# A New Method to prove the Existence, Non-existence, Multiplicity,
Uniqueness, and Orbital Stability/Instability of standing waves for NLS with
partial confinement

Linjie Songa,b, Hichem Hajaiejc

aInstitute of Mathematics, AMSS, Chinese Academy of Science, Beijing 100190,
China
bUniversity of Chinese Academy of Science, Beijing 100049, China
c Department of Mathematics, California State University at Los Angeles, Los
Angeles, CA 90032, USA. The author is supported by CEMS. Email:
songlinjie18<EMAIL_ADDRESS>
###### Abstract

We give a new method to prove the existence, non-existence, multiplicity,
orbital stability/instability of standing waves for NLS with partial
confinement without the subcritical hypothesis, even in the reduction
equation. Using this method, we give an affirmative answer for an open problem
proposed by [7, Remark 1.10] where the authors conjectured the existence of
more than a normalized solution. We also establish uniqueness results of the
ground state solutions depending on the bifurcation parameters. We explain
that when the effect of partial confinement is strong, a dimension reduction
appears for some parameters. We also find different bifurcation phenomena from
the cases with full confinement.

Keywords: Normalized solutions; orbital stability; nonlinear Schrödinger
equations; partial confinement; dimension reduction.

2010 MSC: 35A15, 35B35, 35J20, 35Q55, 35C08

Data availability statement: My manuscript has no associate data.

##  1 Introduction

In [7], Bellazzini and al studied the existence, stability, qualitative, and
symmetry properties of the standing waves associated to the following Cauchy
problem:

$\left\\{\begin{array}[]{cc}i\partial_{t}\Psi+\Delta_{x}\Psi-(x_{1}^{2}+x_{2}^{2})\Psi+|\Psi|^{p-2}\Psi=0,(t,x)\in\mathbb{R}\times\mathbb{R}^{3},\\\
\Psi(0,x)=\Psi_{0}(x),\end{array}\right.$ (1.1)

with $10/3<p<6$. (1.1) models the propagation of the wave functions in Bose-
Einstein condensate (BEC). BEC, the fifth state of matter, is one of the most
remarkable discoveries in the twenty century. It describes a phenomenon that
at very low temperatures all the atoms stop behaving like individual waves and
merge together to form a dense, indistinguishable atomic wave. This phenomenon
was predicted by S. N. Bose and A. Einstein in 1925, but it was first
experimentally realized in dilute alkali gases in 1995 (2001 Nobel Prize in
Physics attributed to E. A. Cornell, C. E Wieman, and W. Ketterle). BEC does
not only provide a new tool to investigate the quantum properties of matter
but also opens new perspectives for understanding the elusive phenomena of
superconductivity and superfluidity. In the experiment, BEC is observed in
presence of a confined potential trap and its macroscopic behavior strongly
depends on the shape of this trap potential. The main goal of this paper is to
provide a new approach to the existence of static solutions to (1.1), and to
provide information about the multiplicity and the stability/ instability of
these solutions. The establishment of such properties is of enormous
importance to the designers of BECs. For example, precise information about
non-existence, or instability or multiplicity give valuable piece of
information about the possibility of the realization of the BECs, and the
validity of the model. Note that the physically relevant cubic nonlinearity
$|\Psi|^{2}\Psi$ is covered by our study. Cubic NLS, often referred as Gross-
Pitaevskii equation (GPE), gives a good description for BEC at temperature
much smaller than the critical temperature, see e.g. [17, 14]. (1.1) covers
the case of attractive interactions and partial confinement, including the
limit case of the so-called cigar-shaped model, see [8].

We emphasize that the partial confinement corresponds to the standard
modelling for magnetic traps in BEC, see [23, 29] for a more detailed account.
Our results provide valuable information to physicists and engineers. For
example, the uniqueness of the ground state solution (Theorem 1.4) is the
ideal situation for realization of a BEC. This finding shows that turning off
the confinement in a direction does not prevent physicists/engineers from the
creation of a stable wavefunction (BEC). This considerably reduces the costs.
Additionally, discussions with the Photonics group of Caltech was the source
of motivation to obtain the asymptotics in Theorem 1.4. Theorem 1.7 has also
an important physical significance, it tells us that under suitable
conditions, the realization of BEC under partial confinement is only possible
when the frequency of the electromagnetic field is low.

Before explaining our contribution and the challenges encountered, let us
first introduce some notations: A standing wave of (1.1) is a solution of the
form $\Psi(t,x)=e^{-i\lambda t}u(x)$ where $u$ solves $\Delta
u-(x_{1}^{2}+x_{2}^{2})u+\lambda u+|u|^{p-2}u=0$ in $\mathbb{R}^{3}$. Without
loss of generality, we always assume that $u$ is real-valued. Thus, we aim to
find solutions of the following equation:

$\left\\{\begin{array}[]{cc}-\Delta u+(x_{1}^{2}+x_{2}^{2})u=\lambda
u+|u|^{p-2}u\ in\ \mathbb{R}^{3},\\\ u(x)\rightarrow 0\ as\
|x|\rightarrow+\infty.\end{array}\right.$ (1.2)

Set

$H=\\{u\in
H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})|u(x_{1},x_{2},x_{3})|^{2}dx<+\infty\\},$

with the inner product

$\langle u,v\rangle=\int_{\mathbb{R}^{3}}(\nabla u\nabla
v+(x_{1}^{2}+x_{2}^{2}+1)uv)dx,$

and the norm

$\|u\|^{2}=\int_{\mathbb{R}^{3}}(|\nabla
u|^{2}+(x_{1}^{2}+x_{2}^{2}+1)u^{2})dx.$

We also introduce the corresponding functional

$J_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{3}}(|\nabla
u|^{2}+(x_{1}^{2}+x_{2}^{2}-\lambda)u^{2})dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|u|^{p}dx$

and

$I(u)=\frac{1}{2}\int_{\mathbb{R}^{3}}(|\nabla
u|^{2}+(x_{1}^{2}+x_{2}^{2}))dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|u|^{p}dx.$

For NLS with a partial confinement, P. Antonelli, R. Carles and J. Drumond
Silva, in [2], studied the scattering phenomena; Bellazzini, Boussaid,
Jeanjean, and Visciglia, in [7], gave the first result of the existence and
stability of standing states in the $L^{2}$ supercritical case. In [19], Y.H.
Hong and S.D. Jin further investigated the properties of these ground states.
In [16], T.X. Gou studied the existence and orbital stability of standing
waves to nonlinear Schrödinger system with partial confinement in the $L^{2}$
subcritical case. Despite all these valuable contributions, many important
open questions regarding (1.1) are still unsolved. Equation (1.1) needs
further understanding. First, as [7, Remark 1.8] pointed, it is natural to ask
whether there is a bifurcation phenomenon from the bottom of the spectrum of
$\Lambda_{0}=-\Delta+(x_{1}^{2}+x_{2}^{2})$. Note that $\Lambda_{0}$ is not an
eigenvalue, it is unclear how to solve this problem via a standard bifurcation
argument. One of our motivations is to provide an answer to this challenging
question. We show that there is a $H^{1}(\mathbb{R}^{3})$-bifurcation
phenomenon from $\Lambda_{0}$. For $H$ norm, we will prove that this is not
true. More precisely, we find a bifurcation phenomenon from infinity that is
different form the case with a full confinement (see Theorem 1.4, Remark 1.6).
Moreover, it is worth noticing that the power of the nonlinearity is mass-
subcritical with respect to the set of variables bearing no potential, which
is an essential requirement in [7, 19]. Though this restriction on $p$ is not
apparent in (1.1) since the mass-critical exponent in one dimension is $6$.
Our second motivation in this paper is to develop a new method, that can be
extended to general dimensions and nonlinearities without the mass-subcritical
condition. Finally, an open problem was proposed in [7, Remark 1.10] whether a
second solution of (1.1) with small prescribed $L^{2}$ mass exists. Our last
motivation is to address this important issue. We will provide an affirmative
answer in Theorem $1.7$. We thank L. Jeanjean for informing us that the open
problem of the existence of the second solution was already solved in [43].
Our method is different, additionally we show that the second solution is
orbitally unstable. Wei and Wu presented a novel method to study normalized
solutions for combined nonlinearities when one is Sobolev critical. They then
applied their method to equation (1.2). On page 6 of [43], they stated that
their method will be helpful in studying normalized solutions of other
elliptic equations. We hope that this claim is true as it can give a second
general approach after the one developed in [20].

In the previous contributions, there are two different approaches to study the
existence of solutions of equation (1.2) using variational methods:

$\bullet$ For fixed and prescribed frequency $\lambda$, we study critical
points of $J_{\lambda}$ but know nothing about the $L^{2}$ mass
$\sqrt{\int_{\mathbb{R}^{3}}u^{2}dx}$.

$\bullet$ For prescribed $L^{2}$ mass $c$, we study critical points of $I$
under the constraint $\int_{\mathbb{R}^{3}}u^{2}dx=c^{2}$. Then $\lambda$ is
determined as a Lagrange multiplier and unknown.

In [7], Bellazzini and al noticed that when $2<p<10/3,$ $I_{c}>-\infty,$ for
any $c\in\mathbb{R}$, and the classical approach by global minimization works.
However, in the supercritical case $p>10/3,I_{c}=-\infty$ for any
$c\in\mathbb{R}$. New ideas and techniques were needed to overcome this
challenging issue. In their transformative paper, [7], the authors obtained
static solutions by considering a suitable localized version of the
minimization problem. More precisely, for every $\rho>0$, they introduced the
following minimization problem

$J_{r}^{\rho}=\inf_{u\in S_{c}\cap B_{\rho}}I(u),$ (BBJV)

where

$S_{c}=\\{u\in H:\int_{\mathbb{R}^{3}}u^{2}dx=c^{2}\\},B_{\rho}=\\{u\in
H:\int_{\mathbb{R}^{3}}(|\nabla
u|^{2}+(x_{1}^{2}+x_{2}^{2})u^{2})dx\leq\rho\\}.$

They showed that for any $\rho>0$, there exists $r_{0}=r_{0}(\rho)>0$ such
that

$\text{(BBJV1)}\qquad S_{c}\cap B_{\rho}\neq 0\quad\forall r<r_{0}$
$\text{(BBJV2)}\qquad\emptyset\neq M_{c}^{\rho}\subset B_{\rho r}\quad\forall
r<r_{0}$

where $M_{c}^{\rho}=\left\\{u\in S_{c}\cap B_{\rho}\ \text{s.t}\
I(u)=J_{r}^{\rho}\right\\}.$ (BBJV1) guarantees that all the minimizing
sequences of $J_{r}^{\rho}$ are compact up to the action of translations w.r.t
$x_{3}$ provided that $r<r_{0}$. (BBJV2) is crucial to show that the
minimizers of (BBJV) are critical points of the energy functional $I$
restricted to the sphere $S_{c},$ where it is also crucial to make sure that
these minimizers do not belong to the boundary of $B_{\rho}\cap S_{c}.$ If
this holds true, then for any minimizer $u,$ there exists a Lagrange
multiplier $\lambda$ such that equation (1.2) is satisfied.

To the best of our knowledge, their method was the fist general approach to
give a general line of attack to establish the existence of normalized
solutions in the supercritical setting. It has then been used by many
colleagues to study the minimization problems associated to various PDEs in
the supercritical case. In this paper, we not only provide a simpler
alternative proof of their main result, namely, Theorem $3$, [7], but we also
derive additional important information about the ground state solutions as
well as their orbital stability/instability. In particular, we will provide
full and complete answers to some conjectures stated in [7]. As discussed in
[17], the abstract framework developed by the two authors of this paper is
applicable to many interesting situations. In this paper we will apply our
general method to equation (1.2) and give an explanation on the effect of this
type of confinement versus the classical one. It is worth mentioning that
continuation argument which is made possible by the study of some limit
problem on which the uniqueness and non-degeneracy is known was developed
independently by several authors recently, see [20, 24, 43]. Let us point out
that [3] obtained results on the existence of ground state solutions of (1.2)
by using a different method, see [3, Lemma 3.2]. However, their approach is
heavily connected to this special case, and does not allow to answer the
conjectures in [7].

######  Theorem 1.1.

[7, Theorems 1-3] Let $10/3<p<6$ and $c>0$ sufficiently small.

$(1)$ (Existence) There exists $u_{c}\in S_{c}$ such that

$I^{\prime}|_{S_{c}}(u_{c})=0\ and\ I(u_{c})=\inf\\{v\in H:v\in
S_{c},I^{\prime}|_{S_{c}}(v)=0\\}.$

Moreover, $u_{c}$ satisfies (1.2) for some $\lambda_{c}$.

$(2)$ (Symmetry and monotonicity) $u_{c}$ is positive and for some
$k\in\mathbb{R}$, $u_{c}(x_{1},x_{2},x_{3}-k)$ is radially symmetric and
decreasing w.r.t. $(x_{1},x_{2})$ and w.r.t. $x_{3}$.

$(3)$ (Stability) The set $M_{c}^{\rho}$ is stable under the flow associated
with (1.1) (the definition can be found in [7]).

$(4)$ (Estimates of $\lambda$) For a universal constant $M>0$ and
$\Lambda_{0}=\inf\sigma(-\Delta+x_{1}^{2}+x_{2}^{2})$,

$(1-Mc^{p-2})\Lambda_{0}\leq\lambda_{c}<\Lambda_{0}.$ (1.3)

In the recent years, the study of normalized solutions (i.e. solutions with
prescribed $L^{2}$ mass) has aroused great interest due to its numerous
applications. There is considerable amount of valuable contributions. It is
impossible to cite all of them, see e.g. [37, 38, 39] for mass subcritical
case on $\mathbb{R}^{N}$, [9, 11, 22] for the mass supercritical case on
$\mathbb{R}^{N}$, [33, 20] for non-autonomous nonlinearities, [28, 30, 34, 20]
for bounded domain cases, and [34, 35] for exterior domain cases. In
particular, in [20], the authors developed a unified and very general method
to study the existence of normalized solutions and orbital
stability/instability of standing waves. The idea of using the behaviour of
the $L^{2}$ norm with respect to $\lambda$ to deduce stability/instability
result can also be found in [26]. For a fixed $\lambda$, we study critical
points of $J_{\lambda}$ and then give the information of the $L^{2}$ mass of
these solutions when $\lambda$ changes. Inspired by [20], we make use of
several tricks for the problem of partial confinement. For the readers’
convenience, provide a short summary of these ideas here.

Step 1:  The existence of a ground state for fixed $\lambda<\Lambda_{0}$.

Define the Nehari manifold with a parameter $\lambda$

$\mathcal{N}_{\lambda}=\\{u\in H:\int_{\mathbb{R}^{3}}(|\nabla
u|^{2}+(x_{1}^{2}+x_{2}^{2}-\lambda)u^{2})dx=\int_{\mathbb{R}^{3}}|u|^{p}dx\\},$

and set

$h(\lambda)=\inf_{u\in\mathcal{N}_{\lambda}}J_{\lambda}(u).$

Step 1 consists in solving this minimization problem.

The main difficulty in the establishment of the ground state is the lack of
compactness, due to the translation invariance w.r.t. $x_{3}$. Using a
concentration-compactness argument similar to [7, Lemma 3.4], for any
$\lambda<\Lambda_{0}$ where
$\Lambda_{0}=\inf\sigma(-\Delta+x_{1}^{2}+x_{2}^{2})$, we can obtain a
minimizer $u_{\lambda}$ such that $J_{\lambda}(u_{\lambda})=h(\lambda)$ (such
solutions are called ground states or least action solutions in this paper).
Furthermore, we will show the positivity, symmetry and monotonicity of
$u_{\lambda}$.

Step 2:  The uniqueness, non-degeneracy and asymptotic behavior of
$u_{\lambda}$ when $\lambda\rightarrow-\infty$.

Let $\mu=1/\lambda^{2}\rightarrow 0^{+}$ and

$v_{\mu}=|\lambda|^{-\frac{1}{p-2}}u_{\lambda}(\frac{x}{\sqrt{|\lambda}|}).$
(1.4)

Then $v_{\mu}$ is the ground state of the following equation

$\left\\{\begin{array}[]{cc}-\Delta v+\mu(x_{1}^{2}+x_{2}^{2})v+v=|v|^{p-2}v\
in\ \mathbb{R}^{3},\\\ v(x)\rightarrow 0\ as\
|x|\rightarrow+\infty,\end{array}\right.$ (1.5)

with the corresponding functional

$\widetilde{J}_{\mu}(v)=\frac{1}{2}\int_{\mathbb{R}^{3}}(|\nabla
v|^{2}+\mu(x_{1}^{2}+x_{2}^{2})v^{2}+v^{2})dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|v|^{p}dx.$

We aim to prove that $v_{\mu}\rightarrow\widetilde{v}$ as $\mu\rightarrow
0^{+}$ where $\widetilde{v}$ is the unique (up to translations), non-
degenerate, positive solution of the limit equation

$\left\\{\begin{array}[]{cc}-\Delta v+v=|v|^{p-2}v\ in\ \mathbb{R}^{3},\\\
v(x)\rightarrow 0\ as\ |x|\rightarrow+\infty.\end{array}\right.$ (1.6)

This implies the uniqueness and non-degeneracy of $v_{\mu}$, which is
equivalent to the uniqueness and non-degeneracy of $u_{\lambda}$.

Difficulties that we will address in this step are the following:

$\bullet$ The space is $H$ when $\mu>0,$ and $H^{1}(\mathbb{R}^{3})$ when
$\mu=0$. Thus the space changes when we take the limit as $\mu\rightarrow
0^{+}$.

$\bullet$ $\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})v^{2}dx$ cannot be
controlled by $\|v\|_{H^{1}(\mathbb{R}^{3})}^{2}$. Therefore, it is difficult
to get the boundedness of $v_{\mu}$ in $H$ before we prove the convergence of
$v_{\mu}$.

The first difficulty invalidates the implicit function methods if $p$ is not
an integer. For this reason, we need new arguments to derive the uniqueness
and non-degeneracy. To overcome the second difficulty, we show the convergence
in $H^{1}(\mathbb{R}^{3})$ rather than in $H$, avoiding to show the
boundedness of $\|v_{\mu}\|$. Then we will prove that
$\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx\rightarrow 0,$ and standing waves
associated to these ground states are orbitally unstable.

Step 3:  The uniqueness, non-degeneracy, and asymptotic behavior of
$u_{\lambda}$ when $\lambda\rightarrow\Lambda_{0}$.

Let $\tau=\Lambda_{0}-\lambda\rightarrow 0^{+}$ and

$w_{\tau}=(\Lambda_{0}-\lambda)^{-\frac{1}{p-2}}u_{\lambda}(x_{1},x_{2},\frac{x_{3}}{\sqrt{\Lambda_{0}-\lambda}}).$
(1.7)

Then $w_{\tau}$ is the ground state of the following equation

$\left\\{\begin{array}[]{cc}\frac{1}{\tau}(-\Delta_{y}+|y|^{2}-\Lambda_{0})w-\partial_{zz}w+w=|w|^{p-2}w\
in\ \mathbb{R}^{3},\\\ w(x)\rightarrow 0\ as\
|x|\rightarrow+\infty,\end{array}\right.$ (1.8)

where $x=(y,z)\in\mathbb{R}^{2}\times\mathbb{R}$, with the corresponding
functional

$\widehat{J}_{\tau}(w)=\frac{1}{2\tau}\int_{\mathbb{R}^{3}}(|\nabla_{y}w|^{2}+|y|^{2}w^{2}-\Lambda_{0}w^{2})dx+\frac{1}{2}\int_{\mathbb{R}^{3}}(|\partial_{z}w|^{2}+w^{2})dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|w|^{p}dx.$

Let

$e_{1}(y)=\frac{1}{\sqrt{\pi}}e^{-\frac{|y|^{2}}{2}}$

be the lowest eigenstate to $-\Delta_{y}+|y|^{2}$, i.e.
$-\Delta_{y}e_{1}(y)+|y|^{2}e_{1}(y)=\Lambda_{0}e_{1}(y)$. When
$\tau\rightarrow 0^{+}$, the effect of the partial confinement is so strong
that $w_{\tau}\rightarrow e_{1}(y)\widehat{w}(z)$ where $\widehat{w}(z)$ is
the ground state of the following one-dimensional problem

$\left\\{\begin{array}[]{cc}-w^{\prime\prime}(z)+w(z)=\frac{2}{p}\pi^{1-\frac{p}{2}}|w(z)|^{p-2}w(z)\
in\ \mathbb{R},\\\ w(z)\rightarrow 0\ as\
|z|\rightarrow+\infty,\end{array}\right.$ (1.9)

Then we show the uniqueness and non-degeneracy of $w_{\tau}$, which is
equivalent to the uniqueness and non-degeneracy of $u_{\lambda}$, and prove
that $\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx\rightarrow 0$ and standing waves
associated to these ground states are orbitally stable.

To summarize our previous discussion, our results read as follows:

######  Theorem 1.2.

Let $2<p<6$. Then for $\lambda<\Lambda_{0}$, $h(\lambda)$ is achieved by a
positive minimizer $u_{\lambda}$, (which is called a ground state and solves
(1.2)). Furthermore, for some $z_{\lambda}\in\mathbb{R}$, the ground state
$u_{\lambda}(x_{1},x_{2},x_{3}-z_{\lambda})$ is radially symmetric and
decreasing w.r.t. $(x_{1},x_{2})$ and w.r.t. $x_{3}$.

######  Theorem 1.3.

Let $2<p<6$. There exists $\Lambda_{1}<0$ such that for $\lambda<\Lambda_{1}$,
the ground state $u_{\lambda}$ of (1.2), given by Theorem 1.2, is unique (up
to translation w.r.t. $x_{3}$) in $H$ and non-degenerate in $H_{s}$, where

$H_{s}=\\{u\in H:u=u(|y|,|z|),x=(y,z)\in\mathbb{R}^{2}\times\mathbb{R}\\}.$

Furthermore, for some $\\{z_{\lambda}\\}\subset\mathbb{R}$, as
$\lambda\rightarrow-\infty$,

$|\lambda|^{-\frac{1}{p-2}}u_{\lambda}(\frac{x_{1}}{\sqrt{|\lambda}|},\frac{x_{2}}{\sqrt{|\lambda}|},\frac{x_{3}}{\sqrt{|\lambda}|}+z_{\lambda})\rightarrow\widetilde{v}\
in\ H^{1}(\mathbb{R}^{3}),$

where $\widetilde{v}$ is the unique (up to translations), non-degenerate,
positive solution of (1.6).

######  Theorem 1.4.

Let $2<p<6$. There exists $\Lambda_{2}<\Lambda_{0}$ such that for
$\Lambda_{2}<\lambda<\Lambda_{0}$, the ground state $u_{\lambda}$ of (1.2),
given by Theorem 1.2, is unique (up to translation w.r.t. $x_{3}$) and non-
degenerate in $H$. Furthermore, for some $\\{z_{\lambda}\\}\subset\mathbb{R}$,
as $\lambda\rightarrow\Lambda_{0}$,

$(\Lambda_{0}-\lambda)^{-\frac{1}{p-2}}u_{\lambda}(y,\frac{z}{\sqrt{\Lambda_{0}-\lambda}}+z_{\lambda})\rightarrow
e_{1}(y)\widehat{w}(z)\ in\ H,$

where $x=(y,z)\in\mathbb{R}^{2}\times\mathbb{R}$ and $\widehat{w}$ is the
unique (up to translations), non-degenerate, positive solution of (1.9).

In particular, if $10/3<p<6$,

$\|u_{\lambda}\|_{H^{1}(\mathbb{R}^{3})}\rightarrow
0,\|u_{\lambda}\|\rightarrow+\infty;$

if $2<p<10/3$,

$\|u_{\lambda}\|_{H^{1}(\mathbb{R}^{3})}\rightarrow
0,\|u_{\lambda}\|\rightarrow 0;$

if $p=10/3$,

$\|u_{\lambda}\|_{H^{1}(\mathbb{R}^{3})}\rightarrow
0,\|u_{\lambda}\|\rightarrow C_{0},$

where
$C_{0}=\sqrt{\int_{\mathbb{R}^{2}}|y|^{2}e_{1}(y)^{2}dy\int_{\mathbb{R}}\widehat{w}(z)^{2}dz}$.

######  Remark 1.5.

Theorems 1.3 and 1.4 tell us that the effect of the partial confinement is so
strong when $\lambda\rightarrow\Lambda_{0}$ that a dimension reduction
appears. On the other hand, this effect is so weak when
$\lambda\rightarrow-\infty$ that we can almost ignore it. The latter is not
trivial since the partial confinement is unbounded. Moreover, the dimension
reduction of Bose-Einstein condensates is important both theoretically and
experimentally, and it has been studied in various settings, see [27, 40] for
cigar-shaped and disk-shaped condensates, and [1, 2, 4, 5, 6, 8, 10, 12, 13,
15, 21, 21] for further references and related results.

######  Remark 1.6.

Theorem 1.4 shows that there is a $H^{1}(\mathbb{R}^{3})$-bifurcation
phenomenon from $\Lambda_{0}$ for all $2<p<6$. For $H$-bifurcation,
$\Lambda_{0}$ is a bifurcation point only when $2<p<10/3$. When $10/3<p<6$,
there is a $H$-bifurcation phenomenon from infinity. We underline that the
appearance of $10/3$ is a coincidence and it is not the mass-critical exponent
for the space dimension. In fact, let us consider

$-\Delta u-|y|^{\alpha}u=\lambda
u+|u|^{p-2}u,x=(y,z)\in\mathbb{R}^{N-d}\times\mathbb{R}^{d},\alpha>0,N\geq
2,1\leq d\leq N-1.$

Remark 1.9 will point that our method is applicable to this equation for all
$2<p<2^{\ast}=2N/(N-2)^{+}$. The critical exponent of $H$-bifurcation is
$2+4/(d+\alpha)$.

######  Theorem 1.7.

Let $10/3<p<6$. Then for sufficiently small $c>0$, (1.2) admits at least two
positive solutions whose $L^{2}$-norm are $c$ for different $\lambda$, one
(for larger $\lambda$) is orbital stable while another (for smaller $\lambda$)
is orbital unstable (the definition will be reviewed in Section 5).

######  Theorem 1.8.

Let $10/3<p<6$ and $u_{\lambda}$ be given by Theorem 1.2. Then

$\sup_{\lambda<\Lambda_{0}}\|u_{\lambda}\|_{L^{2}}\leq C,$

for some $C>0$ independent on $\lambda$. In particular, this implies the non-
existence of ground states with large $L^{2}$-norm.

######  Remark 1.9.

It is worth noticing that our main results can be easily generalized. We can
handle the cases of general dimension $N$ with a general nonlinearity
(including mixed nonlinearity which has gained a lot of interest in the last
year, see e.g. [31, 32, 36, 42]) and a general potential ($V(|y|)$ is strictly
increasing in $|y|$ and unbounded,
$x=(y,z)\in\mathbb{R}^{N-d}\times\mathbb{R}^{d},0\leq d\leq N$).

######  Remark 1.10.

Consider the following equation

$\left\\{\begin{array}[]{cc}i\partial_{t}\Psi+\Delta_{x}\Psi-
x_{1}^{2}\Psi+\Psi^{3}=0,(t,x)\in\mathbb{R}\times\mathbb{R}^{3},\\\
\Psi(0,x)=\psi(x),\end{array}\right.$ (1.10)

which is the limit case of the so-called disk-shaped model. Our method yields
to the existence and asymptotic behavior of the ground state solution when

$\lambda\rightarrow\inf\sigma(-\Delta+x_{1}^{2})=\inf\sigma(-\partial_{x_{1}x_{1}}+x_{1}^{2})=1,$

and there is a two-dimensional reduction of Bose-Einstein condensates from
three dimension. Note that this problem is mass-critical with respect to the
set of variables bearing no potential. It needs more discussion to show the
orbital stability/instability, which is an important, interesting and
challenging issue.

We organize this paper as follows. In Section 2, we study the existence and
properties of $u_{\lambda}$ for a fixed $\lambda<\Lambda_{0}$ and give the
proof of Theorem 1.2. In Sections 3 and 4, we discuss the uniqueness, non-
degeneracy and asymptotic behavior of $u_{\lambda}$ when
$\lambda\rightarrow-\infty$ and $\lambda\rightarrow\Lambda_{0}$ respectively,
proving Theorems 1.3 and 1.4. Finally in Section 5, we obtain the existence
and multiplicity of normalized solutions for small $L^{2}$ norm, show their
orbital stability or instability and complete the proof of Theorem 1.7.
Moreover, in this section, we give the proof of Theorem 1.8 and show the non-
existence result.

##  2 The existence and properties of $u_{\lambda}$ for a fixed
$\lambda<\Lambda_{0}$

######  Lemma 2.1.

Let $\lambda<\Lambda_{0}$ and $\\{u_{n}\\}\subset H$ be a (PS)c sequence of
$J_{\lambda}$ with $c\leq h(\lambda)$. Then there exist $u\in H$ and
$\\{z_{n}\\}\subset\mathbb{R}$ such that
$u_{n}(x_{1},x_{2},x_{3}+z_{n})\rightarrow u$ in $H$ up to a subsequence.

Proof.  Standard arguments yield to the boundedness of $u_{n}$ in $H$. Since

$(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{3}}|u_{n}|^{p}dx\rightarrow c,$

Thus $c\geq 0$. Then two cases will be treated.

Case 1: $c=0$. From

$(\frac{1}{2}-\frac{1}{p})\|u_{n}\|_{\lambda}^{2}\rightarrow c=0,$

we deduce that $u_{n}\rightarrow 0$ in $H$.

Case 2: $0<c\leq h(\lambda)$. Note that

$\liminf_{n\rightarrow+\infty}\int_{\mathbb{R}^{3}}|u_{n}|^{p}dx=\frac{2pc}{p-2}>0.$

Then by [7, Lemma 3.4], for a sequence $\\{z_{n}\\}\subset\mathbb{R}$ and up
to a subsequence, we have

$u_{n}(x_{1},x_{2},x_{3}+z_{n})\rightharpoonup u\neq 0\ in\ H.$

On the one hand, $J^{\prime}_{\lambda}(u)=0$ and thus
$u\in\mathcal{N}_{\lambda}$, yielding that $J_{\lambda}(u)\geq h(\lambda)$. On
the other hand,

$J_{\lambda}(u)=(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{3}}|u|^{p}dx\leq\liminf_{n\rightarrow+\infty}(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{3}}|u_{n}|^{p}dx\rightarrow
c\leq h(\lambda).$

Hence, $J_{\lambda}(u)=c=h(\lambda)$. Moreover, we have
$\int_{\mathbb{R}^{3}}|u_{n}|^{p}dx\rightarrow\int_{\mathbb{R}^{3}}|u|^{p}dx$,
implying that $\|u_{n}\|_{\lambda}\rightarrow\|u\|_{\lambda}$, where
$\|u\|_{\lambda}^{2}=\int_{\mathbb{R}^{3}}(|\nabla
u|^{2}+(x_{1}^{2}+x_{2}^{2})u^{2}-\lambda u^{2})dx$. Noticing the weak
convergence of $u_{n}$ in $H$ and $\|\cdot\|_{\lambda}\sim\|\cdot\|$, we have
$u_{n}\rightarrow u$ in $H$, which completes the proof. ∎

Proof of Theorem 1.2. Let $u_{n}$ be a minimizing sequence for $J_{\lambda}$
at level $h(\lambda)$. By Ekeland variational principle (see [41, Theorem
2.4]), $u_{n}$ can be taken as (PS)h(λ) sequence. Then by Lemma 2.1, we know
that $h(\lambda)$ can be achieved by a minimizer $u_{\lambda}$. Similar to the
proof of [33, Lemma 3.6], we derive that $u_{\lambda}$ does not change sign.
By using the strong maximum principle we get that $u_{\lambda}>0$ (replacing
$u_{\lambda}$ by $-u_{\lambda}$ if necessary). Then we obtain the radial
symmetry and monotonicity properties w.r.t. the $(x_{1},x_{2})$ variables and
$x_{3}$ variable respectively using moving planes techniques as in [25].
(These results can also be shown using the Steiner symmetrization and
reflexion type arguments similar to [7, Section 4].)

######  Remark 2.2.

We think that a second solution of (1.2) can be obtained by using link
theorem, which is sign-changing (more precisely, has two nodal domains), and
its functional value belongs to $(h(\lambda),2h(\lambda))$. It seems quite
challenging to obtain infinite solutions of (1.2) for a fixed $\lambda$.

##  3 The uniqueness, non-degeneracy and asymptotic behavior of $u_{\lambda}$
when $\lambda\rightarrow-\infty$

Let $u_{\lambda}$ be a positive ground state of (1.2) with
$\lambda\rightarrow-\infty$, $u_{\lambda}=u_{\lambda}(|y|,|z|)$, and $v_{\mu}$
be given by (1.4) with $\mu=1/\lambda^{2}\rightarrow 0^{+}$. Define

$\widetilde{\mathcal{N}}_{\mu}=\\{v\in H:\int_{\mathbb{R}^{3}}(|\nabla
v|^{2}+\mu(x_{1}^{2}+x_{2}^{2})v^{2}+v^{2})dx=\int_{\mathbb{R}^{3}}|v|^{p}dx\\},$

and

$\widetilde{h}(\mu)=\inf_{v\in\widetilde{\mathcal{N}}_{\mu}}\widetilde{J}_{\mu}(v).$

######  Lemma 3.1.

$J_{\lambda}(u_{\lambda})=h(\lambda)\Leftrightarrow\widetilde{J}_{\mu}(v_{\mu})=\widetilde{h}(\mu)$.

Proof.  Let

$T(\lambda)u=|\lambda|^{-\frac{1}{p-2}}u(\frac{x}{\sqrt{|\lambda|}}).$

Direct computations imply that:

$u\in N_{\lambda}\Leftrightarrow T(\lambda)u\in\widetilde{\mathcal{N}}_{\mu},$

and

$\widetilde{J}_{\mu}(T(\lambda)u)=|\lambda|^{\frac{3}{2}-\frac{p}{p-2}}J_{\lambda}(u).$

Hence, we can affirm the conclusion. ∎

######  Lemma 3.2.

For any $(\mu,v)\in[0,+\infty)\times H\backslash\\{0\\}$, there exists a
unique function $\widetilde{t}:[0,+\infty)\times
H\backslash\\{0\\}\rightarrow(0,+\infty)$ such that

$(\mu,tv)\in\mathcal{\widetilde{N}},t>0\Leftrightarrow
t=\widetilde{t}(\mu,v),$

and
$\widetilde{J}_{\mu}(\widetilde{t}(\mu,v)v)=\max_{t>0}\widetilde{J}_{\mu}(tv)$,
$\widetilde{t}\in C([0,+\infty)\times H\backslash\\{0\\},(0,\infty))$, where
$\mathcal{\widetilde{N}}:=\cup_{\mu\geq 0}\mathcal{\widetilde{N}}_{\mu}$.

Proof.  Note that

$(\mu,tv)\in\mathcal{\widetilde{N}}\Leftrightarrow\int_{\mathbb{R}^{3}}(|\nabla
v|^{2}+\mu(x_{1}^{2}+x_{2}^{2})v^{2}+v^{2})dx=|t|^{p-2}\int_{\mathbb{R}^{3}}|v|^{p}dx.$

We define

$\widetilde{t}(\mu,v)=(\frac{\int_{\mathbb{R}^{3}}(|\nabla
v|^{2}+\mu(x_{1}^{2}+x_{2}^{2})v^{2}+v^{2})dx}{\int_{\mathbb{R}^{3}}|v|^{p}dx})^{\frac{1}{p-2}}.$

Obviously, $\widetilde{t}\in C((-\infty,0]\times
H\backslash\\{0\\},(0,\infty))$.

Next we will show that
$\widetilde{J}_{\mu}(\widetilde{t}(\mu,v)v)=\max_{t>0}\widetilde{J}_{\mu}(tv)$.
Set

$g(t)=\widetilde{J}_{\mu}(tv),t>0.$

Then

$\displaystyle g^{\prime}(t)$ $\displaystyle=$ $\displaystyle
t\int_{\mathbb{R}^{3}}(|\nabla
v|^{2}+\mu(x_{1}^{2}+x_{2}^{2})v^{2}+v^{2})dx-t^{p-1}\int_{\mathbb{R}^{3}}|v|^{p}dx$
(3.1) $\displaystyle=$ $\displaystyle
t\int_{\mathbb{R}^{3}}|v|^{p}dx(\widetilde{t}(\mu,v)^{p-2}-t^{p-2}).$

Thus we derive that $g^{\prime}(t)>0$ if $0<t<\widetilde{t}(\mu,v)$ and
$g^{\prime}(t)<0$ if $t>\widetilde{t}(\mu,v)$. This completes the proof. ∎

######  Lemma 3.3.

$\lim_{\mu\rightarrow 0^{+}}\widetilde{h}(\mu)=\widetilde{h}(0)$.

Proof.  Step 1: $\liminf_{\mu\rightarrow
0^{+}}\widetilde{h}(\mu)\geq\widetilde{h}(0)$.

By Lemma 3.2, we have

$\widetilde{h}(\mu)=\inf_{\|v\|=1}\sup_{t>0}\widetilde{J}_{\mu}(tv).$

From $\mu\geq 0$, we deduce that

$\sup_{t>0}\widetilde{J}_{\mu}(tv)\geq\sup_{t>0}\widetilde{J}_{0}(tv)\geq\inf_{\|v\|=1}\sup_{t>0}\widetilde{J}_{0}(tv)=\widetilde{h}(0),\forall
v,\|v\|=1.$

Hence, $\widetilde{h}(\mu)\geq\widetilde{h}(0)$, yielding that
$\liminf_{\mu\rightarrow 0^{+}}\widetilde{h}(\mu)\geq\widetilde{h}(0)$.

Step 2: $\limsup_{\mu\rightarrow
0^{+}}\widetilde{h}(\mu)\leq\widetilde{h}(0)$.

Recall that $\widetilde{v}$ is the unique (up to translations), non-
degenerate, positive solution of (1.6). Furthermore, $\widetilde{v}$ is a
ground state solution and decays exponentially at $\infty$. In particular,
$\widetilde{v}\in H$. We consider
$\widetilde{t}(\mu,\widetilde{v})\widetilde{v}\in\mathcal{\widetilde{N}}_{\mu}$.
By Lemma 3.2,
$\widetilde{t}(\mu,\widetilde{v})\widetilde{v}\rightarrow\widetilde{t}(0,\widetilde{v})\widetilde{v}=\widetilde{v}$
in $H$ as $\mu\rightarrow 0^{+}$, implying that:

$\displaystyle\widetilde{h}(\mu)$ $\displaystyle\leq$
$\displaystyle\widetilde{J}_{\mu}(\widetilde{t}(\mu,\widetilde{v})\widetilde{v})$
(3.2) $\displaystyle=$
$\displaystyle\widetilde{J}_{0}(\widetilde{t}(\mu,\widetilde{v})\widetilde{v})+\frac{\widetilde{t}(\mu,\widetilde{v})^{2}\mu}{2}\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})\widetilde{v}^{2}dx$
$\displaystyle=$ $\displaystyle\widetilde{J}_{0}(\widetilde{v})+o(1)$
$\displaystyle=$ $\displaystyle\widetilde{h}(0)+o(1).$

Letting $\mu\rightarrow 0^{+}$, we have $\limsup_{\mu\rightarrow
0^{+}}\widetilde{h}(\mu)\leq\widetilde{h}(0)$. ∎

######  Lemma 3.4.

$v_{\mu}$ is bounded in $H^{1}(\mathbb{R}^{3})$ and
$\mu\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})v_{\mu}^{2}dx$ is bounded as
$\mu\rightarrow 0^{+}$.

Proof.  By Lemma 3.3, we may assume that, as $\mu\rightarrow 0^{+}$,

$\frac{1}{2}\int_{\mathbb{R}^{3}}(|\nabla
v_{\mu}|^{2}+\mu(x_{1}^{2}+x_{2}^{2})v_{\mu}^{2}+v_{\mu}^{2})dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|v_{\mu}|^{p}dx\leq\widetilde{h}(0)+1.$
(3.3)

(3.3), together with

$\int_{\mathbb{R}^{3}}(|\nabla
v_{\mu}|^{2}+\mu(x_{1}^{2}+x_{2}^{2})v_{\mu}^{2}+v_{\mu}^{2})dx=\int_{\mathbb{R}^{3}}|v_{\mu}|^{p}dx,$

imply that

$(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{3}}(|\nabla
v_{\mu}|^{2}+\mu(x_{1}^{2}+x_{2}^{2})v_{\mu}^{2}+v_{\mu}^{2})dx\leq\widetilde{h}(0)+1,$

which completes the proof.

Define

$\widetilde{L}_{\mu}=-\Delta+\mu(x_{1}^{2}+x_{2}^{2})+1-(p-1)|v_{\mu}|^{p-2},$

which is a self-adjoint operator from $L^{2}(\mathbb{R}^{3})$ to
$L^{2}(\mathbb{R}^{3})$, with form domain $H$ if $\mu>0$ and
$H^{1}(\mathbb{R}^{3})$ if $\mu=0$. Let $\widetilde{L}_{\mu,s}$ be the
restriction of $\widetilde{L}_{\mu}$ on

$L_{s}^{2}(\mathbb{R}^{3})=\\{v\in
L^{2}(\mathbb{R}^{3}):v=v(|y|,|z|),x=(y,z)\in\mathbb{R}^{2}\times\mathbb{R}\\}.$

We say that $v_{\mu}$ is non-degenerate in $H_{s}=H\cap L_{s}^{2}$ if
$\ker\widetilde{L}_{\mu,s}=0$.

######  Lemma 3.5.

Let $v_{\mu}\rightarrow v_{0}$ in $H^{1}(\mathbb{R}^{3})$ as $\mu\rightarrow
0^{+}$, where $v_{\mu}\in H_{s}$ solves (1.5), and $v_{0}$ is a non-degenerate
solution of (1.6). Then there exists $\delta>0$ such that $v_{\mu}$ is the
unique and non-degenerate solution of (1.5) in $\\{v\in
H_{s}:\|v-v_{0}\|_{H^{1}(\mathbb{R}^{3})}<\delta\\}$.

Proof.  Step 1: Uniqueness.

We argue by contradiction and assume that there exist $v_{1,n}\neq v_{2,n}$
such that $v_{1,n}\rightarrow v_{0},v_{2,n}\rightarrow v_{0}$ in
$H^{1}(\mathbb{R}^{3})$, $\mu_{n}\rightarrow 0^{+}$. Set

$w_{n}=\frac{v_{1,n}-v_{2,n}}{\|v_{1,n}-v_{2,n}\|_{H^{1}(\mathbb{R}^{3})}}.$

Up to a subsequence, we may assume that $w_{n}\rightharpoonup w_{0}$ in
$H^{1}(\mathbb{R}^{3})$ and $w_{n}\rightarrow w_{0}$ in
$L^{p}_{loc}(\mathbb{R}^{3})$. Note that $w_{n}$ satisfies

$-\Delta
w_{n}+\mu_{n}(x_{1}^{2}+x_{2}^{2})w_{n}+w_{n}=(p-1)|v_{2,n}+\theta_{n}(v_{1,n}-v_{2,n})|^{p-2}w_{n},\theta_{n}\in[0,1].$
(3.4)

For any $\phi\in C_{0}^{\infty}(\mathbb{R}^{3})$, we have

$\int_{\mathbb{R}^{3}}\mu_{n}(x_{1}^{2}+x_{2}^{2})w_{n}\phi
dx=\int_{\mathrm{supp}\ \phi}\mu_{n}(x_{1}^{2}+x_{2}^{2})w_{n}\phi
dx\rightarrow 0.$

This, together with the weak convergence of $w_{n}$ and the strong convergence
of $v_{1,n},v_{2,n}$ in $H^{1}(\mathbb{R}^{3})$, we derive that

$\int_{\mathbb{R}^{3}}(\nabla
w_{0}\nabla\phi+w_{0}\phi)dx=(p-1)\int_{\mathbb{R}^{3}}|v_{0}|^{p-2}w_{0}\phi
dx,\forall\phi\in C_{0}^{\infty}(\mathbb{R}^{3}).$

Since $C_{0}^{\infty}(\mathbb{R}^{3})$ is dense in $H^{1}(\mathbb{R}^{3})$, we
have

$\int_{\mathbb{R}^{3}}(\nabla
w_{0}\nabla\phi+w_{0}\phi)dx=(p-1)\int_{\mathbb{R}^{3}}|v_{0}|^{p-2}w_{0}\phi
dx,\forall\phi\in H^{1}(\mathbb{R}^{3}).$

From the non-degeneracy of $v_{0}$, we derive that $w_{0}=0$. Indeed,
$w_{0}(-x)=-w_{0}(x)$ since
$w_{0}\in\mathrm{span}\\{\partial_{1}v_{0},\partial_{2}v_{0},\partial_{3}v_{0}\\}$.
Noticing that $w_{0}(-x)=w_{0}(x)$ also holds, the proof is completed.

Standard arguments about (3.4) yield that $w_{n}$ exponentially decays
uniformly with respect to $n$. This, together with $w_{n}\rightarrow 0$ in
$L^{p}_{loc}(\mathbb{R}^{3})$, we can deduce that $w_{n}\rightarrow 0$ in
$L^{p}(\mathbb{R}^{3})$. Hence, by (3.4),

$\displaystyle\int_{\mathbb{R}^{3}}(|\nabla w_{n}|^{2}+w_{n}^{2})dx$
$\displaystyle\leq$ $\displaystyle\int_{\mathbb{R}^{3}}(|\nabla
w_{n}|^{2}+\mu_{n}(x_{1}^{2}+x_{2}^{2})w_{n}^{2}+w_{n}^{2})dx$ (3.5)
$\displaystyle=$
$\displaystyle(p-1)\int_{\mathbb{R}^{3}}|v_{2,n}+\theta_{n}(v_{1,n}-v_{2,n})|^{p-2}w_{n}^{2}dx$
$\displaystyle\leq$
$\displaystyle(p-1)\|v_{2,n}+\theta_{n}(v_{1,n}-v_{2,n})\|_{L^{p}(\mathbb{R}^{3})}^{p-2}\|w_{n}\|_{L^{p}(\mathbb{R}^{3})}^{2}\rightarrow
0,$

showing that $w_{n}\rightarrow 0$ in $H^{1}(\mathbb{R}^{3})$, which
contradicts the fact that $\|w_{n}\|_{H^{1}(\mathbb{R}^{3})}=1$.

Step 2: Non-degeneracy.

Suppose on the contrary that there exists $\varphi_{n}\in H_{s}$ satisfying

$-\Delta\varphi_{n}+\mu_{n}(x_{1}^{2}+x_{2}^{2})\varphi_{n}+\varphi_{n}=(p-1)|v_{\mu_{n}}|^{p-2}\varphi_{n},$
(3.6)

where $\mu_{n}\rightarrow 0^{+}$ and
$\|\varphi_{n}\|_{H^{1}(\mathbb{R}^{3})}=1$. Then similar to the proof of Step
1, we can show that $\varphi_{n}\rightarrow 0$ in $H^{1}(\mathbb{R}^{3})$, in
a contradiction with $\|\varphi_{n}\|_{H^{1}(\mathbb{R}^{3})}=1$. ∎

######  Lemma 3.6.

$v_{\mu}\rightarrow\widetilde{v}$ in $H^{1}(\mathbb{R}^{3})$ as
$\mu\rightarrow 0^{+}$.

Proof.  Let $v_{n}=v_{\mu_{n}}$ with $\mu_{n}\rightarrow 0^{+}$. We will show
that $v_{n}\rightarrow\widetilde{v}$ in $H^{1}(\mathbb{R}^{3})$ passing to a
subsequence if necessary. Lemma 3.4 yields the boundedness of $v_{n}$ in
$H^{1}(\mathbb{R}^{3})$. Then two cases will be addressed:

First, we assume that $\int_{\mathbb{R}^{3}}|v_{n}|^{p}dx\rightarrow 0$. Then

$\|v_{n}\|_{H^{1}(\mathbb{R}^{3})}^{2}+\mu_{n}\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})v_{n}^{2}dx=\int_{\mathbb{R}^{3}}|v_{n}|^{p}dx\rightarrow
0,$

implying that $v_{n}\rightarrow 0$ in $H^{1}(\mathbb{R}^{3})$.

Second, we assume that
$\liminf_{n\rightarrow+\infty}\int_{\mathbb{R}^{3}}|v_{n}|^{p}dx=\alpha>0$.
Noticing that $v_{n}(0)=\max v_{n}(x)$ and that $v_{n}=v_{n}(|y|,|z|)$ is
decreasing in $|y|>0,|z|>0$, standard methods imply that, passing to a
subsequence if necessary, there exists $v_{0}\in H^{1}(\mathbb{R}^{3})$ such
that $v_{n}(x)\rightharpoonup v_{0}\neq 0$ in $H^{1}(\mathbb{R}^{3})$,
$v_{n}(x)\rightarrow v_{0}$ in $L^{q}_{loc}(\mathbb{R}^{3}),2\leq q<2^{\ast}$,
$v_{n}(x)\rightarrow v_{0}$ a.e. $x\in\mathbb{R}^{3}$.

By Lemma 3.4

$\displaystyle|\left\langle\widetilde{J}^{\prime}_{0}(v_{n}),\phi\right\rangle|$
(3.7) $\displaystyle=$
$\displaystyle\mu_{n}|\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})v_{n}\phi dx|$
$\displaystyle\leq$
$\displaystyle\sqrt{\mu_{n}\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})v_{n}^{2}dx}\sqrt{\mu_{n}\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})\phi^{2}dx}$
$\displaystyle\rightarrow$ $\displaystyle 0,\forall\phi\in
C_{0}^{\infty}(\mathbb{R}^{3}).$

Since $C_{0}^{\infty}(\mathbb{R}^{3})$ is dense in $H^{1}(\mathbb{R}^{3})$, we
have

$\left\langle\widetilde{J}^{\prime}_{0}(v_{n}),\phi\right\rangle\rightarrow
0,\forall\phi\in H^{1}(\mathbb{R}^{3}),$

implying that $\widetilde{J}^{\prime}_{0}(v_{0})=0$. Thus
$v_{0}\in\widetilde{\mathcal{N}}_{0},$ and we have

$\displaystyle\widetilde{h}(0)$ $\displaystyle\leq$
$\displaystyle\widetilde{J}_{0}(v_{0})$ $\displaystyle=$
$\displaystyle(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{3}}|v_{0}|^{p}dx$
$\displaystyle\leq$
$\displaystyle\liminf_{n\rightarrow+\infty}(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{3}}|v_{n}|^{p}dx$
$\displaystyle\leq$
$\displaystyle\limsup_{n\rightarrow+\infty}(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{3}}|v_{n}|^{p}dx$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\widetilde{J}_{\mu_{n}}(v_{n})$
$\displaystyle=$ $\displaystyle\widetilde{h}(\mu_{n}).$

By Lemma 3.3, $\widetilde{h}(\mu_{n})\rightarrow\widetilde{h}(0)$. Hence,
$\widetilde{J}_{0}(v_{0})=\widetilde{h}(0)$. Moreover, we have
$\int_{\mathbb{R}^{3}}|v_{n}|^{p}dx\rightarrow\int_{\mathbb{R}^{3}}|v_{0}|^{p}dx$,
and then

$\displaystyle\|v_{0}\|_{H^{1}(\mathbb{R}^{3})}^{2}$ $\displaystyle\leq$
$\displaystyle\liminf_{n\rightarrow+\infty}\|v_{n}\|_{H^{1}(\mathbb{R}^{3})}^{2}$
$\displaystyle\leq$
$\displaystyle\limsup_{n\rightarrow+\infty}\|v_{n}\|_{H^{1}(\mathbb{R}^{3})}^{2}$
$\displaystyle\leq$
$\displaystyle\limsup_{n\rightarrow+\infty}(\|v_{n}\|_{H^{1}(\mathbb{R}^{3})}^{2}+\mu_{n}\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})v_{n}^{2}dx)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\int_{\mathbb{R}^{3}}|v_{n}|^{p}dx$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{3}}|v_{0}|^{p}dx$
$\displaystyle=$ $\displaystyle\|v_{0}\|_{H^{1}(\mathbb{R}^{3})}^{2}.$

Concerning the weak convergence of $v_{n}$ in $H^{1}(\mathbb{R}^{3})$, we
obtain that $v_{n}\rightarrow v_{0}$ in $H^{1}(\mathbb{R}^{3})$.

Combining the first and the second case, we prove that $v_{n}\rightarrow
v_{0}$ in $H^{1}(\mathbb{R}^{3})$ passing to a subsequence if necessary, where
$v_{0}=0$ or $v_{0}=\widetilde{v}$. We claim that $v_{0}\neq 0$. If this is
not the case, $0$ is non-degenerate
($\widetilde{J}_{0}^{\prime\prime}(0)=-\Delta+1$) and thus Lemma 3.5 yields
that the solution of (1.6) is unique in $\\{v\in
H_{s}:\|v\|_{H^{1}(\mathbb{R}^{3})}<\delta\\}$. Noticing that $0$ is always a
solution of (1.6) for any $\mu\in\mathbb{R}$, we know that $v_{n}=0$. This
contradicts the nontriviality of $v_{n}$. The claim has been proven, yielding
that $v_{0}=\widetilde{v}$ and the proof is completed. ∎

Proof of Theorem 1.3.  Since $\widetilde{v}$ is non-degenerate, the
combination of Lemma 3.5 and Lemma 3.6 enables us to prove that there exists
$\delta>0$ small enough such that $v_{\mu}$ is the unique and non-degenerate
ground state solution of (1.5) in the neighborhood $\\{(v,\mu)\in
H_{s}\times[0,+\infty):\|v-\widetilde{v}\|_{H^{1}(\mathbb{R}^{3})}+|\mu|<\delta\\}$.

By Lemma 3.1, the uniqueness of $v_{\mu}$ is equivalent to the one of
$u_{\lambda}$. Similarly, the non-degeneracy of $v_{\mu}$ is equivalent to the
one of $u_{\lambda}$. Furthermore, by Theorem 1.2, any ground state
$u_{\lambda}$ of (1.2) satisfying $u_{\lambda}(0)=\max u_{\lambda}(x)$ belongs
to $H_{s}$. Therefore, there exists $\Lambda_{1}<\Lambda_{0}$ such that when
$\lambda<\Lambda_{1}$, the positive ground state of (1.2) is unique in $H$ and
non-degenerate in $H_{s}$. ∎

######  Remark 3.7.

Let $V=V(|y|,|z|)$ be a K-R potential, $L_{\bot}^{2}$ be the orthogonal
complement of $L_{s}^{2}(\mathbb{R}^{3})$ in $L^{2}(\mathbb{R}^{3})$, and
$H_{|L_{\bot}^{2}}$ be the restriction of $-\Delta+|y|^{2}+V(|y|,|z|)$ to
$L_{\bot}^{2}$. It is open whether $H_{|L_{\bot}^{2}}$ enjoys the Perron-
Frobenius property, i.e the first eigenvalue (if it exists) of
$H_{|L_{\bot}^{2}}$ is simple and the corresponding eigenfunction
$\psi=\psi(|y|,|z|)$ satisfies $\psi(y,z)>0$ for $z>0$. If this is true, then
we can get the non-degeneracy of $u_{\lambda}$ in $H$.

##  4 The uniqueness, non-degeneracy and asymptotic behavior of $u_{\lambda}$
when $\lambda\rightarrow\Lambda_{0}$

Let $u_{\lambda}$ be a positive ground state of (1.2) with
$\lambda\rightarrow\Lambda_{0}$, $u_{\lambda}=u_{\lambda}(|y|,|z|)$, and
$w_{\tau}$ be given by (1.7) with $\mu=\Lambda_{0}-\lambda\rightarrow 0^{+}$.
Define

$\widehat{\mathcal{N}}_{\tau}=\\{w\in
H:\frac{1}{\tau}\int_{\mathbb{R}^{3}}(|\nabla_{y}w|^{2}+|y|^{2}w^{2}-\Lambda_{0}w^{2})dx+\int_{\mathbb{R}^{3}}(|\partial_{z}w|^{2}+w^{2})dx=\int_{\mathbb{R}^{3}}|w|^{p}dx\\},$
$\widehat{\mathcal{N}}_{0}=\\{w(z)\in
H^{1}(\mathbb{R}):\int_{\mathbb{R}}(|w^{\prime}(z)|^{2}+w(z)^{2})dx=\frac{2}{p}\pi^{1-\frac{p}{2}}\int_{\mathbb{R}}|w(z)|^{p}dx\\},$
$\widehat{J}_{0}(w)=\frac{1}{2}\int_{\mathbb{R}}(|w^{\prime}(z)|^{2}+w(z)^{2})dx-\frac{2}{p^{2}}\pi^{1-\frac{p}{2}}\int_{\mathbb{R}}|w(z)|^{p}dx$

and

$\widehat{h}(\tau)=\inf_{w\in\widehat{\mathcal{N}}_{\tau}}\widehat{J}_{\tau}(w).$

Similar to Lemma 3.1, we have

######  Lemma 4.1.

$J_{\lambda}(u_{\lambda})=h(\lambda)\Leftrightarrow\widehat{J}_{\tau}(w_{\tau})=\widehat{h}(\tau)$
for any $\tau>0$.

######  Lemma 4.2.

$\widehat{h}(\tau)\leq\widehat{h}(0),\forall\tau>0$.

Proof.  Note that $e_{1}(y)\widehat{w}(z)\in\mathcal{N}_{\tau},\forall\tau>0$.
Hence, we have

$\widehat{h}(\tau)\leq\widehat{J}_{\tau}(e_{1}(y)\widehat{w}(z))=\widehat{h}(0),\forall\tau>0.$

∎

######  Lemma 4.3.

$\int_{\mathbb{R}^{3}}(|\nabla_{y}w_{\tau}|^{2}+|y|^{2}w_{\tau}^{2}-\Lambda_{0}w_{\tau}^{2})dx\lesssim\tau,\int_{\mathbb{R}^{3}}(|\partial_{z}w_{\tau}|^{2}+w_{\tau}^{2})dx\lesssim
1,\int_{\mathbb{R}^{3}}|w_{\tau}|^{p}dx\lesssim 1,\forall\tau>0$.

Proof.  Lemma 4.2 yields that

$\frac{1}{2\tau}\int_{\mathbb{R}^{3}}(|\nabla_{y}w_{\tau}|^{2}+|y|^{2}w_{\tau}^{2}-\Lambda_{0}w_{\tau}^{2})dx+\frac{1}{2}\int_{\mathbb{R}^{3}}(|\partial_{z}w_{\tau}|^{2}+w_{\tau}^{2})dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|w_{\tau}|^{p}dx\leq\widehat{h}(0).$

Then, noticing

$\frac{1}{\tau}\int_{\mathbb{R}^{3}}(|\nabla_{y}w_{\tau}|^{2}+|y|^{2}w_{\tau}^{2}-\Lambda_{0}w_{\tau}^{2})dx+\int_{\mathbb{R}^{3}}(|\partial_{z}w_{\tau}|^{2}+w_{\tau}^{2})dx=\int_{\mathbb{R}^{3}}|w_{\tau}|^{p}dx,$

we can deduce that

$(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{3}}|w_{\tau}|^{p}\leq\widehat{h}(0),$
$(\frac{1}{2}-\frac{1}{p})(\frac{1}{\tau}\int_{\mathbb{R}^{3}}(|\nabla_{y}w_{\tau}|^{2}+|y|^{2}w_{\tau}^{2}-\Lambda_{0}w_{\tau}^{2})dx+\int_{\mathbb{R}^{3}}(|\partial_{z}w_{\tau}|^{2}+w_{\tau}^{2})dx)\leq\widehat{h}(0).$

∎

######  Lemma 4.4.

$w_{\tau}$ is uniformly bounded in $L^{\infty}(\mathbb{R}^{3})$ and in
$H^{2}_{V}(\mathbb{R}^{3})$ with respect to $\tau\leq C$ for any $C>0$, where

$H^{2}_{V}(\mathbb{R}^{3})=\\{w\in
L^{2}(\mathbb{R}^{3}):(-\Delta+(x_{1}^{2}+x_{2}^{2}))w\in
L^{2}(\mathbb{R}^{3})\\}.$

Proof.  From Lemma 4.3 we derive that $w_{\tau}$ is uniformly bounded in $H$.
Note that

$w_{\tau}=(\frac{1}{\tau}(-\Delta_{y}+|y|^{2}-\Lambda_{0})-\partial_{zz}+1)^{-1}|w_{\tau}|^{p-2}w_{\tau}.$

Then, by iterating sufficiently many times, we conclude that
$\|w_{\tau}\|_{L^{\infty}}\lesssim 1$. In particular, we know that
$\|w_{\tau}\|_{L^{2(p-1)}}\lesssim 1$ in this iterating process. Hence,

$\displaystyle\|(-\Delta_{y}+|y|^{2}-\partial_{zz})w_{\tau}\|_{L^{2}}$ (4.1)
$\displaystyle=$
$\displaystyle\|(-\Delta_{y}+|y|^{2}-\partial_{zz})(\frac{1}{\tau}(-\Delta_{y}+|y|^{2}-\Lambda_{0})-\partial_{zz}+1)^{-1}|w_{\tau}|^{p-2}w_{\tau}\|_{L^{2}}$
$\displaystyle\lesssim$ $\displaystyle\||w_{\tau}|^{p-2}w_{\tau}\|_{L^{2}}$
$\displaystyle=$ $\displaystyle\|w_{\tau}\|_{L^{2(p-1)}}^{p-1}$
$\displaystyle\lesssim$ $\displaystyle 1.$

∎

Let $\\{e_{i}\\}_{i=1}^{\infty}\subset L^{2}_{y}(\mathbb{R}^{2})$ be the
collection of $L^{2}_{y}(\mathbb{R}^{2})-$normalized eigenfunctions for
$-\Delta_{y}+|y|^{2}$, i.e.

$(-\Delta_{y}+|y|^{2})e_{i}=\lambda_{i}e_{i},$

with eigenvalues
$\Lambda_{0}=\lambda_{1}<\lambda_{2}\leq\lambda_{3}\leq\cdots$ in a non-
decreasing order. Recall that $\\{e_{i}\\}_{i=1}^{\infty}\subset
L^{2}_{y}(\mathbb{R}^{2})$ forms an orthonormal basis of
$L^{2}_{y}(\mathbb{R}^{2})$, and that the lowest eigenvalue $\Lambda_{0}=2$ is
simple and the corresponding eigenfunction is given by
$e_{1}=\frac{1}{\sqrt{\pi}}e^{-\frac{|y|^{2}}{2}}$. From the spectral
representation, $w\in L^{2}(\mathbb{R}^{3})$ can be written as

$w(x)=\sum_{i=1}^{\infty}\langle
w(y,z),e_{i}(y)\rangle_{L^{2}_{y}(\mathbb{R}^{2})}e_{i}(y),x=(y,z),$

where

$\langle
w(y,z),e_{i}(y)\rangle_{L^{2}_{y}(\mathbb{R}^{2})}=\int_{\mathbb{R}^{2})}w(y,z),e_{i}(y)dy:\mathbb{R}_{z}\rightarrow\mathbb{R}.$

Denote $\langle w(y,z),e_{1}(y)\rangle_{L^{2}_{y}(\mathbb{R}^{2})}$ by
$\rho(z)$ and define

$(Pw)(x)=\rho(z)e_{1}(y):\mathbb{R}^{3}\rightarrow\mathbb{R}.$

Let $Q=1-P$, i.e.

$(Qw)(x)=\sum_{i=2}^{\infty}\langle
w(y,z),e_{i}(y)\rangle_{L^{2}_{y}(\mathbb{R}^{2})}e_{i}(y):\mathbb{R}^{3}\rightarrow\mathbb{R}.$

From now on, we aim to show that $\rho_{\tau}(z)\rightarrow\widehat{w}(z)$ and
$Qw_{\tau}\rightarrow 0$, where $\widehat{w}(z)$ is given by Theorem 1.4 and
$\rho_{\tau}(z)=\langle
w_{\tau}(y,z),e_{i}(y)\rangle_{L^{2}_{y}(\mathbb{R}^{2})}$. By doing that, the
proof of Theorem 1.4 would be complete.

Similar to Lemma 3.2, we have

######  Lemma 4.5.

For any $\rho\in H^{1}_{z}(\mathbb{R})\backslash\\{0\\}$, there exists a
unique function
$\widehat{t}_{0}:H^{1}_{z}(\mathbb{R})\backslash\\{0\\}\rightarrow(0,+\infty)$
such that

$t\rho\in\mathcal{\widehat{N}}_{0},t>0\Leftrightarrow
t=\widehat{t}_{0}(\rho),$

and
$\widehat{J}_{0}(\widehat{t}_{0}(\rho)\rho)=\max_{t>0}\widehat{J}_{0}(t\rho)$,
$\widehat{t}_{0}\in C(H^{1}_{z}(\mathbb{R})\backslash\\{0\\},(0,\infty))$.

######  Lemma 4.6.

For any $C>0$, there exists $\delta=\delta(C)>0$ such that
$\widehat{h}(\tau)\geq\delta$ for $\tau<C$.

Proof.  For any $w\in H$ with $\|w\|=r$, we have

$\displaystyle\widehat{J}_{\tau}(w)\geq\frac{1}{2(C+\Lambda_{0}+1)}\|w\|-\frac{1}{p}\|w\|_{L^{p}(\mathbb{R}^{3})}^{p}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2(C+\Lambda_{0}+1)}r^{2}-\widetilde{C}r^{p},$ (4.2)

where $\widetilde{C}>0$ is a constant. Take $r$ small such that

$\delta=\frac{1}{2(C+\Lambda_{0}+1)}r^{2}-\widetilde{C}r^{p}>0.$

Thus the proof is completed. ∎

######  Theorem 4.7.

$\rho_{\tau}(z)\rightarrow\widehat{w}$ in $H^{1}_{z}(\mathbb{R})$ as
$\tau\rightarrow 0^{+}$.

Proof.  Step 1: $Qw_{\tau}\rightarrow 0$ in $L^{q}(\mathbb{R}^{3}),2\leq
q<2^{\ast}$ as $\tau\rightarrow 0^{+}$.

First, we show that $Qw_{\tau}\rightarrow 0$ in $L^{2}(\mathbb{R}^{3})$.
Indeed,

$\displaystyle\|Qw_{\tau}\|_{L^{2}(\mathbb{R}^{3})}^{2}$
$\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{R}^{3}}(|\nabla_{y}(Qw_{\tau})|^{2}+|y|^{2}(Qw_{\tau})^{2})dx$
(4.3) $\displaystyle\lesssim$
$\displaystyle\int_{\mathbb{R}^{3}}(|\nabla_{y}(Qw_{\tau})|^{2}+|y|^{2}(Qw_{\tau})^{2}-\Lambda_{0}(Qw_{\tau})^{2})dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}(|\nabla_{y}w_{\tau}|^{2}+|y|^{2}w_{\tau}^{2}-\Lambda_{0}w_{\tau}^{2})dx$
$\displaystyle\lesssim$ $\displaystyle\tau.$

Then, noticing that $Qw_{\tau}$ is bounded in $H$, we can conclude.

Step 2: Let $w_{n}=w_{\tau_{n}}$, $\rho_{n}=\rho_{\tau_{n}}$ with
$\tau_{n}\rightarrow 0^{+}$. Then up to a subsequence,
$\rho_{n}(z)\rightarrow\widehat{w}$ in $H^{1}_{z}(\mathbb{R})$.

Lemma 4.6 yields that $\|w_{n}\|_{L^{p}(\mathbb{R}^{3})}\geq\epsilon>0$. By
[7, Lemma 3.4], from the boundedness of $w_{n}$ in $H$ and the fact that
$w_{n}=w_{n}(|y|,|z|)$ is decreasing in $|y|>0,|z|>0$, we can conclude that up
to a subsequence,

$w_{n}(y,z)\rightharpoonup w_{0}\neq 0\ in\ H,w_{n}(y,z)\rightarrow w_{0}\neq
0\ in\ L_{loc}^{p}(\mathbb{R}^{3}).$

By Step 1, we can verify that $w_{0}=\rho_{0}(z)e_{1}(y)$ a.e. on
$\mathbb{R}^{3}$, where $\rho_{0}(z)=\langle
w_{0}(y,z),e_{i}(y)\rangle_{L^{2}_{y}(\mathbb{R}^{2})}$.

For any $\varphi(z)\in C_{0}^{\infty}(\mathbb{R})$, $\varphi(z)e_{1}(y)\in H$.
Thus, we have

$\displaystyle\int_{\mathbb{R}^{3}}(\rho_{0}^{\prime}(z)e_{1}(y)\varphi^{\prime}(z)e_{1}(y)+\rho_{0}(z)e_{1}(y)\varphi(z)e_{1}(y))dx$
(4.4) $\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\int_{\mathbb{R}^{3}}(\partial_{z}w_{n}(y,z)\varphi^{\prime}(z)e_{1}(y)+w_{n}(y,z)\varphi(z)e_{1}(y))dx$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\int_{\mathbb{R}^{3}}|w_{n}(y,z)|^{p-2}w_{n}(y,z)\varphi(z)e_{1}(y)dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}|\rho_{0}(z)|^{p-2}\rho_{0}(z)\varphi(z)e_{1}(y)^{p-1}dx.$

Since $C_{0}^{\infty}(\mathbb{R})$ is dense in $H_{z}^{1}(\mathbb{R})$, we
deduce that

$\widehat{J}_{0}^{\prime}(\rho_{0}(z))=0\ in\ H_{z}^{-1}(\mathbb{R}),$

yielding that $\rho_{0}(z)\in\mathcal{\widehat{N}}_{0}$. Then, from

$\displaystyle\widehat{h}(0)$ $\displaystyle\leq$
$\displaystyle\widehat{J}_{0}(\rho_{0})$ (4.5) $\displaystyle=$
$\displaystyle\frac{p-2}{p^{2}}\pi^{1-\frac{p}{2}}\int_{\mathbb{R}}|\rho_{0}|^{p}dz$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow+\infty}\frac{p-2}{p^{2}}\pi^{1-\frac{p}{2}}\int_{\mathbb{R}}|\rho_{n}(z)|^{p}dz$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\frac{p-2}{2p}\int_{\mathbb{R}^{3}}|\rho_{n}(z)e_{1}(y)|^{p}dx$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\frac{p-2}{2p}\int_{\mathbb{R}^{3}}|w_{n}|^{p}dx$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\widehat{h}(\tau_{n})$
$\displaystyle\leq$ $\displaystyle\widehat{h}(0),$

we obtain that $\widehat{J}_{0}(\rho_{0})=\widehat{h}(0)$, yielding that
$\rho_{0}=\widehat{w}$, and that

$\int_{\mathbb{R}}|\rho_{n}(z)|^{p}dz\rightarrow\int_{\mathbb{R}}|\rho_{0}|^{p}dz,$

implying that

$\int_{\mathbb{R}}(|\rho_{n}^{\prime}(z)|^{2}+\rho_{n}(z)^{2})dz\rightarrow\int_{\mathbb{R}}(|\rho_{0}^{\prime}|^{2}+\rho_{0}^{2})dz.$

Noticing the weak convergence of $\rho_{n}(z)$, we know that
$\rho_{n}(z)\rightarrow\widehat{w}$ in $H^{1}_{z}(\mathbb{R})$. This completes
the proof. ∎

Define

$\widehat{L}_{\tau}=\frac{1}{\tau}(-\Delta_{y}+|y|^{2}-\Lambda_{0})-\partial_{zz}+1-(p-1)|w_{\tau}|^{p-2},\tau>0,$

which is a self-adjoint operator from $L^{2}(\mathbb{R}^{3})$ to
$L^{2}(\mathbb{R}^{3})$, with form domain $H$. We say that $w_{\tau}$ is non-
degenerate in $H$ if $\ker\widehat{L}_{\tau}=\\{\partial_{z}w_{\tau}\\}$.

Proof of Theorem 1.4.  By Theorem 4.7, it suffices to show that

$Qw_{\tau}\rightarrow 0\ in\ H.$

By Lemma 4.5 and Theorem 4.7, $\widehat{t}_{0}(\rho_{\tau}(z))\rightarrow 1$.
Then

$\displaystyle\widehat{J}_{0}(\widehat{w})$ $\displaystyle\leq$
$\displaystyle\widehat{J}_{0}(\widehat{t}_{0}(\rho_{\tau}(z))\rho_{\tau}(z))$
(4.6) $\displaystyle=$ $\displaystyle\widehat{J}_{0}(\rho_{\tau}(z))+o(1)$
$\displaystyle=$
$\displaystyle\widehat{J}_{\tau}(\rho_{\tau}(z)e_{1}(y))+o(1)$
$\displaystyle=$
$\displaystyle\widehat{J}_{\tau}(w_{\tau})-\frac{1}{2\tau}\int_{\mathbb{R}^{3}}(|\nabla_{y}(Qw_{\tau})|^{2}+|y|^{2}(Qw_{\tau})^{2}-\Lambda_{0}(Qw_{\tau})^{2})dx$
$\displaystyle-\frac{1}{2}\int_{\mathbb{R}^{3}}(|\partial_{z}(Qw_{\tau})|^{2}+(Qw_{\tau})^{2})dx+o(1).$

Using Lemma 4.2,
$\widehat{J}_{0}(\widehat{w})\geq\widehat{J}_{\tau}(w_{\tau})$, we obtain
that:

$\frac{1}{\tau}\int_{\mathbb{R}^{3}}(|\nabla_{y}(Qw_{\tau})|^{2}+|y|^{2}(Qw_{\tau})^{2}-\Lambda_{0}(Qw_{\tau})^{2})dx\rightarrow
0\ as\ \tau\rightarrow 0^{+}.$
$\int_{\mathbb{R}^{3}}(|\partial_{z}(Qw_{\tau})|^{2}+(Qw_{\tau})^{2})dx\rightarrow
0\ as\ \tau\rightarrow 0^{+}.$

Thus we show the convergence of $w_{\tau}$ in $H$.

Then, along the lines of [19, Section 5], we can show the non-degeneracy and
uniqueness of $w_{\tau}$ in $H$ for sufficiently small $\tau$, which is
equivalent to the non-degeneracy and uniqueness of $u_{\lambda}$ in $H$ with
$\Lambda_{2}<\lambda<\Lambda_{0}$ for some $\Lambda_{2}<\Lambda_{0}$.

Finally, noticing that:

$\displaystyle(\Lambda_{0}-\lambda)^{\frac{1}{2}-\frac{2}{p-2}}\int_{\mathbb{R}^{3}}(|\nabla_{y}u_{\lambda}|^{2}+u_{\lambda}^{2})dx+(\Lambda_{0}-\lambda)^{-\frac{1}{2}-\frac{2}{p-2}}\int_{\mathbb{R}^{3}}|\partial_{z}u_{\lambda}|^{2}dx$
(4.7) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}(|\nabla_{y}w_{\tau}|^{2}+w_{\tau}^{2}+|\partial_{z}w_{\tau}|^{2})dx$
$\displaystyle\rightarrow$
$\displaystyle\|e_{1}(y)\widehat{w}(z)\|_{H^{1}(\mathbb{R}^{3})}^{2},$

and

$\displaystyle(\Lambda_{0}-\lambda)^{\frac{3}{2}-\frac{2}{p-2}}\int_{\mathbb{R}^{3}}(|\nabla_{y}u_{\lambda}|^{2}+u_{\lambda}^{2})dx$
(4.8) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}|y|^{2}w_{\tau}^{2}dx$
$\displaystyle\rightarrow$
$\displaystyle\int_{\mathbb{R}^{2}}|y|^{2}e_{1}(y)^{2}dy\int_{\mathbb{R}}\widehat{w}(z)^{2}dz,$

we can obtain the asymptotic behaviors of
$\|u_{\lambda}\|_{H^{1}(\mathbb{R}^{3})}$ and $\|u_{\lambda}\|$ as
$\lambda\rightarrow\Lambda_{0}$. This completes the proof. ∎

##  5 The existence, non-existence, multiplicity of normalized solutions, and
orbital stability/instability

###  5.1 Asymptotical behavior and monotonicity of
$\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx$ when $\lambda\rightarrow-\infty$

Similar arguments to the proof of [33, Corollary 2.5] and [20, Lemma 4.23],
yield that $\\{(\lambda,u_{\lambda}):\lambda<\Lambda_{1}\\}$ is a $C^{1}$
curve in $\mathbb{R}\times H$. Let

$\chi_{\lambda}=\partial_{\lambda}u_{\lambda},$
$\widetilde{\chi}_{\mu}=|\lambda|^{-\frac{3-p}{p-2}}\chi_{\lambda}(\frac{x}{\sqrt{|\lambda|}})$

where $\mu=1/\lambda^{2}$.

######  Lemma 5.1.

Recall that $\widetilde{L}_{\mu,s}$ is given by Section 3.

$(i)$ For $\mu<1/\Lambda_{1}^{2}$, there exists $C>0$ such that

$\limsup_{\mu\rightarrow
0^{+}}\|\widetilde{L}_{\mu,s}^{-1}\|_{L^{2}(\mathbb{R}^{3})\rightarrow
H^{2}(\mathbb{R}^{3})}\leq C,$

$(ii)$ For any $v\in L_{s}^{2}(\mathbb{R}^{3})$,
$\widetilde{L}_{\mu,s}^{-1}v\rightarrow\widetilde{L}_{0,s}^{-1}v$ in
$H^{2}(\mathbb{R}^{3})$.

Proof.  $(i)$ Note that $\widetilde{L}_{\mu}$ is
$\widetilde{J}^{\prime\prime}_{\mu}(\widetilde{v})$. By the proof of Theorem
1.3, we may assume that there exists $\epsilon>0$ such that

$\sigma(\widetilde{L}_{\mu,s})\cap(-\epsilon,\epsilon)=\emptyset,\forall\mu\in[0,1/\Lambda_{1}^{2}).$

Hence,

$\|\widetilde{L}_{\mu,s}^{-1}v\|_{H^{2}(\mathbb{R}^{3})}\leq\|\widetilde{L}_{\mu,s}^{-1}v\|_{H_{V}^{2}(\mathbb{R}^{3})}\leq\frac{1}{\epsilon}\|v\|_{L^{2}(\mathbb{R}^{3})}.$

Where

$H_{V}^{2}(\mathbb{R}^{3})=\\{v\in L^{2}(\mathbb{R}^{N}):(-\Delta
v+(x_{1}^{2}+x_{2}^{2})v)\in L^{2}(\mathbb{R}^{N})\\}$

is the domain of $\widetilde{L}_{\mu}$.

$(ii)$ By the proof of $(i)$, $\widetilde{L}_{\mu,s}^{-1}v$ is bounded in
$H^{2}_{V}$ for any $v\in L_{s}^{2}(\mathbb{R}^{3})$. Therefore,

$\widetilde{L}_{0,s}\widetilde{L}_{\mu,s}^{-1}v=v-\mu(x_{1}^{2}+x_{2}^{2})\widetilde{L}_{\mu,s}^{-1}v\rightarrow
v\in L^{2}(\mathbb{R}^{3}),$

implying that

$\|\widetilde{L}_{\mu,s}^{-1}v-\widetilde{L}_{0,s}^{-1}v\|_{H^{2}(\mathbb{R}^{3})}=\|\widetilde{L}_{0,s}^{-1}(\widetilde{L}_{0,s}\widetilde{L}_{\mu,s}^{-1}v-v)\|_{H^{2}(\mathbb{R}^{3})}\leq
C\|\widetilde{L}_{0,s}\widetilde{L}_{\mu,s}^{-1}v-v\|_{L^{2}(\mathbb{R}^{3})}\rightarrow
0.$

∎

######  Lemma 5.2.

Let $10/3<p<6$. Then

$(i)$ $\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx\rightarrow 0$ as
$\lambda\rightarrow-\infty$;

$(ii)$
$\widetilde{\chi}_{\mu}\rightarrow\frac{1}{2-p}\widetilde{v}-\frac{1}{2}x\cdotp\nabla\widetilde{v}$
in $H^{2}(\mathbb{R}^{3})$ as $\mu\rightarrow 0^{+}$.

$(iii)$ for some $\widetilde{\Lambda}_{1}\leq\Lambda_{1}$,
$\frac{d}{d\lambda}\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx>0$ when
$\lambda<\widetilde{\Lambda}_{1}$.

Proof.  $(i)$ From Theorem 1.3, we derive that

$|\lambda|^{\frac{3}{2}-\frac{2}{p-2}}\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx\rightarrow\int_{\mathbb{R}^{3}}\widetilde{v}^{2}dx,$

showing that $\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx\rightarrow 0$, since
$3/2-2/(p-2)>0$.

$(ii)$ Direct computations show that

$\widetilde{L}_{\mu,s}\widetilde{\chi}_{\mu}=v_{\mu}.$

Together with Lemma 5.1 and the convergence of $v_{\mu}$, we have

$\widetilde{\chi}_{\mu}=\widetilde{L}_{\mu,s}^{-1}v_{\mu}=\widetilde{L}_{\mu,s}^{-1}(v_{\mu}-\widetilde{v})+\widetilde{L}_{\mu,s}^{-1}\widetilde{v}\rightarrow\widetilde{L}_{0,s}^{-1}\widetilde{v}\
in\ H^{2}(\mathbb{R}^{3}).$

Noticing that

$\widetilde{L}_{0,s}(\frac{1}{2-p}\widetilde{v}-\frac{1}{2}x\cdotp\nabla\widetilde{v})=\widetilde{v},$

we can conclude.

$(iii)$

$\displaystyle\lim_{\lambda\rightarrow-\infty}|\lambda|^{\frac{p-4}{p-2}+\frac{3}{2}}\frac{d}{d\lambda}\int_{\mathbb{R}^{3}}|u_{\lambda}|^{2}dx$
(5.1) $\displaystyle=$
$\displaystyle\lim_{\lambda\rightarrow-\infty}2|\lambda|^{\frac{p-4}{p-2}+\frac{3}{2}}\int_{\mathbb{R}^{N}}u_{\lambda}\chi_{\lambda}dx$
$\displaystyle=$ $\displaystyle\lim_{\mu\rightarrow
0^{+}}2\int_{\mathbb{R}^{3}}v_{\mu}\widetilde{\chi}_{\mu}dx$ $\displaystyle=$
$\displaystyle
2\int_{\mathbb{R}^{3}}\widetilde{v}(\frac{1}{2-p}\widetilde{v}-\frac{1}{2}x\cdotp\nabla\widetilde{v})dx$
$\displaystyle=$ $\displaystyle
2(\frac{1}{2-p}+\frac{3}{4})\int_{\mathbb{R}^{3}}|\widetilde{v}|^{2}dx>0.$

Therefore, there exists $\widetilde{\Lambda}_{1}\leq\Lambda_{1}$ such that for
any $\lambda<\widetilde{\Lambda}_{1}$,

$\frac{d}{d\lambda}\int_{\mathbb{R}^{3}}|u_{\lambda}|^{2}dx>0.$

∎

###  5.2 Asymptotical behavior and monotonicity of
$\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx$ when $\lambda\rightarrow\Lambda_{0}$

Similar arguments to the proof of [33, Corollary 2.5] and [20, Lemma 4.23],
yield that $\\{(\lambda,u_{\lambda}):\Lambda_{2}<\lambda<\Lambda_{0}\\}$ is a
$C^{1}$ curve in $\mathbb{R}\times H$. Let

$\chi_{\lambda}=\partial_{\lambda}u_{\lambda},$
$\widehat{\chi}_{\tau}=(\Lambda_{0}-\lambda)^{-\frac{3-p}{p-2}}\chi_{\lambda}(y,\frac{z}{\sqrt{\Lambda_{0}-\lambda}})$

where $\tau=\Lambda_{0}-\lambda$.

######  Lemma 5.3.

Recall that $\widehat{L}_{\tau}$ is given by Section 4 and define

$\widehat{L}_{0}=-\frac{d^{2}}{dz^{2}}+1-\frac{2(p-1)}{p}\pi^{1-\frac{p}{2}}|\widehat{w}(z)|^{p-2},$

which is a self-adjoint operator from $L^{2}(\mathbb{R})$ to
$L^{2}(\mathbb{R})$, with form domain $H^{1}(\mathbb{R})$. Then there exists
$\tau_{0}>0$ and $\epsilon>0$ such that

$\|\widehat{L}_{\tau}\phi\|_{L^{2}(\mathbb{R}^{3})}\geq\epsilon\|\phi\|$

for any $\phi=\phi(|y|,|z|)\in H^{2}_{V}$, $\tau\in(0,\tau_{0})$.

Proof.  Suppose on the contrary that there exist $\tau_{n}\rightarrow 0^{+}$
and $\phi_{n}=\phi_{n}(|y|,|z|)\in H^{2}_{V}$ such that $\|\phi_{n}\|=1$ and

$\|\widehat{L}_{\tau_{n}}\phi_{n}\|_{L^{2}(\mathbb{R}^{3})}\rightarrow 0.$

Passing to a subsequence if necessary, we may assume that
$\phi_{n}\rightharpoonup\phi_{\infty}$ in $H$ and
$\phi_{n}\rightarrow\phi_{\infty}$ in $L_{loc}^{p}(\mathbb{R}^{3})$. Similar
to the proof of Theorem 4.7, $\phi_{\infty}=\rho_{\infty}(z)e_{1}(y)$ a.e. on
$\mathbb{R}^{3}$, where
$\rho_{\infty}(z)=\langle\phi_{\infty}(y,z),e_{i}(y)\rangle_{L^{2}_{y}(\mathbb{R}^{2})}$
and for any $\varphi(z)\in C_{0}^{\infty}(\mathbb{R})$, we have

$\displaystyle\int_{\mathbb{R}^{3}}(\rho_{\infty}^{\prime}(z)e_{1}(y)\varphi^{\prime}(z)e_{1}(y)+\rho_{\infty}(z)e_{1}(y)\varphi(z)e_{1}(y))dx$
(5.2) $\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\int_{\mathbb{R}^{3}}(\partial_{z}\phi_{n}(y,z)\varphi^{\prime}(z)e_{1}(y)+\phi_{n}(y,z)\varphi(z)e_{1}(y))dx$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}(\langle\widehat{L}_{\tau_{n}}\phi_{n},\varphi(z)e_{1}(y)\rangle_{L^{2}(\mathbb{R}^{3})}+(p-1)\int_{\mathbb{R}^{3}}|w_{\tau_{n}}|^{p-2}\phi_{n}\varphi(z)e_{1}(y)dx)$
$\displaystyle=$
$\displaystyle(p-1)\int_{\mathbb{R}^{3}}|\widehat{w}(z)|^{p-2}\rho_{\infty}(z)\varphi(z)e_{1}(y)^{p}dx.$

Then the non-degeneracy of $\widehat{w}(z)$ yields that $\rho_{\infty}(z)=0$,
and hence, $\phi_{\infty}=0$. Note that

$1=\|\phi_{n}\|\lesssim\langle\widehat{L}_{\tau_{n}}\phi_{n},\phi_{n}\rangle_{L^{2}(\mathbb{R}^{3})}+(p-1)\int_{\mathbb{R}^{3}}|w_{\tau_{n}}|^{p-2}\phi_{n}^{2}dx.$

However, form the convergence of $w_{\tau_{n}}$ and $\phi_{n}$, we obtain that

$\int_{\mathbb{R}^{3}}|w_{\tau_{n}}|^{p-2}\phi_{n}^{2}dx\rightarrow 0,$

implying that

$\langle\widehat{L}_{\tau_{n}}\phi_{n},\phi_{n}\rangle_{L^{2}(\mathbb{R}^{3})}+(p-1)\int_{\mathbb{R}^{3}}|w_{\tau_{n}}|^{p-2}\phi_{n}^{2}dx\rightarrow
0,$

which is absurd and we complete the proof. ∎

######  Lemma 5.4.

Let $2<p<6$. Then

$(i)$ $\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx\rightarrow 0$ as
$\lambda\rightarrow\Lambda_{0}$;

$(ii)$
$\widehat{\chi}_{\tau}\rightarrow(\frac{1}{2-p}\widehat{w}(z)-\frac{1}{2}z\widehat{w}^{\prime}(z))e_{1}(y)$
in $H$ as $\tau\rightarrow 0^{+}$.

$(iii)$ for some $\widetilde{\Lambda}_{2}\in[\Lambda_{2},\Lambda_{0})$,
$\frac{d}{d\lambda}\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx<0$ when
$\lambda\in(\widetilde{\Lambda}_{2},\Lambda_{0})$.

Proof.  $(i)$ From Theorem 1.4, we derive that

$(\Lambda_{0}-\lambda)^{\frac{1}{2}-\frac{2}{p-2}}\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx\rightarrow\int_{\mathbb{R}}\widehat{w}^{2}dz,$

showing that $\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx\rightarrow 0$, since
$1/2-2/(p-2)<0$.

$(ii)$ Direct computations show that

$\widehat{L}_{\tau}\widehat{\chi}_{\tau}=w_{\tau}.$

By Lemma 5.3,

$\|\widehat{\chi}_{\tau}\|\leq\frac{1}{\epsilon}\|\widehat{L}_{\tau}\widehat{\chi}_{\tau}\|_{L^{2}(\mathbb{R}^{3})}=\frac{1}{\epsilon}\|w_{\tau}\|_{L^{2}(\mathbb{R}^{3})}\lesssim
1.$

For any $\tau_{n}\rightarrow 0$, up to a subsequence, we may assume that
$\widehat{\chi}_{\tau_{n}}\rightharpoonup\widehat{\chi}_{0}$ in $H$ and
$\widehat{\chi}_{\tau_{n}}\rightarrow\widehat{\chi}_{0}$ in
$L_{loc}^{q}(\mathbb{R}^{3}),2\leq q<2^{\ast}$. Standard arguments yields that
$\widehat{\chi}_{0}=\varrho_{0}(z)e_{1}(y)$, where
$\varrho_{0}(z)=\langle\widehat{\chi}_{0}(y,z),e_{i}(y)\rangle_{L^{2}_{y}(\mathbb{R}^{2})}$,
and $\widehat{L}_{0}\varrho_{0}(z)=\widehat{w}(z)$. Noticing that
$\widehat{L}_{0}$ is invertible and

$\widehat{L}_{0}(\frac{1}{2-p}\widehat{w}-\frac{1}{2}z\widehat{w}^{\prime})=\widehat{w},$

we obtain that
$\varrho_{0}=\frac{1}{2-p}\widehat{w}-\frac{1}{2}z\widehat{w}^{\prime}$.

Similar to Step 1 in the proof of Theorem 4.7, we have
$Q\widehat{\chi}_{\tau_{n}}\rightarrow 0$ in $L^{q}(\mathbb{R}^{3}),2\leq
q<2^{\ast}$. Then,

$\displaystyle\frac{1}{\tau_{n}}\int_{\mathbb{R}^{3}}(|\nabla_{y}(Q\widehat{\chi}_{\tau_{n}})|^{2}+|y|^{2}(Q\widehat{\chi}_{\tau_{n}})^{2}-\Lambda_{0}(Q\widehat{\chi}_{\tau_{n}})^{2})dx+\int_{\mathbb{R}^{3}}(|\partial_{z}(Q\widehat{\chi}_{\tau_{n}})|^{2}+(Q\widehat{\chi}_{\tau_{n}})^{2})dx$
(5.3) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}(Q\widehat{\chi}_{\tau_{n}})^{2}dx+(p-1)\int_{\mathbb{R}^{3}}|w_{\tau_{n}}|^{p-2}\widehat{\chi}_{\tau_{n}}Q\widehat{\chi}_{\tau_{n}}dx$
$\displaystyle\rightarrow$ $\displaystyle 0,$

implying that

$\displaystyle\frac{1}{\tau_{n}}\int_{\mathbb{R}^{3}}(|\nabla_{y}(Q\widehat{\chi}_{\tau_{n}})|^{2}+|y|^{2}(Q\widehat{\chi}_{\tau_{n}})^{2})dx$
(5.4) $\displaystyle\lesssim$
$\displaystyle\frac{1}{\tau_{n}}\int_{\mathbb{R}^{3}}(|\nabla_{y}(Q\widehat{\chi}_{\tau_{n}})|^{2}+|y|^{2}(Q\widehat{\chi}_{\tau_{n}})^{2}-\Lambda_{0}(Q\widehat{\chi}_{\tau_{n}})^{2})dx$
$\displaystyle\rightarrow$ $\displaystyle 0,$

and

$\int_{\mathbb{R}^{3}}(|\partial_{z}(Q\widehat{\chi}_{\tau_{n}})|^{2}+(Q\widehat{\chi}_{\tau_{n}})^{2})dx\rightarrow
0.$

Thus $Q\widehat{\chi}_{\tau_{n}}\rightarrow 0$ in $H$. Moreover,

$\displaystyle\|\varrho_{\tau_{n}}\|_{H^{1}(\mathbb{R})}^{2}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}(|\varrho^{\prime}_{\tau_{n}}(z)|^{2}e_{1}(y)^{2}+\varrho_{\tau_{n}}^{2}e_{1}(y)^{2})dx$
(5.5) $\displaystyle=$
$\displaystyle\langle\widehat{L}_{\tau_{n}}\widehat{\chi}_{\tau_{n}},P\widehat{\chi}_{\tau_{n}}\rangle_{L^{2}(\mathbb{R}^{3})}+(p-1)\int_{\mathbb{R}^{3}}|w_{\tau_{n}}|^{p-2}(P\widehat{\chi}_{\tau_{n}})\widehat{\chi}_{\tau_{n}}dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{3}}w_{\tau_{n}}(P\widehat{\chi}_{\tau_{n}})dx+(p-1)\int_{\mathbb{R}^{3}}|w_{\tau_{n}}|^{p-2}(P\widehat{\chi}_{\tau_{n}})\widehat{\chi}_{\tau_{n}}dx$
$\displaystyle\rightarrow$
$\displaystyle\int_{\mathbb{R}}\widehat{w}(z)\varrho_{0}(z)dz+\frac{2(p-1)}{p}\pi^{1-\frac{p}{2}}\int_{\mathbb{R}}|\widehat{w}(z)|^{p-2}\varrho_{0}(z)^{2}dz$
$\displaystyle=$
$\displaystyle\langle\widehat{L}_{0}\varrho_{0},\varrho_{0}\rangle_{L^{2}(\mathbb{R})}+\frac{2(p-1)}{p}\pi^{1-\frac{p}{2}}\int_{\mathbb{R}}|\widehat{w}(z)|^{p-2}\varrho_{0}(z)^{2}dz$
$\displaystyle=$ $\displaystyle\|\varrho_{0}\|_{H^{1}(\mathbb{R})}^{2},$

implying that $\varrho_{\tau_{n}}\rightarrow\varrho_{0}$ in
$H^{1}(\mathbb{R})$. Hence,
$\widehat{\chi}_{\tau_{n}}\rightarrow\varrho_{0}(z)e_{1}(y)$ in $H$ and we can
conclude.

$(iii)$

$\displaystyle\lim_{\lambda\rightarrow\Lambda_{0}}(\Lambda_{0}-\lambda)^{\frac{p-4}{p-2}+\frac{1}{2}}\frac{d}{d\lambda}\int_{\mathbb{R}^{3}}|u_{\lambda}|^{2}dx$
(5.6) $\displaystyle=$
$\displaystyle\lim_{\lambda\rightarrow\Lambda_{0}}2(\Lambda_{0}-\lambda)^{\frac{p-4}{p-2}+\frac{1}{2}}\int_{\mathbb{R}^{N}}u_{\lambda}\chi_{\lambda}dx$
$\displaystyle=$ $\displaystyle\lim_{\tau\rightarrow
0^{+}}2\int_{\mathbb{R}^{3}}w_{\tau}\widehat{\chi}_{\tau}dx$ $\displaystyle=$
$\displaystyle
2\int_{\mathbb{R}}\widehat{w}(\frac{1}{2-p}\widehat{w}-\frac{1}{2}z\widehat{w}^{\prime})dz$
$\displaystyle=$ $\displaystyle
2(\frac{1}{2-p}+\frac{1}{4})\int_{\mathbb{R}}|\widehat{w}|^{2}dz<0.$

Therefore, there exists $\widetilde{\Lambda}_{2}\in[\Lambda_{2},\Lambda_{0})$
such that for any $\lambda\in(\widetilde{\Lambda}_{2},\Lambda_{0})$,

$\frac{d}{d\lambda}\int_{\mathbb{R}^{3}}|u_{\lambda}|^{2}dx<0.$

∎

###  5.3 Proof of Theorem 1.7

Recall that standing waves are called orbitally stable if for each
$\epsilon>0$ there exists $\delta>0$ such that, whenever $\Psi_{0}\in
H^{1}_{V}(\mathbb{R}^{N},\mathbb{C})$ is such that
$\|\Psi_{0}-u_{\lambda}\|_{H^{1}_{V}(\mathbb{R}^{N},\mathbb{C})}<\delta$ and
$\Psi(t,x)$is the solution of (1.1) with $\Psi(0,x)=\Psi_{0}(x)$ in some
interval $[0,t_{0})$, then $\Psi(t,x)$ can be continued to a solution in
$0\leq t<+\infty$ and

$\sup_{t>0}\inf_{w\in\mathbb{R},z\in\mathbb{R}}\|\Psi(t,x)-e^{-i\lambda
w}u_{\lambda}(x_{1},x_{2},x_{3}-z)\|_{H^{1}_{V}(\mathbb{R}^{N},\mathbb{C})}<\epsilon;$

otherwise, they are called unstable. To study the orbital stability, we lean
on the following result, which expresses in our context the abstract theory
developed in [18].

Global well-posedness of the Cauchy problem was established in [2].

######  Proposition 5.5.

If $\frac{d}{d\lambda}\int_{\mathbb{R}^{3}}u_{\lambda}^{2}>0$ (respectively
$<0$), the standing wave $e^{-i\lambda t}u_{\lambda}(x)$ is orbitally unstable
(respectively stable) in

$H_{V}^{1}(\mathbb{R}^{3},\mathbb{C})=\\{v\in
H^{1}(\mathbb{R}^{3},\mathbb{C}):\int_{\mathbb{R}^{3}}(x_{1}^{2}+x_{2}^{2})|v|^{2}dx<+\infty\\}.$

Proof of Theorem 1.7.  Take

$\widehat{c}=\min\\{\sqrt{\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx}:\lambda=\widetilde{\Lambda}_{1}\
or\ \lambda=\widetilde{\Lambda}_{2}\\},$

where $\widetilde{\Lambda}_{1}$ and $\widetilde{\Lambda}_{2}$ are given by
Lemmas 5.2 and 5.4 respectively. Similar arguments to the proof of [33,
Corollary2.5] and [20, Lemma 4.23], yield that

$\lambda\mapsto u_{\lambda},\lambda<\widetilde{\Lambda}_{1}\ or\
\widetilde{\Lambda}_{2}<\lambda<\Lambda_{0}$

is $C^{1}$in $H$, and hence in $L^{2}(\mathbb{R}^{3})$. Thus (1.2) admits a
ground state $u_{\lambda}$ with $\lambda<\widetilde{\Lambda}_{1}$ and a ground
state $u_{\widetilde{\lambda}}$ with
$\widetilde{\Lambda}_{2}<\widetilde{\lambda}<\Lambda_{0}$, where

$\sqrt{\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx}=\sqrt{\int_{\mathbb{R}^{3}}u_{\widetilde{\lambda}}^{2}dx}=c,0<c<\widehat{c}.$

Note that the local well posedness in $H_{V}^{1}(\mathbb{R}^{3},\mathbb{C})$
is established in [2]. Then Lemma 5.2 $(iii)$, Lemma 5.4 $(iii)$, and
Proposition 5.5 yield the orbital stability of
$e^{-i\widetilde{\lambda}t}u_{\widetilde{\lambda}}(x)$ in
$H_{V}^{1}(\mathbb{R}^{3},\mathbb{C})$, and the orbital instability of
$e^{-i\lambda t}u_{\lambda}(x)$ in $H_{V,s}^{1}(\mathbb{R}^{3},\mathbb{C})$,
where

$H_{V,s}^{1}(\mathbb{R}^{3},\mathbb{C})=\\{v\in
H_{V}^{1}(\mathbb{R}^{3},\mathbb{C}):v=v(|y|,|z|)\\}.$

∎

###  5.4 Proof of Theorem 1.8

We show some lemmas first. These results can be proved along the lines of
Lemmas 3.2, 3.3 and we omit the details here.

######  Lemma 5.6.

For any $(\lambda,u)\in(-\infty,\Lambda_{0})\times H\backslash\\{0\\}$, there
exists a unique function $t:(-\infty,\Lambda_{0})\times
H\backslash\\{0\\}\rightarrow(0,+\infty)$ such that

$(\mu,tv)\in\mathcal{N},t>0\Leftrightarrow t=t(\lambda,u),$

and $J_{\lambda}(t(\lambda,u)u)=\max_{t>0}J_{\lambda}(tu)$, $t\in
C((-\infty,\Lambda_{0})\times H\backslash\\{0\\},(0,\infty))$, where
$\mathcal{N}:=\cup_{\lambda<\Lambda_{0}}\mathcal{N}_{\lambda}$.

######  Lemma 5.7.

For any $\lambda_{0}<\Lambda_{0}$,
$\limsup_{\lambda\rightarrow\lambda_{0}}h(\lambda)\leq h(\lambda_{0})$.

Proof of Theorem 1.8.  We will show that

$\sup_{\lambda\in[\widetilde{\Lambda}_{1},\widetilde{\Lambda}_{2}]}\|u_{\lambda}\|_{L^{2}}\leq
C,$ (5.7)

for some $C>0$ independent of $\lambda$ where
$\widetilde{\Lambda}_{1},\widetilde{\Lambda}_{2}$ are given by Lemmas 5.2, 5.4
respectively. This together with results proved in Lemmas 5.2, 5.4, will
complete the proof.

By Lemma 5.7, there exists $\widetilde{C}>0$ independent of $\lambda$ such
that
$\sup_{\lambda\in[\widetilde{\Lambda}_{1},\widetilde{\Lambda}_{2}]}h(\lambda)\leq\widetilde{C}$.
Since $J_{\lambda}(u_{\lambda})=h(\lambda)$, we have

$\frac{1}{2}\int_{\mathbb{R}^{3}}(|\nabla
u_{\lambda}|^{2}+(x_{1}^{2}+x_{2}^{2}-\lambda)u_{\lambda}^{2})dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|u_{\lambda}|^{p}dx\leq\widetilde{C}.$
(5.8)

Moreover, $u_{\lambda}\in\mathcal{N}_{\lambda}$. Hence,

$\int_{\mathbb{R}^{3}}(|\nabla
u_{\lambda}|^{2}+(x_{1}^{2}+x_{2}^{2}-\lambda)u_{\lambda}^{2})dx=\int_{\mathbb{R}^{3}}|u_{\lambda}|^{p}dx.$
(5.9)

From (5.8) and (5.9), we derive that

$(\frac{1}{2}-\frac{1}{p})\int_{\mathbb{R}^{3}}(|\nabla
u_{\lambda}|^{2}+(x_{1}^{2}+x_{2}^{2}-\lambda)u_{\lambda}^{2})dx\leq\widetilde{C}.$
(5.10)

Thus, we have

$(\Lambda_{0}-\widetilde{\Lambda}_{2})\sup_{\lambda\in[\widetilde{\Lambda}_{1},\widetilde{\Lambda}_{2}]}\int_{\mathbb{R}^{3}}u_{\lambda}^{2}dx\leq\sup_{\lambda\in[\widetilde{\Lambda}_{1},\widetilde{\Lambda}_{2}]}\int_{\mathbb{R}^{3}}(|\nabla
u_{\lambda}|^{2}+(x_{1}^{2}+x_{2}^{2}-\lambda)u_{\lambda}^{2})dx\leq\frac{p-2}{2p}\widetilde{C}.$
(5.11)

Take

$C=\sqrt{\frac{p-2}{2p(\Lambda_{0}-\widetilde{\Lambda}_{2})}\widetilde{C}},$

and we show (5.7). Thus we arrive at our conclusion. ∎

Acknowledgement: The authors would like to thank Louis Jeanjean for providing
valuable feedback and important references, and thank the Photonics group at
Caltech for useful discussions and insight. The first author thanks C. Li and
S.J. Li for important discussions.

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