# Remote sensing and control of phase qubits

Dale Li<EMAIL_ADDRESS>Fabio C.S. da Silva  Danielle A. Braje
Raymond W. Simmonds  David P. Pappas  National Institute of Standards and
Technology, Boulder, Colorado 80305, USA

###### Abstract

We demonstrate a remote sensing design of phase qubits by separating the
control and readout circuits from the qubit loop. This design improves
measurement reliability because the control readout chip can be fabricated
using more robust materials and can be reused to test different qubit chips.
Typical qubit measurements such as Rabi oscillations, spectroscopy, and
excited-state energy relaxation are presented.

###### pacs:

Superconducting phase qubits are one of the most promising technologies for a
scalable quantum computer.MartinisQIP  Introduction and improvement of
specialized materials and structures has significantly reduced losses and
improved coherence times.Oh  However, evaluation of these materials creates
challenges in the design and fabrication of qubit circuits primarily because
of variations in material composition and crystalline order.Kline  The ability
to explore different materials would be greatly simplified if the control and
readout circuit to measure the qubit could be fabricated separately from the
qubit devices under investigation. The readout circuit could then be made of
well-established materials and designs, and would operate reliably independent
of materials being used for the qubits. In this letter, we developed a self-
aligning flip-chip technique to separate the qubit circuit from its readout.
The readout chip is inductively coupled to the phase qubit, and contains the
SQUID readout and the superconducting coils for microwave and dc flux control.

Previous superconducting circuits have used flip-chips to perform noise and
remote detection measurements BerggrenRemote ; MayCoil . Flip-chip
implementations of charge qubits operating as interferometers have also been
reported.BornChargeQ  In addition, flip-chips have been used to separate
dissipative single-flux quantum (SFQ) circuits from the temperature-sensitive
qubit circuits.YorozuFlux  Bennett et al. used a separate chip suspended above
an rf-SQUID qubit chip to obtain fast bias pulses.BennettFlip  Steffen et al.
describe a SQUID-less readout scheme that reduces the number of junctions in
the qubit to one (the qubit junction itself). This scheme allows for the
multiplexing of many qubits. However, the overall performance of the system is
affected by the coupling between the microwave feed line and the qubit
circuit.DispReadout  Michotte uses the flip-chip technique to separate the
microstrip line from the SQUID sensor in a microstrip-SQUID
amplifier.MicrostripFlip

Figure 1: (color online). (a) The top chip is self-aligned a distance $z$
above the bottom chip by use of sapphire spheres of diameter $D$. The top chip
pocket diameter $d$ is determined by Eq. (1) with a fixed bottom chip pocket
depth of $h$. (b) The top chip and bottom chip are separated, showing the
alignment sites and the scale of each chip. Note that the top chip is smaller
than the bottom chip to allow space for wire bonding. (c) The assembled flip-
chip.

Our flip-chip design contains the phase qubit loop on the top chip, which
self-aligns, by use of four $200\pm 2.5$ $\mu$m diameter sapphire spheres, to
the bottom chip containing the control/readout circuitry. Sapphire spheres
have a small thermal contraction coefficient, which helps to maintain proper
alignment when the sample is cooled to dilution-refrigerator temperatures. The
spheres sit in pockets etched into the silicon substrates by a deep reactive
ion etcher.

Figure 1(a) shows a cross-sectional drawing of the deeply etched cylindrical
pockets in the top and bottom chips and the self-aligning sapphire spheres.
The diameter of the top chip pocket is given by $d=2\sqrt{(D-h-z)(h+z)}$,
where $D$ is the diameter of the sapphire sphere, $h$ is the depth of the
pocket etched into the bottom chip (with etched diameter equal to $D$), and
$z$ is the desired vacuum gap size. Deep pockets in the bottom chip held the
sapphire spheres in place for reuse, while the shallower pockets in the top
chip were etched deep enough that the sapphire spheres only touch the top chip
at the edges of the pockets. Different pocket diameters for different top
chips were fabricated, giving vacuum gap sizes from 10 $\mu$m to 50 $\mu$m.

Photographs of the fabricated top and bottom chips are shown separately, with
the bottom chip wire-bonded to a test board in Fig. 1(b), and in the flip-chip
configuration in Fig. 1(c). The four positions for the sapphire spheres
facilitate a stable self-alignment, minimize wobble, and place the spheres far
away from the circuit elements. The entire flip-chip assembly is held together
under slight compression by a beryllium-copper leaf spring placed inside a
brass lid, which encloses the two chips and fastens to the circuit board.

Figure 2: (color online). (a) Flip-chip circuit drawing shows the simple qubit
circuit and the three inductively coupled control coils for microwave
excitation, DC flux bias, and DC SQUID bias, as well as the DC three-junction
SQUID for qubit readout. (b) A photograph of qubit loop near the final steps
of fabrication as patterned on the top chip. A final wiring layer connects the
junction and the via (not shown). (c) Photograph of measurement and excitation
circuitry as fabricated on the bottom chip. The dashed large rectangle
indicates where the qubit will align.

Figure 2(a) shows the circuit model for the entire phase qubit including
control and readout (C/R). The C/R circuit consists of a three-junction dc
SQUID (readout), a dc flux bias loop that applies magnetic flux to the qubit
(control), a secondary dc flux bias loop to tune the magnetic flux in the
SQUID (control), and a microwave flux loop that excites the qubit with
microwave frequencies (control). Each inductive loop utilizes a gradiometric
design to minimize both unwanted cross-coupling between coils and the effects
of shifts in background homogeneous magnetic fields by symmetric placement.
Fig. 2(b) shows a photograph of the qubit loop as patterned on the top chip.
To test this flip-chip approach, standard Al/amorphous-Al2O3/Al Josephson
junctions $13$ $\mu$m2 in area were designed and fabricated for qubit
frequencies around 7 GHz. The qubit loop was closed by an Al cross-over wire
connecting the junction and the via (not shown). Fig. 2(c) shows a photograph
of the C/R circuitry above which the qubit loop is placed (dashed rectangle)
when aligned.

For a vacuum gap size $z=20$ $\mu$m, the mutual inductance coupling terms were
calculated between pairs of coils (qubit-SQUID: 71 pH, qubit-flux bias: 5.5
pH, qubit-SQUID bias: $<$1 pH, qubit-microwave line: 5.5 pH, SQUID-SQUID bias:
2 pH, SQUID-flux bias: $<$1 pH). The qubit loop was designed with a self
inductance of 880 pH, while the SQUID was designed with a self-inductance of
341 pH. These large inductances ensured a strong measurable coupling between
the qubit chip and the C/R chip, although smaller inductances could also
provide adequate coupling, depending on the gap size.

Figure 3: (color online). Qubit steps for two different qubit chips (same
readout chip) showing different coupling. (a) $z$=$10$ $\mu$m gap size. The
steps are curved due to the large overlap coupling to the dc SQUID. A flux
quantum in the qubit is observed with the applied voltage $\Phi_{0}$=$44.6$
mV. (b) $z$=$20$ $\mu$m gap size has weaker coupling and samples just the
linear regime of the SQUID. A larger applied voltage is needed to excite a
flux quantum with $\Phi_{0}$=$766$ mV.

We tested the remote sensing and control of the phase qubit with four typical
measurements showing coherent control and reliable readout: qubit steps,
spectroscopy, Rabi oscillations, and $T_{1}$.HistFitting  Additionally, the
response of the SQUID was measured as a function of the applied flux through
the SQUID bias line in order to test the C/R circuit independently of the
qubit. The SQUID bias line also provided the ability to tune the SQUID to a
sensitive, mostly linear regime.

First, we measured the qubit steps by applying a magnetic flux to the qubit
loop and measuring the corresponding value of the SQUID switching current
$I_{s}$. Here, the applied flux is measured in units of the voltage across a
10 k$\Omega$ resistor connected in series with the qubit bias coil. Figure
3(a) shows the behavior of $I_{s}$ versus the applied flux for a gap size
between the bottom and top chips of $10$ $\mu$m. The pronounced nonlinear
behavior of $I_{s}$ arises from a large field change as sensed by the SQUID at
different qubit states, which maps to a larger, less linear regime in the
SQUID response. For this gap size of $10$ $\mu$m, the voltage difference
necessary to induce a quantum of flux ($\Phi_{0}$) variation in the qubit is
44.6 mV. For an increased gap size of $20$ $\mu$m, the flux bias voltage per
flux quantum increased to 766 mV as shown in Fig. 3(b). This change in flux
bias per flux quantum corresponds to a reduction of the coupling by a factor
of 17. Furthermore, the reduction of coupling decreased the amount of qubit
flux sensed by the SQUID so that its response mapped to a more linear regime,
as shown in Fig. 3(b).

Figure 4: (color online). Data collected from $z=20$ $\mu$m gap sized flip-
chip. (a) Spectroscopy data showing the tunability of the qubit resonant
frequency as a function of the applied flux from the bottom chip. The inset
shows a zoom in of one of many splittings due to coupling with parasitic two
level systems in this qubit. (b) Rabi Oscillations in the qubit from microwave
excitation. (c) Relaxation time measurement.

Second, we measured the qubit spectroscopy for a gap size of 20 $\mu$m. The
phase qubit exhibits a tunable absorption spectrum at its transition frequency
($\omega_{01}$) between the ground and first excited state. In Fig. 4(a) the
qubit spectroscopy shows a 2 GHz range of $\omega_{01}$ values centered around
7 GHz. The visibility of only one transition line in the spectroscopy data
indicates that the qubit chip was cooled to low enough temperatures to be
operated as a qubit. The discontinuities in the spectrum are assumed to be due
to parasitic two-level systems in the large-area amorphous-Al2O3 tunnel
barrier.Simmonds2004  A zoom-in of one such discontinuity is shown in the
inset.

Third, Fig. 4(b) shows Rabi oscillations in the same qubit. This experiment is
performed by holding a constant dc flux bias in a region of the spectroscopy
with few discontinuities and applying a microwave pulse for a varied period.
Rabi oscillations demonstrate the ability for state mixing between the ground
and first excited state of the qubit. The oscillation amplitude decays due to
decoherence with a spin bath and should ideally saturate to a $50\%$
occupation probability. In the data, the saturation occurs at about a $33\%$
occupation probability. This discrepancy is due to the measurement process,
which sweeps the coupling of the qubit through many avoided crossings with
parasitic two-level systems that syphon energy from the qubit in Landau-Zener-
like transitions.Cooper2004

Fourth, Fig. 4(c) shows a longitudinal relaxation experiment in the same
qubit. In this experiment, a partially excited qubit state is prepared with a
fixed microwave pulse length of 50 ns, and the qubit state is measured as a
function of time as it decays to its ground state. Our flip-chip test used
similar design considerations, materials, and fabrication techniques as for
integrated chips so we expected the experimental data to agree with previous
results without the introduction of additional noise or loss. Though the
observed relaxation time $T_{1}=23$ ns is short, it matches reported results
for a phase qubit with a $13$ $\mu$m2 thermally oxidized amorphous Al2O3
tunnel barrier on a Silicon substrate.Cooper2004

In conclusion, we demonstrated the remote sensing and control of a phase qubit
by separating the qubit loop and the control/readout (C/R) circuit. Typical
characterization and performance measurements done in several qubit loops with
the same C/R circuit demonstrated reliability and robustness of this design.
The technique has therefore proven to be an adequate candidate for studying
the improvement of specialized materials and structures for superconducting
qubits. Other types of qubits, such as flux qubits could also potentially use
the same flip-chip technique either by direct coupling across a smaller
controlled gap, or by mediated coupling through a resonator circuit or rf-
SQUID.Shane

This research was funded in part by the Office of the Director of National
Intelligence (ODNI) and by Intelligence Advanced Research Projects Activity
(IARPA). All statements of fact, opinion or conclusions contained herein are
those of the authors and should not be construed as representing the official
views or policies of IARPA, the ODNI, or the U.S. Government. Official
contribution of the National Institute of Standards and Technology; not
subject to copyright in the United States.

## References

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Normalized solutions for nonlinear Schrödinger equation involving potential
and Sobolev critical exponent

Zhen-Feng Jin1, Weimin Zhang2,∗

1 School of Mathematics and Computer Science, Shanxi Normal University,

Taiyuan, 030031, P.R. China

2 School of Mathematical Sciences, Key Laboratory of Mathematics and
Engineering Applications (Ministry of Education) & Shanghai Key Laboratory of
PMMP, East China Normal University, Shanghai, 200241, P.R. China

000 ∗Corresponding author.
AMS Subject Classification: 35Q55, 35J60, 35J20.
E-mail addresses<EMAIL_ADDRESS>(Z.-F. Jin),
<EMAIL_ADDRESS>(W. Zhang).

###### Abstract

In this paper, we consider the existence of positive solutions with prescribed
$L^{2}$-norm for the following nonlinear Schrödinger equation involving
potential and Sobolev critical exponent

$\begin{cases}-\Delta u+V(x)u=\lambda
u+\mu|u|^{p-2}u+|u|^{\frac{4}{N-2}}u\;\;\text{ in }\mathbb{R}^{N},\\\
\|u\|_{2}=a>0,\\\ \end{cases}$

where $N\geq 3$, $\mu>0$, $p\in[2+\frac{4}{N},\frac{2N}{N-2})$ and $V\in
C^{1}(\mathbb{R}^{N})$. Under different assumptions on $V$, we derive two
different Pohozaev identities. Based on these two cases, we respectively
obtain the existence of positive solution. As far as we are aware, we did not
find any works on normalized solutions with Sobolev critical growth and
potential $V\not\equiv 0$. Our results extend some results of Wei and Wu [J.
Funct. Anal. 283(2022)] to the potential case.

Keywords: Normalized solutions, Potential, Mass supercritical and Sobolev
critical growth.

##  1 Introduction

This paper is concerned with the existence of positive solutions with
prescribed $L^{2}$-norm for the following nonlinear Schrödinger equation

$\begin{cases}-\Delta u+V(x)u=\lambda u+\mu|u|^{p-2}u+|u|^{2^{*}-2}u\;\;\text{
in }\mathbb{R}^{N},\\\ \|u\|_{2}=a>0,\\\ \end{cases}$ (1.1)

where $N\geq 3$, $\mu>0$, $p\in[2+\frac{4}{N},2^{*})$ with
$2^{*}:=\frac{2N}{N-2}$, and

$\|u\|^{2}_{2}=\int_{\mathbb{R}^{N}}u^{2}\mathrm{d}x.$

The parameter $\lambda\in\mathbb{R}$ arises as a Lagrange multiplier with
respect to the mass constraint $\|u\|_{2}=a$. Here,
$V:\mathbb{R}^{N}\to\mathbb{R}$ is a potential function.

Problem (1.1) comes from the study of standing waves for the nonlinear
Schrödinger equation

$iw_{t}-\Delta w+V(x)w=f(w)\;\;\text{in}\leavevmode\nobreak\
\mathbb{R}^{N}\times(0,\infty),$ (1.2)

where $w$ has the form

$w(x,t)=e^{-i\lambda t}u(x),\;\;(x,t)\in\mathbb{R}^{N}\times(0,\infty),$ (1.3)

and $u$ is a real function. For problem (1.2)-(1.3), the $L^{2}$-norm of $u$
stands for the mass of a particle. If $f(w)=e^{-i\lambda t}f(u)$, (1.2)-(1.3)
will be reduced to

$-\Delta u+V(x)u=\lambda u+f(u)\;\;\text{ in }\mathbb{R}^{N}.$ (1.4)

Such problem for given $\lambda$ is called the fixed frequency problem, which
has been widely studied for the decades. A huge literature exists, we will not
intend to summarize here, the interested readers can refer to [2, 5, 22] and
references therein. Here we are concerned with the case of prescribed mass,
that is, for some fixed $a>0$, we try to find solutions of (1.4) on the
manifold

$S_{a}:=\\{u\in H^{1}(\mathbb{R}^{N}):\|u\|_{2}=a\\},$ (1.5)

with $\lambda\in\mathbb{R}$. Commonly, such solutions are called normalized
solutions. A natural approach is applying variational method to (1.4)-(1.5).
One can derive solutions to (1.4)-(1.5) by looking for critical points of the
associated energy functional on $S_{a}$,

$\displaystyle E_{V}(u)=\frac{1}{2}\|\nabla
u\|_{2}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x-\int_{\mathbb{R}^{N}}F(u)\mathrm{d}x$
(1.6)

where

$F(u)=\int_{0}^{u}f(t)\mathrm{d}t.$

###  1.1 Non-potential case $V\equiv 0$.

The simplest example is the power nonlinearities $f(u)=|u|^{p-2}u$. Consider

$-\Delta u+u=|u|^{p-2}u\leavevmode\nobreak\ \text{ in }\mathbb{R}^{N},\;\;u\in
H^{1}(\mathbb{R}^{N}).$ (1.7)

It is well-known that for $p\in(2,2^{*})$, (1.7) has a unique positive
solution up to a translation, which can be chosen to be radial (see [13]). If
we denote the radial solution by $U_{p}$, one can check that

$U_{\lambda,p}(x):=\lambda^{\frac{1}{p-2}}U_{p}({\sqrt{\lambda}}x),\;\;\lambda>0,$

is the unique positive solution (up to a translation) to

$-\Delta u+\lambda u=|u|^{p-2}u\;\;\text{ in }\mathbb{R}^{N},\;\;u\in
H^{1}(\mathbb{R}^{N}).$ (1.8)

A direct computation shows that

$\|U_{\lambda,p}\|_{2}^{2}=\lambda^{\frac{4-(p-2)N}{2(p-2)}}\|U_{p}\|_{2}^{2}.$

If $p\neq\hat{p}:=2+\frac{4}{N}$, for any $a>0$, there exists a unique
$\lambda>0$ such that $\|U_{\lambda,p}\|_{2}=a$, which means that (1.8) has a
unique positive radial normalized solution $u\in S_{a}$ for any
$p\in(2,\hat{p})\cup(\hat{p},2^{*})$. While for $p=\hat{p}$, the equation
(1.8) has a positive normalized solution $u\in S_{a}$ if and only if
$a=\|U_{p}\|_{2}$. Therefore, $\hat{p}=2+\frac{4}{N}$ is called the mass
critical exponent or $L^{2}$-critical exponent to (1.8).

Moreover, the mass critical exponent plays an important role in the geometry
of energy functional. In order to preserve the $L^{2}$-norm, usually we use
the scaling

$u_{h}(x):=h^{\frac{N}{2}}u(hx),\;\;h>0,$ (1.9)

which gives readily

$\|\nabla u_{h}\|_{2}^{2}=h^{2}\|\nabla
u\|_{2}^{2},\;\;\|u_{h}\|_{p}^{p}=h^{\frac{N(p-2)}{2}}\|u\|_{p}^{p}.$

Remark that to have $h^{2}=h^{\frac{N(p-2)}{2}}$ for $h\neq 1$, one needs
$p=\hat{p}$. Thus, for fixed $u\in S_{a}$, the sign of $p-\hat{p}$ decides the
shape of $h\mapsto E_{0}(u_{h})$, seeing the Gagliardo-Nirenberg inequality
(see [18]): For any $N\geq 3$ and $p\in[2,2^{*}]$, there exists a constant
$C_{N,\,p}$ depending on $N$, $p$ such that

$\|u\|_{p}\leq C_{N,\,p}\|u\|_{2}^{1-\gamma_{p}}\|\nabla
u\|_{2}^{\gamma_{p}},\;\;\forall\leavevmode\nobreak\ u\in
H^{1}(\mathbb{R}^{N})$ (1.10)

where $\gamma_{p}=\frac{N(p-2)}{2p}$. The inequality holds true also in $N\leq
2$ for any $2\leq p<\infty$.

We can observe that

$p\gamma_{p}\begin{cases}<2\;\;\text{if}\leavevmode\nobreak\ 2<p<\hat{p}\\\
=2\;\;\text{if}\leavevmode\nobreak\ p=\hat{p}\\\
>2\;\;\text{if}\leavevmode\nobreak\ \hat{p}<p<2^{*},\\\
\end{cases}\;\;\text{and}\;\;\gamma_{2^{*}}=1.$

The sign of $p-\hat{p}$ decides whether the functional $E_{0}$ is bounded from
below on $S_{a}$. For $p=2^{*}$, we have a particular case of (1.10), the
Sobolev inequality

$\mathcal{S}\|u\|_{2^{*}}^{2}\leq\|\nabla
u\|_{2}^{2},\;\;\forall\leavevmode\nobreak\ u\in
D^{1,2}(\mathbb{R}^{N}),\leavevmode\nobreak\ N\geq 3.$ (1.11)

The sharp constant $\mathcal{S}$ for (1.11) is called the best Sobolev
constant. It is well known from [23] that

$\widetilde{U}_{\varepsilon,y}(x)=[N(N-2)]^{\frac{N-2}{4}}\left(\frac{\varepsilon}{\varepsilon^{2}+|x-y|^{2}}\right)^{\frac{N-2}{2}},\;\;\varepsilon>0,\,y\in\mathbb{R}^{N},$

are the unique functions to achieve (1.11), which are also the only classical
solutions (see [8]) to

$-\Delta u=u^{2^{*}-1},\;\;u>0\;\;\text{in}\leavevmode\nobreak\
\mathbb{R}^{N}.$

Clearly, $\widetilde{U}_{\varepsilon,y}\in L^{2}(\mathbb{R}^{N})$ if and only
if $N\geq 5$. Thus, for the critical case $p=2^{*}$, (1.8) admits a positive
radial normalized solution $\widetilde{U}_{\varepsilon,0}\in S_{a}$ with
$\lambda=0$ and a unique choice of $\varepsilon>0$ if $N\geq 5$, and no
positive solution exists any more if $N\leq 4$.

For nonhomogeneous nonlinearities, the scaling method does not work. Jeanjean
[10] did a seminal work and considered a class of mass supercritical and
Sobolev subcritical problem where $f(u)$ can be chosen as $\sum_{1\leq j\leq
k}a_{j}|u|^{\sigma_{j}-2}u$ with $k\geq 1,a_{j}>0$ and

$\hat{p}<\sigma_{j}<2^{*}\;\;\text{if}\leavevmode\nobreak\ N\geq
3;\;\;\hat{p}<\sigma_{j}\;\;\text{if}\leavevmode\nobreak\ N=1,2.$

He showed that the energy functional $E_{0}$ possesses a mountain pass
geometry on $S_{a}$. A crucial step in [10] is to construct an augmented
functional

$\widetilde{E_{0}}(u,h):=E_{0}(u_{h}),\;\;(u,h)\in S_{a}\times\mathbb{R}^{+},$

in order to obtain a Palais-Smale sequence approaching the Pohozaev manifold

$\mathcal{P}_{0}:=\left\\{u\in
S_{a}:\frac{\mathrm{d}}{\mathrm{d}h}E_{0}(u_{h})\Big{|}_{h=1}=0\right\\}.$

For more references on normalized solution problem with Sobolev subcritical
growth, we refer to [20, 12] and references therein. A lot of works have been
done following Jeanjean’s method, see for instance [3, 17, 12, 21, 20]. Here
we are mainly concerned with the Sobolev critical growth problem.

Recently, Soave [21] considered the combined nonlinearities involving Sobolev
critical exponent

$\begin{cases}-\Delta u=\lambda u+\mu|u|^{p-2}u+|u|^{2^{*}-2}u\;\;\text{ in
}\mathbb{R}^{N},\\\ \|u\|_{2}=a>0,\\\ \end{cases}$ (1.12)

where $N\geq 3$ and $\mu\in\mathbb{R}$. The corresponding energy functional to
(1.12) is

$\displaystyle J_{\mu}(u):=\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{\mu}{p}\|u\|_{p}^{p}-\frac{1}{2^{*}}\|u\|_{2^{*}}^{2^{*}},\;\;u\in
S_{a}.$

Recall that a ground state of (1.12) is a solution having the minimal energy
among all solutions. The ground state of (1.12) can be obtained by finding the
minimizers of $J_{\mu}$ on the Pohozaev manifold

$\mathcal{P}_{\mu}:=\\{u\in S_{a}:\|\nabla
u\|_{2}^{2}-\mu\gamma_{p}\|u\|_{p}^{p}-\|u\|_{2^{*}}^{2^{*}}=0\\}.$

Let

$m_{a}:=\underset{u\in\mathcal{P}_{\mu}}{\inf}{J_{\mu}(u)}.$ (1.13)

Soave investigated the existence of ground state solutions to (1.12) for
different exponent $p$. He proved that for $p\in(2,2^{*})$, (1.13) can be
attained provided $N\geq 3$, $\mu$, $a>0$ and

$\mu a^{{p(1-\gamma_{p})}}<\alpha_{N,\,p}$ (1.14)

for some $\alpha_{N,\,p}>0$, see [21, Theorem 1.1]. Also, he proposed some
natural questions, including

  * ($Q_{1}$)

: Does $J_{\mu}|_{S_{a}}$ have a critical point of mountain-pass type in the
case $p\in(2,\hat{p})$?

  * ($Q_{2}$)

: Does $J_{\mu}|_{S_{a}}$ have a ground state if $\mu>0$ and $\mu
a^{{p(1-\gamma_{p})}}$ is large?

Jeanjean and Le [11] and Wei and Wu [24] answered question $(Q_{1})$ for
$N\geq 4$ and $N=3$ respectively. Under the condition (1.14), the authors in
[11, 24] showed that $J_{\mu}$ has a mountain pass geometry around the ground
state, and the mountain pass level $M_{a}$ can be upper bounded by

$M_{a}<m_{a}+\frac{1}{N}\mathcal{S}^{\frac{N}{2}}.$ (1.15)

It was verified in [11, Proposition 1.11] that for $N\geq 3$, if the mountain
pass level $M_{a}$ satisfies (1.15), then any radial Palais-Smale sequence
with level $M_{a}$ approaching the Pohozaev manifold $P_{\mu}$ is relatively
compact in $H^{1}(\mathbb{R}^{N})$. Moreover, Wei and Wu also answered
$(Q_{2})$ in [24] for $p\in[\hat{p},2^{*})$. Now we summarize the existence
results of [21, 24] for (1.12) with $p\in[\hat{p},2^{*})$:

######  Theorem 1.1.

Let $N\geq 3$, $\hat{p}\leq p<2^{*}$ and let $\mu,a>0$.

  * $(i)$

If $p=\hat{p}$, then for

$\mu a^{\frac{4}{N}}<\alpha:=\frac{\hat{p}}{2(C_{N,\hat{p}})^{\hat{p}}},$

$m_{a}$ can be attained by some $u$ which is positive, radially symmetric, and
a solution to (1.12) for some $\lambda<0$. Here $C_{N,\hat{p}}$ is the best
constant for Gagliardo-Nirenberg inequality given by (1.10). Moreover, $m_{a}$
can not be attained if $\mu a^{\frac{4}{N}}\geq\alpha$.

  * $(ii)$

If $\hat{p}<p<2^{*}$, then for any $\mu,a>0$, $m_{a}$ is attained by some $u$
which is positive, radially symmetric, and a solution to (1.12) with
$\lambda<0$.

######  Remark 1.2.

When $p=\hat{p}$, we can get the precise value of $\alpha$ owing to [21,
(5.1)] and [24, (3.13)].

###  1.2 Potential case $V\not\equiv 0$.

When $V\not\equiv 0$, many works were done with variational method to find
solutions of (1.4)-(1.5).

####  1.2.1 Mass-subcritical growth

When $f$ has mass-subcritical growth, for example, $f(u)=\sum_{1\leq j\leq
k}a_{j}|u|^{\sigma_{j}-2}u$ with $k\geq 1,a_{j}>0$, $N\geq 1$ and

$2<\sigma_{j}<\hat{p},$

the associated energy functional $E_{V}$ (see (1.6)) is bounded from below on
$S_{a}$. In this case, to get a minimizer, the main difficulty is to prove
that the minimum level is sub-additive with respect to the mass. For example:

  * •

In [9], Ikoma and Miyamoto assumed that

$V\in
C(\mathbb{R}^{N}),\;\;\underset{|x|\to\infty}{\lim}V(x)=\underset{x\in\mathbb{R}^{N}}{\sup}V(x)=0;$

if $N\geq 5$, the following additional assumption is imposed

$V\in C^{0,1}(\mathbb{R}^{N}),\;\;\nabla V(x)\cdot
x\leq\frac{(N-2)^{2}}{2|x|^{2}}\leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ a.e.\leavevmode\nobreak\
x\in\mathbb{R}^{N}\backslash\\{0\\}.$

They applied the concentration-compactness arguments of Lions [15, 16].

  * •

Zhong and Zou [26] considered $V\in C(\mathbb{R}^{N})$ satisfying

$\underset{|x|\to\infty}{\lim}V(x)=\underset{x\in\mathbb{R}^{N}}{\sup}V(x)\in(0,\infty],\;\;V(0)=\underset{x\in\mathbb{R}^{N}}{\min}V(x).$

They presented a new approach based on iteration to obtain the strictly sub-
additive inequality.

  * •

Alves and Ji [1] considered $f(u)=|u|^{p-2}u$ with $p\in(2,\hat{p})$, and a
positive, ${\mathbb{Z}}^{N}$-periodic (or asymptotically periodic) potential
$V$. In their proof, ${\mathbb{Z}}^{N}$-periodicity prevents the vanishing of
the minimizing sequence at infinity.

####  1.2.2 Mass-supercritical and Sobolev subcritical growth

When $f$ has mass-supercritical and Sobolev subcritical growth, for example
$f(u)=\sum_{1\leq j\leq k}a_{j}|u|^{\sigma_{j}-2}u$ with $k\geq 1,a_{j}>0$ and

$\hat{p}<\sigma_{j}<2^{*}\;\;\text{if}\leavevmode\nobreak\ N\geq
3;\;\;\hat{p}<\sigma_{j}\;\;\text{if}\leavevmode\nobreak\ N=1,2,$

the energy functional is unbounded from below on $S_{a}$. We often use the
mountain pass geometry to handle the existence issue of solutions to
(1.4)-(1.5).

  * •

Molle et al. [17] investigated the case $f(u)=|u|^{p-2}u$ with
$p\in(\hat{p},2^{*})$. They constructed a splitting lemma to obtain the
compactness of Palais-Smale sequence. If moreover

$\max\\{\|V\|_{N/2},\left\|V(x)|x|\right\|_{N}\\}<L,\;\;V(x)\leq 0,$

for some $L=L(N,p)>0$, they derived a mountain-pass solution at a positive
level.

  * •

Bartsch et al. [3] discussed the non-trapping potential case, with

$V(x)\geq\underset{|x|\to+\infty}{\lim\inf}\,V(x)>-\infty,$

where the mountain pass structure by Jeanjean is destroyed. They constructed a
linking geometry developed by [4], and using a minimax argument, they obtained
the existence of solutions with high Morse index.

  * •

In [6], Ding and Zhong treated the case for more general $f$ with mass super-
critical and Sobolev subcritical growth, satisfying some Ambrosetti-Rabinowitz
type condition. They assumed that $V$ is negative, twice differentiable a.e.
in $\mathbb{R}^{N}$ and required some compactness conditions or Poincaré
inequality on $V$, which yields that the Pohozaev manifold is a natural
constraint. By considering a minimizing sequence on the Pohozaev manifold,
they derived existence results.

  * •

The Lyapunov-Schmidt reduction approach also has been applied to problem
(1.4)-(1.5), see [19, Section 3].

###  1.3 Existence results with Sobolev critical growth

It seems that all the above works can not be directly extended to the
nonlinearities involving Sobolev critical growth. As far as we are aware, we
did not find any works on normalized solution with Sobolev critical growth and
potential $V\not\equiv 0$. Motivated by [24, 21] for the non-potential case,
we consider (1.1) with mass-critical or mass-supercritical, and Sobolev
critical nonlinearities. For potential $V\in C^{1}(\mathbb{R}^{N})$, we will
use the following assumptions:

  * $(V_{1})$

$\lim\limits_{|x|\rightarrow\infty}V(x)=\underset{x\in\mathbb{R}^{N}}{\sup}V(x)=:V_{\infty}<\infty$
and there exists $\sigma_{1}>0$ such that

$\int_{\mathbb{R}^{N}}\left|V-V_{\infty}\right|u^{2}\mathrm{d}x\leq\sigma_{1}\|\nabla
u\|^{2}_{2},\;\;\forall\,u\in H^{1}(\mathbb{R}^{N}).$ (1.16)

  * $(V_{2})$

Let $W(x):=\frac{1}{2}\langle\nabla V(x),x\rangle$, there holds
$\lim\limits_{|x|\rightarrow\infty}W(x)=0$ and there is $\sigma_{2}>0$ such
that

$\left|\int_{\mathbb{R}^{N}}\left[\frac{1}{2}(V-V_{\infty})+\frac{1}{p\gamma_{p}}W\right]u^{2}\mathrm{d}x\right|\leq\sigma_{2}\|\nabla
u\|^{2}_{2},\;\;\forall\,u\in H^{1}(\mathbb{R}^{N}).$ (1.17)

  * $(\widetilde{V}_{2})$

There exists $\widetilde{\sigma}_{2}>0$ such that

$\left|\int_{\mathbb{R}^{N}}Wu^{2}\mathrm{d}x\right|\leq\widetilde{\sigma}_{2}\|u\|^{2}_{H^{1}},\;\;\forall\,u\in
H^{1}(\mathbb{R}^{N}).$ (1.18)

  * $(V_{3})$

$\underset{|x|\to\infty}{\lim}|x|(V-V_{\infty})(x)=0$. There exists
${\sigma_{3}}>0$ such that

$\int_{\mathbb{R}^{N}}\left(V-V_{\infty}\right)^{2}|x|^{2}u^{2}\mathrm{d}x\leq{\sigma_{3}^{2}}\|\nabla
u\|^{2}_{2},\;\;\forall\,u\in H^{1}(\mathbb{R}^{N}).$ (1.19)

  * $(V_{4})$

$V+W\leq V_{\infty}$ for any $x\in\mathbb{R}^{N}.$

######  Remark 1.3.

The condition (1.19) implies (1.16). In fact, using Hardy’s inequality, we
have

$\int_{\mathbb{R}^{N}}\left|V-V_{\infty}\right|u^{2}\mathrm{d}x\leq\|(V-V_{\infty})|x|u\|_{2}\left\|\frac{u}{|x|}\right\|_{2}\leq\frac{2}{N-2}\sigma_{3}\|\nabla
u\|_{2}^{2}.$

Thus $\sigma_{1}\leq\frac{2}{N-2}\sigma_{3}$.

######  Remark 1.4.

In $(V_{1})$, if $V_{\infty}\neq 0$, one may replace $(V,\lambda)$ by
$(V-V_{\infty},\lambda+V_{\infty})$. Hence, without loss of generality, we can
assume that $V_{\infty}=0$.

By virtue of Remark 1.4, from now on, we assume that $V_{\infty}=0$. Problem
(1.1) can be regarded as a perturbation problem of [21, 24], whose energy
functional is given by

$\displaystyle J_{\mu,V}(u)=\frac{1}{2}\|\nabla
u\|_{2}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x-\frac{\mu}{p}\|u\|_{p}^{p}-\frac{1}{2^{*}}\|u\|_{2^{*}}^{2^{*}}.$

Recall that for $p\in[\hat{p},2^{*})$, the solutions of [21, Theorem 1.1] and
[24, Theorem 1.1] are mountain pass type solutions of Jeanjean. In present
paper, we seek for $V$ to preserve the mountain pass geometry of $J_{\mu,V}$
on $S_{a}$. When different integrability assumptions are required on $V$, the
Pohozaev identity for (1.1) could have two different type of forms, that is,

  * $(a)$

if $V\in L_{loc}^{\frac{N}{2}}(\mathbb{R}^{N})$ satisfies (1.19), any solution
to (2.3) will satisfy

$P_{\mu,V}(u):=\|\nabla
u\|_{2}^{2}+\frac{N}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x+\int_{\mathbb{R}^{N}}V(x)u\nabla
u\cdot x\mathrm{d}x-\mu\gamma_{p}\|u\|_{p}^{p}-\|u\|_{2^{*}}^{2^{*}}=0;$
(1.20)

  * $(b)$

if $V\in C^{1}(\mathbb{R}^{N})$ and (1.16), (1.18) hold, any solution to (2.3)
will satisfy

$\widetilde{P}_{\mu,V}(u):=\|\nabla
u\|_{2}^{2}-\int_{\mathbb{R}^{N}}W(x)u^{2}\mathrm{d}x-\mu\gamma_{p}\|u\|_{p}^{p}-\|u\|_{2^{*}}^{2^{*}}=0.$
(1.21)

We will work with either (1.20) or (1.21) following different assumptions on
$V$. Our main results are the following.

######  Theorem 1.5.

Assume that $N\geq 3$, $\mu,a>0$ and $V\in C^{1}(\mathbb{R}^{N})$ satisfies
$(V_{1})$, $(\widetilde{V}_{2})$, $(V_{3})$, $(V_{4})$.

  * $(i)$

If $p=\hat{p}$, $\sigma_{1}<1$ and

$\sigma_{3}<\frac{2}{N-2}\;\;\text{if}\leavevmode\nobreak\ N\geq
4;\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{or}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{3\sigma_{1}+2{\sigma_{3}}}<4\;\;\text{if}\leavevmode\nobreak\ N=3,$ (1.22)

then there exists some $a_{0}=a_{0}(\mu,p,N)>0$ such that for any
$a\in(0,a_{0})$, problem (1.1) has a positive solution.

  * $(ii)$

If $p\in(\hat{p},2^{*})$, $\sigma_{1}<1$ and

${\sigma_{3}}<\frac{p\gamma_{p}}{2}-1\;\;\text{if}\leavevmode\nobreak\ N\geq
4;$ (1.23)
$(p\gamma_{p}-3)\sigma_{1}+2\sigma_{3}<p\gamma_{p}-2\;\;\text{if}\leavevmode\nobreak\
N=3,\,4<p<6,$ (1.24)

then for any $\mu,\,a>0$, problem (1.1) has a positive solution.

######  Theorem 1.6.

Assume that $N\geq 3$, $\mu,a>0$ and $V\in C^{1}(\mathbb{R}^{N})$ satisfies
$(V_{1})$, $(V_{2})$, $(V_{4})$.

  * $(i)$

If $p=\hat{p}$, $\sigma_{1}<1$ and

$\sigma_{2}<\frac{1}{N},$ (1.25)

there exists some $a_{0}=a_{0}(\mu,p,N)>0$ such that for any $a\in(0,a_{0})$,
problem (1.1) has a positive solution.

  * $(ii)$

If $p\in(\hat{p},2^{*})$, $\sigma_{1}<1$ and

$\sigma_{2}<\frac{1}{2}-\frac{1}{p\gamma_{p}},$ (1.26)

then for any $\mu,\,a>0$, problem (1.1) has a positive solution.

To prove Theorems 1.5, 1.6, we will follow Jeanjean’s method [10]. First, the
condition $\sigma_{1}<1$ in Theorems 1.5-1.6 is to preserve the mountain pass
geometry of $J_{\mu,V}$ on $S_{a}$ (see Lemma 3.2). Note that for the Sobolev
critical problems, the compactness of a Palais-Smale sequence often relies on
its energy level. For this reason, in section 3 we select the mountain pass
end points $\\{e_{0},e_{1}\\}$ located on the fiber of $v^{a}$, where $v^{a}$
is the solution of Theorem 1.1. Thus, the mountain pass level of $J_{\mu,\,V}$
has an upper bound $\frac{1}{N}\mathcal{S}^{\frac{N}{2}}$ (see Lemmas 2.1,
3.3).

Next, considering the augmented functional $\widetilde{J}_{\mu,V}$ given in
(3.4), it also has a mountain pass geometry on $S_{a}\times\mathbb{R}^{+}$with
the same mountain pass level as ${J}_{\mu,V}$, seeing Lemma 3.4. Applying
Proposition 2.7 to $\widetilde{J}_{\mu,V}|_{S_{a}\times\mathbb{R}^{+}}$, we
get a Palais-Smale sequence approaching the Pohozaev manifold

$\mathcal{P}_{\mu,V}:=\big{\\{}u\in
S_{a}:{P}_{\mu,V}(u)=0\big{\\}}\;\;\text{or}\;\;\widetilde{\mathcal{P}}_{\mu,V}:=\big{\\{}u\in
S_{a}:\widetilde{P}_{\mu,V}(u)=0\big{\\}}.$

In contrast with [6], we do not require the Pohozaev manifold as a natural
constraint. Thus the Pohozaev manifold is not required to be in $C^{1}$, hence
we just assume $V\in C^{1}(\mathbb{R}^{N})$. Following the method of [3, 17],
(1.22)-(1.26) are used to guarantee the boundedness of the Palais-Smale
sequence, and the nonnegativity of the energy for solutions to (1.1) (see
Lemmas 2.5, 2.6, 4.2). Due to the critical term and the potential term, the
sign of $\lambda$ cannot simply be concluded. Different from [17] and [6], we
will use the assumption $(V_{4})$ to compute the sign of $\lambda$ (see Lemma
2.4).

Furthermore, when considering the compactness of the Palais-Smale sequence in
$H^{1}(\mathbb{R}^{N})$, we cannot always work in the radial subspace
$H_{rad}^{1}(\mathbb{R}^{N})$ since the potential is not supposed to be
radial, which brings some new difficulties. At last, the Sobolev critical
growth yields further difficulties since the usual Lions’ lemma (see [25,
Theorem 1.34]) is no longer suitable. To overcome all these obstacles, we will
prove a splitting lemma (see Lemma 5.1) inspired by [3, 17] and [21,
Proposition 3.1].

When these parameters $\sigma_{1}$-$\sigma_{3}$, $\widetilde{\sigma}_{2}$ are
0, the equation (1.1) becomes the non-potential case, and Theorems 1.5, 1.6
will come back to Theorem 1.1.

######  Remark 1.7.

Notice that (1.16)-(1.19) are a kind of compactness assumptions or Poincaré
inequalities. Let $\alpha,\beta>0$ and

$V_{\alpha,\beta}=\left(\frac{\beta}{\alpha^{4}}|x|^{2}-\frac{2\beta}{\alpha^{2}}\right)\chi_{\\{|x|\leq\alpha\\}}-\frac{\beta}{|x|^{2}}\chi_{\\{|x|>\alpha\\}},$

where $\chi$ is the characteristic function. Whenever $\alpha$, $\beta$ are
positive but small enough, all the assumptions $(V_{1})$–$(V_{4})$ and
$(\widetilde{V}_{2})$ will be satisfied.

######  Remark 1.8.

Theorems 1.5 and 1.6 respectively use the Pohozaev identities (1.20) and
(1.21). When $p=\hat{p}$, the values of $a_{0}$ in Theorems 1.5, 1.6 can be
given from (2.13), (2.14), (2.17) and (3.1). By Lemma 2.4, we also know that
$\lambda<-V_{\infty}$ in Theorems 1.5, 1.6.

######  Remark 1.9.

When $V\in C_{loc}^{0,\alpha}(\mathbb{R}^{N},\mathbb{R})$ for some
$\alpha\in(0,1)$, by standard elliptic theory ([22, B.3 Lemma, B.2 Theorem]),
any solution in $H^{1}(\mathbb{R}^{N})$ of (1.1) is a classical solution.

This paper is organized as follows. In Section 2, we introduce some notations
and preliminary results. In Section 3, we verify the mountain pass structure
of $J_{\mu,V}$ on $S_{a}$. In section 4, we derive the existence of a Palais-
Smale sequence approaching Pohozaev manifold. In section 5, we complete the
proof of Theorem 1.5 and Theorem 1.6.

##  2 Notations and preliminary results

For $p\in[1,\infty]$, we denote by $L^{p}(\mathbb{R}^{N})$ the Lebesgue’s
space with norm $\|\cdot\|_{p}$ and by $H^{1}(\mathbb{R}^{N})$,
$D^{1,2}(\mathbb{R}^{N})$ the usual Sobolev spaces. We always denote
$\gamma_{p}=\frac{N(p-2)}{2p}$. Let $\mathcal{S}$ be the best Sobolev constant
given in (1.11), $C_{N,p}$ be the optimal constant for Gagliardo-Nirenberg
inequality given in (1.10). For $u\in H^{1}(\mathbb{R}^{N})$, $u_{h}$ denotes
the scaling transformation (1.9). $o_{n}(1)$ denotes a real sequence with
$o_{n}(1)\to 0$ as $n\to\infty$.

For $2\leq p\leq 2^{*}$, the Pohozaev identity of the equation

$-\Delta u=\lambda u+\mu|u|^{p-2}u+|u|^{2^{*}-2}u\leavevmode\nobreak\
\;\;\text{ in }\mathbb{R}^{N},$ (2.1)

is $P_{\mu}(u)=0$, where for any $u\in H^{1}(\mathbb{R}^{N})$,

$P_{\mu}(u):=\|\nabla
u\|_{2}^{2}-\mu\gamma_{p}\|u\|_{p}^{p}-\|u\|_{2^{*}}^{2^{*}}.$ (2.2)

We state a series of preliminary results. We begin with giving the Pohozaev
identity for $u\in H^{1}(\mathbb{R}^{N})$, solution to

$-\Delta u+V(x)u=\lambda u+\mu|u|^{p-2}u+|u|^{2^{*}-2}u\;\;\text{ in
}\mathbb{R}^{N}.$ (2.3)

Let $h>0,u\in S_{a}$, we denote a fiber function of $u$ as

$\displaystyle\begin{split}\Psi_{u}(h)&=J_{\mu,V}(u_{h})=\frac{h^{2}}{2}\|\nabla
u\|_{2}^{2}+\frac{h^{N}}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}(hx)\mathrm{d}x-\frac{\mu
h^{p\gamma_{p}}}{p}\|u\|_{p}^{p}-\frac{h^{2^{*}}}{2^{*}}\|u\|_{2^{*}}^{2^{*}}\\\
&=\frac{h^{2}}{2}\|\nabla
u\|_{2}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(h^{-1}x)u^{2}\mathrm{d}x-\frac{\mu
h^{p\gamma_{p}}}{p}\|u\|_{p}^{p}-\frac{h^{2^{*}}}{2^{*}}\|u\|_{2^{*}}^{2^{*}}.\end{split}$
(2.4)

Using Lemma A.1 (see Appendix), the condition (1.19) yields
$\Psi^{\prime}_{u}(1)={P}_{\mu,V}(u)$ where ${P}_{\mu,V}$ is given in (1.20).
On the other hand, if $V\in C^{1}(\mathbb{R}^{N})$ and (1.16), (1.18) hold,
then $\Psi^{\prime}_{u}(1)=\widetilde{P}_{\mu,V}(u)$ where
$\widetilde{P}_{\mu,V}$ is defined in (1.21). Using Proposition A.2, any
solution of (2.3) satisfies the Pohozaev identity

$\begin{cases}{P}_{\mu,V}(u)=0\;\;\text{if \leavevmode\nobreak\ }V\in
L_{loc}^{\frac{N}{2}}(\mathbb{R}^{N})\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \eqref{2301271916}\leavevmode\nobreak\
\text{hold},\\\ \widetilde{P}_{\mu,V}(u)=0\;\;\text{if \leavevmode\nobreak\
}V\in C^{1}(\mathbb{R}^{N})\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\eqref{2301262110},\,\eqref{2301271518}\leavevmode\nobreak\
\text{hold}.\end{cases}$ (2.5)

The following result comes from [24, Lemma 3.3], it showed the monotonicity
and the bound of $m_{a}$.

######  Lemma 2.1.

Let $N\geq 3$ and $\hat{p}\leq p<2^{*}$. Then there exists $\alpha_{N,p}$ only
depending on $N,p$, such that $m_{a}$ is strictly decreasing for
$0<\mu<a^{{p\gamma_{p}-p}}\alpha_{N,p}$ and is nonincreasing for $\mu\geq
a^{{p\gamma_{p}-p}}\alpha_{N,p}$, where $m_{a}$ is given by (1.13). Moreover,
$0<m_{a}<\frac{1}{N}\mathcal{S}^{\frac{N}{2}}$ for all $\mu>0$ if
$\hat{p}<p<2^{*}$, while $m_{a}=0$ for $\mu\geq
a^{{p\gamma_{p}-p}}\alpha_{N,p}$ if $p=\hat{p}$.

As already mentioned, when $p=\hat{p}$, there holds
$\alpha_{N,\hat{p}}=\frac{\hat{p}}{2(C_{N,\hat{p}})^{\hat{p}}}$, see the
formula[24, (3.13)]. Note that when $\mu,a$ satisfy

$\mu,\leavevmode\nobreak\ a>0\;\;\text{if}\leavevmode\nobreak\
p\in(\hat{p},2^{*});\;\;\mbox{or }\;\;0<\mu
a^{\frac{4}{N}}<\frac{\hat{p}}{2(C_{N,\hat{p}})^{\hat{p}}}\;\;\text{if}\leavevmode\nobreak\
p=\hat{p},$ (2.6)

the functional $J_{\mu}$ has a mountain pass geometry on $S_{a}$. In
particular, for any $u\in S_{a}$, $J_{\mu}(u_{h})$ has a unique maximal point
on $\mathbb{R}^{+}$ and

$m_{a}=\underset{u\in
S_{a}}{\inf}\underset{h\in\mathbb{R}^{+}}{\max}J_{\mu}(u_{h}).$ (2.7)

Here $u_{h}$ is given in (1.9). We have the following monotonicity result.

######  Lemma 2.2.

Assume that $N\geq 3$, $\hat{p}\leq p<2^{*}$, $\mu,\,a$ satisfy (2.6) and
$m_{a}$ is defined in (1.13). Then $m_{a}$ is non-increasing with respect to
$a$.

###### Proof.

Let $a_{2}>a_{1}>0$, and $\theta=\frac{a_{2}}{a_{1}}>1$. For any $u_{1}\in
S_{a_{1}}$, let

$u_{2}(x):=\theta^{\frac{2-N}{2}}u_{1}(\theta^{-1}x).$

By a direct computation, we have $\|\nabla u_{2}\|_{2}=\|\nabla u_{1}\|_{2}$,
$\|u_{2}\|_{2}=\theta\|u_{1}\|_{2}$,
$\|u_{2}\|_{p}^{p}=\theta^{\frac{p(2-N)+2N}{2}}\|u_{1}\|_{p}^{p}$ and
$\|u_{2}\|_{2^{*}}=\|u_{1}\|_{2^{*}}$. By (2.7), there exists some $h_{u_{2}}$
such that

$\displaystyle m_{a_{2}}$
$\displaystyle\leq\underset{h\in\mathbb{R}^{+}}{\max}J_{\mu}((u_{2})_{h})=J_{\mu}((u_{2})_{h_{u_{2}}})$
$\displaystyle=\frac{1}{2}\|\nabla(u_{1})_{h_{u_{2}}}\|_{2}^{2}-\frac{\mu}{p}\theta^{\frac{p(2-N)+2N}{2}}\|(u_{1})_{h_{u_{2}}}\|_{p}^{p}-\frac{1}{2^{*}}\|(u_{1})_{h_{u_{2}}}\|_{2^{*}}^{2^{*}}$
$\displaystyle\leq\frac{1}{2}\|\nabla(u_{1})_{h_{u_{2}}}\|_{2}^{2}-\frac{\mu}{p}\|(u_{1})_{h_{u_{2}}}\|_{p}^{p}-\frac{1}{2^{*}}\|(u_{1})_{h_{u_{2}}}\|_{2^{*}}^{2^{*}}$
$\displaystyle\leq\underset{h\in\mathbb{R}^{+}}{\max}J_{\mu}((u_{1})_{h}).$

By the definition of $m_{a_{1}}$, we have $m_{a_{1}}\geq m_{a_{2}}$. ∎

In the sequel, let $P_{\mu}$ be given in (2.2). We have the following lemma.

######  Lemma 2.3.

Assume that $N\geq 3$, $\hat{p}\leq p<2^{*}$ and $\mu,\,a$ satisfy (2.6). Let
$\\{u_{n}\\}\subset S_{a}$ be a bounded sequence in $H^{1}(\mathbb{R}^{N})$
such that $P_{\mu}(u_{n})\rightarrow 0$. If $\|\nabla u_{n}\|_{2}^{2}>c>0$,
then

$m_{a}\leq\underset{n\rightarrow\infty}{\lim\inf}J_{\mu}(u_{n}).$ (2.8)

###### Proof.

Since $P_{\mu}(u_{n})\rightarrow 0$, we have

$\|\nabla
u_{n}\|_{2}^{2}-\mu\gamma_{p}\|u_{n}\|_{p}^{p}-\|u_{n}\|_{2^{*}}^{2^{*}}=o_{n}(1).$
(2.9)

There exists a sequence $h_{n}>0$ such that
$(u_{n})_{h_{n}}\in\mathcal{P}_{\mu}$, that is,

$\|\nabla(u_{n})_{h_{n}}\|_{2}^{2}-\mu\gamma_{p}\|(u_{n})_{h_{n}}\|_{p}^{p}-\|(u_{n})_{h_{n}}\|_{2^{*}}^{2^{*}}=0,$

where $(u_{n})_{h_{n}}$ is given in (1.9). By direct computation,

$\displaystyle h_{n}^{2}\|\nabla
u_{n}\|_{2}^{2}-\mu\gamma_{p}h_{n}^{p\gamma_{p}}\|u_{n}\|_{p}^{p}-h_{n}^{2^{*}}\|u_{n}\|_{2^{*}}^{2^{*}}=0.$
(2.10)

Combining (2.9) with (2.10), we obtain

$\mu\gamma_{p}(h_{n}^{p\gamma_{p}-2}-1)\|u_{n}\|_{p}^{p}+(h_{n}^{2^{*}-2}-1)\|u_{n}\|_{2^{*}}^{2^{*}}=o_{n}(1).$

From $\|\nabla u_{n}\|_{2}^{2}>c>0$ and (2.9), it follows that
$\\{\|u_{n}\|_{2^{*}}^{2^{*}}\\}$ has a positive lower bound, which deduces
that $h_{n}\rightarrow 1$. So

$J_{\mu}(u_{n})=J_{\mu}((u_{n})_{h_{n}})+o_{n}(1)\geq m_{a}+o_{n}(1).$

Thus (2.8) holds. ∎

Now, we claim a non-existence result of solutions to (2.3), which yields the
sign of the Lagrange multiplier $\lambda$ in (1.1).

######  Lemma 2.4.

Suppose that $N\geq 3$, $p\in(2,2^{*})$, and $V\in C^{1}(\mathbb{R}^{N})$
satisfies (1.16), $(\widetilde{V}_{2})$, $(V_{4})$, then the equation (2.3)
has no nontrivial solution provided $\lambda\geq 0$.

###### Proof.

Assume that $u$ is a nontrivial solution to (2.3). Multiplying (2.3) by $u$,
we obtain

$\|\nabla
u\|_{2}^{2}+\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x=\lambda\|u\|_{2}^{2}+\mu\|u\|_{p}^{p}+\|u\|_{2^{*}}^{2^{*}}.$
(2.11)

By Proposition A.2, the Pohozaev identity (1.21) holds. Combining (2.11) with
(1.21), we have

$\int_{\mathbb{R}^{N}}(V(x)+W(x))u^{2}\mathrm{d}x=\mu(1-\gamma_{p})\|u\|_{p}^{p}+\lambda\|u\|^{2}_{2}.$
(2.12)

From $(V_{4})$ and $\gamma_{p}<1$, it follows that $\lambda<0$. ∎

Next, we are going to prove that under suitable condition, the functional
$J_{\mu,\,V}$ for any solution to (2.3) is nonnegative.

######  Lemma 2.5.

Suppose that $N\geq 3$, $p\in[\hat{p},2^{*})$, and $V\in
L_{loc}^{\frac{N}{2}}(\mathbb{R}^{N})$ satisfies $(V_{1})$ and $(V_{3})$ with

  * $(i)$

either $p=\hat{p}$,

${\sigma_{3}}<\frac{2}{N-2}\left(1-\frac{2\mu}{\hat{p}}(C_{N,\hat{p}})^{\hat{p}}a^{\frac{4}{N}}\right)\;\;\text{if}\leavevmode\nobreak\
N\geq 4;$ (2.13)
$3\sigma_{1}+2\sigma_{3}<4\left(1-\frac{2\mu}{\hat{p}}(C_{3,\hat{p}})^{\hat{p}}a^{\frac{4}{3}}\right)\;\;\text{if}\leavevmode\nobreak\
N=3;$ (2.14)

  * $(ii)$

or $p\in(\hat{p},2^{*})$,

${\sigma_{3}}<\frac{p\gamma_{p}}{2}-1\;\;\text{if}\leavevmode\nobreak\ N\geq
4;$ (2.15)

$(p\gamma_{p}-3)\sigma_{1}+2\sigma_{3}<p\gamma_{p}-2\;\;\text{if}\leavevmode\nobreak\
N=3,\,4<p<6.$ (2.16)

Then any solution to (2.3) satisfies $J_{\mu,V}(u)\geq 0$.

###### Proof.

Since $u$ is a solution to (2.3), by Proposition A.2, $u$ satisfies the
Pohozaev identity (1.20). First, suppose $p\in(2+\frac{4}{N},2^{*})$. Taking
(1.20) into $J_{\mu,V}$, we get

$\displaystyle J_{\mu,V}(u)$ $\displaystyle=\frac{1}{2}\|\nabla
u\|_{2}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x-\frac{\mu}{p}\|u\|_{p}^{p}-\frac{1}{2^{*}}\|u\|_{2^{*}}^{2^{*}}$
$\displaystyle\geq\frac{1}{2}\|\nabla
u\|_{2}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x-\frac{1}{p\gamma_{p}}\left(\mu\gamma_{p}\|u\|_{p}^{p}+\|u\|_{2^{*}}^{2^{*}}\right)$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{p\gamma_{p}}\right)\|\nabla
u\|_{2}^{2}+\left(\frac{1}{2}-\frac{N}{2p\gamma_{p}}\right)\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x-\frac{1}{p\gamma_{p}}\int_{\mathbb{R}^{N}}V(x)u\nabla
u\cdot x\mathrm{d}x$

If $N\geq 4$ (hence $N\geq p\gamma_{p}$ for all $p\geq\hat{p}$), by Hölder
inequality, $(V_{3})$, $V(x)\leq 0$ and (2.15), we have

$\displaystyle
J_{\mu,V}(u)\geq\left(\frac{1}{2}-\frac{1}{p\gamma_{p}}\right)\|\nabla
u\|_{2}^{2}-\frac{1}{p\gamma_{p}}{\sigma_{3}}\|\nabla u\|^{2}_{2}\geq 0.$

If $N=3$, $4<p<6$, we get $N<p\gamma_{p}$. Using Hölder inequality, $(V_{1})$,
$(V_{3})$ and (2.16), there holds

$\displaystyle
J_{\mu,V}(u)\geq\left(\frac{1}{2}-\frac{1}{p\gamma_{p}}\right)\|\nabla
u\|_{2}^{2}-\left(\frac{1}{2}-\frac{3}{2p\gamma_{p}}\right)\sigma_{1}\|\nabla
u\|_{2}^{2}-\frac{1}{p\gamma_{p}}{\sigma_{3}}\|\nabla u\|^{2}_{2}\geq 0.$

Suppose now $p=\hat{p}$. Similarly as above, taking (1.20) into $J_{\mu,V}$
and eliminating the term $\|u\|_{2^{*}}^{2^{*}}$, we get

$\displaystyle J_{\mu,V}(u)=$
$\displaystyle\left(\frac{1}{2}-\frac{1}{2^{*}}\right)\|\nabla
u\|_{2}^{2}+\frac{1}{2}\left(1-\frac{N}{2^{*}}\right)\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x-\left(\frac{1}{{\hat{p}}}-\frac{\gamma_{\hat{p}}}{2^{*}}\right)\mu\|u\|_{\hat{p}}^{\hat{p}}$
$\displaystyle-\frac{1}{2^{*}}\int_{\mathbb{R}^{N}}V(x)u\nabla u\cdot
x\mathrm{d}x$ $\displaystyle\geq$
$\displaystyle\left[\frac{1}{2}-\frac{1}{2^{*}}-\left(\frac{1}{{\hat{p}}}-\frac{\gamma_{\hat{p}}}{2^{*}}\right)\mu(C_{N,\,{\hat{p}}})^{{\hat{p}}}a^{\frac{4}{N}}-\frac{{\sigma_{3}}}{2^{*}}\right]\|\nabla
u\|_{2}^{2}+\frac{1}{2}\left(1-\frac{N}{2^{*}}\right)\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x$

If $N\geq 4$ (so $N\geq 2^{*}$), by $V(x)\leq 0$ and (2.13), we get

$J_{\mu,V}(u)\geq\frac{1}{2^{*}}\left[\frac{2}{N-2}\left(1-\mu\gamma_{\hat{p}}(C_{N,{\hat{p}}})^{{\hat{p}}}a^{\frac{4}{N}}\right)-\sigma_{3}\right]\|\nabla
u\|_{2}^{2}\geq 0.$

If $N=3$, then $N<2^{*}$. By $(V_{1})$ and (2.14), there holds

$J_{\mu,V}(u)\geq\frac{1}{2}\left[\frac{2}{3}\left(1-\mu\gamma_{\hat{p}}(C_{3,\hat{p}})^{\hat{p}}a^{\frac{4}{3}}\right)-\frac{1}{3}{\sigma_{3}}-\frac{1}{2}\sigma_{1}\right]\|\nabla
u\|_{2}^{2}\geq 0.$

The proof is completed. ∎

######  Lemma 2.6.

Suppose that $N\geq 3$, $p\in[\hat{p},2^{*})$ and $V\in C^{1}(\mathbb{R}^{N})$
satisfies $(V_{1})$, $(V_{2})$ with

  * $(i)$

either $p=\hat{p}$,

$\sigma_{2}<\frac{1}{N}\left(1-\frac{2\mu}{\hat{p}}(C_{N,\hat{p}})^{\hat{p}}a^{\frac{4}{N}}\right);$
(2.17)

  * $(ii)$

or $p\in(\hat{p},2^{*})$,

$\sigma_{2}<\frac{1}{2}-\frac{1}{p\gamma_{p}}.$ (2.18)

If $u$ is a solution to (2.3), then $J_{\mu,V}(u)\geq 0$.

###### Proof.

Since $u$ is a solution to (2.3), by Proposition A.2, $u$ satisfies the
Pohozaev identity (1.21). Suppose $p\in(\hat{p},2^{*})$. Taking (1.21) into
$J_{\mu,V}$ and using $(V_{1})$, $(V_{2})$ and (2.18), we have

$\displaystyle J_{\mu,V}(u)$ $\displaystyle\geq\frac{1}{2}\|\nabla
u\|_{2}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x-\frac{1}{p\gamma_{p}}\left(\mu\gamma_{p}\|u\|_{p}^{p}+\|u\|_{2^{*}}^{2^{*}}\right)$
$\displaystyle=\frac{1}{2}\|\nabla
u\|_{2}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x-\frac{1}{p\gamma_{p}}\left(\|\nabla
u\|_{2}^{2}-\int_{\mathbb{R}^{N}}W(x)u^{2}\mathrm{d}x\right)$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{p\gamma_{p}}\right)\|\nabla
u\|_{2}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x+\frac{1}{p\gamma_{p}}\int_{\mathbb{R}^{N}}W(x)u^{2}\mathrm{d}x$
$\displaystyle\geq\left(\frac{1}{2}-\frac{1}{p\gamma_{p}}-\sigma_{2}\right)\|\nabla
u\|_{2}^{2}$ $\displaystyle\geq 0.$

If now $p=\hat{p}$. Taking (1.21) into $J_{\mu,V}$ and eliminating the term
$\|u\|_{2^{*}}^{2^{*}}$ and by $(V_{1})$, $(V_{2})$, (2.17), there holds

$\displaystyle J_{\mu,V}(u)$
$\displaystyle=\left(\frac{1}{2}-\frac{1}{2^{*}}\right)\|\nabla
u\|_{2}^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x+\frac{1}{2^{*}}\int_{\mathbb{R}^{N}}W(x)u^{2}\mathrm{d}x-\left(\frac{1}{p}-\frac{\gamma_{\hat{p}}}{2^{*}}\right)\mu\|u\|_{\hat{p}}^{\hat{p}}$
$\displaystyle\geq\left(\frac{1}{2}\left(1-\frac{2}{2^{*}}\right)\left(1-\mu\gamma_{\hat{p}}C_{N,\,{\hat{p}}}^{{\hat{p}}}a^{\frac{4}{N}}\right)-\sigma_{2}\right)\|\nabla
u\|_{2}^{2}$ $\displaystyle\geq 0.$

The proof is completed. ∎

To find a Palais-Smale sequence approaching Pohozaev manifold, the following
frame will be often used, which is a special case of [7, Theorem 4.5] (see
also [3, Lemma 3.1]):

######  Proposition 2.7.

Let $X$ be a Hilbert manifold and $J\in C^{1}(X,\mathbb{R})$. Let $K\subset X$
be compact and consider a family

$\mathcal{E}\subset\\{E\subset X:E\leavevmode\nobreak\
\text{is}\leavevmode\nobreak\ \text{compact},\leavevmode\nobreak\ K\subset
E\\}$

which is invariant with respect to all deformations leaving $K$ fixed. Assume
that

$\underset{u\in K}{\max}J(u)<c:=\underset{E\in\mathcal{E}}{\inf}\underset{u\in
E}{\max}J(u)\in\mathbb{R}.$

Let $\sigma_{n}\in\mathbb{R}$, $\sigma_{n}\to 0$ and $E_{n}\in\mathcal{E}$ be
a sequence such that

$c\leq\underset{u\in E_{n}}{\max}J(u)<c+\sigma_{n}.$

Then there exists a sequence $v_{n}\in X$ such that for some $C^{\prime}>0$,

$(i)\;c\leq
J(v_{n})<c+\sigma_{n};\;\;(ii)\;\|\nabla_{X}J(v_{n})\|<C^{\prime}\sqrt{\sigma_{n}};\;\;(iii)\;\text{dist}(v_{n},E_{n})<C^{\prime}\sqrt{\sigma_{n}}.$

##  3 Mountain pass structure

In this section, we are devoted to construct the mountain pass geometry of
$J_{\mu,V}$ on $S_{a}$. Let $u_{h}$ be given in (1.9).

######  Lemma 3.1.

Assume that $N\geq 3$, $p\in(2,2^{*})$, and $(V_{1})$ holds. Then for any
$u\in S_{a}$, $\underset{h\to 0^{+}}{\lim}J_{\mu,V}(u_{h})=0$ and
$\underset{h\to\infty}{\lim}J_{\mu,V}(u_{h})=-\infty$.

###### Proof.

Let $u\in S_{a}$. By direct computation, we have

$\displaystyle J_{\mu,V}(u_{h})\leq\frac{h^{2}}{2}\|\nabla
u\|_{2}^{2}-\frac{\mu
h^{p\gamma_{p}}}{p}\|u\|_{p}^{p}-\frac{h^{2^{*}}}{2^{*}}\|u\|_{2^{*}}^{2^{*}}.$

For $p\in(2,2^{*})$, we deduce that
$\underset{h\to\infty}{\lim}J_{\mu,V}(u_{h})=-\infty$. On the other hand, by
$(V_{1})$,

$\displaystyle\left|J_{\mu,V}(u_{h})\right|\leq\left(\frac{1}{2}+\sigma_{1}\right)h^{2}\|\nabla
u\|_{2}^{2}+\frac{\mu
h^{p\gamma_{p}}}{p}\|u\|_{p}^{p}+\frac{h^{2^{*}}}{2^{*}}\|u\|_{2^{*}}^{2^{*}}.$

Thus $\underset{h\to 0^{+}}{\lim}J_{\mu,V}(u_{h})=0$. ∎

Now, we consider for $k>0$,

$A_{k}=\\{u\in S_{a}:\|\nabla u\|_{2}<k\\}\;\;\text{and}\;\;\partial
A_{k}=\\{u\in S_{a}:\|\nabla u\|_{2}=k\\}.$

A key observation is that under $(V_{1})$, $J_{\mu,V}$ is lower bounded by a
positive constant on $\partial A_{k}$ for $k>0$ small enough.

######  Lemma 3.2.

Assume that $N\geq 3$, $p\in[\hat{p},2^{*})$, and $(V_{1})$ holds with

$\sigma_{1}<1\;\;\text{if}\leavevmode\nobreak\
p\in(\hat{p},2^{*});\;\;\text{or}\;\;\sigma_{1}<{1}-\frac{2\mu}{\hat{p}}(C_{N,\hat{p}})^{\hat{p}}a^{\frac{4}{N}}\;\;\,\text{if}\leavevmode\nobreak\
p=\hat{p}.\\\ $ (3.1)

Then there is some $k_{0}>0$ such that $J_{\mu,V}$ has a positive lower bound
$\beta$ on $\partial A_{k_{0}}$.

###### Proof.

By Gagliardo-Nirenberg inequality and Sobolev inequality,

$\displaystyle J_{\mu,V}(u)$ $\displaystyle\geq\frac{1}{2}\|\nabla
u\|_{2}^{2}-\frac{1}{2}\sigma_{1}\|\nabla
u\|_{2}^{2}-\frac{\mu}{p}\|u\|_{p}^{p}-\frac{1}{2^{*}}\|u\|_{2^{*}}^{2^{*}}$
$\displaystyle\geq\frac{1-\sigma_{1}}{2}\|\nabla
u\|_{2}^{2}-\frac{\mu(C_{N,p})^{p}}{p}a^{{p(1-\gamma_{p})}}\|\nabla
u\|_{2}^{p\gamma_{p}}-\frac{1}{2^{*}}\mathcal{S}^{-\frac{2^{*}}{2}}\|\nabla
u\|_{2}^{2^{*}}.$

Then there exist $k_{0}>0$ and $\beta>0$ such that $J_{\mu,V}(u)>\beta$ for
$u\in\partial A_{k_{0}}$. ∎

Note that by Lemmas 3.1 and 3.2, the condition (3.1) can preserve the mountain
pass geometry of $J_{\mu,V}$. Moreover, (3.1) implies (2.6), which yields that
$m_{a}$ is well defined. Let $v^{a}$ be the solution of Theorem 1.1. That is,

$J_{\mu}(v^{a})=m_{a}.$ (3.2)

According to Lemma 3.1, we have that $\underset{h\to
0^{+}}{\lim}J_{\mu,V}(v^{a}_{h})=0$ and
$\underset{h\to\infty}{\lim}J_{\mu,V}(v^{a}_{h})=-\infty$. Hence there exist
$h_{1}>h_{0}>0$ such that $e_{0}:=v^{a}_{h_{0}}\in A_{k_{0}}$,
$e_{1}:=v^{a}_{h_{1}}\in S_{a}\backslash A_{k_{0}}$, $J_{\mu,V}(e_{0})<\beta$
and $J_{\mu,V}(e_{1})<0$, where $\beta$ is given in Lemma 3.2. Thus, the
mountain pass paths are given by

$\Gamma:=\\{\xi\in C([0,1],S_{a}):\xi(0)=e_{0},\,\xi(1)=e_{1}\\},$

and the mountain pass level is

$m_{V,\,a}:=\underset{\xi\in\Gamma}{\inf}\underset{t\in[0,1]}{\max}{J_{\mu,V}(\xi(t))}.$

Obviously,

$m_{V,\,a}\geq\beta>0.$ (3.3)

Moreover, $m_{V,\,a}$ has an upper bound $m_{a}$ given in (1.13).

######  Lemma 3.3.

Assume that $N\geq 3$, $p\in[\hat{p},2^{*})$, $(V_{1})$ holds with (3.1) and
$V\not\equiv 0$. Then $m_{a}>m_{V,\,a}$.

###### Proof.

Let $v^{a}$ be given in (3.2). By Theorem 1.1, $v^{a}$ is positive. By Lemma
3.1, there exists some $\widetilde{h}$ such that
$\underset{h\in\mathbb{R}^{+}}{\max}J_{\mu,V}(v^{a}_{h})=J_{\mu,V}(v^{a}_{\widetilde{h}})$.
Therefore, using (2.7), we have

$\displaystyle m_{V,\,a}$
$\displaystyle\leq\underset{h\in\mathbb{R}^{+}}{\max}J_{\mu,V}(v^{a}_{h})=J_{\mu,V}(v^{a}_{\widetilde{h}})<J_{\mu}(v^{a}_{\widetilde{h}})\leq\underset{h\in\mathbb{R}^{+}}{\max}J_{\mu}(v^{a}_{h})=m_{a}.$

So we are done. ∎

In order to derive a bounded Palais-Smale sequence of $J_{\mu,V}$ at
$m_{V,\,a}$, we follow the idea in [10] (see also [3, 17]) to construct an
augmented functional

$\widetilde{J}_{\mu,V}(u,h):=J_{\mu,V}(u_{h}),\;\;(u,h)\in
H^{1}(\mathbb{R}^{N})\times\mathbb{R}^{+}$ (3.4)

where $u_{h}$ is defined in (1.9). Actually $\widetilde{J}_{\mu,V}$ also
possesses the mountain pass structure on $S_{a}\times\mathbb{R}^{+}$. The new
mountain pass paths on $S_{a}\times\mathbb{R}^{+}$ is given by

$\widetilde{\Gamma}:=\\{\widetilde{\xi}\in
C([0,1],S_{a}\times\mathbb{R}^{+}):\widetilde{\xi}(0)=(e_{0},1),\,\widetilde{\xi}(1)=(e_{1},1)\\}.$

Any path $\widetilde{\xi}\in\widetilde{\Gamma}$ can be represented by

$\widetilde{\xi}(t)=(\xi(t),s(t))\;\;\text{with}\leavevmode\nobreak\
\xi\in\Gamma,\leavevmode\nobreak\ s\in
C([0,1],\mathbb{R}^{+})\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
s(0)=s(1)=1,$

which will intersect the set

$\widetilde{\partial A_{k_{0}}}:=\\{(u,h)\in
S_{a}\times\mathbb{R}^{+}:u_{h}\in\partial A_{k_{0}}\\}.$

Moreover, $\widetilde{J}_{\mu,V}(u)\geq\beta$ for any $u\in\widetilde{\partial
A_{k_{0}}}$. The new mountain pass level is

$\widetilde{m}_{V,\,a}:=\underset{\widetilde{\xi}\in\widetilde{\Gamma}}{\inf}\underset{t\in[0,1]}{\max}{\widetilde{J}_{\mu,V}(\widetilde{\xi}(t))}.$

######  Lemma 3.4.

Assume that $N\geq 3$, $p\in[\hat{p},2^{*})$ and $(V_{1})$ holds with (3.1).
Then $\widetilde{m}_{V,\,a}={m}_{V,\,a}$.

###### Proof.

Clearly, $\widetilde{m}_{V,\,a}$ and ${m}_{V,\,a}$ are well defined. For any
$\xi\in\Gamma$, we have $(\xi,1)\in\widetilde{\Gamma}$. Thus

$\widetilde{m}_{V,\,a}\leq\underset{\xi\in\Gamma}{\inf}\underset{t\in[0,1]}{\max}{\widetilde{J}_{\mu,V}(\xi(t),1)}=\underset{\xi\in\Gamma}{\inf}\underset{t\in[0,1]}{\max}{J_{\mu,V}(\xi(t))}={m}_{V,\,a}.$

On the other hand, for any $\widetilde{\xi}\in\widetilde{\Gamma}$, it can be
represented by $\widetilde{\xi}=(\xi,s)$ with
$\xi\in\Gamma,\leavevmode\nobreak\ s\in
C([0,1],\mathbb{R}^{+})\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
s(0)=s(1)=1$. One can get $\xi_{s}\in\Gamma$ where $\xi_{s}$ is given by
(1.9). Hence

${m}_{V,\,a}\leq\underset{t\in[0,1]}{\max}J_{\mu,V}(\xi_{s}(t))=\underset{t\in[0,1]}{\max}\widetilde{J}_{\mu,V}(\xi(t),s(t)).$

Since $\widetilde{\xi}$ is arbitrary, we conclude
${m}_{V,\,a}\leq\widetilde{m}_{V,\,a}$. So
$\widetilde{m}_{V,\,a}={m}_{V,\,a}$. ∎

##  4 Palais-Smale sequence

In this section, we are dedicated to find a bounded Palais-Smale sequence
$\\{u_{n}\\}$ approaching the Pohozaev manifold, that is,
$P_{\mu}(u_{n})=o_{n}(1)$ or $\widetilde{P}_{\mu}(u_{n})=o_{n}(1)$. Recall
that a Palais-Smale sequence $\\{u_{n}\\}$ of $J_{\mu,\,V}|_{S_{a}}$ at $c$,
means that $u_{n}\in S_{a}$,

$J_{\mu,\,V}(u_{n})=c+o_{n}(1),$

and

$\|J^{\prime}_{\mu,\,V}(u_{n})\|_{T^{*}_{u_{n}}S_{a}}=o_{n}(1),$

where

$T_{u_{n}}S_{a}=\left\\{v\in
H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}u_{n}v\mathrm{d}x=0\right\\}$

is a subspace of $H^{1}(\mathbb{R}^{N})$, and $T^{*}_{u_{n}}S_{a}$ is the dual
space of $T_{u_{n}}S_{a}$. By means of Proposition 2.7, we will obtain the
following Lemma.

######  Lemma 4.1.

Assume that $N\geq 3$, $p\in[\hat{p},2^{*})$. We have the following
assertions:

  * $(i)$

If $(V_{1})$, $(V_{3})$ hold with (3.1), then there exists a Palais-Smale
sequence $\\{u_{n}\\}$ of $J_{\mu,V}|_{S_{a}}$ at the mountain pass level
${m}_{V,\,a}$, which satisfies

$\displaystyle P_{\mu,V}(u_{n})\to 0\;\;\text{and}\;\;\|u_{n}^{-}\|_{2}\to
0\;\;\text{as}\leavevmode\nobreak\ n\to\infty.$ (4.1)

  * $(ii)$

If $V\in C^{1}(\mathbb{R}^{N})$ satisfies $(V_{1})$, $(V_{2})$ with (3.1),
then there exists a Palais-Smale sequence $\\{u_{n}\\}$ of
$J_{\mu,V}|_{S_{a}}$ at the mountain pass level ${m}_{V,\,a}$, which satisfies

$\displaystyle\widetilde{P}_{\mu,V}(u_{n})\to
0\;\;\text{and}\;\;\|u_{n}^{-}\|_{2}\to 0\;\;\text{as}\leavevmode\nobreak\
n\to\infty.$ (4.2)

###### Proof.

According to the definition of ${m}_{V,\,a}$ and Lemma 3.4, there is a
minimizing sequence $\\{\xi_{n}\\}\subset\Gamma$ for ${m}_{V,\,a}$, which
satisfies

$\widetilde{m}_{V,\,a}={m}_{V,\,a}\leq\underset{t\in[0,1]}{\max}J_{\mu,V}(\xi_{n}(t))<{m}_{V,\,a}+\frac{1}{n}=\widetilde{m}_{V,\,a}+\frac{1}{n}.$

Here the minimizing sequence $\\{\xi_{n}\\}$ can be replaced by
$\\{|\xi_{n}|\\}$ since $J_{\mu,V}(u)\geq J_{\mu,V}(|u|)$ for all $u\in
S_{a}$. Therefore we can assume that $\xi_{n}\geq 0$. We will prove this lemma
by applying Proposition 2.7 to $\widetilde{J}_{\mu,V}$ with

$X=S_{a}\times\mathbb{R}^{+},\;\;K=\\{(e_{0},1),(e_{1},1)\\},\;\;\mathcal{E}=\\{{\rm
Im}({\widetilde{\xi}}):{\widetilde{\xi}}\in\widetilde{\Gamma}\\},\;\;E_{n}=\\{(\xi_{n}(t),1):t\in[0,1]\\},$

and $\sigma_{n}=\frac{1}{n}$. It results that there exist a sequence
$\\{(v_{n},h_{n})\\}\subset S_{a}\times\mathbb{R}^{+}$ and $c>0$ such that

$\begin{cases}\begin{aligned}
\widetilde{m}_{V,\,a}\leq\widetilde{J}_{\mu,V}(v_{n},h_{n})<\widetilde{m}_{V,\,a}+\frac{1}{n},\\\
\|\nabla_{S_{a}\times\mathbb{R}^{+}}\widetilde{J}_{\mu,V}(v_{n},h_{n})\|_{T^{*}_{(v_{n},h_{n})}(S_{a}\times\mathbb{R}^{+})}<c\frac{1}{\sqrt{n}},\\\
\underset{t\in[0,1]}{\min}\|\xi_{n}(t)-v_{n}\|_{H^{1}}+|h_{n}-1|<c\frac{1}{\sqrt{n}},\\\
\end{aligned}\end{cases}$ (4.3)

where

$T_{(v_{n},h_{n})}(S_{a}\times\mathbb{R}^{+})=\left\\{(u,t):u\in
H^{1}(\mathbb{R}^{N}),\,t\in\mathbb{R},\,\int_{\mathbb{R}^{N}}uv_{n}\mathrm{d}x=0\right\\}$

which is a subspace of $H^{1}(\mathbb{R}^{N})\times\mathbb{R}$, and
$T^{*}_{(v_{n},h_{n})}(S_{a}\times\mathbb{R}^{+})$ is the dual space of
$T_{(v_{n},h_{n})}(S_{a}\times\mathbb{R}^{+})$. In virtue of Lemma A.1, if
$(V_{1})$, $(V_{3})$ hold, by differentiating $\widetilde{J}_{\mu,V}$ with
respect to second variable, we get from (4.3) that

$\left|\frac{\partial\widetilde{J}_{\mu,V}}{\partial
h}(v_{n},h_{n})\right|=|{P}_{\mu,V}((v_{n})_{h_{n}})|\to 0;$ (4.4)

if $V\in C^{1}(\mathbb{R}^{N})$ satisfies $(V_{1})$, $(V_{2})$, then

$\left|\frac{\partial\widetilde{J}_{\mu,V}}{\partial
h}(v_{n},h_{n})\right|=|\widetilde{P}_{\mu,V}((v_{n})_{h_{n}})|\to 0,$ (4.5)

as $n\to\infty$. We differentiate $\widetilde{J}_{\mu,V}$ with respect to the
first variable. It follows that

$\|J^{\prime}_{\mu,\,V}((v_{n})_{h_{n}})\|_{T^{*}_{(v_{n})_{h_{n}}}S_{a}}\to
0,$

as $n\to\infty$, see [10]. Setting $u_{n}:=(v_{n})_{h_{n}}$ and by $h_{n}\to
1$, we conclude from (4.3), (4.4), (4.5) that (4.1), (4.2) hold, and
$\\{u_{n}\\}$ is a Palais-Smale sequence of $J_{\mu,V}$ at ${m}_{V,\,a}$. ∎

Under the assumptions $(V_{1})$, $(V_{2})$ or $(V_{3})$, we are ready to prove
the existence of bounded Palais-Smale sequence.

######  Lemma 4.2.

Let $N\geq 3$, $p\in[\hat{p},2^{*})$. Suppose that $V\in
C^{1}(\mathbb{R}^{N})$ satisfies

  * $(i)$

either $(V_{1})$, $(V_{3})$ hold with (3.1), (2.13)-(2.16);

  * $(ii)$

or $(V_{1})$, $(V_{2})$ hold with (3.1), (2.17), (2.18).

Let $\\{u_{n}\\}$ be a Palais-Smale sequence of $J_{\mu,V}$ at ${m}_{V,\,a}$
such that $P_{\mu,V}(u_{n})\to 0$ if $(i)$ holds, or
$\widetilde{P}_{\mu,V}(u_{n})\to 0$ if $(ii)$ holds. Then $\\{u_{n}\\}$ is
bounded in $H^{1}(\mathbb{R}^{N})$.

###### Proof.

In order to simplify the symbols in this proof, we write

$\displaystyle a_{n}:=\|\nabla
u_{n}\|_{2}^{2},\;\;b_{n}:=\int_{\mathbb{R}^{N}}V(x)u^{2}_{n}\mathrm{d}x,\;\;c_{n}:=\int_{\mathbb{R}^{N}}V(x)u_{n}\nabla
u_{n}\cdot x\mathrm{d}x,$ $\displaystyle
d_{n}:=\|u_{n}\|_{p}^{p},\;\;e_{n}:=\|u_{n}\|_{2^{*}}^{2^{*}},\;\;f_{n}:=\int_{\mathbb{R}^{N}}Wu_{n}^{2}\mathrm{d}x.$

Case (i). According to the assumptions of this lemma, we have

$a_{n}+b_{n}-\frac{2\mu}{p}d_{n}-\frac{2}{2^{*}}e_{n}=2{m}_{V,\,a}+o_{n}(1),$
(4.6) $a_{n}+\frac{N}{2}b_{n}+c_{n}-\mu\gamma_{p}d_{n}-e_{n}=o_{n}(1).$ (4.7)

When $p\in(\hat{p},2^{*})$, it follows from (4.7) that

$\frac{2\mu}{p}d_{n}+\frac{2}{2^{*}}e_{n}\leq\frac{2}{p\gamma_{p}}\left(\mu\gamma_{p}d_{n}+e_{n}\right)=\frac{2}{p\gamma_{p}}\left(a_{n}+\frac{N}{2}b_{n}+c_{n}\right)+o_{n}(1).$
(4.8)

Combining (4.6) with (4.8), we get

$\displaystyle
a_{n}+b_{n}-\frac{2}{p\gamma_{p}}\left(a_{n}+\frac{N}{2}b_{n}+c_{n}\right)\leq
a_{n}+b_{n}-\frac{2\mu}{p}d_{n}-\frac{2}{2^{*}}e_{n}+o_{n}(1)=2{m}_{V,\,a}+o_{n}(1).$

By the assumptions $(V_{1})$ and $(V_{3})$, we know that $0\leq-
b_{n}\leq\sigma_{1}a_{n}$ and $|c_{n}|\leq{\sigma_{3}}a_{n}$. From the above
computations, it follows that if $N\geq 4$ (so $N\geq p\gamma_{p}$),

$\displaystyle 2{m}_{V,\,a}+o_{n}(1)\geq$
$\displaystyle(1-\frac{2}{p\gamma_{p}})a_{n}+(1-\frac{N}{p\gamma_{p}})b_{n}-\frac{2}{p\gamma_{p}}c_{n}$
$\displaystyle\geq$
$\displaystyle(1-\frac{2}{p\gamma_{p}})a_{n}-\frac{2}{p\gamma_{p}}{\sigma_{3}}a_{n}$
$\displaystyle=$
$\displaystyle\left(1-\frac{2}{p\gamma_{p}}-\frac{2}{p\gamma_{p}}{\sigma_{3}}\right)a_{n}.$

Hence $\\{a_{n}\\}$ is bounded by (2.15).

If $N=3$ (hence $N<p\gamma_{p}$),

$\displaystyle 2{m}_{V,\,a}+o_{n}(1)\geq$
$\displaystyle(1-\frac{2}{p\gamma_{p}})a_{n}-(1-\frac{3}{p\gamma_{p}})\sigma_{1}a_{n}-\frac{2}{p\gamma_{p}}{\sigma_{3}}a_{n}$
$\displaystyle=$
$\displaystyle\left(1-\frac{2}{p\gamma_{p}}-(1-\frac{3}{p\gamma_{p}})\sigma_{1}-\frac{2}{p\gamma_{p}}{\sigma_{3}}\right)a_{n}.$

Hence $\\{a_{n}\\}$ is bounded by (2.16).

When $p=\hat{p}$, by (4.7), we have

$e_{n}=a_{n}+\frac{N}{2}b_{n}+c_{n}-\mu\gamma_{\hat{p}}d_{n}+o_{n}(1).$ (4.9)

Combining (4.6) with (4.9), we get

$\displaystyle
2{m}_{V,\,a}+o_{n}(1)=\left(1-\frac{2}{2^{*}}\right)\left(a_{n}-\mu\gamma_{\hat{p}}d_{n}\right)-\frac{2}{2^{*}}c_{n}+\left(1-\frac{N}{2^{*}}\right)b_{n}+o_{n}(1).$

If $N\geq 4$ (we get $N\geq 2^{*}$), then

$2{m}_{V,\,a}+o_{n}(1)\geq\left(1-\frac{2}{2^{*}}\right)\left(1-\mu\gamma_{\hat{p}}C^{\hat{p}}_{N,\,{\hat{p}}}a^{\frac{4}{N}}\right)a_{n}-\frac{2}{2^{*}}{\sigma_{3}}a_{n}+o_{n}(1).$
(4.10)

Hence $\\{a_{n}\\}$ is bounded by (2.13).

If $N=3$ (so $N<2^{*}$), we obtain from $d_{n}\leq
C^{\hat{p}}_{3,\,{\hat{p}}}a_{n}^{{\hat{p}}\gamma_{\hat{p}}/2}a^{\frac{4}{3}}$
and $b_{n}\leq 0$ that

$2{m}_{V,\,a}+o_{n}(1)\geq\left(1-\frac{2}{6}\right)\left(1-\mu\gamma_{p}C^{\hat{p}}_{3,\,{\hat{p}}}a^{\frac{4}{3}}\right)a_{n}-\frac{2}{6}{\sigma_{3}}a_{n}-\left(1-\frac{3}{6}\right)\sigma_{1}a_{n}+o_{n}(1).$

Hence $\\{a_{n}\\}$ is bounded by (2.14).

Case (ii). By a similar argument as above, (4.6) holds and (4.7) is replaced
by

$a_{n}-f_{n}-\mu\gamma_{p}d_{n}-e_{n}=o_{n}(1).$ (4.11)

When $p\in(\hat{p},2^{*})$, it follows from (4.6) that

$\frac{2\mu}{p}d_{n}+\frac{2}{2^{*}}e_{n}\leq\frac{2}{p\gamma_{p}}\left(\mu\gamma_{p}d_{n}+e_{n}\right)=\frac{2}{p\gamma_{p}}\left(a_{n}-f_{n}\right)+o_{n}(1).$
(4.12)

Combining (4.6) with (4.12), we get

$\displaystyle a_{n}+b_{n}-\frac{2}{p\gamma_{p}}\left(a_{n}-f_{n}\right)\leq$
$\displaystyle
a_{n}+b_{n}-\frac{2\mu}{p}d_{n}-\frac{2}{2^{*}}e_{n}+o_{n}(1)=2{m}_{V,\,a}+o_{n}(1).$

By $(V_{1})$ and $(V_{2})$, we have $|b_{n}|\leq\sigma_{1}a_{n}$ and
$|f_{n}|\leq\sigma_{2}a_{n}$, then

$\displaystyle 2{m}_{V,\,a}+o_{n}(1)\geq$
$\displaystyle\left(1-\frac{2}{p\gamma_{p}}\right)a_{n}+b_{n}+\frac{2}{p\gamma_{p}}f_{n}$
$\displaystyle\geq$
$\displaystyle\left(1-\frac{2}{p\gamma_{p}}\right)a_{n}-2\sigma_{2}a_{n}$
$\displaystyle=$
$\displaystyle\left(1-\frac{2}{p\gamma_{p}}-2\sigma_{2}\right)a_{n}.$

Hence $\\{a_{n}\\}$ is bounded by (2.18).

When $p=\hat{p}$, it follows from (4.11) that

$e_{n}=a_{n}-f_{n}-\mu\gamma_{\hat{p}}d_{n}+o_{n}(1).$ (4.13)

Taking (4.13) into (4.6), we obtain

$\displaystyle 2{m}_{V,\,a}+o_{n}(1)=$
$\displaystyle\left(1-\frac{2}{2^{*}}\right)\left(a_{n}-\mu\gamma_{\hat{p}}d_{n}\right)+b_{n}+\frac{2}{2^{*}}f_{n}+o_{n}(1)$
$\displaystyle\geq$
$\displaystyle\left(1-\frac{2}{2^{*}}\right)\left(1-\mu\gamma_{\hat{p}}C^{\hat{p}}_{N,\,{\hat{p}}}a^{\frac{4}{N}}\right)a_{n}-2\sigma_{2}a_{n}.$

Hence $\\{a_{n}\\}$ is bounded by (2.17). ∎

##  5 Proof of main theorems

In previous section, we have obtained a bounded Palais-Smale sequence
$\\{u_{n}\\}$ at level $m_{V,\,a}$, which approach the Pohozaev manifold
$\mathcal{P}_{\mu,V}$ or $\widetilde{\mathcal{P}}_{\mu,V}$. To finish our
theorems, a key step is to prove that $\\{u_{n}\\}$ has a convergent
subsequence in $H^{1}(\mathbb{R}^{N})$. We can not work in the radial space
$H_{rad}^{1}(\mathbb{R}^{N})$ due to the fact that the potential function is
not assumed to be radial, so we need to deal with the problem that
$H^{1}(\mathbb{R}^{N})\subset L^{p}(\mathbb{R}^{N})$ is not compact. The
Sobolev critical exponent brings another difficulties. The strategy in this
section is using Lions’ lemma and the estimate
$m_{V,\,a}<\frac{1}{N}\mathcal{S}^{\frac{N}{2}}$ to get the following
splitting result.

######  Lemma 5.1.

Let $N\geq 3$, $p\in[\hat{p},2^{*})$. Suppose that $V\in
C^{1}(\mathbb{R}^{N})$ satisfies

  * $(i)$

either $(V_{1})$, $(\widetilde{V}_{2})$, $(V_{3})$, $(V_{4})$ hold with (3.1),
(2.13)-(2.16);

  * $(ii)$

or $(V_{1})$, $(V_{2})$, $(V_{4})$ hold with (3.1), (2.17), (2.18).

Let $\\{u_{n}\\}$ be a Palais-Smale sequence of $J_{\mu,V}$ at ${m}_{V,\,a}$
such that $P_{\mu,V}(u_{n})\to 0$ if $(i)$ holds, or
$\widetilde{P}_{\mu,V}(u_{n})\to 0$ if $(ii)$ holds. In the sense of
subsequence, we can assume that for some $u\in H^{1}(\mathbb{R}^{N})$,
$u_{n}\rightharpoonup u$ weakly in $H^{1}(\mathbb{R}^{N})$. If this
convergence is not strong, then there exist $k$ ( for some
$k\in\mathbb{N}^{+}$) nontrivial solutions $w^{1},\dots,w^{k}$ to (2.1) for
some $\lambda<0$ and $k$ sequences $\\{y_{n}^{j}\\}\subset\mathbb{R}^{N}$ with
$1\leq j\leq k$ such that $|y_{n}^{j}|\rightarrow\infty$,
$|y_{n}^{j}-y_{n}^{i}|\to\infty$ for $i\neq j$ as $n\to\infty$, and

$u_{n}=u+\sum_{1\leq j\leq
k}w^{j}(\cdot+y_{n}^{j})+o_{n}(1)\;\;\text{in}\;\;H^{1}(\mathbb{R}^{N}).$
(5.1)

Moreover, we have

$\displaystyle\|u_{n}\|_{2}^{2}=\|u\|_{2}^{2}+\sum_{1\leq j\leq
k}\|w^{j}\|_{2}^{2},\;\;J_{\mu,V}(u_{n})$
$\displaystyle=J_{\mu,V}(u)+\sum_{1\leq j\leq k}J_{\mu}(w^{j})+o_{n}(1).$
(5.2)

###### Proof.

In this proof, we argue up to suitable subsequences. Since $\\{u_{n}\\}$ is a
Palais-Smale sequence of $J_{\mu,V}|_{S_{a}}$ at level ${m}_{V,\,a}$, there
exists a sequence $\\{\lambda_{n}\\}\subset\mathbb{R}$ such that

$-\Delta
u_{n}+V(x)u_{n}-\mu|u_{n}|^{p-2}u_{n}-|u_{n}|^{2^{*}-2}u_{n}-\lambda_{n}u_{n}\rightarrow
0\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ H^{-1}(\mathbb{R}^{N}),$
(5.3)

so

$\lambda_{n}a^{2}=\|\nabla
u_{n}\|_{2}^{2}+\int_{\mathbb{R}^{N}}V(x)u_{n}^{2}\mathrm{d}x-\mu\|u_{n}\|_{p}^{p}-\|u_{n}\|_{2^{*}}^{2^{*}}+o_{n}(1).$
(5.4)

It follows from Lemma 4.2 that $\\{u_{n}\\}$ is bounded in
$H^{1}(\mathbb{R}^{N})$, so $\\{\lambda_{n}\\}$ is also a bounded sequence. We
can assume $\lambda_{n}\rightarrow\lambda\in\mathbb{R}$. Because
$u_{n}\rightharpoonup u$ weakly in $H^{1}(\mathbb{R}^{N})$, $u$ is a solution
to (2.3). Therefore, Lemma 2.4 leads to either $\lambda<0$ or $u=0$.

Case (i). Suppose that $u=0$. Because $\\{u^{2}_{n}\\}$ is bounded in
$L^{\frac{N}{N-2}}(\mathbb{R}^{N})$ and $u^{2}_{n}(x)\rightarrow 0$ for a.e.
$x\in\mathbb{R}^{N}$, we obtain that $u^{2}_{n}\rightharpoonup 0$ weakly in
$L^{\frac{N}{N-2}}(\mathbb{R}^{N})$. For any $R>0$, we put

$\int_{\mathbb{R}^{N}}V(x)u_{n}^{2}\mathrm{d}x=\int_{B_{R}(0)}V(x)u_{n}^{2}\mathrm{d}x+\int_{\mathbb{R}^{N}\backslash
B_{R}(0)}V(x)u_{n}^{2}\mathrm{d}x.$ (5.5)

From $V\in L^{\frac{N}{2}}_{loc}(\mathbb{R}^{N})$ and
$\lim\limits_{|x|\rightarrow\infty}V(x)=0$ and taking $R$ large enough, it
holds that

$\int_{\mathbb{R}^{N}}V(x)u_{n}^{2}\mathrm{d}x=o_{n}(1).$ (5.6)

Similarly, it follows from $(V_{3})$ that

$\int_{\mathbb{R}^{N}}V(x)u_{n}\nabla u_{n}\cdot x\mathrm{d}x=o_{n}(1).$ (5.7)

Hence

${m}_{V,\,a}=J_{\mu,V}(u_{n})+o_{n}(1)=\frac{1}{2}\|\nabla
u_{n}\|_{2}^{2}-\frac{\mu}{p}\|u_{n}\|_{p}^{p}-\frac{1}{2^{*}}\|u_{n}\|_{2^{*}}^{2^{*}}+o_{n}(1),$

and

$o_{n}(1)=P_{\mu,V}(u_{n})=\|\nabla
u_{n}\|_{2}^{2}-\mu\gamma_{p}\|u_{n}\|_{p}^{p}-\|u_{n}\|_{2^{*}}^{2^{*}}+o_{n}(1).$
(5.8)

So $P_{\mu}(u_{n})\rightarrow 0$. If $\|\nabla u_{n}\|_{2}^{2}\to 0$, then
$\|u_{n}\|_{p}^{p}\to 0$ and $\|u_{n}\|_{2^{*}}^{2^{*}}\to 0$. Hence
${m}_{V,\,a}=0$, that contradicts (3.3). As a result, up to a subsequence, we
can assume that $\|\nabla u_{n}\|_{2}^{2}>c$ for some $c>0$. By Lemma 2.3, we
conclude

${m}_{V,\,a}=\underset{n\rightarrow\infty}{\lim}J_{\mu}(u_{n})\geq m_{a},$

which contradicts Lemma 3.3.

From the above, we know that $u\neq 0$ and $\lambda<0$. Letting
$w^{1}_{n}:=u_{n}-u$, we obtain that $w^{1}_{n}\rightharpoonup 0$ weakly in
$H^{1}(\mathbb{R}^{N})$, $w^{1}_{n}\rightarrow 0$ strongly in
$L^{2}_{loc}(\mathbb{R}^{N})$, $L^{p}_{loc}(\mathbb{R}^{N})$, and a.e. in
$\mathbb{R}^{N}$. Set

$L:=\underset{n\rightarrow\infty}{\lim\inf}\underset{y\in\mathbb{R}^{n}}{\sup}\int_{B_{1}(y)}|w^{1}_{n}|^{2}\mathrm{d}x.$

If $L=0$, up to a subsequence, it follows from Lions’ lemma (see [25, Theorem
1.34]) that $\|w_{n}^{1}\|_{p}^{p}\rightarrow 0$. By Brezis-Lieb lemma (see
[25, Lemma 1.32]), we obtain

$\displaystyle{m}_{V,\,a}$ $\displaystyle=J_{\mu,V}(u_{n})+o_{n}(1)$
$\displaystyle=J_{\mu,V}(u)+J_{\mu}(w_{n}^{1})+o_{n}(1)$
$\displaystyle=J_{\mu,V}(u)+\frac{1}{2}\|\nabla
w^{1}_{n}\|_{2}^{2}-\frac{1}{2^{*}}\|w_{n}^{1}\|_{2^{*}}^{2^{*}}+o_{n}(1).$

In virtue of the boundedness of $\\{w^{1}_{n}\\}$ in $H^{1}(\mathbb{R}^{N})$,
we can assume that $\|\nabla w^{1}_{n}\|_{2}^{2}\rightarrow l\in\mathbb{R}$.
By $P_{\mu,V}(u)=0$, $P_{\mu,V}(u_{n})=o_{n}(1)$, Brezis-Lieb lemma and
similar arguments in (5.6), (5.7), we get

$\displaystyle o_{n}$ $\displaystyle(1)=P_{\mu,V}(u_{n})-P_{\mu,V}(u)$
$\displaystyle=\|\nabla
w_{n}^{1}\|_{2}^{2}+\frac{N}{2}\int_{\mathbb{R}^{N}}V(x)(w_{n}^{1})^{2}\mathrm{d}x+\int_{\mathbb{R}^{N}}V(x)w_{n}^{1}\nabla
w_{n}^{1}\cdot
x\mathrm{d}x-\mu\gamma_{p}\|w^{1}_{n}\|_{p}^{p}-\|w^{1}_{n}\|_{2^{*}}^{2^{*}}+o_{n}(1)$
$\displaystyle=\|\nabla
w_{n}^{1}\|_{2}^{2}-\|w^{1}_{n}\|_{2^{*}}^{2^{*}}+o_{n}(1).$

Thus

$l=\|\nabla
w^{1}_{n}\|_{2}^{2}+o_{n}(1)\geq\mathcal{S}\|w^{1}_{n}\|_{2^{*}}^{2}+o_{n}(1)=\mathcal{S}l^{\frac{2}{2^{*}}}+o_{n}(1),$

which implies either $l=0$ or $l\geq\mathcal{S}^{\frac{N}{2}}$. However, if
$l\geq\mathcal{S}^{\frac{N}{2}}$, using Lemma 2.5, then

${m}_{V,\,a}=J_{\mu,V}(u)+\frac{1}{2}\|\nabla
w^{1}_{n}\|_{2}^{2}-\frac{1}{2^{*}}\|w^{1}_{n}\|_{2^{*}}^{2^{*}}+o_{n}(1)\geq\frac{1}{N}\mathcal{S}^{\frac{N}{2}}+o_{n}(1),$

which contradicts the fact that
${m}_{V,\,a}<m_{a}<\frac{1}{N}\mathcal{S}^{\frac{N}{2}}$ by Lemma 2.1 and
Lemma 3.3. Consequently, $l=0$ and we deduce that $\|\nabla
w^{1}_{n}\|_{2}\rightarrow 0$. Multiplying (5.3), (2.3) by $w^{1}_{n}$, we
have

$\displaystyle\|\nabla
w^{1}_{n}\|_{2}^{2}+\int_{\mathbb{R}^{N}}V(x)|w^{1}_{n}|^{2}\mathrm{d}x=$
$\displaystyle\int_{\mathbb{R}^{N}}(\lambda_{n}u_{n}-\lambda
u)w^{1}_{n}\mathrm{d}x+\mu\int_{\mathbb{R}^{N}}(|u_{n}|^{p-2}u_{n}-|u|^{p-2}u)w^{1}_{n}\mathrm{d}x$
$\displaystyle+\int_{\mathbb{R}^{N}}(|u_{n}|^{2^{*}-2}u_{n}-|u|^{2^{*}-2}u)w^{1}_{n}\mathrm{d}x+o_{n}(1).$

By $\|\nabla w^{1}_{n}\|_{2}\rightarrow 0$, we obtain

$\displaystyle\lambda\int_{\mathbb{R}^{N}}(u_{n}-u)^{2}\mathrm{d}x\rightarrow
0.$

From $\lambda<0$, it follows that $u_{n}\rightarrow u$ in
$H^{1}(\mathbb{R}^{N})$. A contradiction gives $L>0$. As a consequence, there
exists a sequence $\\{y_{n}^{1}\\}\subset\mathbb{R}^{N}$ such that
$|y_{n}^{1}|\to\infty$ and

$\int_{B_{1}(y^{1}_{n})}|w^{1}_{n}|^{2}\mathrm{d}x\geq\frac{L}{2}.$

Letting $u^{1}_{n}(\cdot):=w^{1}_{n}(\cdot+y^{1}_{n})$, there exists some
$w^{1}\in H^{1}(\mathbb{R}^{N})\backslash\\{0\\}$ such that
$u^{1}_{n}\rightharpoonup w^{1}$ weakly in $H^{1}(\mathbb{R}^{N})$ and
$u^{1}_{n}(x)\rightharpoonup w^{1}(x)$ a.e. in $\mathbb{R}^{N}$. Setting
$w^{2}_{n}:=u^{1}_{n}-w^{1}$, by Brezis-Lieb lemma again, we have

$\displaystyle{m}_{V,\,a}$ $\displaystyle=J_{\mu,V}(u_{n})+o_{n}(1)$
$\displaystyle=J_{\mu,V}(u)+J_{\mu}(w_{n}^{1})+o_{n}(1)$
$\displaystyle=J_{\mu,V}(u)+J_{\mu}(w^{1})+J_{\mu}(w^{2}_{n})+o_{n}(1).$

It is clear that there exists some $C>0$ such that $|J_{\mu}(v)|\leq C$ for
any

$v\in\\{v\in H^{1}(\mathbb{R}^{N}):\|v\|_{2}\leq a,\,\|\nabla
v\|_{2}\leq\underset{n}{\sup}\|\nabla u_{n}\|_{2}\\}.$

Using Lemma 2.2, we get $J_{\mu}(w^{j})\geq m_{a}$. By induction, we deduce
that this process will terminate after a finite of times. Consequently, there
exists some $k\in\mathbb{N}^{+}$ and $k$ nontrivial solutions
$w^{1},\dots,w^{k}$ to (2.1) and $k$ sequences
$\\{y_{n}^{j}\\}\subset\mathbb{R}^{N}$ such that (5.1), (5.2) hold.

Case (ii). We only point out some differences to the above proof. First, by
$(V_{2})$, (5.7) can be replaced by

$\int_{\mathbb{R}^{N}}W(x)u_{n}^{2}\mathrm{d}x=o_{n}(1),$

and therefore (5.8) becomes

$o_{n}(1)=\widetilde{P}_{\mu,V}(u_{n})=\|\nabla
u_{n}\|_{2}^{2}-\mu\gamma_{p}\|u_{n}\|_{p}^{p}-\|u_{n}\|_{2^{*}}^{2^{*}}+o_{n}(1).$

Replace $P_{\mu,V}$ by $\widetilde{P}_{\mu,V}$. Similarly as the proof of Case
(i), one can complete the proof of Case (ii). ∎

###### Proof of Theorem 1.5 completed.

According to the assumptions of Theorem 1.5, when $p=\hat{p}$, if the mass $a$
is small enough, then the conditions of Lemma 2.5 (i), Lemma 3.3, Lemma 4.1
(i), Lemma 4.2 (i) and Lemma 5.1 (i) will be satisfied. While for
$p\in(\hat{p},2^{*})$, the mass $a$ is not required to be small.

By Lemma 4.1, there exists a Palais-Smale sequence $\\{u_{n}\\}$ of
$J_{\mu,V}|_{S_{a}}$ at level ${m}_{V,\,a}$, which satisfies (4.1). Lemma 4.2
leads to the boundedness of $\\{u_{n}\\}$ in $H^{1}(\mathbb{R}^{N})$. Up to a
subsequence, we can assume that there is some $u\in H^{1}(\mathbb{R}^{N})$
such that $u_{n}\rightharpoonup u$ weakly in $H^{1}(\mathbb{R}^{N})$, and a.e.
in $\mathbb{R}^{N}$. If this convergence is not strong, then by Lemma 5.1,
there exist $\lambda<0,\,k\in\mathbb{N}^{+}$ and $k$ nontrivial solutions
$w^{1},\dots,w^{k}$ of (2.1) such that (5.2) holds. By Lemma 2.5, we have
$J_{\mu,V}(u)\geq 0$. Furthermore, it follows from Lemma 2.2 and Lemma 3.3
that $J_{\mu}(w^{j})\geq m_{a}>m_{V,\,a}$, which contradicts (5.2). Thus
$u_{n}\rightarrow u$ strongly in $H^{1}(\mathbb{R}^{N})$.

In the sequel, we shall prove $u>0$. In virtue of $V\in
C^{1}(\mathbb{R}^{N})$, we can obtain from Remark 1.9 that $u$ is a classical
solution. Due to $\|u_{n}^{-}\|_{2}\to 0$, it follows that $u\geq 0$. By the
strong maximum principle, we get $u>0$ in $\mathbb{R}^{N}$. ∎

###### Proof of Theorem 1.6 completed.

This proof is very similar to the proof of Theorem 1.5 where we need to use
Lemma 2.5 instead of Lemma 2.6. ∎

##  Appendix A Appendix

Here we will show the differential property for the potential term of
$J_{\mu,\,V}$. In addition, we will give some Pohozaev identities for (2.3),
that is, any $H^{1}$ solution to (2.3) will satisfy (2.5).

######  Lemma A.1.

Assume that $N\geq 1$. We have the following assertions:

  * $(i)$

If the potential $V$ satisfies (1.19), then for $u\in H^{1}(\mathbb{R}^{N})$,

$\frac{d}{dt}\int_{\mathbb{R}^{N}}V(x)u^{2}(tx)\mathrm{d}x\Big{|}_{t=1}=2\int_{\mathbb{R}^{N}}V(x)u(x)\nabla
u(x)\cdot x\,\mathrm{d}x.$ (A.1)

  * $(ii)$

If the potential $V\in C^{1}(\mathbb{R}^{N})$ satisfies (1.16) and (1.18),
then for $u\in H^{1}(\mathbb{R}^{N})$,

$\frac{d}{dt}\int_{\mathbb{R}^{N}}V\left(\frac{x}{t}\right)u^{2}\mathrm{d}x\Big{|}_{t=1}=-2\int_{\mathbb{R}^{N}}W(x)u^{2}\mathrm{d}x.$
(A.2)

###### Proof.

We only prove (i), because (ii) is similar. From Remak 1.3, (1.16) holds and
$\sigma_{1}$ can be chosen to be $\frac{2}{N-2}\sigma_{3}$. If $u\in
C_{c}^{\infty}(\mathbb{R}^{N})$, there holds

$\displaystyle\quad\int_{\mathbb{R}^{N}}V(x)u^{2}(tx)\mathrm{d}x-\int_{\mathbb{R}^{N}}V(x)u^{2}(x)\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}^{N}}\int_{1}^{t}2V(x)u(sx)\nabla u(sx)\cdot
x\,\mathrm{d}s\mathrm{d}x$ (A.3)
$\displaystyle=\int_{1}^{t}\int_{\mathbb{R}^{N}}2V(x)u(sx)\nabla u(sx)\cdot
x\,\mathrm{d}x\mathrm{d}s.$

Since $C_{c}^{\infty}(\mathbb{R}^{N})$ is dense in $H^{1}(\mathbb{R}^{N})$,
for $u\in H^{1}(\mathbb{R}^{N})$, there exists a sequence $\\{u_{n}\\}\subset
C_{c}^{\infty}(\mathbb{R}^{N})$ such that $u_{n}\to u$ in
$H^{1}(\mathbb{R}^{N})$. By (1.16) and Hölder inequality, we get

$\displaystyle\left|\int_{\mathbb{R}^{N}}V(x)u_{n}^{2}\mathrm{d}x-\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x\right|$
$\displaystyle\leq\left|\int_{\mathbb{R}^{N}}V(x)(u_{n}-u)^{2}\mathrm{d}x\right|+2\left|\int_{\mathbb{R}^{N}}V(x)(u_{n}-u)u\mathrm{d}x\right|$
$\displaystyle\leq
C\sigma_{3}\|\nabla(u_{n}-u)\|_{2}^{2}+2C\sigma_{3}\|\nabla(u_{n}-u)\|_{2}\|\nabla
u\|_{2}.$

On the other hand, for $s\in(0,\infty)$, by (1.19) and Hölder inequality, we
have

$\displaystyle\quad\left|\int_{\mathbb{R}^{N}}V(x)u_{n}(sx)\nabla
u_{n}(sx)\cdot x\mathrm{d}x-\int_{\mathbb{R}^{N}}V(x)u(sx)\nabla u(sx)\cdot
x\,\mathrm{d}x\right|$
$\displaystyle\leq\left|\int_{\mathbb{R}^{N}}V(x)u_{n}(sx)(\nabla
u_{n}(sx)-\nabla u(sx))\cdot
x\,\mathrm{d}x\right|+\left|\int_{\mathbb{R}^{N}}V(x)(u_{n}(sx)-u(sx))\nabla
u(sx)\cdot x\,\mathrm{d}x\right|$
$\displaystyle\leq{\sigma_{3}}s^{1-{N}}\|\nabla(u_{n}-u)\|_{2}\|\nabla
u_{n}\|_{2}+{\sigma_{3}}s^{1-{N}}\|\nabla(u_{n}-u)\|_{2}\|\nabla u\|_{2}.$

Thus, (A.3) holds for $u\in H^{1}(\mathbb{R}^{N})$ and $t\in(0,\infty)$. Next,
we claim that

$g(s):=\int_{\mathbb{R}^{N}}2V(x)u(sx)\nabla u(sx)\cdot x\,\mathrm{d}x$ (A.4)

is continuous for $s\in(0,\infty)$. Without loss of generality we only prove
the continuity of $g$ at $s=1$. It is seen from (1.19) that

$\displaystyle|g(s)-g(1)|$ $\displaystyle\leq
2\int_{\mathbb{R}^{N}}\left|V(x)u(sx)\nabla u(sx)\cdot x-V(x)u(x)\nabla
u(x)\cdot x\right|\,\mathrm{d}x$ $\displaystyle\leq
2\int_{\mathbb{R}^{N}}\left|V(x)u(sx)\nabla u(sx)\cdot x-V(x)u(sx)\nabla
u(x)\cdot x\right|\,\mathrm{d}x$
$\displaystyle\quad+2\int_{\mathbb{R}^{N}}\left|V(x)u(sx)\nabla u(x)\cdot
x-V(x)u(x)\nabla u(x)\cdot x\right|\,\mathrm{d}x$ $\displaystyle\leq 2\|\nabla
u(sx)-\nabla u(x)\|_{2}\|V(x)|x|u(sx)\|_{2}+2\|\nabla
u\|_{2}\|V(x)|x|(u(sx)-u(x))\|_{2}$ $\displaystyle\leq
2{\sigma_{3}}s^{{1-\frac{N}{2}}}\|\nabla u(sx)-\nabla u(x)\|_{2}\|\nabla
u\|_{2}+2\|\nabla u\|_{2}\|\nabla(u(sx)-u(x))\|_{2}.$

It is clear from Brezis-Lieb lemma (see [25, Lemma 1.32]) that

$\underset{s\to 1}{\lim}\|\nabla u(sx)-\nabla u(x)\|_{2}=\underset{s\to
1}{\lim}\|\nabla(u(sx)-u(x))\|_{2}=0,$

So we get the claim. Finally, by (A.3) and (A.4),

$\frac{d}{dt}\int_{\mathbb{R}^{N}}V(x)u^{2}(tx)\mathrm{d}x\Big{|}_{t=1}=\underset{t\to
1}{\lim}\frac{1}{t-1}\int_{1}^{t}g(s)\mathrm{d}s=g(1).$

The proof is completed. ∎

Next, we prove the Pohozaev identities.

######  Proposition A.2.

Let $N\geq 3$, $p\in(2,2^{*})$. Then (2.5) holds true for any $u\in
H^{1}({\mathbb{R}}^{N})$ solution to (2.3).

###### Proof.

Suppose that $u$ solves (2.3). Use $u$ as a test function to (2.3), there
holds

$\|\nabla
u\|_{2}^{2}+\int_{\mathbb{R}^{N}}V(x)u^{2}\mathrm{d}x=\lambda\|u\|_{2}^{2}+\mu\|u\|_{p}^{p}+\|u\|_{2^{*}}^{2^{*}}.$
(A.5)

By [14, Proposition 2.1], we know that if $V\in C^{1}(\mathbb{R}^{N})$ and
(1.16), (1.18) are satisfied, any solution $u\in H^{1}({\mathbb{R}}^{N})$ to
(2.3) will satisfy $\widetilde{P}_{\mu,V}(u)=0$.

Now, it suffices to check that if (1.19) is valid, any solution to (2.3)
satisfies $P_{\mu,V}(u)=0$. We follow the proof of [14, Proposition 2.1] and
[25, Theorem B.3]. Let $\eta\in C_{c}^{\infty}(\mathbb{R})$ be a cut-off
function satisfying $0\leq\eta\leq 1$, $\eta(r)=1$ for $r\leq 1$ and
$\eta(r)=0$ for $r\geq 2$. Let

$\eta_{n}(x)=\eta\left(\frac{|x|^{2}}{n^{2}}\right),\quad\forall\;n\geq 1.$

Thus, $0\leq\eta_{n}\leq 1$ and there exists $c>0$ such that

$|x||\nabla\eta_{n}(x)|\leq c,\quad\forall\;x\in{\mathbb{R}}^{N},n\geq 1.$

By [22, B.3 Lemma], $u\in W_{loc}^{2,q}(\mathbb{R}^{N})\cap
H^{1}(\mathbb{R}^{N})$ for any $q\geq 1$, since $V\in
L_{loc}^{\frac{N}{2}}(\mathbb{R}^{N})$. Hence

$\displaystyle\eta_{n}\Delta u(x\cdot\nabla u)$ $\displaystyle={\rm
div}\left[(x\cdot\nabla u)\eta_{n}\nabla u-\frac{\eta_{n}|\nabla
u|^{2}}{2}x\right]$ (A.6) $\displaystyle\quad-(x\cdot\nabla u)(\nabla
u\cdot\nabla\eta_{n})+\frac{N-2}{2}\eta_{n}|\nabla u|^{2}+\frac{|\nabla
u|^{2}}{2}\nabla\eta_{n}\cdot x.$

Let

$g(x,u):=-V(x)u+f(u),$

with $f(u):=\lambda u+\mu|u|^{p-2}u+|u|^{2^{*}-2}u$. Therefore

$g(x,u)\eta_{n}x\cdot\nabla u=-V(x)u\eta_{n}x\cdot\nabla u+{\rm
div}\big{[}\eta_{n}xF(u)\big{]}-N\eta_{n}F(u)-F(u)\nabla\eta_{n}\cdot x,$
(A.7)

where

$F(u)=\int_{0}^{u}f(t)\mathrm{d}t=\frac{\lambda}{2}|u|_{2}^{2}+\frac{\mu}{p}|u|_{p}^{p}+\frac{1}{2^{*}}|u|_{2^{*}}^{2^{*}}.$

By (A.6) and (A.7), under (1.19), applying divergence theorem and Lebesgue’s
dominated convergence theorem, taking $n$ tend to $\infty$, we arrive at

$\frac{N-2}{2}\|\nabla u\|_{2}^{2}=\int_{\mathbb{R}^{N}}V(x)ux\cdot\nabla
u\mathrm{d}x+\frac{N\lambda}{2}\|u\|_{2}^{2}+\frac{N\mu}{p}\|u\|_{p}^{p}+\frac{N}{2^{*}}\|u\|_{2^{*}}^{2^{*}}.$
(A.8)

By (A.5) and (A.8), we get $P_{\mu,V}(u)=0$. ∎

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