations are the optical model potential parameters (OMP) and
different level density models. The effect of OMP is less than 10$\%$ on the
$S$ factor. However, the choice of different models for level densities in
TALYS can significantly impact the value of the $S$ factor inside the Gamow
window. The normalized $S$ factor inside the Gamow window can vary from
$0.001-0.04$ MeV b due to different level density models.

In our earlier work, the ($d,p$) measurements AL22  including the contribution
of the 16.63 MeV state led to a maximum total $S$ factor $\sim$ 167 MeV b.
However, this has less than 1$\%$ effect in reducing the 7Li abundance from
BBN calculations. From Fig. 4, we see that the ($d$,3He) $S$ factor
extrapolated to BBN energies is more than three orders of magnitude lower than
the ($d,p$) $S$ factor. Thus, it can be inferred that the contribution of the
($d$,3He) channel is negligible as compared to the ($d,p$) channel in
affecting the lithium abundance anomaly. This agrees very well with the
calculations in BO10 . We also find that the total $S$ factor from the present
measurements is almost 50$\%$ lower than earlier measurements LI18  at nearby
energies. The reason for this discrepancy may be lower statistics and
contribution of other channels in the work of Li et al LI18 .

Now, to study the impact of the above cross sections on the lithium
production, we calculate the astrophysical reaction rate $N_{A}<\sigma~{}v>$
IL07 , where

$\displaystyle<\sigma~{}v>~{}=\bigg{(}\frac{8}{\pi\mu}\bigg{)}^{1/2}\frac{1}{(kT)^{3/2}}\int_{0}^{\infty}\sigma(E)Ee^{-E/kT}dE.$
(1)

Here, $N_{A}$ is the Avogadro number, $\sigma$ is the total cross section, $v$
is the velocity, $\sigma(E)$ is the excitation function, $\mu$ is the reduced
mass of the 7Be + d system and $k$ is the Boltzmann constant. In Fig. 5, the
reaction rates of ($d,p$) and ($d$,3He) considering the maximum $S$ factor of
167 MeV b AL22  and 0.04 MeV b (present work) at Ecm = 0.22 MeV (Gamow peak at
T = 0.5 GK), are compared with previous experimental and theoretical works.
Although the total $S$ factor from the present work is lower as compared to
that in LI18 , the reaction rate at BBN energies is greater since we have
considered the reaction rate with the maximum possible $S$ factor taking into
account all the level density models in TALYS. The ($d$,3He) reaction rate
from Boyd et al. BO10 , shown in Fig. 5, corresponds to 7Be∗($d$,3He)6Li
reaction proceeding through the thermally populated first excited state of 7Be
as mentioned in the introduction.

Figure 5:  The 7Be($d$,3He)6Li reaction rate from the present work (solid red
line) as a function of temperature. For comparison, the rates from previous
experimental and theoretical works are also shown. The 7Be($d,p$)8Be∗ rate
from AL22  is shown with black dotted lines. The blue band indicates the
temperature range of interest for BBN ($0.5\leq T\leq 1$ GK).

To have an estimate of the abundance ratios of the lithium isotopes, the code
PArthENoPE 3.0 GA22  was used to carry out the standard BBN calculations. It
can take the input of reaction networks of up to 100 reactions. The code was
run with the following conditions : the neutron average lifetime
$\tau_{n}=879.4$ s ZY20 , $N_{\nu}=3$ neutrino species ZY20  and the baryon-
to-photon ratio $\eta=6.10\times 10^{-10}$ AL22 . If we consider the ($d,p$)
reaction rate from CF88 CF88  and do not incorporate ($d$,3He) reaction, the
BBN calculations provide 6Li/H $=$ $1.8\times 10^{-14}$ at BBN energies.
Incorporating the ($d,p$) reaction rate from Ali et al. AL22  and ($d$,3He)
rate from the present work into the code, the 6Li and 7Li abundances change
respectively by $\sim 0.01\%$ and $\sim 0.1\%$. The 6Li/7Li abundance ratio
from the BBN calculations is $\sim 10^{-4}$ which is well below the observed
ratio of $0.01-0.10$ MO17 ; GO19 ; WA22 . A thousand fold increase of both the
($d,p$) and ($d$,3He) reaction rates from our work can give rise to 35$\%$
decrease in the abundance of 7Li and only a 0.9$\%$ increase in abundance of
6Li. So it can be concluded that the effect of the 7Be($d$,3He)6Li reaction to
any of the lithium problems is insignificant.

Table 1: Potential parameters used in the present work. $V$ and $W$ are the real and imaginary depths in MeV, $r$ and $a$ are the radius and diffuseness in fm. $R_{x}=r_{x}A^{1/3}$ fm ($x=V,W,S,SO,C$). Channel | $V$ | $r_{V}$ | $a_{V}$ | $W_{V}$ | $r_{W}$ | $a_{W}$ | $W_{S}$ | $r_{S}$ | $a_{S}$ | $V_{SO}$ | $r_{SO}$ | $a_{SO}$ | $r_{C}$ | Ref.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$d$ \+ 7Be | 80.98 | 1.35 | 0.83 | $-$ | $-$ | $-$ | 36.91 | 2.21 | 0.10 | 2.08 | 0.49 | 0.42 | 1.30 | AL22
| 79.47 | 1.32 | 0.83 | $-$ | $-$ | $-$ | 31.46 | 2.37 | 0.10 | 2.08 | 0.49 | 0.42 | 1.30 | This work
3He + 6Li | 140.0 | 1.20 | 0.75 | 30.0 | 2.30 | 0.50 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1.30 | LU68
$d$ \+ 6Li | 78.0 | 1.04 | 0.95 | 30.0 | 0.85 | 0.85 | $-$ | $-$ | $-$ | 12.50 | 1.04 | 0.95 | 1.04 | BI65
$d$ \+ 3He | 80.2 | 2.36 | 0.50 | 3.7 | 2.36 | 0.50 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1.30 | PE76
$\alpha$ \+ $d$ (g.s) | 77.47 | 1.05 | 0.65 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1.05 | FE15
$\alpha$ \+ $d$ (3+) | 85.39 | 1.05 | 0.65 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1.05 | FE15
$p$ \+ 6Li | 111Varied to match separation energy | 1.25 | 0.65 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | BA80
$d$ \+ $p$ | 11footnotemark: 1 | 1.25 | 0.65 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | SA20
$\alpha$ \+ 3He222$V(r)=-(V+4\alpha V_{SO}\vec{l}\cdot\vec{s})\exp{(-\alpha r^{2})},\alpha=0.15747$ fm-2 | 83.78 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $1.003$ | $-$ | $-$ | 3.095 | BU88
Table 2: Spectroscopic factors ($C^{2}S$) for $A=B+x$ systems used in the CCBA calculations of the present work.  $A$ | $B$ | $x$ | $nlj$ | $C^{2}S$ | Ref.
---|---|---|---|---|---
7Be |  6Li | $p$ | $1p_{3/2}$ | $0.438$ | BR11
|  |  | $1p_{1/2}$ | $0.225$ | BR11
7Be |  6Li∗ | $p$ | $1p_{3/2}$ | $0.457$ | BR11
3He | $d$ | $p$ | $1s_{1/2}$ | $1.16$ | KI20
7Be | $\alpha$ |  3He | $2P_{1}$ | 1.19 | RU96
6Li (g.s) | $\alpha$ | $d$ | $2S_{1}$ | 0.70 | WA15
6Li (3+) | $\alpha$ | $d$ | $2S_{1}$ | 0.72 | WA15
Table 3: The total cross section obtained from CCBA calculations $(\sigma_{\textrm{CCBA}})$ and the experimental data $(\sigma_{\textrm{exp}})$ for 7Be($d$,3He)6Li*. Ex (MeV) | $J^{\pi}$ |  $\sigma_{\textrm{CCBA}}$ (mb) |  $\sigma_{\textrm{exp}}$ (mb)
---|---|---|---
0.0 | $1^{+}$ | $17.83$ | $21.2\pm 4.2$
2.186 | $3^{+}$ | $22.40$ | $18.6\pm 3.7$

The present work is a significant improvement on the only experimental work
LI18  involving the 7Be($d$,3He)6Li reaction. We studied the $(d,p)$ and
($d$,3He) reactions at the same center of mass energy, quantitatively
describing their relative contribution in the context of the lithium
abundance. Although sensitivity studies have shown that the reaction does not
influence the lithium abundance problem BO10 , they did not involve any
experimental constraints. We also measured, for the first time, transfer
reaction cross sections to the 2.186 MeV excited state of 6Li, breaking up
into $\alpha$ and $d$. This may have important consequences in similar studies
involving the resonance states of the unbound 6Be nucleus GU03 ; CH12 . The
angular distributions of the ground state and the excited state can be used to
constrain the Asymptotic Normalization Coefficients (ANC) of
7Be$\rightarrow$3He+4He and 7Be$\rightarrow$6Li+$p$ systems, useful in the
study of astrophysically relevant reactions such as 3He($\alpha$,$\gamma$)7Be
and 6Li($p$,$\gamma$)7Be LI18b ; KI20 ; KI21 .

##  IV Conclusion

We report the measurement of the 7Be($d$,3He)6Li* transfer cross section at 5
MeV/u, observing the 2.186 MeV excited state of 6Li for the first time in this
reaction. The effect of the 7Be($d$,3He)6Li reaction on the lithium abundances
was investigated at the relevant BBN energies. The excitation function was
determined by normalizing the TALYS calculations to the experimental total
cross section of 21.2 $\pm$ 4.2 mb (present work) for the ground state. The
$S$ factor of the ($d$,3He) channel is found to be $\sim$ 50$\%$ lower than
extant data at nearby energies of 4.0 and 6.7 MeV. It is also smaller by three
orders of magnitude than ($d,p$) AL22  at BBN energies. The ($d$,3He) channel
is found to have a negligible influence ($\leq$ 0.1 $\%$) on the lithium
abundances. The present work provides a substantially improved cross section
measurement of the 7Be($d$,3He)6Li* reaction, important for similar studies in
6Be as well as constraining ANC values in astrophysically relevant reactions.

Acknowledgement

The authors thank the ISOLDE engineers in charge, RILIS team and Target Group
at CERN for their support. D. Gupta acknowledges research funding from the
European Union’s Horizon 2020 research and innovation programme under grant
agreement no. 654002 (ENSAR2) and ISRO, Government of India under grant no.
ISRO/RES/2/378/15$-$16\. O. Tengblad would like to acknowledge the support by
the Spanish Funding Agency (AEI / FEDER, EU) under the project
PID2019-104390GB-I00. I. Martel would like to acknowledge the support by the
Ministry of Science, Innovation and Universities of Spain (Grant No.
PGC2018-095640-B-I00). J. Cederkall acknowledges grants from the Swedish
Research Council (VR) under contract numbers VR-2017-00637 and VR-2017-03986
as well as grants from the Royal Physiographical Society. J. Park would like
to acknowledge the support by Institute for Basic Science (IBS-R031-D1).
A.M.M. is supported by the I+D+i project PID2020-114687GB-I00 funded by
MCIN/AEI/10.13039/501100011033, by the grant Group FQM-160 and by project
P20_01247, funded by the Consejería de Economía, Conocimiento, Empresas y
Universidad, Junta de Andalucía (Spain) and by “ERDF A way of making Europe”.

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# Normalized bound state solutions for the fractional Schrödinger equation
with potential111Supported by National Natural Science Foundation of China
(No.11971393) and Natural Science Foundation of Chongqing, China
(cstc2020jcyjjqX0029).

Xin Bao, Ying Lv, Zeng-Qi Ou222Corresponding author. E-mail address:
<EMAIL_ADDRESS>(X. Bao<EMAIL_ADDRESS>(Y. Lv<EMAIL_ADDRESS>(Z.-Q.
Ou).
School of Mathematics and Statistics, Southwest University, Chongqing, 400715,
People’s Republic of China

###### Abstract

In this paper, we study the following fractional Schrödinger equation with
prescribed mass

$\left\\{\begin{aligned} &(-\Delta)^{s}u=\lambda
u+a(x)|u|^{p-2}u,\quad\text{in $\mathbb{R}^{N}$},\\\
&\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\quad u\in
H^{s}(\mathbb{R}^{N}),\end{aligned}\right.$

where $0<s<1$, $N>2s$, $2+\frac{4s}{N}<p<2_{s}^{*}:=\frac{2N}{N-2s}$, $c>0$,
$\lambda\in\mathbb{R}$ and $a(x)\in C^{1}(\mathbb{R}^{N},\mathbb{R}^{+})$ is a
potential function. By using a minimax principle, we prove the existence of
bounded state normalized solution under various conditions on $a(x)$.

_Keywords:_ Fractional Schrödinger equation; Normalized solutions; Bound
state; Potential

##  1 Introduction and main results

In this paper, we consider the following fractional Schrödinger equation

$\displaystyle(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u,$ (1.1)

and possessing prescribed mass

$\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},$ (1.2)

where $0<s<1$, $N>2s$, $2+\frac{4s}{N}<p<2_{s}^{*}:=\frac{2N}{N-2s}$, $c>0$,
and the function $a(x)$ satisfies the following assumptions:

$(A_{1})$ $a\in C^{1}(\mathbb{R}^{N},\mathbb{R}^{+})$, and
$a_{*}:=\inf\limits_{x\in\mathbb{R}^{N}}a(x)$ with

$a_{*}>\max\Big{\\{}\frac{N(N-2s)}{N^{2}-2s(N-2s)},\frac{2N}{Np-4s}\Big{\\}};$

$(A_{2})$ $\lim\limits_{|x|\rightarrow\infty}a(x)=1$;

$(A_{3})$ $x\cdot\nabla a(x)\geqslant 0$ for all $x\in\mathbb{R}^{N}$, with
the strict inequality holds on a subset of positive Lebesgue

measure of $\mathbb{R}^{N}$;

$(A_{4})$ $Na(x)+x\cdot\nabla a(x)\leq N$ for all $x\in\mathbb{R}^{N}$;

$(A_{5})$ The map $W:x\mapsto x\cdot\nabla a(x)$ is in
$L^{\infty}(\mathbb{R}^{N})$, and

$\|W\|_{\infty}\leq\frac{(2N+p(2s-N))((Np-4s)a_{*}-2N)}{8(p-2)s};$ (1.3)

$(A_{6})$ $(1-a(x))\in L^{t_{1}}(\mathbb{R}^{N})$, and

$\|1-a(x)\|_{t_{1}}<\frac{(2^{1-\frac{4s}{N(p-2)}}-1)Np(p-2)}{(N(p-2)-4s)\|w_{c}\|_{t_{2}p}^{p}}m_{c},$
(1.4)

where $t_{1}=\frac{2N}{2N-Np+4s}$, $t_{2}=\frac{2N}{Np-4s},$ $w_{c}$ and
$m_{c}$ are defined in (2.2) and (2.5) respectively.

The fractional Laplacian operator $(-\Delta)^{s}$ is defined by

$(-\Delta)^{s}u(x)=C_{N,s}\
P.V.\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy,\quad\text{in
$\mathbb{R}^{N}$}$

for $u\in C_{0}^{\infty}(\mathbb{R}^{N})$, where $C_{N,s}$ is a suitable
positive normalizing constant and $P.V.$ denotes the Cauchy principle value,
see [16].

When we are looking for solutions of problem (1.1), a possible choice is to
consider that $\lambda\in\mathbb{R}$ is fixed, and to look for critical points
of the functional $F_{\lambda}:H^{s}(\mathbb{R}^{N})\rightarrow\mathbb{R}$
(see e.g. [13, 15, 34, 40])

$F_{\lambda}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}a(x)|u|^{p}dx$
(1.5)

with

$\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy,$

where $H^{s}(\mathbb{R}^{N})$ is a Hilbert space with the inner product and
norm respectively

$\langle
u,v\rangle=\int_{\mathbb{R}^{N}}(-\Delta)^{s/2}u(-\Delta)^{s/2}vdx+\int_{\mathbb{R}^{N}}uvdx,$
$\|u\|=\Big{(}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx+\int_{\mathbb{R}^{N}}|u|^{2}dx\Big{)}^{1/2}.$

Another interesting way is to search for solutions with prescribed mass, that
is, (1.2) holds, and $\lambda\in\mathbb{R}$ appears as a lagrange multiplier.
This type of solution is called normalized solution, and can be obtained by
looking for critical points of the functional

$F(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}a(x)|u|^{p}dx$
(1.6)

on the constraint

$S_{c}:=\\{u\in H^{s}(\mathbb{R}^{N}):\|u\|_{2}=c\\}.$

From the physical point of view, the normalized solution is particularly
meaningful, since in addition to there is a conservation of mass, the mass has
often an important physical meaning.

Considering the following Schrödinger equation with a normalization constraint

$\left\\{\begin{aligned} &(-\Delta)^{s}u+V(x)u=\lambda u+f(u),\quad\text{in
$\mathbb{R}^{N}$},\\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\quad u\in
H^{s}(\mathbb{R}^{N}).\end{aligned}\right.$ (1.7)

If $s=1$, in recent years, much attention has been paid to study problem
(1.7), see [3, 5, 7, 24, 26, 32, 33, 36] for $V(x)=1$ and [2, 22, 29, 11, 4,
17, 35] for $V(x)\neq$ constant. In particular, Ikoma and Miyamoto [22]
studied normalized ground state solutions of problem (1.7) with the vanishing
potential $V$ in the $L^{2}$-subcritical case. In the $L^{2}$-supercritical
case, by using a new minimax argument, Bartsch et al. [4] proved the existence
of normalized solutions of problem (1.7) with the vanishing potential.

For $s\in(0,1)$ and $f(u)=|u|^{p-2}u$, the associated energy functional of
(1.7) is given by

$F_{V}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)|u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx.$

From the variational point of view, $F_{V}$ is bounded from below on $S_{c}$
for $p\in(2,2+\frac{4s}{N})$ ($L^{2}$-subcritical), and unbounded from below
on $S_{c}$ for $p\in(2+\frac{4s}{N},2^{*}_{s})$ ($L^{2}$-supercritical). Here
$2+\frac{4s}{N}$ is called the $L^{2}$-critical exponent, which comes from the
Gagliardo-Nirenberg inequality [30]. There are many results about problem
(1.7), see [28, 39, 41] for normalized solutions without potential, [31] for
normalized solutions with the vanishing potential, [19] for normalized
solutions with the trapping potential, [27] for normalized solutions with a
ring-shaped potential, [42] for normalized solutions with a weak form of the
steep well potential.

Note that, all the results mentioned above are concerned with the problem
(1.7) with autonomous nonlinearities. For the study of normalized solutions to
the nonautonomous equations, i.e., the following equation

$\left\\{\begin{aligned} &(-\Delta)^{s}u=\lambda u+a(x)f(u),\quad\text{in
$\mathbb{R}^{N}$},\\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\quad u\in
H^{s}(\mathbb{R}^{N}).\end{aligned}\right.$ (1.8)

Chen and Tang [11] first considered (1.8) with $s=1$, and established the
existence of normalized ground state solutions under suitable assumptions on
$a(x)$ and $f$. Compared with the autonomous problems with a prescribed
$L^{2}$-norm constraint, the main differences and difficulties of [11] are
finding a new method to construct a Pohožaev-Palais-Smale sequence, and
recover the compactness for this sequence. By using a technical condition
about $a(x)$ which was introduced in [8], Alves [1] considered the
multiplicity of normalized solutions to problem (1.8) with $s=1$ in the
$L^{2}$-subcritical case. Moreover, the above results are later generalized to
the Kirchhoff type in $\mathbb{R}^{3}$ by Chen et al. [10] and the fractional
case by Yang et al. [38]. For more related results, see [12, 18, 37] for more
details.

Nowadays, the study of problem (1.8) has attracted a lot interest. However, as
far as we know, there are only a few papers dealing with problem (1.8) (even
for $s=1$) besides the one already mentioned above [1, 10, 11, 38]. With
regard to this point, inspired by [11] and [25], in this work, we study
normalized bound state solutions of problem (1.1)-(1.2).

######  Theorem 1.1.

Assume that $(A_{1})$-$(A_{5})$ hold, and the supremum of $|1-a(x)|$ is
sufficiently small, then problem (1.1)-(1.2) has a solution $(u,\lambda)\in
H^{s}(\mathbb{R}^{N})\times\mathbb{R}^{-}$.

######  Theorem 1.2.

Assume that $(A_{1})$-$(A_{6})$ hold, then problem (1.1)-(1.2) has a solution
$(u,\lambda)\in H^{s}(\mathbb{R}^{N})\times\mathbb{R}^{-}$.

######  Remark 1.1.

We point out that the supremum of $|1-a(x)|$ in Theorem 1.1 can be made
explicit, see (3.10).

######  Remark 1.2.

Different from [11], where the authors considered $a(x)\geq a_{\infty}$ and
the existence of ground state solutions, while, in this paper, we focus on the
case $a(x)\leq a_{\infty}$, and obtain a bound state solution.

######  Remark 1.3.

Let us give a brief illustration of our proof.

Step 1: Under the conditions of $a(x)$, we prove that $F$ has a linking
geometry;

Step 2: Borrowing the idea of [23], and using a new minimax principle
introduced in [21], we construct a bounded Pohožaev-Palais-Smale sequence;

Step 3: By using the splitting Lemma, we prove the convengence.

The paper is organized as follows. Section 2 contains some preliminaries. In
Section 3, we use a minimax principle to construct a Pohožaev-Palais-Smale
sequence. Section 4 is devoted to the proofs of Theorems 1.1 and 1.2.
Throughout the paper, we will use the notation
$\|\cdot\|_{q}:=\|\cdot\|_{L^{q}(\mathbb{R}^{N})}$, $q\in(1,\infty)$,
$B_{R}(y)=\\{x\in\mathbb{R}^{N}:|x-y|\leq R\\}$ is the closed ball of radius
$R$ around $y$, $C,C_{i},i\in\mathbb{N}^{+}$ denote positive constants
possibly different from line to line.

##  2 Preliminaries

By [20], for $p\in(2+\frac{4s}{N},2_{s}^{\ast})$, the following fractional
Schrödinger equation:

$(-\Delta)^{s}u=-u+u^{p-1},\quad x\in\mathbb{R}^{N},\quad u>0,$ (2.1)

has an unique solution $w\in H^{s}(\mathbb{R}^{N})$, which is radial and
radially decreasing. Then for $c>0$, one obtains a solution $(w_{c}$,
$\lambda_{c})\in H_{\text{rad }}^{s}(\mathbb{R}^{N})\times\mathbb{R}^{-}$ of
the following equation

$\left\\{\begin{array}[]{l}(-\Delta)^{s}w_{c}=\lambda_{c}w_{c}+|w_{c}|^{p-2}w_{c}\\\
w_{c}>0,\quad\|w_{c}\|_{2}=c,\end{array}\right.$ (2.2)

by scaling:

$w_{c}(x):=(-\lambda_{c})^{\frac{1}{p-2}}w((-\lambda_{c})^{\frac{1}{2s}}x),$

where $\lambda_{c}$ is determined by

$-\lambda_{c}=\left(\frac{c}{c_{0}}\right)^{-\theta-2}>0,\quad
c_{0}:=\|w\|_{2},\quad\theta:=\frac{4N-2p(N-2s)}{N(p-2)-4s}>0,$ (2.3)

and $H_{\mathrm{rad}}^{s}(\mathbb{R}^{N})$ is the subset of the radially
symmetric functions in $H^{s}(\mathbb{R}^{N})$. The solution $w_{c}$ of (2.2)
is a critical point, in fact a mountain pass critical point of

$F_{\infty}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx$
(2.4)

constrained to $S_{c}$. For any $c>0$, setting

$m_{c}:=F_{\infty}(w_{c})\ \ \mbox{and}\ \ m_{c_{0}}=F_{\infty}(w),$ (2.5)

one has

$\displaystyle m_{c}$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}w_{c}|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|w_{c}|^{p}dx$
(2.6)
$\displaystyle=(-\lambda_{c})^{\frac{p}{p-2}-\frac{N}{2s}}\Big{(}\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}w|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|w|^{p}dx\Big{)}$
$\displaystyle=(-\lambda_{c})^{\frac{p}{p-2}-\frac{N}{2s}}m_{c_{0}}$
$\displaystyle=\left(\frac{c}{c_{0}}\right)^{-\theta}m_{c_{0}}.$

Now we recall the notion of barycentre of a function $u\in
H^{s}(\mathbb{R}^{N})\backslash\\{0\\}$ which has been introduced in [6, 9].
Setting

$\nu(u)(x)=\frac{1}{|B_{1}(0)|}\int_{B_{1}(x)}|u(y)|dy.$

We can observe that $\nu(u)$ is bounded and continuous, and then, the function

$\widehat{u}(x)=\Big{(}\nu(u)(x)-\frac{1}{2}\max\nu(u)\Big{)}^{+}$

is well defined. Moreover, $\widehat{u}$ is continuous and has a compact
support. Therefore, we can define
$\beta:H^{s}(\mathbb{R}^{N})\backslash\\{0\\}\rightarrow\mathbb{R}^{N}$ as

$\beta(u)=\frac{1}{\|\widehat{u}\|_{1}}\int_{\mathbb{R}^{N}}\widehat{u}(x)xdx.$

The map $\beta$ is well defined, since $\widehat{u}$ has a compact support,
and one can verify that it satisfies the following properties:

$(i)$ $\beta$ is continuous in $H^{s}(\mathbb{R}^{N})\backslash\\{0\\}$;

$(ii)$ if $u$ is a radial function, then $\beta(u)=0$;

$(iii)$ $\beta(tu)=\beta(u)$ for any $t\neq 0$ and $u\in
H^{s}(\mathbb{R}^{N})\backslash\\{0\\}$;

$(iv)$ letting $u_{z}(x)=u(x-z)$ for any $z\in\mathbb{R}^{N}$ and $u\in
H^{s}(\mathbb{R}^{N})\backslash\\{0\\}$, there holds
$\beta\left(u_{z}\right)=\beta(u)+z$.

######  Lemma 2.1.

[21, Lemma 4.5] Let $M$ be a Hilbert manifold and $\mathcal{I}\in
C^{1}(M,\mathbb{R})$ be a given functional. Let $K\subset M$ be compact and
consider a subset

$\mathcal{G}\subset\\{G\subset M:G\text{ is compact, }K\subset G\\}$

which is homotopy-stable, that is, it is invariant with respect to
deformations leaving $K$ fixed. Assume that

$\max_{u\in K}\mathcal{I}(u)<e:=\inf_{G\in\mathcal{G}}\max_{u\in
G}\mathcal{I}(u)\in\mathbb{R}.$

If $\tau_{n}\in\mathbb{R}$ such that $\tau_{n}\rightarrow 0$ and
$G_{n}\in\mathcal{G}$ is a sequence such that

$0\leq\max_{u\in G_{n}}\mathcal{I}(u)-e\leq\tau_{n}.$

Then there exists a sequence $\\{u_{n}\\}\subset M$ such that

$(1)$ $|\mathcal{I}(u_{n})-e|\leq\tau_{n}$,

$(2)$ $\|\nabla_{M}\mathcal{I}(u_{n})\|\leq C\sqrt{\tau_{n}}$,

$(3)$ $\operatorname{dist}(u_{n},G_{n})\leq C\sqrt{\tau_{n}}$,
for some constant $C>0$.

To study the behavior of a Palais-Smale sequence, we introduce a splitting
Lemma. For $\lambda<0$, let

$F_{\lambda}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}a(x)|u|^{p}dx$
(2.7)

and

$F_{\infty,\lambda}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx.$
(2.8)

######  Lemma 2.2.

(Splitting Lemma) Let $\\{v_{n}\\}\subset H^{s}(\mathbb{R}^{N})$ be a Palais-
Smale sequence for $F_{\lambda}$ such that $v_{n}\rightharpoonup v$ in
$H^{s}(\mathbb{R}^{N})$. Then there exist an integer $k\geq 0$, $k$ non-
trivial solutions $u^{1}$, $u^{2}$, $\cdots$, $u^{k}\in H^{s}(\mathbb{R}^{N})$
to the limit equation

$(-\Delta)^{s}u=\lambda u+|u|^{p-2}u\quad\text{in $\mathbb{R}^{N}$}$

and $k$ sequences $\\{y_{n}^{j}\\}\subset\mathbb{R}^{N}$, $1\leq j\leq k$,
such that $|y_{n}^{j}|\rightarrow\infty$ as $n\rightarrow\infty$, and

$v_{n}=v+\sum_{j=1}^{k}u^{j}(\cdot-y_{n}^{j})+o(1)\quad\text{in
$H^{s}(\mathbb{R}^{N})$}.$ (2.9)

Moreover, we have

$\|v_{n}\|_{2}^{2}=\|v\|_{2}^{2}+\sum_{j=1}^{k}\|u^{j}\|_{2}^{2}+o(1)$ (2.10)

and

$F_{\lambda}(v_{n})\rightarrow
F_{\lambda}(v)+\sum_{j=1}^{k}F_{\infty,\lambda}(u^{j})\ \ \text{as
}n\rightarrow\infty.$ (2.11)

###### Proof.

The proof of Lemma 2.2 can be found in [14, Lemma 3.1]. The only difference is
that [14] deals with exterior domains, not with in the whole space, however
the proof is exactly the same with $\lambda<0$ and thus we omit it here. ∎

##  3 The minimax approach

For $h\in\mathbb{R}$ and $u\in H^{s}(\mathbb{R}^{N})$, we introduce the
scaling

$h\star u(x):=e^{\frac{Nh}{2}}u(e^{h}x),$

which preserves the $L^{2}$-norm: $\|h\star u\|_{2}=\|u\|_{2}$ for all
$h\in\mathbb{R}$. For $R>0$ and $h_{1}<0<h_{2}$, which will be determined
later, we set

$Q:=B_{R}(0)\times[h_{1},h_{2}]\subset\mathbb{R}^{N}\times\mathbb{R}.$

For $c>0$ we define

$\Gamma_{c}:=\left\\{\gamma\in C(Q,S_{c}):\gamma(y,h)=h\star
w_{c}(\cdot-y)\text{ for all }(y,h)\in\partial Q\right\\}.$

We want to find a solution of problem (1.1)-(1.2) whose energy is given by

$m_{a,c}:=\inf_{\gamma\in\Gamma_{c}}\max_{(y,h)\in Q}F(\gamma(y,h)).$ (3.1)

In order to develop a min-max argument, we need to prove that

$\sup_{\gamma\in\Gamma_{c}}\max_{(y,h)\in\partial Q}F(\gamma(y,h))<m_{a,c}$

at least for some suitable choice of $Q$. For this purpose, we will prove
Lemma 3.3, which gives a lower bound of $m_{a,c}$, and Lemma 3.4, which gives
an upper bound of $F\circ\gamma$ on the boundary $\partial Q$, for any given
$\gamma\in\Gamma_{c}$. The values of $R>0$ and $h_{1}<0<h_{2}$ will be
determined in Lemma 3.4.

Next, for $h_{1}<0<h_{2}$ determined in Lemma 3.4, we define

$\displaystyle\mathcal{D}:=\\{D\subset S_{c}:D\text{ is compact and connected,
}h_{1}\star w_{c},\ \ h_{2}\star w_{c}\in D\\},$
$\displaystyle\mathcal{D}_{0}:=\\{D\in\mathcal{D}:\beta(u)=0\text{ for all
}u\in D\\},$ $\displaystyle\mathcal{D}_{r}:=\mathcal{D}\cap
H_{\mathrm{rad}}^{s}(\mathbb{R}^{N}),$

and

$\displaystyle l_{c}$ $\displaystyle:=\inf_{D\in\mathcal{D}}\max_{u\in
D}F_{\infty}(u),$ $\displaystyle l_{c}^{0}$
$\displaystyle:=\inf_{D\in\mathcal{D}_{0}}\max_{u\in D}F_{\infty}(u),$
$\displaystyle l_{c}^{r}$ $\displaystyle:=\inf_{D\in\mathcal{D}_{r}}\max_{u\in
D}F_{\infty}(u).$

By a similar argument as [28], we obtain

$m_{c}=\inf_{\sigma\in\Sigma_{c}}\max_{t\in(0,1)}F_{\infty}(\sigma(t)),$

where

$\Sigma_{c}=\\{\sigma\in C([0,1],S_{c}):\sigma(0)=h_{1}\star w_{c},\ \
\sigma(1)=h_{2}\star w_{c}\\}.$

######  Lemma 3.1.

$l_{c}^{r}=l_{c}^{0}=l_{c}=m_{c}$.

###### Proof.

Since $\mathcal{D}_{r}\subset\mathcal{D}_{0}\subset\mathcal{D}$, we have
$l_{c}^{r}\geq l_{c}^{0}\geq l_{c}$. In order to prove that $l_{c}\geq m_{c}$
and $m_{c}\geq l_{c}^{r}$, arguing by contradiction we assume that
$l_{c}<m_{c}$. Then $\max\limits_{u\in D}F_{\infty}(u)<m_{c}$ for some
$D\in\mathcal{D}$, hence $\max\limits_{u\in U_{\delta}(D)}F_{\infty}(u)<m_{c}$
for some $\delta>0$, where $U_{\delta}(D)$ is a $\delta$-neighborhood of $D$.
Observe that $U_{\delta}(D)$ is open and connected, so it is path-connected.
Therefore there exists a path $\sigma\in\Sigma_{c}$ such that
$\max\limits_{t\in(0,1)}F_{\infty}(\sigma(t))<m_{c}$, which is a
contradiction.

On the other hand, let $\tilde{D}:=\\{h\star
w_{c}:h\in[h_{1},h_{2}]\\}\in\mathcal{D}_{r}$, then

$\max_{u\in\tilde{D}}F_{\infty}(u)=\max_{h\in[h_{1},h_{2}]}F_{\infty}(h\star
w_{c})=F_{\infty}(w_{c})=m_{c}.$

Hence, $m_{c}\geq l_{c}^{r}$. ∎

######  Lemma 3.2.

$L_{c}:=\inf\limits_{D\in\mathcal{D}_{\mathrm{0}}}\max\limits_{u\in
D}F(u)>m_{c}$.

###### Proof.

By $(A_{3})$, $(A_{4})$ and Lemma 3.1, we know

$\max_{u\in D}F(u)\geq\max_{u\in D}F_{\infty}(u)\geq
l_{c}^{0}=m_{c}\quad\text{for all $D\in\mathcal{D}_{0}$}.$

Now we argue by contradiction and assume that there exists a sequence
$\\{D_{n}\\}\subset\mathcal{D}_{0}$ such that

$\max_{u\in D_{n}}F(u)\rightarrow m_{c}.$

Then we have

$\max_{u\in D_{n}}F_{\infty}(u)\rightarrow m_{c}.$ (3.2)

Borrowing the idea of [23, Lemma 2.4], we consider the functional

$\tilde{F}_{\infty}:H^{s}(\mathbb{R}^{N})\times\mathbb{R}\rightarrow\mathbb{R},\quad\tilde{F}_{\infty}(u,h):=F_{\infty}(h\star
u)$

constrained to $M:=S_{c}\times\mathbb{R}$. We apply Lemma 2.1 with

$K:=\\{(h_{1}\star w_{c},0),(h_{2}\star w_{c},0)\\}$

and

$\mathcal{G}:=\\{G\subset M:G\text{ is compact, connected, }K\subset G\\}.$

Observe that

$\tilde{l}_{c}:=\inf_{G\in\mathcal{G}}\max_{(u,h)\in
G}\tilde{F}_{\infty}(u,h)=l_{c}=m_{c}.$

In fact, since $\mathcal{D}\times\\{0\\}\subset\mathcal{G}$, hence
$l_{c}\geq\tilde{l}_{c}$, and for any $G\in\mathcal{G}$, we have $D:=\\{h\star
u:(u,h)\in G\\}\in\mathcal{D}$ and

$\max_{(u,h)\in G}\tilde{F}_{\infty}(u,h)=\max_{(u,h)\in G}F_{\infty}(h\star
u)=\max_{\bar{u}\in D}F_{\infty}(\bar{u}),$

hence $l_{c}\leq\tilde{l}_{c}$. Therefore, Lemma 2.1 yields a sequence
$(u_{n},h_{n})\in S_{c}\times\mathbb{R}$ such that

$(1)$ $|\tilde{F}_{\infty}(u_{n},h_{n})-m_{c}|\rightarrow 0$ as
$n\rightarrow\infty$,

$(2)$
$\|\nabla_{S_{c}\times\mathbb{R}}\tilde{F}_{\infty}(u_{n},h_{n})\|\rightarrow
0$ as $n\rightarrow\infty$,

$(3)$ $\operatorname{dist}((u_{n},h_{n}),D_{n}\times\\{0\\})\rightarrow 0$ as
$n\rightarrow\infty$.

In particular, let $v_{n}:=h_{n}\star u_{n}$, differentiation shows
that$\\{v_{n}\\}\subset S_{c}$ is a Palais-Smale sequence for $F_{\infty}$ on
$S_{c}$ at level $m_{c}$ satisfying the Pohožaev identity for $F_{\infty}$,
that is there exist Lagrange multipliers $\mu_{n}\in\mathbb{R}$ such that

$\displaystyle\frac{1}{2}\|(-\Delta)^{s/2}v_{n}\|_{2}^{2}-\frac{1}{p}\|v_{n}\|_{p}^{p}\rightarrow
m_{c},$
$\displaystyle\|F_{\infty}^{\prime}(v_{n})-\mu_{n}H^{\prime}(v_{n})\|_{(H^{s}(\mathbb{R}^{N}))^{*}}\rightarrow
0,\quad\text{ where }H(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx,$
$\displaystyle
s\|(-\Delta)^{s/2}v_{n}\|_{2}^{2}-\frac{N(p-2)}{2p}\|v_{n}\|_{p}^{p}\rightarrow
0,$

as $n\rightarrow\infty$. Moreover, $\\{v_{n}\\}$ is bounded in
$H^{s}(\mathbb{R}^{N})$. In fact, from the first and third relations above, we
can infer that

$\|(-\Delta)^{s/2}v_{n}\|_{2}^{2}\rightarrow\frac{2N(p-2)}{N(p-2)-4s}m_{c}$

and

$\|v_{n}\|_{p}^{p}\rightarrow\frac{4ps}{N(p-2)-4s}m_{c}.$

Thus $\\{v_{n}\\}$ is bounded in $H^{s}(\mathbb{R}^{N})$, and the second
relation implies that

$\|(-\Delta)^{s/2}v_{n}\|_{2}^{2}-\mu_{n}\|v_{n}\|_{2}^{2}-\|v_{n}\|_{p}^{p}\rightarrow
0,$

which is equivalent to

$\mu_{n}=\frac{\|(-\Delta)^{s/2}v_{n}\|_{2}^{2}-\|v_{n}\|_{p}^{p}}{\|v_{n}\|_{2}^{2}}+o(1)\rightarrow\frac{N(p-2)-2ps}{N(p-2)-4s}\cdot\frac{2m_{c}}{c^{2}}=:\mu<0.$

Therefore, after passing to subsequences, $\\{v_{n}\\}$ converges weakly in
$H^{s}(\mathbb{R}^{N})$ to a solution $v\in H^{s}(\mathbb{R}^{N})$ of
$(-\Delta)^{s}v=\mu v+|v|^{p-2}v$. We claim that

$v_{n}\rightarrow\pm w_{c}\quad\text{strongly in $H^{s}(\mathbb{R}^{N})$}.$
(3.3)

In order to see this we first observe that, using again the second relation
above,

$\int_{\mathbb{R}^{N}}(-\Delta)^{s/2}v_{n}(-\Delta)^{s/2}\varphi
dx-\mu_{n}\int_{\mathbb{R}^{N}}v_{n}\varphi
dx-\int_{\mathbb{R}^{N}}|v_{n}|^{p-2}v_{n}\varphi dx=o(1)\|\varphi\|$

for every $\varphi\in H^{s}(\mathbb{R}^{N})$, hence $\\{v_{n}\\}$ is also a
Palais-Smale sequence for $F_{\infty,\mu}$ at the level
$m_{c}-\frac{\mu}{2}c^{2}$. As a consequence, Lemma 2.2 implies

$v_{n}=v+\sum_{j=1}^{m}u^{j}(\cdot-y_{n}^{j})+o(1)$

in $H^{s}(\mathbb{R}^{N})$, where $m\geq 0$ and $u^{j}\neq 0$ are solutions to

$(-\Delta)^{s}u^{j}=\mu u^{j}+|u^{j}|^{p-2}u^{j}$

and $|y_{n}^{j}|\rightarrow\infty$. Moreover, setting $\gamma:=\|v\|_{2}$ and
$\alpha_{j}:=\|u^{j}\|_{2}$, then there holds

$c^{2}=\gamma^{2}+\sum_{j=1}^{m}\alpha_{j}^{2}$

and thus at least one of the limit functions must be non-trivial. In addition,
we have

$F_{\infty}(v_{n})-\frac{\mu}{2}c^{2}=F_{\infty}(v)-\frac{\mu}{2}\gamma^{2}+\sum_{j=1}^{m}(F_{\infty}(u^{j})-\frac{\mu}{2}\alpha_{j}^{2})+o(1),$

which yields that

$F_{\infty}(v_{n})=F_{\infty}(v)+\sum_{j=1}^{m}F_{\infty}(u^{j})+o(1).$

Using Appendix in [31] and (2.6), we have that, if $v$ is non-trivial, then
$F_{\infty}(v)\geq m_{\gamma}\geq m_{c}$. Therefore, if $m=0$, then $v\neq 0$;
if $m=1$, since $u^{j}\neq 0$, we have $F_{\infty}(u^{j})\geq
m_{\alpha_{j}}\geq m_{c}$, then

$m_{c}+o(1)=F_{\infty}(v_{n})=F_{\infty}(v)+\sum_{j=1}^{m}F_{\infty}(u^{j})+o(1)\geq
2m_{c}+o(1),$

which implies that $v=0$.

If $m=1$ and $v=0$, then $v_{n}=u^{1}(x-y_{n}^{1})+o(1)$, that is,
$\tilde{v}_{n}:=v_{n}(\cdot+y_{n}^{1})=u^{1}+o(1)$. On the other hand, due to
point $(3)$, we have that $v_{n}=\varphi_{n}+o(1)$ with $\varphi_{n}\in
D_{n}$, so that $\beta(\varphi_{n})=0$. Moreover, by the continuity of $\beta$
in $H^{s}(\mathbb{R}^{N})$, $u^{1}=\tilde{v}_{n}+o(1)$ and
$\tilde{v}_{n}=\varphi_{n}(\cdot+y_{n}^{1})+o(1)$, we obtain

$\beta(u^{1})=\beta(\tilde{v}_{n})+o(1)=\beta(\varphi_{n}(\cdot+y_{n}^{1}))+o(1)=y_{n}^{1}+o(1),$

which contradicts $|y_{n}^{1}|\rightarrow\infty$. So far we deduced that $m=0$
and $v_{n}\rightarrow v$ strongly in $H^{s}(\mathbb{R}^{N})$. Using again
point $(3)$, we have $\beta(v)=0$. Hence, by uniqueness, (3.3) follows. This
implies

$F(v_{n})=F_{\infty}(v_{n})+\frac{1}{p}\int_{\mathbb{R}^{N}}(1-a(x))|v_{n}|^{p}dx\rightarrow
m_{c}+\frac{1}{p}\int_{\mathbb{R}^{N}}(1-a(x))|w_{c}|^{p}dx>m_{c}$

as $n\rightarrow\infty$, which contradicts (3.2). ∎

######  Lemma 3.3.

For any $c>0$ there holds $m_{a,c}\geq L_{c}$.

###### Proof.

Given a function $\gamma:Q=B_{R}(0)\times[h_{1},h_{2}]\rightarrow S_{c}$ and
$h\in[h_{1},h_{2}]$ we consider the mapping

$f_{h}:B_{R}(0)\rightarrow\mathbb{R}^{N},\quad y\mapsto\beta\circ\gamma(y,h).$

We note that $f_{h_{i}}(y)=0$ if and only if $y=0$ for $i=1,2$, and
$f_{h}(y)=y\neq 0$ for any $y\in\partial B_{R}(0)$, so that
$\operatorname{deg}(f_{h},B_{R}(0),0)=1$ for all $h\in[h_{1},h_{2}]$.
Therefore, by degree theory, there exists a connected compact set
$Q_{0}\subset Q$ such that $(0,h_{i})\in Q_{0}$ for $i=1,2$, and
$\beta\circ\gamma(y,h)=0$ for any $(y,h)\in Q_{0}$. Hence the set
$D_{0}:=\gamma(Q_{0})\in\mathcal{D}_{0}$ satisfies

$\max_{(y,h)\in Q}F(\gamma(y,h))\geq\max_{u\in D_{0}}F(u),$

which concludes the proof. ∎

######  Lemma 3.4.

For any $c>0$ and $\varepsilon>0$ there exist $\bar{R}>0$ and
$\bar{h}_{1}<0<\bar{h}_{2}$ such that for $Q=B_{R}(0)\times[h_{1},h_{2}]$ with
$R\geq\bar{R}$, $h_{1}\leq\bar{h}_{1}$, $h_{2}\geq\bar{h}_{2}$ the following
holds:

$\max_{(y,h)\in\partial Q}F(h\star w_{c}(\cdot-y))<m_{c}+\varepsilon.$ (3.4)

###### Proof.

Since

$F(h\star
w_{c}(\cdot-y))=\frac{e^{2sh}}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}w_{c}|^{2}dx-\frac{e^{\frac{p-2}{2}Nh}}{p}\int_{\mathbb{R}^{N}}a(e^{-h}x+y)|w_{c}|^{p}dx,$

from $a_{*}\leq a(x)\leq 1$ for any $x\in\mathbb{R}^{N}$ and
$p>2+\frac{4s}{N}$, we deduce that

$F(h\star w_{c}(\cdot-y))\rightarrow-\infty,\quad\text{as
$h\rightarrow\infty$}$

uniformly in $y\in B_{R}(0)$ for any $R>0$, and

$F(h\star w_{c}(\cdot-y))\rightarrow 0,\quad\text{as $h\rightarrow-\infty$}$

uniformly in $y\in\mathbb{R}^{N}$. Thus,

$\max_{y\in B_{R}(0),h\in\\{h_{1},h_{2}\\}}F(h\star w_{c}(\cdot-y))<m_{c},$
(3.5)

provided that $h_{1}<0$ is small enough and $h_{2}>0$ is large enough.
Moreover, for $|y|=R$ large enough and $h\in[h_{1},h_{2}]$, choosing
$\alpha\in(0,1)$ such that $\alpha(1+e^{-h_{1}})<1$, then we have

$\displaystyle\int_{\mathbb{R}^{N}}(1-a(x))|(h\star w_{c})(x-y)|^{p}dx$ (3.6)
$\displaystyle\leq$ $\displaystyle\int_{|x|>\alpha R}(1-a(x))|(h\star
w_{c})(x-y)|^{p}dx+\int_{|x-y|>\alpha Re^{-h}}(1-a(x))|(h\star
w_{c})(x-y)|^{p}dx.$

The first integral is bounded by

$\int_{|x|>\alpha R}(1-a(x))|(h\star w_{c})(x-y)|^{p}dx\leq
e^{\frac{p-2}{2}Nh}\|1-a(x)\|_{L^{\infty}(|x|>\alpha
R)}\|w_{c}\|_{p}^{p}\rightarrow 0$ (3.7)

as $R\rightarrow\infty$ and

$\displaystyle\int_{|x-y|>\alpha Re^{-h}}(1-a(x))|(h\star w_{c})(x-y)|^{p}dx$
(3.8) $\displaystyle=$ $\displaystyle e^{\frac{p}{2}Nh}\int_{|x-y|>\alpha
Re^{-h}}(1-a(x))|w_{c}(e^{h}(x-y))|^{p}dx$ $\displaystyle=$ $\displaystyle
e^{\frac{p-2}{2}Nh}\int_{|\xi|>\alpha R}(1-a(e^{-h}\xi+y))|w_{c}|^{p}d\xi$
$\displaystyle\leq$ $\displaystyle
e^{\frac{p-2}{2}Nh}\|1-a(e^{-h}\xi+y)\|_{\infty}\|w_{c}\|_{L^{p}(|\xi|>\alpha
R)}^{p}\rightarrow 0$

as $R\rightarrow\infty$. Note that

$F(h\star w_{c}(\cdot-y))=F_{\infty}(h\star
w_{c})+\frac{1}{p}\int_{\mathbb{R}^{N}}(1-a(x))|(h\star w_{c})(x-y)|^{p}dx,$

form (3.6), (3.7) and (3.8), it follows that

$\max_{|y|=R,h\in[h_{1},h_{2}]}F(h\star w_{c}(\cdot-y))<m_{c}+\varepsilon,$
(3.9)

as $R\rightarrow\infty$. Combining (3.5) with (3.9), (3.4) is proved. ∎

In the following, we always assume $|h_{1}|$, $h_{2}$ and $R$ large enough but
fixed. (3.4) implies that $F$ has a linking geometry and there exists a
Palais-Smale sequence of $F$ at level $m_{a,c}$. The aim of the next parts is
to prove that $m_{a,c}$ is a critical value for $F$. Moreover, for future
purposes, we prove the following lemmas.

######  Lemma 3.5.

Assume that

$\sup_{x\in\mathbb{R}^{N}}|1-a(x)|<\frac{pm_{c}}{e^{\frac{p-2}{2}Nh_{2}}\|w_{c}\|_{p}^{p}},$
(3.10)

where $h_{2}$ is given by Lemma 3.4, then $m_{a,c}<2m_{c}$.

###### Proof.

This follows from

$\displaystyle m_{a,c}$ $\displaystyle\leq\max_{(y,h)\in
Q}\Big{\\{}F_{\infty}(h\star
w_{c}(\cdot-y))+\frac{1}{p}\int_{\mathbb{R}^{N}}(1-a(x))|(h\star
w_{c})(x-y)|^{p}dx\Big{\\}}$ $\displaystyle\leq
m_{c}+\frac{1}{p}e^{\frac{p-2}{2}Nh_{2}}\sup_{x\in\mathbb{R}^{N}}|1-a(x)|\cdot\|w_{c}\|_{p}^{p}$
$\displaystyle<2m_{c},$

provided that $|h_{1}|$, $h_{2}$ are large enough. ∎

Now we will construct a bounded Palais-Smale sequence $\\{v_{n}\\}\subset
H^{s}(\mathbb{R}^{N})$ of $F$ at level $m_{a,c}$. Adapting the approach of
[23], we introduce the following $C^{1}$-functional

$\tilde{F}(u,h):=F(h\star u)\ \text{for all $(u,h)\in
H^{s}(\mathbb{R}^{N})\times\mathbb{R}$},$

and define

$\tilde{\Gamma}_{c}:=\\{\tilde{\gamma}\in
C(Q,S_{c}\times\mathbb{R}):\tilde{\gamma}(y,h):=(h\star
w_{c}(\cdot-y),0)\text{ for all }(y,h)\in\partial Q\\}$

and

$\tilde{m}_{a,c}:=\inf_{\tilde{\gamma}\in\tilde{\Gamma}_{c}}\max_{(y,h)\in
Q}\tilde{F}(\tilde{\gamma}(y,h)).$

######  Lemma 3.6.

$\tilde{m}_{a,c}=m_{a,c}$.

###### Proof.

Since $\Gamma_{c}\times\\{0\\}\subset\tilde{\Gamma}_{c}$, then
$m_{a,c}\geq\tilde{m}_{a,c}$. On the other hand, for any
$\tilde{\gamma}=(u,h)\in\tilde{\Gamma}_{c}$, the function $\gamma:=h\star
u\in\Gamma_{c}$ satisfies

$\max_{(y,h)\in Q}\tilde{F}(\tilde{\gamma}(y,h))=\max_{(y,h)\in
Q}F(\gamma(y,h)),$

so that $m_{a,c}\leq\tilde{m}_{a,c}$. ∎

######  Lemma 3.7.

If $\\{(u_{n},h_{n})\\}$ is a $(PS)_{c^{\prime}}$ sequence for $\tilde{F}$ and
$h_{n}\rightarrow 0$, then $\\{(h_{n}\star u_{n})\\}$ is a $(PS)_{c^{\prime}}$
sequence for $F$.

###### Proof.

Let $w_{n}:=h_{n}\star u_{n}$, we have $F(w_{n})=F(h_{n}\star
u_{n})=\tilde{F}(u_{n},h_{n})\rightarrow c^{\prime}$. We claim that
$F^{\prime}(w_{n})\rightarrow 0$ in $(H^{s}(\mathbb{R}^{N}))^{*}$.

First of all, for any $\tilde{\varphi}\in T_{u_{n}}:=\\{\tilde{\varphi}\in
H^{s}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}u_{n}\tilde{\varphi}dx=0\\}$, one
has

$\langle\partial_{u}\tilde{F}(u_{n},h_{n}),\tilde{\varphi}\rangle\rightarrow
0\ \ \mbox{as}\ n\rightarrow\infty.$

For any $\varphi\in T_{w_{n}}:=\\{\varphi\in
H^{s}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}w_{n}\varphi dx=0\\}$, let
$\tilde{\varphi}(x)=e^{-\frac{N}{2}h_{n}}\varphi(e^{-h_{n}}x)$, we have

$\int_{\mathbb{R}^{N}}u_{n}\tilde{\varphi}dx=e^{-\frac{Nh_{n}}{2}}\int_{\mathbb{R}^{N}}u_{n}(x)\varphi(e^{-h_{n}}x)dx=\int_{\mathbb{R}^{N}}w_{n}\varphi
dx=0,$

that is, $\tilde{\varphi}\in T_{u_{n}}$. Hence, from $h_{n}\rightarrow 0$, we
have

$\displaystyle\langle F^{\prime}(w_{n}),\varphi\rangle$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}(-\Delta)^{s/2}w_{n}(-\Delta)^{s/2}\varphi
dx-\int_{\mathbb{R}^{N}}a(x)|w_{n}|^{p-2}w_{n}\varphi dx$ $\displaystyle=$
$\displaystyle
e^{-\frac{N}{2}h_{n}+2sh_{n}}\int_{\mathbb{R}^{N}}(-\Delta)^{s/2}u_{n}(-\Delta)^{s/2}\varphi(e^{-h_{n}}x)dx-e^{\frac{p-3}{2}Nh_{n}}\int_{\mathbb{R}^{N}}a(e^{-h_{n}}x)|u_{n}|^{p-2}u_{n}\varphi(e^{-h_{n}}x)dx,$
$\displaystyle=$ $\displaystyle
e^{2sh_{n}}\int_{\mathbb{R}^{N}}(-\Delta)^{s/2}u_{n}(-\Delta)^{s/2}\tilde{\varphi}dx-e^{\frac{p-2}{2}Nh_{n}}\int_{\mathbb{R}^{N}}a(e^{-h_{n}}x)|u_{n}|^{p-2}u_{n}\tilde{\varphi}dx$
$\displaystyle=$
$\displaystyle\langle\partial_{u}\tilde{F}(u_{n},h_{n}),\tilde{\varphi}\rangle\rightarrow
0,$

as $n\rightarrow\infty$. the conclusion is proved. ∎

Applying Lemma 2.1, we immediately have the following proposition.

######  Lemma 3.8.

Let $\tilde{g}_{n}\in\tilde{\Gamma}_{c}$ be a sequence such that

$\max_{(y,h)\in Q}\tilde{F}(\tilde{g}_{n}(y,h))\leq m_{a,c}+\frac{1}{n}.$

Then there exist a sequence $(u_{n},h_{n})\in S_{c}\times\mathbb{R}$ and $C>0$
such that

$m_{a,c}-\frac{1}{n}\leq\tilde{F}(u_{n},h_{n})\leq m_{a,c}+\frac{1}{n},$
(3.11) $\min_{(y,h)\in
Q}\|(u_{n},h_{n})-\tilde{g}_{n}(y,h)\|_{H^{s}(\mathbb{R}^{N})\times\mathbb{R}}\leq\frac{C}{\sqrt{n}},$
(3.12)
$\|\nabla_{S_{c}\times\mathbb{R}}\tilde{F}(u_{n},h_{n})\|\leq\frac{C}{\sqrt{n}}.$
(3.13)

The last inequality means:

$|D\tilde{F}(u_{n},h_{n})((z,s))|\leq\frac{C}{\sqrt{n}}(\|z\|+|s|)$

for all

$(z,s)\in\Big{\\{}(z,s)\in
H^{s}(\mathbb{R}^{N})\times\mathbb{R}:\int_{\mathbb{R}^{N}}zu_{n}dx=0\Big{\\}}.$

######  Lemma 3.9.

There exists a bounded sequence $\\{v_{n}\\}$ in $S_{c}$ such that

$F(v_{n})\rightarrow m_{a,c},\quad\nabla_{S_{c}}F(v_{n})\rightarrow 0$ (3.14)

and

$s\|(-\Delta)^{s/2}v_{n}\|_{2}^{2}-\frac{N(p-2)}{2p}\int_{\mathbb{R}^{N}}a(x)|v_{n}|^{p}dx+\frac{1}{p}\int_{\mathbb{R}^{N}}(x\cdot\nabla
a(x))|v_{n}|^{p}dx\rightarrow 0$ (3.15)

as $n\rightarrow\infty$. Moreover, the sequence of of Lagrange multipliers

$\lambda_{n}:=\frac{\langle F^{\prime}(v_{n}),v_{n}\rangle}{c^{2}}$ (3.16)

admits a subsequence $\lambda_{n}\rightarrow\lambda$ with

$-\frac{\delta_{0}}{c^{2}}<\lambda<0,$ (3.17)

where

$\delta_{0}:=\Big{(}\frac{4(p(2s-N)+2N)}{N(p-2)-4s}+\frac{(p-2)\|w\|_{\infty}}{N(p-2)-4s}\cdot\frac{16s}{(Np-4s)a_{\ast}-2N}\Big{)}m_{c}.$
(3.18)

###### Proof.

By the definition of $m_{a,c}$, we choose a sequence $g_{n}\in\Gamma_{c}$ such
that

$\max_{(y,h)\in Q}F(g_{n}(y,h))\leq m_{a,c}+\frac{1}{n}.$

Since $F(u)\geq F(|u|)$ for any $u\in H^{s}(\mathbb{R}^{N})$ and
$|\gamma|\in\Gamma_{c}$ for any $\gamma\in\Gamma_{c}$, we can assume that
$g_{n}(y,h)\geq 0$ almost everywhere in $\mathbb{R}^{N}$. Define
$\tilde{g_{n}}(y,h):=(g_{n}(y,h),0)$, then
$\tilde{g_{n}}(y,h)\in\tilde{\Gamma}_{c}$, and

$\max_{(y,h)\in Q}\tilde{F}(\tilde{g_{n}}(y,h))=\max_{(y,h)\in
Q}\tilde{F}((g_{n}(y,h),0)=\max_{(y,h)\in Q}{F}({g_{n}}(y,h))\leq
m_{a,c}+\frac{1}{n}.$

Applying Lemma 3.8, we can prove the existence of a sequence
$\\{(u_{n},h_{n})\\}\subset H^{s}(\mathbb{R}^{N})\times\mathbb{R}$ such that
$F(h_{n}\star u_{n})\rightarrow m_{a,c}$. We also note that

$\min_{(y,h)\in
Q}\|(u_{n},h_{n})-\tilde{g}_{n}(y,h)\|_{H^{s}(\mathbb{R}^{N})\times\mathbb{R}}\leq\frac{C}{\sqrt{n}},$

so that $h_{n}\rightarrow 0$ as $n\rightarrow\infty$ and there exists
$(y_{n},\bar{h}_{n})\in B_{R}(0)\times[h_{1},h_{2}]$ such that
$u_{n}-g_{n}(y_{n},\bar{h}_{n})\rightarrow 0$ in $H^{s}(\mathbb{R}^{N})$ as
$n\rightarrow+\infty$. We define

$v_{n}:=h_{n}\star u_{n}.$

Observe that, since $g_{n}(y_{n},\bar{h}_{n})\geq 0$ a.e. in $\mathbb{R}^{N}$,
then $\|u_{n}^{-}\|_{2}\leq\|u_{n}-g_{n}(y_{n},\bar{h}_{n})\|_{2}=o(1)$ and we
can deduce that $u_{n}^{-}\rightarrow 0$ a.e. in $\mathbb{R}^{N}$. So

$\|v_{n}^{-}\|_{2}\rightarrow 0\quad\text{as }n\rightarrow\infty.$

Moreover, by Lemma 3.7, $\\{v_{n}\\}$ is a Palais-Smale sequence for $F$, that
is

$\frac{1}{2}\|(-\Delta)^{s/2}v_{n}\|_{2}^{2}-\frac{1}{p}\int_{\mathbb{R}^{N}}a(x)|v_{n}|^{p}dx\rightarrow
m_{a,c}.$ (3.19)

and

$\|F^{\prime}(v_{n})-\lambda_{n}H^{\prime}(v_{n})\|_{(H^{s}(\mathbb{R}^{N}))^{*}}\rightarrow
0,$ (3.20)

where $H(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx$.

Since $h_{n}\rightarrow 0$ as $n\rightarrow\infty$ and

$\displaystyle\partial_{h}\tilde{F}(u_{n},h_{n})$ $\displaystyle=$
$\displaystyle\partial_{h}\Big{(}\frac{1}{2}e^{2sh_{n}}\|(-\Delta)^{s/2}u_{n}\|_{2}^{2}-\frac{1}{p}\int_{\mathbb{R}^{N}}a(x)|e^{\frac{Nh_{n}}{2}}u_{n}(e^{h_{n}}x)|^{p}dx\Big{)}$
$\displaystyle=$ $\displaystyle
s\|(-\Delta)^{s/2}e^{\frac{Nh_{n}}{2}}u_{n}(e^{h_{n}}x)\|_{2}^{2}-\frac{N(p-2)}{2p}\int_{\mathbb{R}^{N}}a(x)|e^{\frac{Nh_{n}}{2}}u_{n}(e^{h_{n}}x)|^{p}dx$
$\displaystyle+\frac{1}{p}\int_{\mathbb{R}^{N}}(x\cdot\nabla
a(x))|e^{\frac{Nh_{n}}{2}}u_{n}(e^{h_{n}}x)|^{p}dx,$

we obtain (3.15).

By $(A_{4})$, (3.15) and (3.19), we have

$\displaystyle m_{a,c}+o(1)$ $\displaystyle=$
$\displaystyle\frac{Np-4s}{4ps}\int_{\mathbb{R}^{N}}a(x)|v_{n}|^{p}dx-\frac{1}{2ps}\int_{\mathbb{R}^{N}}(Na(x)+x\cdot\nabla
a(x))|v_{n}|^{p}dx$ $\displaystyle\geq$
$\displaystyle\frac{Np-4s}{4ps}\int_{\mathbb{R}^{N}}a_{\ast}|v_{n}|^{p}dx-\frac{N}{2ps}\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx$
$\displaystyle=$
$\displaystyle\frac{1}{2ps}\Big{(}\frac{Np-4s}{2}a_{\ast}-N\Big{)}\|v_{n}\|_{p}^{p}.$

Condition $(A_{1})$ implies that $\\{\|v_{n}\|_{p}^{p}\\}$ is bounded, and

$\|v_{n}\|_{p}^{p}\leq\frac{4psm_{a,c}}{(Np-4s)a_{\ast}-2N}+o(1).$ (3.21)

This together with $(A_{2})$, $(A_{3})$ and $(A_{4})$ yields that

$\displaystyle\|(-\Delta)^{s/2}v_{n}\|_{2}^{2}$
$\displaystyle=2m_{a,c}+\frac{2}{p}\int_{\mathbb{R}^{N}}a(x)|v_{n}|^{p}dx+o(1)$
(3.22) $\displaystyle\leq
2m_{a,c}+\frac{2}{p}\int_{\mathbb{R}^{N}}|v_{n}|^{p}dx+o(1)$
$\displaystyle\leq\Big{(}2+\frac{8s}{(Np-4s)a_{\ast}-2N}\Big{)}m_{a,c}+o(1),$

which implies that $\\{v_{n}\\}$ is bounded in $H^{s}(\mathbb{R}^{N})$.
Moreover, we have

$\int_{\mathbb{R}^{N}}x\cdot\nabla
a(x)|v_{n}|^{p}dx\leq\|W\|_{\infty}\|v_{n}\|_{p}^{p}.$ (3.23)

In order to see (3.17), we set

$a_{n}:=\|(-\Delta)^{s/2}v_{n}\|_{2}^{2},\quad
b_{n}:=\int_{\mathbb{R}^{N}}a(x)|v_{n}|^{p}dx,\quad
d_{n}:=\int_{\mathbb{R}^{N}}x\cdot\nabla a(x)|v_{n}|^{p}dx.$

Then from (3.21), (3.22), (3.23) and the definition of $\lambda_{n}$, up to a
subsequence, we can assume that

$a_{n}\rightarrow a\geq 0,\quad b_{n}\rightarrow b\geq 0,\quad
d_{n}\rightarrow d\geq 0,\quad\lambda_{n}\rightarrow\lambda\in\mathbb{R}.$

Passing to the limit in (3.15), (3.19) and (3.20), then

$a=\frac{2}{p}b+2m_{a,c},$ (3.24)
$sa+(\frac{N}{p}-\frac{N}{2})b+\frac{1}{p}d=0,$ (3.25) $a=\lambda c^{2}+b.$
(3.26)

Now, we claim that $\lambda<0$. In fact, from the above three equalities, we
have

$\displaystyle-\lambda c^{2}$ $\displaystyle=b-a$
$\displaystyle=\frac{p-2}{p}b-2m_{a,c}$
$\displaystyle=\frac{p-2}{p}\cdot\frac{2ps}{N(p-2)-4s}(2m_{a,c}+\frac{1}{ps}d)-2m_{a,c}$
$\displaystyle=\frac{p(2s-N)+2N}{N(p-2)-4s}2m_{a,c}+\frac{2(p-2)}{p(N(p-2)-4s)}d$
$\displaystyle>\frac{p(2s-N)+2N}{N(p-2)-4s}2m_{c}+\frac{2(p-2)}{p(N(p-2)-4s)}d,$

where we have used Lemmas 3.2 and 3.3, from $2+\frac{4s}{N}<p<2_{s}^{*}$, we
have $\lambda<0$. On the other hand,

$\displaystyle-\lambda c^{2}$
$\displaystyle=\frac{p(2s-N)+2N}{N(p-2)-4s}2m_{a,c}+\frac{2(p-2)}{p(N(p-2)-4s)}d$
$\displaystyle<\frac{p(2s-N)+2N}{N(p-2)-4s}4m_{c}+\frac{(p-2)\|W\|_{\infty}}{N(p-2)-4s}\cdot\frac{16s}{(Np-4s)a_{\ast}-2N}m_{c}=\delta_{0}.$

Thus, we have (3.17) holds. Moreover, recalling the definition of $\theta$, we
obtain

$\displaystyle\frac{\delta_{0}}{m_{c}}=$
$\displaystyle\frac{4p(2s-N)+8N}{N(p-2)-4s}+\frac{16s(p-2)\|W\|_{\infty}}{(N(p-2)-4s)((Np-4s)a_{\ast}-2N)}$
(3.27) $\displaystyle\leq$
$\displaystyle\frac{12N-6p(N-2s)}{N(p-2)-4s}=3\theta.$

∎

##  4 The Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1: Since $\\{v_{n}\\}$ is bounded in
$H^{s}(\mathbb{R}^{N})$, after passing to a subsequence it converges weakly in
$H^{s}(\mathbb{R}^{N})$ to $v\in H^{s}(\mathbb{R}^{N})$. By Lemma 3.9, $v\geq
0$, and by weak convergence, $v$ is a weak solution of

$(-\Delta)^{s}v=\lambda v+a(x)|v|^{p-2}v,$

such that

$\|v\|_{2}\leq\liminf_{n\rightarrow 0}\|v_{n}\|_{2}=c.$

It remains to prove that $\|v\|_{2}=c$. Since

$\int_{\mathbb{R}^{N}}(-\Delta)^{s/2}v_{n}(-\Delta)^{s/2}\varphi
dx-\lambda_{n}\int_{\mathbb{R}^{N}}v_{n}\varphi
dx-\int_{\mathbb{R}^{N}}a(x)|v_{n}|^{p-2}v_{n}\varphi dx=o(1)\|\varphi\|$

for every $\varphi\in H^{s}(\mathbb{R}^{N})$, then $\\{v_{n}\\}$ is a Palais-
Smale sequence for $F_{\lambda}$ at level $m_{a,c}-\frac{\lambda}{2}c^{2}$.
Therefore, by Lemma 2.2, we have

$v_{n}=v+\sum_{j=1}^{k}u^{j}(\cdot-y_{n}^{j})+o(1),$

with $u^{j}$ being solutions to

$(-\Delta)^{s}u^{j}=\lambda u^{j}+|u^{j}|^{p-2}u^{j}$

and $|y_{n}^{j}|\rightarrow\infty$. We note that, if $k=0$, then
$v_{n}\rightarrow v$ strongly in $H^{s}(\mathbb{R}^{N})$, hence $\|v\|_{2}=c$
and we are done, thus we can assume that $k\geq 1$, or equivalently
$\rho:=\|v\|_{2}<c$.

First, we exclude the case $v=0$. In fact, if $v=0$ and $k=1$, we would have
$u^{1}>0$ and $\|u^{1}\|_{2}=c$, so that (2.11) gives $m_{a,c}=m_{c}$, which
is not possible due to Lemma 3.3. On the other hand, if $k\geq 2$, by Lemma
2.2, we have

$F_{\lambda}(v_{n})\to\sum_{j=1}^{k}F_{\infty,\lambda}(u^{j})=\sum_{j=1}^{k}F_{\infty}(u^{j})-\sum_{j=1}^{k}\frac{\lambda}{2}\|u^{j}\|^{2}_{2},\quad
c^{2}=\sum_{j=1}^{k}\|u^{j}\|_{2}^{2}.$

Let $\alpha^{2}_{j}=\|u^{j}\|_{2}^{2}$, since $F_{\infty}(u^{j})\geq
m_{\alpha_{j}}$ and

$m_{\alpha}>m_{\beta},\quad\text{if $\alpha<\beta$},$ (4.1)

we have $2m_{c}\leq m_{a,c}$, which contradicts Lemma 3.5.

Therefore, from now on we assume that $v\neq 0$. From $F(v_{n})\rightarrow
m_{a,c}$, we deduce that

$m_{a,c}-\frac{\lambda}{2}c^{2}=F(v)-\frac{\lambda}{2}\rho^{2}+\sum_{j=1}^{k}F_{\infty}(u^{j})-\frac{\lambda}{2}\sum_{j=1}^{k}\alpha_{j}^{2}$

Using $F_{\infty}(u^{j})\geq m_{\alpha_{j}}$, (4.1) and

$c^{2}=\rho^{2}+\sum_{j=1}^{k}\alpha_{j}^{2},$

we have

$2m_{c}>m_{a,c}\geq F(v)+\sum_{j=1}^{k}m_{\alpha_{j}}\geq F(v)+m_{\alpha}\geq
F(v)+m_{c},$

where $\alpha=\max\limits_{j}\alpha_{j}$. Moreover, using the equation for
$v$, it is easy to check that

$F_{\lambda}(v)=\max_{t>0}F_{\lambda}(tv).$

Let $\beta>0$ be such that

$\lambda_{\beta}=\lambda,$

according to (2.3). By Appendix in [31], $w_{\beta}$ satisfies the limit
equation with multiplier $\lambda$ and $\beta\leq\alpha$. Using Lemma 3.5 and
the above arguments, we have

$\displaystyle 2m_{c}$ $\displaystyle>m_{a,c}\geq
m_{\alpha}+F(v)=m_{\alpha}+F_{\lambda}(v)+\frac{\lambda}{2}\rho^{2}$
$\displaystyle=m_{\alpha}+\max_{t>0}F_{\lambda}(tv)+\frac{\lambda}{2}\rho^{2}$
$\displaystyle\geq
m_{\alpha}+\max_{t>0}F_{\infty,\lambda}(tv)+\frac{\lambda}{2}(c^{2}-\alpha^{2})$
$\displaystyle\geq
m_{\alpha}+F_{\infty,\lambda}(w_{\beta})+\frac{\lambda}{2}(c^{2}-\alpha^{2})$
$\displaystyle\geq
m_{\alpha}+m_{\beta}+\frac{\lambda}{2}(c^{2}-\alpha^{2}-\beta^{2}).$

Since $m_{\beta}\geq m_{\alpha}>m_{c}$ and $\lambda<0$, we deduce that

$c^{2}-\alpha^{2}-\beta^{2}>0.$

Using Lemma 3.9 and (2.6), we obtain

$\displaystyle 2m_{c}$
$\displaystyle>\frac{c^{\theta}}{\alpha^{\theta}}m_{c}+\frac{c^{\theta}}{\beta^{\theta}}m_{c}+\frac{\lambda}{2}(c^{2}-\alpha^{2}-\beta^{2})$
$\displaystyle>\frac{c^{\theta}}{\alpha^{\theta}}m_{c}+\frac{c^{\theta}}{\beta^{\theta}}m_{c}-\frac{\lambda_{0}m_{c}}{2c^{2}}(c^{2}-\alpha^{2}-\beta^{2}),$

where $\lambda_{0}:=\frac{\delta_{0}}{m_{c}}.$ Hence, one gets

$2+\frac{\lambda_{0}}{2}>\frac{c^{\theta}}{\alpha^{\theta}}+\frac{c^{\theta}}{\beta^{\theta}}+\frac{\lambda_{0}}{2}(\frac{\alpha^{2}}{c^{2}}+\frac{\beta^{2}}{c^{2}}).$
(4.2)

Note that the following inequality holds for $A>\theta$ (see [4])

$\min\Big{\\{}x^{-\theta/2}+y^{-\theta/2}+A(x+y):x,y>0,x+y\leq
1\Big{\\}}\geq\frac{3}{2}\theta+2,$

and by using $(A_{3})$ and (3.27), we have

$\lambda_{0}=\frac{\delta_{0}}{m_{c}}>\frac{4(p(2s-N)+2N)}{N(p-2)-4s}=2\theta,$

so (4.2) implies that

$2+\frac{\lambda_{0}}{2}>\frac{3\theta}{2}+2,$

which is a contradiction to (3.27).

Proof of Theorem 1.2: The strategy is similar to the proof of Theorem 1.1. We
observe that the results obtained in Section 3 excepting Lemma 3.4 and Lemma
3.5 still hold true without any changes in the proof.

First, we claim that if $(A_{6})$ holds, then $m_{a,c}<2m_{c}$.

From the definition of $m_{c}$, we have

$m_{c}=\frac{1}{2}\|(-\Delta)^{s/2}w_{c}\|_{2}^{2}-\frac{1}{p}\|w_{c}\|_{p}^{p}$

and

$s\|(-\Delta)^{s/2}w_{c}\|_{2}^{2}-\frac{N(p-2)}{2p}\int_{\mathbb{R}^{N}}a(x)|w_{c}|^{p}dx=0,$

which implies that

$\|(-\Delta)^{s/2}w_{c}\|_{2}^{2}=\frac{2N(p-2)}{N(p-2)-4s}m_{c},\quad\|w_{c}\|_{p}^{p}=\frac{4sp}{N(p-2)-4s}.$
(4.3)

Note that

$\displaystyle m_{a,c}$
$\displaystyle\leq\max_{h\in\mathbb{R},y\in\mathbb{R}^{N}}\Big{\\{}F_{\infty}(h\star
w_{c})+\frac{1}{p}\int_{\mathbb{R}^{N}}(1-a(x))|(h\star
w_{c})(x-y)|^{p}dx\Big{\\}}$ (4.4)
$\displaystyle\leq\max_{h\in\mathbb{R}}\Big{\\{}F_{\infty}(h\star
w_{c})+\frac{e^{2sh}}{p}\|1-a(x)\|_{t_{1}}\|w_{c}\|_{t_{2}p}^{p}\Big{\\}}$
$\displaystyle=\max_{h\in\mathbb{R}}\Big{(}e^{2sh}\Big{(}\frac{1}{2}\|(-\Delta)^{s/2}w_{c}\|_{2}^{2}+\frac{1}{p}\|1-a(x)\|_{t_{1}}\|w_{c}\|_{t_{2}p}^{p}\Big{)}-\frac{e^{\frac{p-2}{2}Nh}}{p}\|w_{c}\|_{p}^{p}\Big{)}.$

Differentiating with respect to $h$, then there exists $h_{0}$ such that

$2se^{2sh_{0}}\Big{(}\frac{1}{2}\|(-\Delta)^{s/2}w_{c}\|_{2}^{2}+\frac{1}{p}\|1-a(x)\|_{t_{1}}\|w_{c}\|_{t_{2}p}^{p}\Big{)}-\frac{N(p-2)}{2p}e^{\frac{p-2}{2}Nh_{0}}\|w_{c}\|_{p}^{p}=0.$

Hence, we obtain that

$e^{\frac{p-2}{2}Nh_{0}}=\left\\{\frac{N(p-2)-4s}{N(p-2)m_{c}}\Big{(}\frac{1}{2}\|(-\Delta)^{s/2}w_{c}\|_{2}^{2}+\frac{1}{p}\|1-a(x)\|_{t_{1}}\|w_{c}\|_{t_{2}p}^{p}\Big{)}\right\\}^{\frac{N(p-2)}{N(p-2)-4s}}.$
(4.5)

Combining $(A_{6})$ with (4.3) and (4.5), we obtain

$e^{\frac{p-2}{2}Nh_{0}}<2.$

From (4.3) and (4.4), we have

$m_{a,c}\leq e^{\frac{p-2}{2}Nh_{0}}m_{c}.$

This ends the proof.

Second, if $(A_{6})$ holds, then Lemma 3.4 still holds true.

Same as the proof of (3.5), we can fix $h_{1}<0<h_{2}$ such that

$\sup_{(y,h)\in\mathbb{R}^{N}\times\\{h_{1},h_{2}\\}}F(h\star
w_{c}(\cdot-y))\leq m_{c}.$ (4.6)

For all $(y,h)\in\mathbb{R}^{N}\times\mathbb{R}$, we have

$\displaystyle\limsup_{|y|\to+\infty}\max_{h\in[h_{1},h_{2}]}F(h\star
w_{c}(x-y))$ $\displaystyle=$
$\displaystyle\limsup_{|y|\to+\infty}\max_{h\in[h_{1},h_{2}]}\Big{(}F_{\infty}(h\star
w_{c})+\frac{1}{p}\int_{\mathbb{R}^{N}}(1-a(x))|(h\star
w_{c})(x-y)|^{p}dx\Big{)}.$

Since $w_{c}\in L^{t_{2}p}(\mathbb{R}^{N})$ and $(1-a(x))\in
L^{t_{1}}(\mathbb{R}^{N})$, there exists $R>0$ large enough for any
$\varepsilon>0$ such that

$\|w_{c}\|_{L^{t_{2}p}(B_{R}(y))}^{p}<\varepsilon,\quad\|1-a(x)\|_{L^{t_{1}}(\mathbb{R}^{N}\backslash
B_{R}(0))}<\varepsilon,\quad\text{for}\ |y|\rightarrow\infty.$

Therefore,

$\displaystyle\max_{h\in[h_{1},h_{2}]}\int_{\mathbb{R}^{N}}(1-a(x))|(h\star
w_{c})(x-y)|^{p}dx$ $\displaystyle=$
$\displaystyle\max_{h\in[h_{1},h_{2}]}\left(\int_{B_{R}(0)}(1-a(x))|(h\star
w_{c})(x-y)|^{p}dx+\int_{\mathbb{R}^{N}\backslash B_{R}(0)}(1-a(x))|(h\star
w_{c})(x-y)|^{p}dx\right)$ $\displaystyle\leq$
$\displaystyle\|1-a(x)\|_{t_{1}}\max_{h\in[h_{1},h_{2}]}e^{2sh}\|w_{c}\|_{L^{t_{2}p}(B_{R}(y))}^{p}+\|1-a(x)\|_{L^{t_{1}}(\mathbb{R}^{N}\backslash
B_{R}(0))}\max_{h\in[h_{1},h_{2}]}\|h\star w_{c}\|_{t_{2}p}^{p}\rightarrow 0.$

Using $(A_{3})$, $(A_{4})$ and $(A_{6})$,

$\limsup_{|y|\to+\infty}\max_{h\in[h_{1},h_{2}]}F(h\star w_{c}(x-y))\leq
m_{c}.$

Moreover,

$\limsup_{|y|\to+\infty}F(w_{c}(x-y))\geq\limsup_{|y|\to+\infty}F_{\infty}(w_{c}(x-y))=F_{\infty}(w_{c})=m_{c},$

which implies that

$\limsup_{|y|\to+\infty}\max_{h\in[h_{1},h_{2}]}F(h\star w_{c}(x-y))=m_{c}.$
(4.7)

This together with (4.6) and (4.7) yields that Lemma 3.4 holds.

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