# A General and Unified Method to Prove the Uniqueness of Ground State
Solutions and the Existence/Non-existence, and Multiplicity of Normalized
Solutions with applications to various NLS

Hichem Hajaiej  Department of Mathematics, California State University at Los
Angeles, Los Angeles, CA 90032, USA<EMAIL_ADDRESS>and  Linjie Song
Institute of Mathematics, AMSS, Academia Sinica, Beijing 100190, China
University of Chinese Academy of Science, Beijing 100049, China
<EMAIL_ADDRESS>
###### Abstract.

We first give an abstract framework to show the uniqueness of Ground State
Solutions (GSS) for a large class of PDEs. To the best of our knowledge, all
the existing results in the literature only addressed particular cases.
Moreover, our self-contained approach offers a general framework to study the
existence/non-existence and multiplicity of normalized solutions. We will
exhibit concrete examples to which our method applies, and verify all the
assumptions we need. Our approach is applicable to a wide range of operators
and domains provided that our hypotheses are verified. Additionally, we prove
new results about the non-degeneracy and uniqueness of positive GSS in a
general setting. Our findings are applicable to fractional nonlinear
Schrödinger equations with non-autonomous nonlinearities. In particular, we
were able to extend the main results of [13, 14] to general non-autonomous and
mixed nonlinearities. This does not seem possible by using the approach
developed by the authors of the above breakthrough papers. The orbital
stability/instability of the standing waves will be addressed thanks to the
non-degeneracy.

Data availability statement: My manuscript has no associate data.

###### Key words and phrases:

Normalized solutions; non-degeneracy and uniqueness; orbital stability;
abstract method

###### 2010 Mathematics Subject Classification:

35A15, 35B09, 35B35

Linjie Song is supported by CEMS

##  1\. Introduction

Since the breakthrough paper of Lions in 1985, [33], on the existence of
normalized solutions in the sub-critical case using the concentration-
compactness method, and the one of Cazenave and Lions in 1986 on the orbital
stability of standing waves, [6], there were no additional contributions in
these two directions involving a heuristic or general methods until quite
recently. In 2017, Bellazzini, Boussaid, Jeanjean, and Visciglia developed a
general approach to show the existence of normalized solutions in the super-
critical case, [1]. Many mathematicians borrowed their idea to address the
existence of normalized solutions and to study the orbital stability of
standing waves in the super-critical case. To the best of our knowledge, there
are no general approaches addressing the uniqueness of GSS or the existence of
normalized solutions in any regime.

The main goal of this work is to address these two questions and to provide a
unified and very general method that establishes the uniqueness of GSS and
applies to study normalized solutions in the sub-critical, critical or super-
critical cases, bounded, exterior or entire space, Laplacian, fractional
Laplacian, $p$-Laplacian, $p$-fractional Laplacian, for autonomous or non-
autonomous nonlinearities and with or without the presence of a potential. We
will exhibit some concrete examples of the latter and discuss all the details.
Our abstract framework applies to many other situations and our method is
quite flexible as the reader will notice it.

Before introducing the notation and our main results (Theorem 4.1, Theorem
4.4, and Theorem 4.7), let us point out that we believe that the ideas and
ingredients used in this paper are novel. The subtle interplay between the
bifurcation theory, spectral theory, and variational methods turned out to be
powerful to show the uniqueness of GSS. Let us also mention that some of the
results in Section 3 can seem stated under strong assumptions (non-degeneracy,
convergence of (P.S) sequence). We count on the reader’s patience to see how
all these assumptions were satisfied in our concrete examples. Additionally,
these hypotheses are optimal in this framework as it is shown in a ongoing
work of the authors. This means that our framework is fruitful as it addresses
many difficult challenges. In particular, we extend the breakthrough results
of [13, 14] to non-autonomous nonlinearities using our self-containing
approach. In what follows, we will describe our approach:

Assume that $W$ is a Banach space with the norm $\|\cdot\|$,
$\Phi_{\lambda}\in C^{2}(W,\mathbb{R})$ is a Fréchet-differentiable functional
with derivative $D_{u}\Phi_{\lambda}:W\rightarrow W^{\ast}$, where the space
$W^{\ast}$ is the dual space of $W$ and $\lambda\in\mathbb{R}$. Consider a
particular form of $\Phi_{\lambda}$, (so that (1.1) has an underlying
variational structure):

(1.1) $D_{u}S(u)+D_{u}G(u)-D_{u}F(u)-\lambda D_{u}Q(u)=0,\quad\text{i.e.}\
u\in W,$

where $\Phi_{\lambda}=S(u)+G(u)-F(u)-\lambda Q(u)$, $S(u)$ and $Q(u)$ are
positive for any $u\in W\backslash\\{0\\}$, $G(u)$ is nonnegative for any
$u\in W\backslash\\{0\\}$. Assume that $D_{u}\Phi_{\lambda}(0)=0$, ( $0$ is
always a trivial solution of (1.1) for any $\lambda$). In this paper, we are
interested in the nontrivial solutions. In previous contributions, there are
two different approaches to study the existence of solutions of equation (1.1)
using variational methods: When $\lambda$ is prescribed, we study
$\Phi_{\lambda}$ directly and find its critical points $u_{\lambda}$. However,
we don’t have information about $Q(u_{\lambda})$. When $\rho>0$ is prescribed,
we study $I(u)=S(u)+G(u)-F(u)$ under the constraint $Q(u)=\rho$ and get
critical points $u_{\rho}$ of $I_{|\\{Q(u)=\rho\\}}$. Then
$(\lambda_{\rho},u_{\rho})$ is a solution of (1.1), where $\lambda_{\rho}$ is
determined as a Lagrange multiplier and unknown, $u_{\rho}$ is a normalized
solution that satisfies the normalized condition $Q(u_{\rho})=\rho$.
Normalized solutions are the most important ones because they are known to be
the most stable. The latter makes them very attractive especially that they
also usually have nice symmetry properties. In this paper, we give an abstract
framework in which $\Phi_{\lambda}$ is studied directly and the information of
$Q(u)$ is prescribed.

Our method is self-contained and novel. It provides a unified and general
framework to prove the existence of normalized solutions. To the best of our
knowledge, all previous works had completely different approaches for the sub-
critical and the super-critical cases. Moreover, our method applies to
equations on the entire space, bounded domains, and exterior domains provided
that some assumptions are satisfied. Our unified approach is also novel in
this sense. Surprisingly, there are few papers studying the existence of
normalized solutions on bounded domains. As far as we know, the result in
Section 7 is the first result establishing the existence of normalized
solutions for $p$-Laplacian equations on bounded domains. Moreover, the
literature is silent regarding the existence of normalized solutions on
exterior domains. To the best of our knowledge, the first result in this
direction has been proven using the method we developed in this paper ( see
[52, 53]). Additionally, we will also address the orbital stability of
standing waves by exploring non-degeneracy results that we also needed to
achieve other purposes - namely the uniqueness of ground state solutions-.

Our abstract framework contains three parts. They are all independent and
complementary. First, let us introduce the abstract framework 2. For
$\lambda\in(a,b)$, we assume that $\mathcal{N}_{\lambda}(\subset W)$ is a
natural constraint, which means that any critical point of $\Phi_{\lambda}$
restricted to $\mathcal{N}_{\lambda}$ is a critical point of $\Phi_{\lambda}$
in $W$, and $h(\lambda)=\inf_{u\in\mathcal{N}_{\lambda}}\Phi_{\lambda}(u)$ can
be achieved by $u_{\lambda}$, which is called a ground state solution. Set

$\mathcal{K}_{\lambda}:=\\{u\in\mathcal{N}_{\lambda}:D_{u}\Phi_{\lambda}(u)=0,\Phi_{\lambda}(u)=h(\lambda)\\}.$

If $\Phi_{\lambda|\mathcal{N}_{\lambda}}$ satisfies $(PS)_{h(\lambda)}$
condition (i.e. any sequence
$\\{u_{n}\\}_{n=1}^{+\infty}\subset\mathcal{N}_{\lambda}$ such that
$\Phi_{\lambda}(u_{n})\rightarrow h(\lambda)$ and
$D_{u}\Phi_{\lambda}(u_{n})\rightarrow 0$ has a convergent subsequence in
$\mathcal{N}_{\lambda}$), then $\mathcal{K}_{\lambda}$ is compact and for
$\lambda\in(a,b)$, we can define

$d(\lambda):=\inf_{u\in\mathcal{K}_{\lambda}}Q(u)=\min_{u\in\mathcal{K}_{\lambda}}Q(u),$
$\hat{d}(\lambda):=\sup_{u\in\mathcal{K}_{\lambda}}Q(u)=\max_{u\in\mathcal{K}_{\lambda}}Q(u).$

When $d\equiv\hat{d}$ in $(a,b)$, we can prove that $d(\lambda)$ is
continuous. This property will play a crucial role in our discussion. The
behavior of $d(\lambda)$ is studied when $\lambda\rightarrow a$ and
$\lambda\rightarrow b$ later. Then for any
$\rho\in(\inf_{\lambda\in(a,b)}d(\lambda),\sup_{\lambda\in(a,b)}d(\lambda))$,
there exists $u_{\lambda}\in\mathcal{K}_{\lambda}$ such that
$Q(u_{\lambda})=\rho$ and $D_{u}\Phi_{\lambda}(u_{\lambda})=0$.

######  Remark 1.1.

Usually, the Nehari manifold can be chosen as the natural constraint. Define

$\mathcal{N}_{\lambda}:=\\{u\in
W\backslash\\{0\\}:D_{u}\Phi_{\lambda}(u)(u)=0\\}.$

Then any solution of (1.1) belongs to $\mathcal{N}_{\lambda}$. Suppose that
$\Phi_{\lambda}(u_{\lambda})=\inf_{u\in U}\Phi_{\lambda}(u)$ and
$\Phi_{\lambda}\in C^{2}(U,\mathbb{R})$ where $U$ is some neighbourhood of
$u_{\lambda}$ in $W$. Then by the Lagrange multiplier rule, one has
$D_{u}\Phi_{\lambda}(u_{\lambda})=\mu(D_{u}\Phi_{\lambda}(u_{\lambda})+D_{uu}\Phi_{\lambda}(u_{\lambda})(u_{\lambda},\cdot))$
for some $\mu$. Plugging into this equality $u_{\lambda}$, we obtain $\mu
D_{uu}\Phi_{\lambda}(u_{\lambda})(u_{\lambda},u_{\lambda})=0$. Hence, if

$D_{uu}\Phi_{\lambda}(u_{\lambda})(u_{\lambda},u_{\lambda})\neq 0,$

then $\mu=0$ and therefore $D_{u}\Phi_{\lambda}(u_{\lambda})=0$. The Nehari
manifold method was introduced in [39, 40], [26, 61, 65] and the references
therein provide more details.

A key ingredient of the framework 2 is to show that the quantity
$d(\lambda)=\hat{d}(\lambda)$, which holds true if $d$ is a continuous
function. In some cases, this is ensured by the uniqueness of the ground state
solution ( see Section 5.1, Section 7). In Section 7.3, we provide some
examples in which this uniqueness is known. In other cases, we can show the
continuity of $d(\lambda)$ and consequently establish the existence of
normalized solutions even if we do not know that the positive ground state is
unique ( see Section 5.5). The fNLS with a Hardy potential falls into this
category. For such case, we will only provide examples for NLS that can be
straightforwardly extended to other PDEs (fNLS, for example).

In the abstract framework 3, $a=-\infty$, $b=0$, and we are mainly concerned
with the situations when $\lambda\rightarrow-\infty$ and $\lambda\rightarrow
0^{-}$. Under some abstract settings, we can prove that the local uniqueness
of the ground state solution when $\lambda\rightarrow-\infty$ or
$\lambda\rightarrow 0^{-}$, and obtain the existence of normalized solutions.
Some scaling methods are used and we assume that the limit equation has a
unique, non-degenerate ground state solution. We will give an application to
mixed nonlinearities, this kind of nonlinearity has gained a lot of interest
in the last years ( see e.g. [49, 50, 55, 66]), and we cover the most
attractive cases, i.e, cubic-quintic ( see Section 6). Moreover, we will show
how our approach applies to mixed fractional Laplacians ( see Section 8). This
type of equations arises in various fields ranging from biophysics to
population dynamics and has very recently received an increasing interest.
Non-autonomous mixed nonlinearities will be considered.

In the abstract framework 2, we assume the uniqueness of the ground state
solution. Note that the abstract framework 3 does not provide any information
when the $\lambda$ is away from $-\infty$ and $0$. The abstract framework 1
will complete the picture. More precisely, it will establish a $C^{1}$ global
branch parameterized by $\lambda$ under a non-degeneracy assumption.
Furthermore, once we show that the ground state is unique when
$\lambda\rightarrow-\infty$, we can prove the uniqueness of the global branch.
Thus, the abstract framework 1 provides a framework to prove the uniqueness of
all ground states. We will exhibit concrete examples showing that our abstract
framework plays an important role for non-local operators. Regarding scalar
field equations involving a fractional Laplacian, it is more difficult to show
the uniqueness since $(-\Delta)^{s}$ is a nonlocal operator. When the
nonlinearity is pure power $|u|^{q-2}u$, [13, 14] showed the uniqueness of
positive ground state, leaving the problem open for general nonlinearities. To
the best of our knowledge, the problem is still open when the nonlinearity is
non-autonomous. We will provide a class of non-autonomous nonlinearities for
which the positive ground state solution is unique and provide two different
methods to prove it, one is based on our abstract framework and one is
inspired with some ideas developed in [13]. Furthermore, our general method
also applies to the fractional nonlinear Schrödinger equations (fNLS) with a
potential ( see Section 5.4). To solve the uniqueness problem, we first show
the non-degeneracy. As far as we know, these non-degeneracy ( see Lemmas 5.13,
5.24) and uniqueness ( see Theorems 5.16, 5.25) results are completely new,
even in the case of the Laplacian. They present interest by themselves.

The rest of this paper is organized as follows. We present our general method
and our main results in Sections 2 \- 4. We then discuss some applications for
scalar field equations involving a fractional Laplacian with non-autonomous
nonlinearities in Section 5. A fractional Laplacian with mixed nonlinearities
is considered in section 6, p-laplacian in Section 7, and mixed fractional
Laplacians in Section 8.

##  2\. Abstract Framework 1

In this section, we give the first part of our abstract framework to establish
a $C^{1}$ global branch parameterized by $\lambda\in(a,b)$ under a non-
degeneracy assumption. The readers will see that $a=-\infty$ under our setting
in this section. Furthermore, once we show that the ground state is unique
when $\lambda\rightarrow-\infty$, we can prove the uniqueness of the global
branch. The existence of a $C^{1}$ global branch is enough to obtain the
existence of normalized solutions. Nevertheless, the uniqueness of ground
state has its own meaning and importance in various fields. Hence, this
abstract framework 1 can also be viewed as a framework to prove the uniqueness
of ground states. In particular, it is worth noting that the ODE techniques
(which are essential to show the uniqueness in many cases) are not applicable
in some important situations, such as non-local operators. We will exhibit
concrete examples showing that our abstract framework plays an important role
in these settings.

Throughout this section, we assume

$(H_{1})$ $W$ is a Hilbert space, $D_{u}Q(u)=u$, the linear operator
$D_{u}S+D_{u}G$ is self-adjoint and bounded below on a Hilbert space $E$ with
operator domain $\widehat{W}$ and form domain $W$.

Then $S(u)=\frac{1}{2}\left\langle
D_{u}S(u),u\right\rangle,G(u)=\frac{1}{2}\left\langle
D_{u}G(u),u\right\rangle,Q(u)=\frac{1}{2}\left\langle u,u\right\rangle$.
Moreover, if $\lambda<\lambda_{1}:=\inf\sigma(D_{u}S+D_{u}G)$ where
$\sigma(D_{u}S+D_{u}G)$ denotes the spectrum of
$D_{u}S+D_{u}G:\widehat{W}\rightarrow E$, then

$\|u\|_{\lambda}=\sqrt{\left\langle D_{u}S(u)+D_{u}G(u)-\lambda
u,u\right\rangle}$

is equivalent to the norm of $W$.

Let $G$ be a topological group, and assume that the action of $G$ on $E$ is
isometric. Define

(2.1) $E_{G}=\\{u\in E:gu=u,\forall g\in G\\},\widehat{W}_{G}=\widehat{W}\cap
E_{G},W_{G}=W\cap E_{G}.$

We assume the following:

$(H_{2})$ $\Phi_{\lambda}(gu)=\Phi_{\lambda}(u),\forall g\in G$.

$(H_{3})$ $\Phi_{\lambda}|_{W_{G}}$ satisfies (PS) condition in $W_{G}$ for
all $\lambda<\lambda_{1}$.

$(H_{4})$ $D_{uu}F(0)=0$, $\left\langle
D_{uu}F(u)v,w\right\rangle=\left\langle v,D_{uu}F(u)w\right\rangle$. And there
exists some $p>2$ such that

(2.2) $\left\langle D_{uu}F(u)u,u\right\rangle\geq(p-1)\left\langle
D_{u}F(u),u\right\rangle>0,\forall u\in W\setminus\\{0\\}.$

By the principle of symmetric criticality (see [65, Theorem 1.28]), $(H_{2})$
ensures that any critical point of $\Phi_{\lambda}$ restricted to
$\widehat{W}_{G}$ is a critical point of $\Phi_{\lambda}$. To show the
existence of solutions, we define

$\mathcal{N}_{G,\lambda}=\\{u\in W_{G}\setminus\\{0\\}:\left\langle
D_{u}S(u)+D_{u}G(u)-\lambda u,u\right\rangle=\left\langle
D_{u}F(u),u\right\rangle.$

######  Definition 2.1.

Set $h_{G}(\lambda)=\inf_{u\in\mathcal{N}_{G,\lambda}}\Phi_{\lambda}(u)$. We
say that $u\in W_{G}\setminus\\{0\\}$ is a $G$-ground state solution if $u$
solves (1.1) and achieves $h_{G}(\lambda)$.

If $\mathcal{N}_{G,\lambda}$ is a $C^{1}$ manifold with codimension $1$ in
$W_{G}$, $h_{G}(\lambda)$ is well-defined and $h_{G}(\lambda)\neq 0$ for all
$\lambda<\lambda_{1}$, by $(H_{3})$, standard arguments yield the existence of
a $G$-ground state solution with $G$-Morse index $1$ (the definition will be
given by (2.3)). Then, we establish a $C^{1}$ global branch in
$\widehat{W}_{G}$ under the following assumptions:

$(H_{5})$ If $\Phi_{\lambda}^{\prime}(u_{\lambda})=0$ and
$\mu_{G}(u_{\lambda})=1$, then $\ker
D_{uu}\Phi_{\lambda}(u_{\lambda})|_{E_{G}}=\\{0\\}$, where $\mu_{G}(u)$, the
$G$-Morse index of $u$, is defined as

(2.3) $\mu_{G}(u):=\sharp\\{e<0:e\ is\ an\ eigenvalue\ of\
D_{uu}\Phi_{\lambda}(u)|_{E_{G}}\\}.$

$(H_{6})$ If $u\in W$ solves (1.1). Then $u\in\widehat{W}$.

$(H_{7})$ $D_{u}F(u)\in E,\forall u\in W$, and $D_{uu}F(u)$ maps $\widehat{W}$
to $E$ for all $u\in\widehat{W}$.

Our main result in this section reads as follows:

######  Theorem 2.2.

Assume that $(H_{1})$ \- $(H_{7})$ hold, and that $\mathcal{N}_{G,\lambda}$ is
a $C^{1}$ manifold with codimension $1$ in $W_{G}$, $h_{G}(\lambda)$ is well-
defined and $h_{G}(\lambda)\neq 0$ for all $\lambda<\lambda_{1}$. Then there
exists

$\lambda\mapsto u_{\lambda}\in
C^{1}((-\infty,\lambda_{1}),W_{G}\setminus\\{0\\}),$

such that $u_{\lambda}$ solves (1.1), and $\mu_{G}(u_{\lambda})=1$.

For the readers’ convenience, the proof of our main result will be divided
into three lemmas.

######  Lemma 2.3.

Assume that $(H_{1})$ and $(H_{4})$ hold. If

$\lambda\mapsto u_{\lambda}\in
C^{1}((\lambda^{\ast},\lambda_{\ast}],W\setminus\\{0\\})$

for some $\lambda^{\ast}<\lambda_{\ast}<\lambda_{1}$ where $u_{\lambda}$
solves (1.1). Then $u_{\lambda}$ is uniformly bounded with respect to
$\lambda\in(\lambda^{\ast},\lambda_{\ast}]$.

Proof. Since $u_{\lambda}$ solves (1.1), we obtain

(2.4) $\left\langle D_{u}S(u_{\lambda})+D_{u}G(u_{\lambda})-\lambda
u_{\lambda},u_{\lambda}\right\rangle=\left\langle
D_{u}F(u_{\lambda}),u_{\lambda}\right\rangle.$

Hence, we have

(2.5) $(\lambda_{1}-\lambda)\left\langle
u_{\lambda},u_{\lambda}\right\rangle\leq\left\langle
D_{u}F(u_{\lambda}),u_{\lambda}\right\rangle.$

Differentiating $D_{u}S(u_{\lambda})+D_{u}G(u_{\lambda})=\lambda
u_{\lambda}+D_{u}F(u_{\lambda})$ with respect to $\lambda$, we get

(2.6)
$D_{u}S(\partial_{\lambda}u_{\lambda})+D_{u}G(\partial_{\lambda}u_{\lambda})=u_{\lambda}+\lambda\partial_{\lambda}u_{\lambda}+D_{uu}F(u_{\lambda})\partial_{\lambda}u_{\lambda},$

i.e.

(2.7) $D_{uu}\Phi(u_{\lambda})\partial_{\lambda}u_{\lambda}=u_{\lambda}.$

Set

$g(\lambda)=\left\langle
D_{u}F(u_{\lambda}),u_{\lambda}\right\rangle-2F(u_{\lambda}).$

Note that

(2.8)
$D_{uu}\Phi(u_{\lambda})u_{\lambda}=D_{u}F(u_{\lambda})-D_{uu}F(u_{\lambda})u_{\lambda}.$

Then

(2.9) $\displaystyle g^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle\left\langle
D_{uu}F(u_{\lambda})\partial_{\lambda}u_{\lambda},u_{\lambda}\right\rangle-\left\langle
D_{u}F(u_{\lambda}),\partial_{\lambda}u_{\lambda}\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
D_{uu}\Phi(u_{\lambda})^{-1}(D_{uu}F(u_{\lambda})u_{\lambda}-D_{u}F(u_{\lambda})),u_{\lambda}\right\rangle$
$\displaystyle=$ $\displaystyle-\left\langle
u_{\lambda},u_{\lambda}\right\rangle$ $\displaystyle\geq$
$\displaystyle-\frac{1}{\lambda_{1}-\lambda}\left\langle
D_{u}F(u_{\lambda}),u_{\lambda}\right\rangle$ $\displaystyle\geq$
$\displaystyle-\frac{p}{p-2}\frac{1}{\lambda_{1}-\lambda}g(\lambda).$

Let

$\widetilde{g}(\lambda)=(\lambda_{1}-\lambda)^{-\frac{p}{p-2}}g(\lambda),\lambda\in(\lambda^{\ast},\lambda_{\ast}].$

Then

$\widetilde{g}^{\prime}(\lambda)\geq 0,$

yielding that $\widetilde{g}(\lambda)\leq\widetilde{g}(\lambda_{\ast})$ for
any $\lambda\in(\lambda^{\ast},\lambda_{\ast}]$ and hence

$g(\lambda)\leq(\frac{\lambda_{1}-\lambda^{\ast}}{\lambda_{1}-\lambda_{\ast}})^{\frac{p}{p-2}}g(\lambda_{\ast}).$

By (2.4) and $(H_{4})$,
$g(\lambda)\sim\|u_{\lambda}\|_{\lambda}\sim\|u_{\lambda}\|$. Thus
$u_{\lambda}$ is uniformly bounded with respect to
$\lambda\in(\lambda^{\ast},\lambda_{\ast}]$. ∎

######  Lemma 2.4.

Assume that $(H_{1})$ \- $(H_{4})$ hold and that

$\lambda\mapsto u_{\lambda}\in
C^{1}((\lambda^{\ast},\lambda_{\ast}],W_{G}\setminus\\{0\\})$

for some $\lambda^{\ast}<\lambda_{\ast}<\lambda_{1}$ where $u_{\lambda}$ is a
solution of (1.1). Let
$\\{\lambda_{n}\\}\subset(\lambda^{\ast},\lambda_{\ast})$ be a sequence such
that $\lambda_{n}\rightarrow\lambda^{\ast}$, and $\mu_{G}(u_{\lambda_{n}})=1$.
Then, after possibly passing to a subsequence, we have
$u_{\lambda_{n}}\rightarrow u_{\lambda^{\ast}}$ in $W\setminus\\{0\\}$ where
$\mu_{G}(u_{\lambda^{\ast}})=1$ and $u_{\lambda^{\ast}}$ satisfies

(2.10) $D_{u}\Phi_{\lambda^{\ast}}(u_{\lambda^{\ast}})=0.$

Proof. From Lemma 2.3, we obtain the boundedness of $u_{\lambda_{n}}$ in $W$
and the boundedness of $\left\langle
u_{\lambda_{n}},u_{\lambda_{n}}\right\rangle$. Then we have that
$\Phi_{\lambda^{\ast}}(u_{\lambda_{n}})=\Phi_{\lambda_{n}}(u_{\lambda_{n}})+\frac{1}{2}(\lambda_{n}-\lambda^{\ast})\left\langle
u_{\lambda_{n}},u_{\lambda_{n}}\right\rangle$ is bounded and

$D_{u}\Phi_{\lambda^{\ast}}(u_{\lambda_{n}})=D_{u}\Phi_{\lambda_{n}}(u_{\lambda_{n}})+(\lambda_{n}-\lambda^{\ast})u_{\lambda_{n}}\rightarrow
0.$

Since $\Phi_{\lambda^{\ast}}$ satisfies (PS) condition in $W_{G}$, up to a
subsequence, there exists $u_{\lambda^{\ast}}\in W_{G}$ such that
$u_{\lambda_{n}}\rightarrow u_{\lambda^{\ast}}$ in $W$ and
$u_{\lambda^{\ast}}$ satisfies (2.10).

We claim that $u_{\lambda^{\ast}}\neq 0$. If it is not the case,
$D_{uu}\Phi_{\lambda^{\ast}}(0)=D_{u}S+D_{u}G-\lambda^{\ast}$ is invertible.
Since $0$ is always a solution of (1.1) for any $\lambda$, the implicit
function theorem yields that $u_{\lambda_{n}}$ must be $0$ for $n$ large
enough, contradicting the fact that $u_{\lambda_{n}}\neq 0$.

Finally, we show that $\mu_{G}(u_{\lambda^{\ast}})=1$. On the one hand,
$\mu_{G}(u_{\lambda^{\ast}})\geq 1$ since

$\langle
D_{uu}\Phi_{\lambda^{\ast}}(u_{\lambda^{\ast}})u_{\lambda^{\ast}},u_{\lambda^{\ast}}\rangle=\langle
D_{u}F(u_{\lambda^{\ast}})-D_{uu}F(u_{\lambda^{\ast}})u_{\lambda^{\ast}},u_{\lambda^{\ast}}\rangle<0.$

One the other hand, since the Morse index is lower semi-continuous with
respect to the norm-resolvent topology, we conclude that

$\mu_{G}(u_{\lambda^{\ast}})\leq\liminf_{n\rightarrow+\infty}\mu_{G}(u_{\lambda_{n}})=1,$

which completes the proof. ∎

######  Lemma 2.5.

Assume that $(H_{1})$ \- $(H_{7})$ hold, and that for some
$\lambda_{\ast}<\lambda_{1}$, there exists $u_{\lambda_{\ast}}\in
W_{G}\setminus\\{0\\}$ such that $u_{\lambda_{\ast}}$ solves (1.1), and
$\mu_{G}(u_{\lambda_{\ast}})=1$. Then there exists

$\lambda\mapsto u_{\lambda}\in
C^{1}((-\infty,\lambda_{\ast}],W_{G}\setminus\\{0\\}),$

such that $u_{\lambda}$ solves (1.1), and $\mu_{G}(u_{\lambda})=1$.

Proof. We use an implicit function argument for the map

$\Psi:\widehat{W}_{G}\times(-\infty,\lambda_{1})\rightarrow\widehat{W}_{G},\Psi(u,\lambda)=u-(D_{u}S+D_{u}G-\lambda)^{-1}D_{u}F(u).$

It is not difficult to see that $\Psi$ is a well-defined map of class $C^{1}$.
By $(H_{6})$, $u\in W_{G}$ solves (1.1) if and only if $\Psi(u,\lambda)=0$.
$u_{\lambda_{\ast}}$ solves (1.1), thus
$\Psi(u_{\lambda_{\ast}},\lambda_{\ast})=0$. Next, we consider the Fréchet
derivative

(2.11)
$\Psi_{u}(u_{\lambda_{\ast}},\lambda_{\ast})=1-K,K=(D_{u}S+D_{u}G-\lambda_{\ast})^{-1}D_{uu}F(u_{\lambda_{\ast}}).$

By $(H_{7})$, $D_{uu}F(u_{\lambda_{\ast}})$ maps $\widehat{W}_{G}$ to $E_{G}$.
Hence, $K$ maps $\widehat{W}_{G}$ to $\widehat{W}_{G}$. If $v-Kv=0$ for some
$v\in\widehat{W}_{G}$, then
$D_{uu}\Phi_{\lambda_{\ast}}(u_{\lambda_{\ast}})v=0$. By $(H_{5})$,

$\ker D_{uu}\Phi_{\lambda_{\ast}}(u_{\lambda_{\ast}})|_{E_{G}}=0.$

Thus we deduce that $v=0$ and thus
$1-K:\widehat{W}_{G}\rightarrow\widehat{W}_{G}$ is injective. By standard
arguments, the operator $D_{uu}\Phi_{\lambda_{\ast}}(u_{\lambda_{\ast}})$ has
a bounded inverse on $E_{G}$ since
$D_{uu}\Phi_{\lambda_{\ast}}(u_{\lambda_{\ast}})$ has trivial kernel on
$E_{G}$. Then for any $w\in\widehat{W}_{G}$,
$D_{uu}\Phi_{\lambda_{\ast}}(u_{\lambda_{\ast}})^{-1}(D_{u}S+D_{u}G-\lambda_{\ast})w\in\widehat{W}_{G}$.
Therefore,

(2.12)
$\displaystyle(1-K)D_{uu}\Phi_{\lambda_{\ast}}(u_{\lambda_{\ast}})^{-1}Bw$
$\displaystyle=$ $\displaystyle
B^{-1}D_{uu}\Phi_{\lambda_{\ast}}(u_{\lambda_{\ast}})D_{uu}\Phi_{\lambda_{\ast}}(u_{\lambda_{\ast}})^{-1}Bw$
$\displaystyle=$ $\displaystyle w,$

where $B=D_{u}S+D_{u}G-\lambda_{\ast}$, showing that
$1-K:\widehat{W}_{G}\rightarrow\widehat{W}_{G}$ is surjective. Using the
implicit function theorem for $\Psi(u,\lambda)$ at
$(u_{\lambda_{\ast}},\lambda_{\ast})$, we deduce that for some $\delta>0$,
there exists a map $u(\lambda)\in
C^{1}((\lambda_{\ast}-\delta,\lambda_{\ast}],\widehat{W}_{G}\setminus\\{0\\})$
such that the following holds true:

$(i)$ $u(\lambda)$ solves $\Psi(u,\lambda)=0$ for all
$\lambda\in(\lambda_{\ast}-\delta,\lambda_{\ast}]$;

$(ii)$ there exists $\epsilon>0$ such that $u(\lambda)$ is the unique solution
of $\Psi(u,\lambda)=0$ for $\lambda\in(\lambda_{\ast}-\delta,\lambda_{\ast}]$
in the neighborhood
$\\{u\in\widehat{W}_{G}:\|u-u(\lambda_{\ast})\|_{\widehat{W}}<\epsilon\\}$; in
particular, we have that $u(\lambda_{\ast})=u_{\lambda_{\ast}}$ holds.

Note that $D_{uu}\Phi_{\lambda}(u_{\lambda})\rightarrow
D_{uu}\Phi_{\lambda_{\ast}}(u_{\lambda_{\ast}})$ in the norm-resolvent sense
as $\lambda\rightarrow\lambda_{\ast}$. Then, along the line of the proof of
Lemma A.6 $(iii)$ below, we can deduce that $\mu_{G}(u(\lambda))=1$.

We claim that this branch $u(\lambda)$ can be extend to $-\infty$. Otherwise,
if this branch $u(\lambda)$ can only be extend to $\lambda^{\ast}$, where
$\mu_{G}(u(\lambda))=1$ for $\lambda\in(\lambda^{\ast},\lambda_{\ast}]$. By
Lemma 2.4, there exists $u_{\lambda^{\ast}}\in W_{G}\setminus\\{0\\}$ such
that $\mu_{G}(u_{\lambda^{\ast}})=1$ and $u_{\lambda^{\ast}}$ satisfies
(2.10). By $(H_{5})$,
$D_{uu}\Phi_{\lambda^{\ast}}(u_{\lambda^{\ast}})|_{E_{G}}=\\{0\\}$. Then
similar implicit function arguments yields that this branch $u(\lambda)$ can
be extend to $\lambda^{\ast}-\delta<\lambda^{\ast}$, in a contradiction with
the minimality of $\lambda^{\ast}$. Thus we complete the proof. ∎

Completion of the Proof for Theorem 2.2. Note that $\mathcal{N}_{G,\lambda}$
is a $C^{1}$ manifold with codimension $1$ in $W_{G}$, $h_{G}(\lambda)$ is
well-defined and $h_{G}(\lambda)\neq 0$ for all $\lambda\neq\lambda_{1}$. Then
there exists a (PS) sequence $\\{u_{n}\\}\subset W_{G}$ at level
$h_{G}(\lambda)$. By $(H_{3})$, there exists $u_{\lambda}\in W_{G}$ such that
$u_{n}\rightarrow u_{\lambda}$ in $W$. Hence,
$\Phi_{\lambda}^{\prime}(u_{\lambda})=0$ and
$\Phi_{\lambda}(u_{\lambda})=h_{G}(\lambda)$, showing the existence of a
nontrivial $G$-ground state solution. Next we prove that
$\mu_{G}(u_{\lambda})=1$. On the one hand, from

(2.13) $\left\langle
D_{uu}\Phi_{\lambda}(u_{\lambda})u_{\lambda},u_{\lambda}\right\rangle=\left\langle
D_{u}F(u_{\lambda})-D_{uu}F(u_{\lambda})u_{\lambda},u_{\lambda}\right\rangle<0,$

we derive that $\mu_{G}(u_{\lambda})\geq 1$. On the other hand, noticing that
the codimension of $\mathcal{N}_{\lambda}$ is $1$ and $u_{\lambda}$ achieves
the minimum of $h(\lambda)$, we have that $\mu_{G}(u_{\lambda})\leq 1$. Hence,
$\mu_{G}(u_{\lambda})=1$. Then by Lemma 2.5, we can take
$\lambda_{\ast}<\lambda_{1}$ such that there exists

$\lambda\mapsto u_{\lambda}\in
C^{1}((-\infty,\lambda_{\ast}],W_{G}\setminus\\{0\\}),$

and $u_{\lambda}$ solves (1.1), $\mu_{G}(u_{\lambda})=1$. Noticing that
$\lambda_{\ast}$ can be chosen arbitrarily close to $\lambda_{1}$, we know
that there exists

$\lambda\mapsto u_{\lambda}\in
C^{1}((-\infty,\lambda_{1}),W_{G}\setminus\\{0\\}),$

such that $u_{\lambda}$ solves (1.1), and $\mu_{G}(u_{\lambda})=1$. ∎

######  Corollary 2.6.

(Uniqueness) Under the hypotheses of Theorem 2.2. If we also assume that there
exists $\Lambda_{1}<\lambda_{1}$ such that (1.1) admits a unique solution with
$G$-Morse index $1$ for all $\lambda<\Lambda_{1}$. Then for all
$\lambda<\lambda_{1}$, (1.1) admits a unique solution with $G$-Morse index
$1$.

In particular, (1.1) admits a unique $G$-ground state solution for all
$\lambda<\lambda_{1}$.

Furthermore, if all ground states of (1.1) belong to $W_{G}$. Then a
$G$-ground state is indeed a ground state, and (1.1) admits a unique ground
state solution for all $\lambda<\lambda_{1}$.

Proof. The existence of a solution with $G$-Morse index $1$ has been
established in the proof of Theorem 2.2. Next we will show its uniqueness.
Suppose on the contrary that (1.1) has two solutions
$u_{\lambda_{0}},\widetilde{u}_{\lambda_{0}}$ with $G$-Morse index $1$ in
$W_{G}$ for some $\lambda_{0}<\lambda_{1}$. By the proof of Theorem 2.2, we
obtain two global branches
$u(\lambda),\widetilde{u}(\lambda),\lambda\in[\Lambda_{1}-1,\lambda_{0}]$
stemming from
$u(\lambda_{0})=u_{\lambda_{0}},\widetilde{u}(\lambda_{0})=\widetilde{u}_{\lambda_{0}}$
respectively. Furthermore, both $u(\Lambda_{1}-1)$ and
$\widetilde{u}(\Lambda_{1}-1)$ are solutions of (1.1) with
$\lambda=\Lambda_{1}-1$ and
$\mu_{G}(u(\Lambda_{1}-1))=\mu_{G}(\widetilde{u}(\Lambda_{1}-1))=1$. The
uniqueness when $\lambda<\Lambda_{1}$ implies that
$u(\Lambda_{1}-1)=\widetilde{u}(\Lambda_{1}-1)$. By $(H_{5})$,
$u(\Lambda_{1}-1)$ is non-degenerate in $W_{G}$. However, there are two
different local branches $u(\lambda)$ and $\widetilde{u}(\lambda)$,
$\lambda\in[\Lambda_{1}-1,\Lambda_{1}-1+\delta)$ stemming form
$u(\Lambda_{1}-1)$, contradicting the non-degeneracy of $u(\Lambda_{1}-1)$.
Hence the proof is complete. ∎

##  3\. Abstract Framework 2

In Section 2, we established a $C^{1}$ global branch $u_{\lambda}$
parameterized by $\lambda\in(-\infty,\lambda_{1})$. To get the existence of
normalized solutions, we need abstract frameworks to show the behaviors of
$Q(u_{\lambda})$ as $\lambda\rightarrow-\infty$ and
$\lambda\rightarrow\lambda_{1}$. This section and the next one provide two
different approaches. In this section, we merely need that $\Phi_{\lambda}\in
C^{1}$.

Throughout this section, we always assume that $\lambda$,
$\lambda_{0}\in(a,b)$, $\mathcal{N}_{\lambda}$, $\mathcal{K}_{\lambda}$,
$h(\lambda)$, $d(\lambda)$ and $\hat{d}(\lambda)$ are given in Section 1, and
we always assume that $h(\lambda)\neq 0$, ensuring that
$0\notin\mathcal{K}_{\lambda}$. Therefore, all elements in
$\mathcal{K}_{\lambda}$ are nontrivial solutions of (1.1). Our goals are to
obtain the continuity of $d(\lambda)$ if $d(\lambda)\equiv\hat{d}(\lambda)$
and to study the asymptotical behaviors of $d(\lambda)$ and
$\hat{d}(\lambda)$. It is worth noting that the abstract framework 2 in this
section is not just a simple continuation of abstract framework 1, but a
framework in its own right. For example, classical arguments are invalid to
get the existence of normalized solutions of the equation $\Delta u+\lambda
u+|u|^{p-2}u=0$ on $\mathbb{R}^{N}\setminus B_{1}$, particularly in the
$L^{2}$-supercritical case. Relying on ODE techniques, the uniqueness of
positive radial solution can be proved, ensuring that
$d(\lambda)\equiv\hat{d}(\lambda)$ (in fact, the uniqueness is enough to
obtain a continuous global branch $u_{\lambda}$ parameterized by $\lambda$).
Therefore, our framework 2 is applicable. It helps to obtain threshold results
for the existence, non-existence and multiplicity of normalized solutions for
semi-linear elliptic equations in the exterior of a ball, see [53]. In some
cases, even if we do not know the uniqueness of ground state, the abstract
framework 2 can be used, see Section 5.5 for an example with a Hardy
potential.

First, we study the properties of $h(\lambda)$, $d(\lambda)$ and
$\hat{d}(\lambda)$. To do this, we assume

$(F_{1})$ For any $(\lambda,u)\in(a,b)\times W\backslash\\{0\\}$, there exists
an unique function $t:(a,b)\times W\backslash\\{0\\}\rightarrow(0,\infty)$
such that

$when\ t>0,\ then\ (\lambda,tu)\in\mathcal{N}\Leftrightarrow t=t(\lambda,u),$

and $\Phi_{\lambda}(t(\lambda,u)u)=\max\limits_{t>0}\Phi_{\lambda}(tu)$, $t\in
C((a,b)\times W\backslash\\{0\\},(0,\infty))$, where
$\mathcal{N}:=\cup_{a<\lambda<b}\mathcal{N}_{\lambda}$.

$(F_{2})$ $Q(u_{\lambda})$ is bounded in $\mathbb{R}$ and
$D_{u}Q(u_{\lambda})$ is bounded in $W^{\ast}$ whenever
$u_{\lambda}\in\mathcal{K}_{\lambda}$ and $\lambda\rightarrow\lambda_{0}$.

$(F_{3})$ $\Phi_{\lambda_{0}}$ satisfies $(PS)_{c}$ condition for $c\leq
h(\lambda_{0})$, $\forall\lambda_{0}\in(a,b)$.

$(F_{4})$ $\liminf_{\lambda\rightarrow\lambda_{0}}h(\lambda)>0$,
$\forall\lambda_{0}\in(a,b)$.

######  Theorem 3.1.

Suppose $(F_{1})$ \- $(F_{4})$.

$(i)$ $d(\lambda_{0})\leq\liminf_{\lambda\rightarrow\lambda_{0}}d(\lambda)$,
$\hat{d}(\lambda_{0})\geq\limsup_{\lambda\rightarrow\lambda_{0}}\hat{d}(\lambda)$.

$(ii)$ $\lim_{\lambda\rightarrow\lambda_{0}}h(\lambda)=h(\lambda_{0})$.

$(iii)$ $h(\lambda)$ is differentiable at almost everywhere $\lambda\in(a,b)$.
Furthermore,

$\lim\limits_{\lambda\rightarrow\lambda_{0}^{+}}\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}=-\hat{d}(\lambda_{0}),\lim\limits_{\lambda\rightarrow\lambda_{0}^{-}}\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}=-d(\lambda_{0}).$

In particular, $d(\lambda_{0})=\hat{d}(\lambda_{0})\Leftrightarrow h(\lambda)$
is differentiable at $\lambda_{0}$ and
$h^{\prime}(\lambda_{0})=-d(\lambda_{0})$.

Proof.  Step 1: $\limsup_{\lambda\rightarrow\lambda_{0}}h(\lambda)\leq
h(\lambda_{0})$.

Take $u_{\lambda_{0}}\in\mathcal{K}_{\lambda_{0}}$, then
$\Phi_{\lambda_{0}}(u_{\lambda_{0}})=h(\lambda_{0})$. Consider
$t(\lambda,u_{\lambda_{0}})u_{\lambda_{0}}\in\mathcal{N}_{\lambda}$. By
$(F_{1})$,

$t(\lambda,u_{\lambda_{0}})u_{\lambda_{0}}\rightarrow
t(\lambda_{0},u_{\lambda_{0}})u_{\lambda_{0}}=u_{\lambda_{0}}\ in\ W\ as\
\lambda\rightarrow\lambda_{0},$

yielding that

(3.1) $\displaystyle h(\lambda)$ $\displaystyle\leq$
$\displaystyle\Phi_{\lambda}(t(\lambda,u_{\lambda_{0}})u_{\lambda_{0}})$
$\displaystyle=$
$\displaystyle\Phi_{\lambda_{0}}(t(\lambda,u_{\lambda_{0}})u_{\lambda_{0}})+(\lambda_{0}-\lambda)Q(t(\lambda,u_{\lambda_{0}})u_{\lambda_{0}})$
$\displaystyle=$ $\displaystyle\Phi_{\lambda_{0}}(u_{\lambda_{0}})+o(1).$

Letting $\lambda\rightarrow\lambda_{0},$ we have
$\limsup_{\lambda\rightarrow\lambda_{0}}h(\lambda)\leq h(\lambda_{0})$.

Step 2: If $\lambda_{n}\rightarrow\lambda_{0}$,
$u_{\lambda_{n}}\in\mathcal{K}_{\lambda_{n}}$, then
$u_{\lambda_{n}}\rightarrow u_{\lambda_{0}}$ in $W$ up to a subsequence and
$u_{\lambda_{0}}\in\mathcal{K}_{\lambda_{0}}$.

By $(F_{2})$, $(\lambda_{n}-\lambda_{0})Q(u_{\lambda_{n}})\rightarrow 0$.
Noticing that

$\Phi_{\lambda_{0}}(u_{\lambda_{n}})=\Phi_{\lambda_{n}}(u_{\lambda_{n}})+(\lambda_{n}-\lambda_{0})Q(u_{\lambda_{n}})=h(\lambda_{n})+o(1),$

we obtain that
$\limsup_{n\rightarrow\infty}\Phi_{\lambda_{0}}(u_{\lambda_{n}})\leq
h(\lambda_{0})$. By $(F_{4})$, we may assume that
$\Phi_{\lambda_{0}}(u_{\lambda_{n}})\rightarrow c\in(0,h(\lambda_{0})]$
passing to a subsequence if necessary. Furthermore, since
$D_{u}Q(u_{\lambda_{n}})$ is bounded in $W^{\ast}$,
$D_{u}\Phi_{\lambda_{0}}(u_{\lambda_{n}})=D_{u}\Phi_{\lambda_{n}}(u_{\lambda_{n}})+(\lambda_{n}-\lambda_{0})D_{u}Q(u_{\lambda_{n}})\rightarrow
0$ as $n\rightarrow\infty$.

By $(F_{3})$, we may assume that $u_{\lambda_{n}}\rightarrow u_{\lambda_{0}}$
up to a subsequence. Using $(F_{4})$, $\Phi_{\lambda_{0}}(u_{\lambda_{0}})>0$,
implying that $u_{\lambda_{0}}\neq 0$. Since $\Phi_{\lambda_{0}}\in C^{1}$,
$D_{u}\Phi_{\lambda_{0}}(u_{\lambda_{0}})=0$. Thus $u_{\lambda_{0}}\in
N_{\lambda_{0}}$.

Next, we will show that $u_{\lambda_{0}}\in\mathcal{K}_{\lambda_{0}}$.
$\forall\epsilon>0$, $\exists X>0$, $\forall n>X$, there holds that

(3.2) $\displaystyle h(\lambda_{0})$ $\displaystyle\leq$
$\displaystyle\Phi_{\lambda_{0}}(u_{\lambda_{0}})$ $\displaystyle\leq$
$\displaystyle\Phi_{\lambda_{0}}(u_{\lambda_{n}})+\frac{\epsilon}{2}$
$\displaystyle=$
$\displaystyle\Phi_{\lambda_{n}}(u_{\lambda_{n}})+(\lambda_{n}-\lambda_{0})Q(u_{\lambda_{n}})+\frac{\epsilon}{2}$
$\displaystyle\leq$
$\displaystyle\Phi_{\lambda_{n}}(u_{\lambda_{n}})+\epsilon.$

Letting $n\rightarrow\infty,$ we get that
$h(\lambda_{0})\leq\limsup_{\lambda\rightarrow\lambda_{0}}h(\lambda)$.
Together with Step 1, we can conclude that
$\limsup_{\lambda\rightarrow\lambda_{0}}h(\lambda)=h(\lambda_{0})$. We can
derive that $\Phi_{\lambda_{0}}(u_{\lambda_{0}})=h(\lambda_{0})$ by (3.2)
immediately. Thus $u_{\lambda_{0}}\in\mathcal{K}_{\lambda_{0}}$.

Step 3: The Proof of Theorem 3.1 $(i)$ and $(ii)$.

Take $\\{\lambda_{n}\\}_{n=1}^{+\infty}$ such that
$\lambda_{n}\rightarrow\lambda_{0}$,
$\liminf_{\lambda\rightarrow\lambda_{0}}d(\lambda)=\lim_{n\rightarrow+\infty}d(\lambda_{n})$.
Since $\mathcal{K}_{\lambda}$ is compact, we can take
$u_{\lambda_{n}}\in\mathcal{K}_{\lambda_{n}}$ such that
$d(\lambda_{n})=Q(u_{\lambda_{n}})$. By Step 2, passing to a subsequence if
necessary, $u_{\lambda_{n}}\rightarrow
u_{\lambda_{0}}\in\mathcal{K}_{\lambda_{0}}$. Therefore,

$\displaystyle d(\lambda_{0})$ $\displaystyle\leq$ $\displaystyle
Q(u_{\lambda_{0}})$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}Q(u_{\lambda_{n}})$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}d(\lambda_{n})$ $\displaystyle=$
$\displaystyle\liminf_{\lambda\rightarrow\lambda_{0}}d(\lambda).$

Similarly, we can take $\\{\lambda_{n}\\}_{n=1}^{+\infty}$ such that
$\lambda_{n}\rightarrow\lambda_{0}$,
$\limsup_{\lambda\rightarrow\lambda_{0}}\hat{d}(\lambda)=\lim_{n\rightarrow\infty}\hat{d}(\lambda_{n})$
and take $u_{\lambda_{n}}\in\mathcal{K}_{\lambda_{n}}$ such that
$\hat{d}(\lambda_{n})=Q(u_{\lambda_{n}})$. Then

$\displaystyle\hat{d}(\lambda_{0})$ $\displaystyle\geq$ $\displaystyle
Q(u_{\lambda_{0}})$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}Q(u_{\lambda_{n}})$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\hat{d}(\lambda_{n})$ $\displaystyle=$
$\displaystyle\limsup_{\lambda\rightarrow\lambda_{0}}\hat{d}(\lambda).$

Next we prove $(ii)$. According to Step 2, for any $\\{\lambda_{n}\\}$ with
$\lambda_{n}\rightarrow\lambda_{0}$, passing to a subsequence if necessary,
there exists $u_{\lambda_{n}}\in K_{\lambda_{n}}$ s.t.
$\lim_{n\rightarrow\infty}h(\lambda_{n})=\lim_{n\rightarrow\infty}\Phi_{\lambda_{n}}(u_{\lambda_{n}})=h(\lambda_{0})$.
Hence, we conclude that
$\lim_{\lambda\rightarrow\lambda_{0}}h(\lambda)=h(\lambda_{0}),$ which
completes the proof.

Step 4: Suppose that $u_{\lambda}\in\mathcal{K}_{\lambda}$,
$u_{\lambda_{0}}\in\mathcal{K}_{\lambda_{0}}$. Let
$\widetilde{u}_{\lambda}=t(\lambda_{0},u_{\lambda})u_{\lambda}$ and
$\widetilde{u}_{\lambda_{0}}=t(\lambda,u_{\lambda_{0}})u_{\lambda_{0}}$, then

(3.3)
$\Phi_{\lambda}(u_{\lambda})\leq(\lambda_{0}-\lambda)Q(\widetilde{u}_{\lambda_{0}})+\Phi_{\lambda_{0}}(u_{\lambda_{0}}),$
(3.4)
$\Phi_{\lambda_{0}}(u_{\lambda_{0}})\leq(\lambda-\lambda_{0})Q(\widetilde{u}_{\lambda})+\Phi_{\lambda}(u_{\lambda}).$

By $(F_{1})$,
$\Phi_{\lambda}(u_{\lambda})=\max\limits_{t>0}\Phi_{\lambda}(tu_{\lambda})$
and
$\Phi_{\lambda_{0}}(u_{\lambda_{0}})=\max\limits_{t>0}\Phi_{\lambda_{0}}(tu_{\lambda_{0}})$.
Therefore,

$\Phi_{\lambda}(\widetilde{u}_{\lambda})\leq\Phi_{\lambda}(u_{\lambda}),\Phi_{\lambda_{0}}(\widetilde{u}_{\lambda_{0}})\leq\Phi_{\lambda_{0}}(u_{\lambda_{0}}).$

Since $u_{\lambda}\in\mathcal{K}_{\lambda}$,
$\widetilde{u}_{\lambda_{0}}\in\mathcal{N}_{\lambda},$
$u_{\lambda_{0}}\in\mathcal{K}_{\lambda_{0}}$, and $\widetilde{u}_{\lambda}\in
N_{\lambda_{0}}$, then

$\Phi_{\lambda}(\widetilde{u}_{\lambda_{0}})\geq\Phi_{\lambda}(u_{\lambda}),\Phi_{\lambda_{0}}(\widetilde{u}_{\lambda})\geq\Phi_{\lambda_{0}}(u_{\lambda_{0}}).$

It follows that:

(3.5) $\displaystyle\Phi_{\lambda}(u_{\lambda})$ $\displaystyle\leq$
$\displaystyle\Phi_{\lambda}(\widetilde{u}_{\lambda_{0}})$ $\displaystyle=$
$\displaystyle(\lambda_{0}-\lambda)Q(\widetilde{u}_{\lambda_{0}})+\Phi_{\lambda_{0}}(\widetilde{u}_{\lambda_{0}})$
$\displaystyle\leq$
$\displaystyle(\lambda_{0}-\lambda)Q(\widetilde{u}_{\lambda_{0}})+\Phi_{\lambda_{0}}(u_{\lambda_{0}}),$
(3.6) $\displaystyle\Phi_{\lambda_{0}}(u_{\lambda_{0}})$ $\displaystyle\leq$
$\displaystyle\Phi_{\lambda_{0}}(\widetilde{u}_{\lambda})$ $\displaystyle=$
$\displaystyle(\lambda-\lambda_{0})Q(\widetilde{u}_{\lambda})+\Phi_{\lambda}(\widetilde{u}_{\lambda})$
$\displaystyle\leq$
$\displaystyle(\lambda-\lambda_{0})Q(\widetilde{u}_{\lambda})+\Phi_{\lambda}(u_{\lambda}).$

Step 5: The Proof of Theorem 3.1 $(iii)$.

By Step 4,

(3.7) $(\lambda_{0}-\lambda)Q(\widetilde{u}_{\lambda})\leq
h(\lambda)-h(\lambda_{0})\leq(\lambda_{0}-\lambda)Q(\widetilde{u}_{\lambda_{0}}),$

where $\widetilde{u}_{\lambda}$ and $\widetilde{u}_{\lambda_{0}}$ are given by
Step 4. Thus $h(\lambda)$ is monotonically decreasing in $(a,b)$. Then we know
that $h(\lambda)$ is differentiable at almost every $\lambda\in(a,b)$.

Next, we want to estimate

$\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}.$

First, we assume that $\lambda>\lambda_{0}$. It follows that:

(3.8)
$-Q(\widetilde{u}_{\lambda})\leq\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}\leq-Q(\widetilde{u}_{\lambda_{0}}).$

On the one hand,

(3.9)
$\displaystyle\limsup_{\lambda\rightarrow\lambda_{0}^{+}}\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}$
$\displaystyle\leq$
$\displaystyle-\liminf_{\lambda\rightarrow\lambda_{0}^{+}}Q(\widetilde{u}_{\lambda_{0}})$
$\displaystyle=$
$\displaystyle-\lim_{\lambda\rightarrow\lambda_{0}^{+}}Q(t(\lambda,u_{\lambda_{0}})u_{\lambda_{0}})$
$\displaystyle=$ $\displaystyle-Q(u_{\lambda_{0}}).$

Using the arbitrariness of $u_{\lambda_{0}}$, we know that

(3.10)
$\displaystyle\limsup_{\lambda\rightarrow\lambda_{0}^{+}}\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}$
$\displaystyle\leq$
$\displaystyle\min_{u_{\lambda_{0}}\in\mathcal{K}_{\lambda_{0}}}(-Q(u_{\lambda_{0}}))$
$\displaystyle=$
$\displaystyle-\max\limits_{u_{\lambda_{0}}\in\mathcal{K}_{\lambda_{0}}}Q(u_{\lambda_{0}})$
$\displaystyle=$ $\displaystyle-\hat{d}(\lambda_{0}).$

On the other hand,

(3.11)
$\liminf_{\lambda\rightarrow\lambda_{0}^{+}}\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}\geq\liminf_{\lambda\rightarrow\lambda_{0}^{+}}(-Q(\widetilde{u}_{\lambda}))=-\limsup_{\lambda\rightarrow\lambda_{0}^{+}}Q(\widetilde{u}_{\lambda}).$

Take $\\{\lambda_{n}\\}_{n=1}^{+\infty}$ such that
$\lambda_{n}\rightarrow\lambda_{0}^{+}$ (i.e. $\lambda_{n}\geq\lambda_{0}$ and
$\lim_{n\rightarrow\infty}\lambda_{n}=\lambda_{0}$) and

$\lim_{n\rightarrow\infty}Q(\widetilde{u}_{\lambda_{n}})=\limsup_{\lambda\rightarrow\lambda_{0}^{+}}Q(\widetilde{u}_{\lambda}).$

By Step 2, passing to a subsequence if necessary, there exists some
$u_{0}\in\mathcal{K}_{\lambda_{0}}$ s.t. $u_{\lambda_{n}}\rightarrow u_{0}$ in
$W$ as $n\rightarrow+\infty$. Thus
$\widetilde{u}_{\lambda_{n}}=t(\lambda_{0},u_{\lambda_{n}})u_{\lambda_{n}}\rightarrow
u_{0}$ in $W$ as $n\rightarrow+\infty$ and
$\lim_{n\rightarrow\infty}Q(\widetilde{u}_{\lambda_{n}})=Q(u_{0})$. Therefore,

(3.12)
$\displaystyle\liminf_{\lambda\rightarrow\lambda_{0}^{+}}\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}$
$\displaystyle\geq$
$\displaystyle-\limsup_{\lambda\rightarrow\lambda_{0}^{+}}Q(\widetilde{u}_{\lambda})$
$\displaystyle=$
$\displaystyle-\lim_{n\rightarrow\infty}Q(\widetilde{u}_{\lambda_{n}})$
$\displaystyle=$ $\displaystyle-Q(u_{0})$ $\displaystyle\geq$
$\displaystyle-\hat{d}(\lambda_{0}).$

By (3.10) and (3.12), we have

$\lim_{\lambda\rightarrow\lambda_{0}^{+}}\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}=-\hat{d}(\lambda_{0}).$

Similar arguments enable as to conclude that

$\lim_{\lambda\rightarrow\lambda_{0}^{-}}\frac{h(\lambda)-h(\lambda_{0})}{\lambda-\lambda_{0}}=-d(\lambda_{0}).$

∎

Next, we study the asymptotical behavior of $d(\lambda)$ and
$\hat{d}(\lambda)$. Assume

$(F_{5})$ There exists $l>0$ such that $\Phi_{\lambda}(u)\leq-l\lambda Q(u)$
for any $u\in\mathcal{K}_{\lambda}$.

$(F_{6})$ There exists $k>0$ such that $\Phi_{\lambda}(u)\geq-k\lambda Q(u)$
for any $u\in\mathcal{K}_{\lambda}$.

######  Theorem 3.2.

Suppose $(F_{1})$ \- $(F_{4})$.

$(i)$ Assume that $(F_{5})$ holds with $0<l<1$. If $h(\lambda)$ is well-
defined in $(-\infty,b)$, then
$\lim_{\lambda\rightarrow-\infty}d(\lambda)=+\infty$.

$(ii)$ Assume $(F_{6})$ holds with $k>1$. If $h(\lambda)$ is well-defined in
$(-\infty,b)$, then $\lim_{\lambda\rightarrow-\infty}\hat{d}(\lambda)=0$.

$(iii)$ Assume $(F_{5})$ and $(F_{6})$ hold with $l>k>0$ and $h(\lambda)$ is
well-defined in $(a,0)$. If $0<k<l<1$, then $\lim_{\lambda\rightarrow
0^{-}}\hat{d}(\lambda)=0$; if $l>k>1$, then $\lim_{\lambda\rightarrow
0^{-}}d(\lambda)=+\infty$.

$(iv)$ Assume $(F_{5})$ and $(F_{6})$ hold with $k=l$ and $\lambda<0$. Then
$d(\lambda)\equiv\hat{d}(\lambda)$ and
$d(\lambda)=\frac{C}{k}(-\lambda)^{\frac{1}{k}-1}$ where $C>0$ is a constant.

Proof.  For $\lambda<0$, we define

$c(\lambda)=\frac{h(\lambda)}{\lambda},$
$c^{\prime}_{-}(\lambda)=\lim_{\Delta\lambda\rightarrow
0^{-}}\frac{c(\lambda+\Delta\lambda)-c(\lambda)}{\Delta\lambda},$
$c^{\prime}_{+}(\lambda)=\lim_{\Delta\lambda\rightarrow
0^{+}}\frac{c(\lambda+\Delta\lambda)-c(\lambda)}{\Delta\lambda}.$

We may assume that $b>-1$ and $\lambda<0$. By Theorem 3.1 $(iii)$,
$h^{\prime}_{+}(\lambda)=-\hat{d}(\lambda)$,
$h^{\prime}_{-}(\lambda)=-d(\lambda)$.

By $(F_{5})$, $h(\lambda)\leq-l\lambda d(\lambda)$. Since
$h^{\prime}_{\pm}(\lambda)=c^{\prime}_{\pm}(\lambda)\lambda+c(\lambda)$,

(3.13) $c^{\prime}_{+}(\lambda)\lambda+c(\lambda)\leq
c^{\prime}_{-}(\lambda)\lambda+c(\lambda)\leq\frac{1}{l}c(\lambda),$

i.e.

(3.14) $c^{\prime}_{+}(\lambda)\lambda\leq
c^{\prime}_{-}(\lambda)\lambda\leq(\frac{1}{l}-1)c(\lambda).$

Set $g(\lambda)=|c(-1)|(-\lambda)^{\frac{1}{l}-1}$ for $\lambda<0$. Then
$g^{\prime}(\lambda)\lambda=(\frac{1}{l}-1)g(\lambda)$ and $g(\lambda)>0$.
Now, set $\psi(\lambda)=\frac{c(\lambda)}{g(\lambda)}$, then

(3.15)
$\psi^{\prime}_{\pm}(\lambda)=\frac{g(\lambda)c^{\prime}_{\pm}(\lambda)-g^{\prime}(\lambda)c(\lambda)}{g^{2}(\lambda)}\geq
0.$

Therefore, $\psi$ is monotonic increasing in $(-\infty,-1]$, then

$\psi(\lambda)\leq\psi(-1)=\frac{c(-1)}{|c(-1)|},$

i.e.

$c(\lambda)\leq c(-1)(-\lambda)^{\frac{1}{l}-1},\forall\lambda<-1.$

When $h(\lambda)>0$ and $\lambda<0$, $c(\lambda)<0$. When $l\in(0,1)$,

$c(\lambda)\leq c(-1)(-\lambda)^{\frac{1}{l}-1}\rightarrow-\infty\ as\
\lambda\rightarrow-\infty.$

By $(F_{5})$, $d(\lambda)\geq-\frac{1}{l}c(\lambda)$. Thus
$\lim_{\lambda\rightarrow-\infty}d(\lambda)=+\infty$ and we prove $(i)$.

By $(F_{6})$, $h(\lambda)\geq-k\lambda\hat{d}(\lambda)$. A similar argument
implies that:

$c(\lambda)\geq c(-1)(-\lambda)^{\frac{1}{k}-1},\forall\lambda<-1.$

By $(F_{4})$ and Theorem 3.1 $(ii)$, for any $\lambda_{0}\in(a,b)$,
$h(\lambda_{0})=\lim_{\lambda\rightarrow\lambda_{0}}h(\lambda)>0$, i.e.
$h(\lambda)>0$ on its definition domain. Hence, when $\lambda<0$,
$c(\lambda)<0$. When $k>1$,

$c(\lambda)\geq c(-1)(-\lambda)^{\frac{1}{k}-1}\rightarrow 0\ as\
\lambda\rightarrow-\infty.$

By the definition of $\mathcal{K}_{\lambda}$,
$\Phi_{\lambda}(u)=h(\lambda),\forall u\in\mathcal{K}_{\lambda}$. Then by
$(F_{6})$, $\Phi_{\lambda}(u)\geq-k\lambda Q(u),\forall
u\in\mathcal{K}_{\lambda}$. Therefore,
$-k\lambda\hat{d}(\lambda)=-k\lambda\sup_{u\in\mathcal{K}_{\lambda}}Q(u)\leq\sup_{u\in\mathcal{K}_{\lambda}}\Phi_{\lambda}(u)=\sup_{u\in\mathcal{K}_{\lambda}}h(\lambda)=h(\lambda)$,
i.e. $\hat{d}(\lambda)\leq-\frac{1}{k}c(\lambda)$. Thus
$\lim\limits_{\lambda\rightarrow-\infty}\hat{d}(\lambda)=0$ and $(ii)$ is
proved.

Next we prove $(iii)$. We may assume that $a<-1$. By the proof of $(i)$ and
$(ii)$, we know that:

$c(-1)(-\lambda)^{\frac{1}{l}-1}\leq c(\lambda)\leq
c(-1)(-\lambda)^{\frac{1}{k}-1},\forall\lambda\in(-1,0).$

$h(\lambda)\geq-k\lambda Q(u)>0$, thus $c(\lambda)<0$. If $0<k<l<1$, then

$0>c(\lambda)\geq c(-1)(-\lambda)^{\frac{1}{l}-1}\rightarrow 0\ as\
\lambda\rightarrow 0^{-}.$

By $(F_{6})$, $\hat{d}(\lambda)\leq-\frac{1}{k}c(\lambda)$. Thus
$\lim_{\lambda\rightarrow 0^{-}}\hat{d}(\lambda)=0$. If $l>k>1$, then

$c(\lambda)\leq c(-1)(-\lambda)^{\frac{1}{k}-1}\rightarrow-\infty\ as\
\lambda\rightarrow 0^{-}.$

By $(F_{5})$, $d(\lambda)\geq-\frac{1}{l}c(\lambda)$. Thus
$\lim_{\lambda\rightarrow 0^{-}}d(\lambda)=+\infty$.

Finally, if $(F_{5})$ and $(F_{6})$ hold with $k=l$, then
$h(\lambda)=-k\lambda Q(u)$ for any $u\in\mathcal{K}_{\lambda}$. Thus $Q(u)$
is a constant for $u\in\mathcal{K}_{\lambda}$ and we conclude that
$d\equiv\hat{d}$. By Theorem 3.1 $(iii)$, $h(\lambda)$ is differentiable and
$h^{\prime}(\lambda)=-d(\lambda)$. Therefore, $h(\lambda)=k\lambda
h^{\prime}(\lambda)$. Since $h(\lambda)=-k\lambda Q(u)>0$ when $\lambda<0$,
$h(\lambda)=C(-\lambda)^{\frac{1}{k}}$ for some $C>0$, showing
$d(\lambda)=\frac{C}{k}(-\lambda)^{\frac{1}{k}-1}$. ∎

To get stronger results when $\lambda\rightarrow 0^{-}$, we have the following
assumptions and discussions.

$(F_{7})$ $u$ is a solution of (1.1), then $S(u)\leq dF(u)$ and $-\lambda
Q(u)\geq eF(u)$ for some $d>0$, $e>0$.

$(F_{8})$ For $u\in W$, $F(u)\leq
C[S(u)^{\theta_{1}}Q(u)^{\tau_{1}}+S(u)^{\theta_{2}}Q(u)^{\tau_{2}}]$ where
$C>0$ is a constant and $\theta_{i}>1$, $\tau_{i}>0$, $i=1$, $2$.

######  Theorem 3.3.

Suppose $(F_{7})$ and $(F_{8})$. If $(\lambda_{n},u_{n})$ are solutions of
(1.1) with $\lambda_{n}\rightarrow 0^{-}$, then $Q(u_{n})\rightarrow+\infty$.
In particular, $\lim_{\lambda\rightarrow 0^{-}}d(\lambda)=+\infty$.

proof.  Arguing by contradiction, let us assume that
$Q(u_{n})\nrightarrow+\infty$. We can then suppose that
$\\{Q(u_{n})\\}_{n=1}^{+\infty}$ is bounded passing to a subsequence if
necessary.

By $(F_{7})$,

$F(u_{n})\geq\frac{1}{d}S(u_{n})>0,\ \ \ -\lambda_{n}Q(u_{n})\geq
eF(u_{n})>0.$

Thus $\lambda_{n}<0$. Sending $n$ to $+\infty$, $F(u_{n})\rightarrow 0$. Thus
$S(u_{n})\rightarrow 0$.

On the other hand, by $(F_{8})$,

$F(u_{n})\leq
C[S(u_{n})^{\theta_{1}}Q(u_{n})^{\tau_{1}}+S(u_{n})^{\theta_{2}}Q(u_{n})^{\tau_{2}}].$

Hence,

$1\leq
dC[S(u_{n})^{\theta_{1}-1}Q(u_{n})^{\tau_{1}}+S(u_{n})^{\theta_{2}-1}Q(u_{n})^{\tau_{2}}].$

But $\theta_{i}-1>0$, $i=1$, $2$, contradicting the fact $S(u_{n})\rightarrow
0$ and $\\{Q(u_{n})\\}_{n=1}^{+\infty}$ is bounded. ∎

######  Corollary 3.4.

$(i)$ Assume $(F_{1})$ \- $(F_{5})$ hold with $l\in(0,1)$. If $h(\lambda)>0$
is well-defined and $d(\lambda)\equiv\hat{d}(\lambda)$ in $(-\infty,b)$, then
$\exists\rho_{1}>0$ s.t. for any $\rho\in(\rho_{1},+\infty)$, (1.1) admits a
solution $(\lambda,u_{\lambda})$ with $Q(u_{\lambda})=\rho$.

$(ii)$ Assume $(F_{1})$ \- $(F_{4})$ and $(F_{6})$ hold with $k>1$. If
$h(\lambda)$ is well-defined and $d(\lambda)\equiv\hat{d}(\lambda)$ in
$(-\infty,b)$, then $\exists\rho_{2}>0$ s.t. for any $\rho\in(0,\rho_{2})$,
(1.1) admits a solution $(\lambda,u_{\lambda})$ with $Q(u_{\lambda})=\rho$.

$(iii)$ Assume $(F_{7})$ and $(F_{8})$ hold. If $h(\lambda)$ is well-defined
and can be achieved and $d(\lambda)\equiv\hat{d}(\lambda)$ in $(a,0)$, then
$\exists\rho_{3}>0$ s.t. for any $\rho\in(\rho_{3},+\infty)$, (1.1) admits a
solution $(\lambda,u_{\lambda})$ with $Q(u_{\lambda})=\rho$.

$(iv)$ Assume $(F_{1})$ \- $(F_{6})$ hold with $0<k<l<1$ or $l>k>1$. If
$h(\lambda)$ is well-defined and $d(\lambda)\equiv\hat{d}(\lambda)$ in
$(-\infty,0)$, then for any $\rho\in(0,+\infty)$, (1.1) admits a solution
$(\lambda,u_{\lambda})$ with $Q(u_{\lambda})=\rho$.

$(v)$ Assume $(F_{1})$ \- $(F_{6})$ hold with $k=l\neq 1$. If $h(\lambda)$ is
well-defined in $(-\infty,0)$, then for any $\rho\in(0,+\infty)$, (1.1) admits
a solution $(\lambda,u_{\lambda})$ with $Q(u_{\lambda})=\rho$.

$(vi)$ Assume $(F_{1})$ \- $(F_{4})$, $(F_{6})$ \- $(F_{8})$ hold with $k>1$.
If $h(\lambda)$ is well-defined and $d(\lambda)\equiv\hat{d}(\lambda)$ in
$(-\infty,0)$, then for any $\rho\in(0,+\infty)$, (1.1) admits a solution
$(\lambda,u_{\lambda})$ with $Q(u_{\lambda})=\rho$.

Proof.  We prove $(i)$. Standard arguments yield that $h(\lambda)$ can be
achieved by $u_{\lambda}$ and thus $\mathcal{K}_{\lambda}$ is not empty
($u_{\lambda}\in\mathcal{K}_{\lambda}$). By Theorem 3.1 $(i)$, $d(\lambda)$ is
continuous in $(-\infty,b)$ since $d\equiv\hat{d}$. By Theorem 3.2 $(i)$,
$\lim_{\lambda\rightarrow-\infty}d(\lambda)=+\infty$. Thus for any
$\rho\in(\rho_{1},\infty)$ where $\rho_{1}=\inf_{\lambda<b}d(\lambda)$, there
exists some $\lambda<b$ such that $d(\lambda)=\rho$. Since
$d(\lambda)\equiv\hat{d}(\lambda)$, $Q(u)=d(\lambda),\forall
u\in\mathcal{K}_{\lambda}$. Take $u_{\lambda}\in\mathcal{K}_{\lambda}$ and we
complete the proof of $(i)$. Similar arguments apply to $(ii)$ \- $(vi)$. ∎

######  Remark 3.5.

If there exists a global branch

$\lambda\mapsto u_{\lambda}\in C^{1}((-\infty,0),W\setminus\\{0\\}),$

such that $u_{\lambda}$ solves (1.1), it is easy to verify that
$\partial_{\lambda}\Phi_{\lambda}(u_{\lambda})=-Q(u_{\lambda})$. Then
arguments in Theorems 3.2, 3.3 and Corollary 3.4 also hold if we replace
$d(\lambda),\hat{d}(\lambda)$ with $Q(u_{\lambda})$.

##  4\. Abstract framework 3

Throughout this section, we always assume that $h(\lambda)$ is well-defined
and achieved by $u_{\lambda}$ in $(-\infty,0)$. Under some abstract settings,
we will prove the uniqueness of the ground state solution when
$\lambda\rightarrow-\infty$ or $\lambda\rightarrow 0^{-}$ and obtain the
existence of normalized solutions. This section contains two subsections. We
will need the following assumptions:

$(S_{1})$ $D_{u}S(tu)=tD_{u}S(u)$, $D_{u}G(tu)=tD_{u}G(u)$,
$D_{u}Q(tu)=tD_{u}Q(u)$.

$(S_{2})$ There exists $T(\lambda):W\rightarrow W,W^{\ast}\rightarrow
W^{\ast}$ such that

$D_{u}S(T(\lambda)u)=\frac{1}{\lambda}T(\lambda)D_{u}S(u),$
$D_{u}Q(T(\lambda)u)=T(\lambda)D_{u}Q(u),$
$D_{u}F(T(\lambda)u)=T(\lambda)D_{u}F(u).$

$(G_{1})$ $T(\lambda)$ is given by $(S_{2})$ and

$\lim_{\lambda\rightarrow-\infty}\frac{1}{\lambda}T(\lambda)D_{u}G(u)=0.$

$(Q_{1})$ $T(\lambda)$ is given by $(S_{2})$ and

$Q(T(\lambda)u)=\lambda^{\gamma}Q(u).$

$(N_{1})$ There exists $p>2$ such that

$\lim_{|t|\rightarrow
0}\frac{D_{u}F(tu)}{(tu)^{p-1}}=m_{1}\in(0,+\infty),\lim_{|t|\rightarrow
0}\frac{pF(tu)}{(tu)^{p}}=m_{1}\in(0,+\infty).$

$(N_{2})$ There exists $q>2$ such that

$\lim_{|t|\rightarrow+\infty}\frac{D_{u}F(tu)}{(tu)^{q-1}}=m_{2}\in(0,+\infty),\lim_{|t|\rightarrow+\infty}\frac{qF(tu)}{(tu)^{q}}=m_{2}\in(0,+\infty).$

###  4.1. The case when $\lambda\rightarrow 0^{-}$

We always assume that $D_{u}G=0$ in this subsection. Let $u_{\lambda}$ be the
ground state of (1.1) as $\lambda\rightarrow 0^{-}$. We consider

(4.1) $w_{\lambda}=|\lambda|^{-\frac{1}{p-2}}T(\lambda)u_{\lambda},$

which satisfies the following equation

(4.2)
$D_{w}S(w)-|\lambda|^{-\frac{p-1}{p-2}}D_{w}F(|\lambda|^{\frac{1}{p-2}}w)+D_{w}Q(w)=0\
in\ W^{\ast}.$

Let

$\widehat{\Phi}_{\lambda}(w)=S(w)-|\lambda|^{-\frac{p}{p-2}}F(|\lambda|^{\frac{1}{p-2}}w)+Q(w),$
$\widehat{\Phi}_{0}(w)=S(w)-m_{1}F_{0,p}(w)+Q(w),\ where\
D_{w}F_{0,p}(w)=w^{p-1}.$

Set

$\widehat{\mathcal{N}}_{\lambda}=|\lambda|^{-\frac{1}{p-2}}T(\lambda)\mathcal{N}_{\lambda}.$

We assume that $T(\lambda)$ is invertible. Then
$\widehat{\mathcal{N}}_{\lambda}$ is a natural constraint of
$\widehat{\Phi}_{\lambda}$. Let $\widehat{\mathcal{N}}_{0}$ be a natural
constraint of $\widehat{\Phi}_{0}$ and set

$\widehat{h}(\lambda)=\inf_{w\in\widehat{\mathcal{N}}_{\lambda}}\widehat{\Phi}_{\lambda}(w),$
$\mathcal{\widehat{K}}_{\lambda}=\\{w\in\widehat{\mathcal{N}}_{\lambda}:D_{w}\widehat{\Phi}_{\lambda}(w)=0,\widehat{\Phi}_{\lambda}(w)=\widehat{h}(\lambda)\\}.$

We have the following assumptions:

$(A_{1})$ There exists $\delta>0$ such that for any
$(\lambda,w)\in(-\delta,0]\times W\backslash\\{0\\}$, there exists a unique
function $\widehat{t}:(-\delta,0]\times
W\backslash\\{0\\}\rightarrow(0,\infty)$ such that

$when\ t>0,\ then\ (\lambda,tw)\in\mathcal{\widehat{N}}\Leftrightarrow
t=\widehat{t}(\lambda,w),$

$\widehat{\Phi}_{\lambda}(\widehat{t}(\lambda,w)w)=\max_{t>0}\widehat{\Phi}_{\lambda}(tw)$,
$\widehat{t}\in C((-\delta,0]\times W\backslash\\{0\\},(0,\infty))$, where
$\mathcal{\widehat{N}}:=\cup_{-\delta<\lambda\leq
0}\mathcal{\widehat{N}}_{\lambda}$.

$(A_{2})$ $w_{\lambda}$ is bounded in $W$ whenever
$w_{\lambda}\in\mathcal{\widehat{K}}_{\lambda}$ and $\lambda\rightarrow
0^{-}$.

$(A_{3})$ $D_{w}\widehat{\Phi}_{0}(w)=0$ admits an unique and non-degenerate
ground state $w^{\ast}$.

$(A_{4})$ $\widehat{\Phi}_{0}$ satisfies (PS)c condition in $W$ for
$c\leq\widehat{h}(0)$.

$(A_{5})$ $\liminf_{\lambda\rightarrow 0^{-}}\widehat{h}(\lambda)>0$.

######  Theorem 4.1.

Assume that $(S_{1})$, $(S_{2})$, $(N_{1})$, $(A_{1})$ \- $(A_{5})$ hold true.
Then there exists $\Lambda_{1}<0$ such that when $\lambda\in(\Lambda_{1},0)$,
the ground state of (1.1) is unique and non-degenerate.

Proof.  Step 1: $\limsup_{\lambda\rightarrow
0^{-}}\widehat{h}(\lambda)\leq\widehat{h}(0)$.

Consider
$\widehat{t}(\lambda,w^{\ast})w^{\ast}\in\mathcal{\widehat{N}}_{\lambda}$. By
$(A_{1})$,
$\widehat{t}(\lambda,w^{\ast})w^{\ast}\rightarrow\widehat{t}(0,w^{\ast})w^{\ast}=w^{\ast}$
in $W$ as $\lambda\rightarrow 0^{-}$. Together with $(N_{1})$, we know that

$\displaystyle\widehat{h}(\lambda)$ $\displaystyle\leq$
$\displaystyle\widehat{\Phi}_{\lambda}(\widehat{t}(\lambda,w^{\ast})w^{\ast})$
$\displaystyle=$
$\displaystyle\widehat{\Phi}_{0}(\widehat{t}(\lambda,w^{\ast})w^{\ast})+m_{1}F_{0,p}(\widehat{t}(\lambda,w^{\ast})w^{\ast})-|\lambda|^{-\frac{p}{p-2}}F(|\lambda|^{\frac{1}{p-2}}\widehat{t}(\lambda,w^{\ast})w^{\ast})$
$\displaystyle=$ $\displaystyle\widehat{\Phi}_{0}(w^{\ast})+o(1)$
$\displaystyle=$ $\displaystyle\widehat{h}(0)+o(1).$

Letting $\lambda\rightarrow 0^{-},$ we obtain that
$\limsup_{\lambda\rightarrow 0^{-}}\widehat{h}(\lambda)\leq\widehat{h}(0)$.

Step 2: Let $u_{\lambda_{n}}=u_{n}$ be the ground state of (1.1) as
$\lambda_{n}\rightarrow 0^{-}$ and $w_{n}=w_{\lambda_{n}}$ is given by (4.1).
Then $w_{n}\rightarrow w^{\ast}$ in $W$ up to a subsequence.

Since $\Phi_{\lambda_{n}}(u_{n})=h(\lambda_{n})$, we know that
$\widehat{\Phi}_{\lambda_{n}}(w_{n})=\widehat{h}(\lambda_{n})$. By $(A_{2})$,
$w_{n}$ is bounded in $W$. Then by $(N_{1})$,

$|\lambda_{n}|^{-\frac{p}{p-2}}F(|\lambda_{n}|^{\frac{1}{p-2}}w_{n})-m_{1}F_{0,p}(w_{n})\rightarrow
0.$

Noticing that

$\widehat{\Phi}_{0}(w_{n})=\widehat{\Phi}_{\lambda_{n}}(w_{n})+|\lambda_{n}|^{-\frac{p}{p-2}}F(|\lambda_{n}|^{\frac{1}{p-2}}w_{n})-m_{1}F_{0,p}(w_{n})=\widehat{h}(\lambda_{n})+o(1),$

we conclude that
$\limsup_{n\rightarrow\infty}\widehat{\Phi}_{0}(w_{n})\leq\widehat{h}(0)$.
This, together with $(A_{5})$, we may assume that
$\widehat{\Phi}_{0}(w_{n})\rightarrow c\in(0,\widehat{h}(0)],$ passing to a
subsequence if necessary. Then by $(N_{1})$, as $n\rightarrow\infty$,

$D_{w}\widehat{\Phi}_{0}(w_{n})=D_{w}\widehat{\Phi}_{\lambda_{n}}(w_{n})+|\lambda_{n}|^{-\frac{p-1}{p-2}}D_{w}F(|\lambda_{n}|^{\frac{1}{p-2}}w_{n})-m_{1}D_{w}F_{0,p}(w_{n})\rightarrow
0.$

By $(A_{4})$, up to a subsequence, $w_{n}\rightarrow w_{0}$ in $W$. Since
$\widehat{\Phi}_{0}\in C^{1}$, $D_{w}\widehat{\Phi}_{0}(w_{0})=0$. Noticing
that

$\widehat{\Phi}_{0}(w_{0})=\lim_{n\rightarrow+\infty}\widehat{\Phi}_{0}(w_{n})>0,$

we have that $w_{0}\neq 0$. Thus $w_{0}\in\widehat{\mathcal{N}}_{0}$.

Next, we will show that $w_{0}\in\widehat{\mathcal{K}}_{0}$.
$\forall\epsilon>0$, $\exists X>0$, $\forall n>X$, we have that:

(4.3) $\displaystyle\widehat{h}(0)$ $\displaystyle\leq$
$\displaystyle\widehat{\Phi}_{0}(w_{0})$ $\displaystyle\leq$
$\displaystyle\widehat{\Phi}_{0}(w_{n})+\frac{\epsilon}{2}$ $\displaystyle=$
$\displaystyle\widehat{\Phi}_{\lambda_{n}}(w_{n})+|\lambda_{n}|^{-\frac{p}{p-2}}F(|\lambda_{n}|^{\frac{1}{p-2}}w_{n})-m_{1}F_{0,p}(w_{n})+\frac{\epsilon}{2}$
$\displaystyle\leq$
$\displaystyle\widehat{\Phi}_{\lambda_{n}}(w_{n})+\epsilon.$

Letting $n\rightarrow\infty,$ we get
$\widehat{h}(0)\leq\limsup_{\lambda\rightarrow 0^{-}}\widehat{h}(\lambda)$.
Therefore,

$\limsup_{\lambda\rightarrow 0^{-}}\widehat{h}(\lambda)=\widehat{h}(0).$

We can derive that $\widehat{\Phi}_{0}(w_{0})=\widehat{h}(0)$ by (4.3)
immediately. Thus $w_{0}\in\widehat{\mathcal{K}}_{0}$. Finally, using
$(A_{3})$, $w_{0}=w^{\ast}$.

Step 3: There exists $\Lambda_{1}<0$ such that when
$\lambda\in(\Lambda_{1},0)$, the positive ground state of (1.1) is unique and
non-degenerate.

Since $w^{\ast}$ is non-degenerate, we can use the implicit function theorem.
Consider

$P(w,\lambda)=D_{w}\widehat{\Phi}_{\lambda}(w)=0.$

Then $P(w^{\ast},0)=0$,
$P_{w}(w^{\ast},0)=D_{ww}\widehat{\Phi}_{\lambda}(w^{\ast})$ is invertible. By
applying the implicit function theorem to the map $P$ at $(w^{\ast},0)$, we
derive that there exists $\delta>0$ small enough such that $w_{\lambda}$ is
the unique and non-degenerate solution of $P(w,\lambda)=0$ in the neighborhood
$\\{(w,\lambda)\in W\times(-\infty,0]:\|w-w^{\ast}\|+|\lambda|<\delta\\}$. In
particular, $w_{0}=w^{\ast}$. The by Step 1, for $\lambda<0$ with $|\lambda|$
small enough, $w_{\lambda}$ is the unique and non-degenerate ground state of
$D_{w}\widehat{\Phi}_{\lambda}(w)=0$. Note that the uniqueness and non-
degeneracy of $w_{\lambda}$ is equivalent to the uniqueness and non-degeneracy
of $u_{\lambda}$. Therefore, there exists $\Lambda_{1}<0$ with $|\Lambda_{1}|$
small such that when $\lambda\in(\Lambda_{1},0)$, the positive ground state of
(1.1) is unique and non-degenerate. ∎

######  Corollary 4.2.

Assume that $(S_{1})$, $(S_{2})$, $(Q_{1})$, $(N_{1})$, $(A_{1})$ \- $(A_{5})$
hold true. Let $u_{\lambda}$ be the unique ground state of (1.1). Then

$(i)$ $\lim_{\lambda\rightarrow 0^{-}}Q(u_{\lambda})=+\infty$ if
$\gamma-2/(p-2)>0$;

$(ii)$ $\lim_{\lambda\rightarrow 0^{-}}Q(u_{\lambda})=Q(w^{\ast})$ if
$\gamma-2/(p-2)=0$;

$(iii)$ $\lim_{\lambda\rightarrow 0^{-}}Q(u_{\lambda})=0$ if
$\gamma-2/(p-2)<0$.

Proof.  Using $(S_{1})$, $(Q_{1}),$ and the proof of Theorem 4.1, we have

$Q(w_{\lambda})=|\lambda|^{\gamma-\frac{2}{p-2}}Q(u_{\lambda})\rightarrow
Q(w^{\ast})\ as\ \lambda\rightarrow 0^{-}.$

Then the claims follow. ∎

######  Corollary 4.3.

Assume that $(S_{1})$, $(S_{2})$, $(Q_{1})$, $(N_{1})$, $(A_{1})$ \- $(A_{5})$
hold true. Then

$(i)$ if $\gamma-2/(p-2)>0$, there exists $c_{1}>0$ large such that for any
$c>c_{1}$, (1.1) admits a ground state $u_{\lambda}$ with $\lambda<0$ and
$|\lambda|$ small such that $Q(u_{\lambda})=c$;

$(ii)$ if $\gamma-2/(p-2)<0$, there exists $c_{2}>0$ small such that for any
$c<c_{2}$, (1.1) admits a ground state $u_{\lambda}$ with $\lambda<0$ and
$|\lambda|$ small such that $Q(u_{\lambda})=c$.

###  4.2. The case when $\lambda\rightarrow-\infty$

First, we assume that $D_{u}G=0$. Let $u_{\lambda}$ be a ground state of (1.1)
as $\lambda\rightarrow-\infty$ and $\mu=1/\lambda$. We consider

(4.4) $v_{\mu}=|\mu|^{\frac{1}{q-2}}T(\lambda)u_{\lambda},$

which satisfies the following equation

(4.5)
$D_{v}S(v)-|\mu|^{\frac{q-1}{q-2}}D_{v}F(|\mu|^{-\frac{1}{q-2}}v)+D_{v}Q(v)=0\
in\ W^{\ast}.$

Let

$\widetilde{\Phi}_{\mu}(v)=S(v)-|\mu|^{\frac{q}{q-2}}F(|\mu|^{-\frac{1}{q-2}}v)+Q(v),$
$\widetilde{\Phi}_{0}(v)=S(v)-m_{2}F_{0,q}(v)+Q(v),\ where\
D_{v}F_{0,q}(v)=v^{q-1}.$

Set

$\widetilde{\mathcal{N}}_{\mu}=|\mu|^{\frac{1}{q-2}}T(1/\mu)\mathcal{N}_{1/\mu}.$

We assume that $T_{\lambda}$ is invertible. Then
$\widetilde{\mathcal{N}}_{\mu}$ is a natural constraint of
$\widetilde{\Phi}_{\mu}$. Let $\widetilde{\mathcal{N}}_{0}$ be a natural
constraint of $\widetilde{\Phi}_{0}$ and set

$\widetilde{h}(\mu)=\inf_{v\in\widetilde{\mathcal{N}}_{\mu}}\widetilde{\Phi}_{\mu}(v),$
$\mathcal{\widetilde{K}}_{\mu}=\\{v\in\widetilde{\mathcal{N}}_{\mu}:D_{v}\widetilde{\Phi}_{\mu}(v)=0,\widetilde{\Phi}_{\mu}(v)=\widetilde{h}(\mu)\\}.$

Let us assume the following:

$(B_{1})$ There exists $\delta>0$ such that for any
$(\mu,v)\in(-\delta,0]\times W\backslash\\{0\\}$, there exists a unique
function $\widetilde{t}:(-\delta,0]\times
W\backslash\\{0\\}\rightarrow(0,\infty)$ such that

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

$\widetilde{\Phi}_{\mu}(\widetilde{t}(\mu,v)v)=\max_{t>0}\widetilde{\Phi}_{\mu}(tv)$,
$\widetilde{t}\in C((-\delta,0]\times W\backslash\\{0\\},(0,\infty))$, where
$\mathcal{\widetilde{N}}:=\cup_{-\delta<\mu\leq
0}\mathcal{\widetilde{N}}_{\mu}$.

$(B_{2})$ $v_{\mu}$ is bounded in $W$ whenever
$v_{\mu}\in\mathcal{\widetilde{K}}_{\mu}$ and $\mu\rightarrow 0^{-}$.

$(B_{3})$ $D_{v}\widetilde{\Phi}_{0}(v)=0$ admits an unique and non-degenerate
ground state $v^{\ast}$.

$(B_{4})$ $\widetilde{\Phi}_{0}$ satisfies (PS)c condition in $W$ for
$c\leq\widetilde{h}(0)$.

$(B_{5})$ $\liminf_{\mu\rightarrow 0^{-}}\widetilde{h}(\mu)>0$.

######  Theorem 4.4.

Assume that $(S_{1})$, $(S_{2})$, $(N_{2})$, $(B_{1})$ \- $(B_{5})$ hold true.
Then there exists $\Lambda_{2}<0$ such that when $\lambda<\Lambda_{2}$, the
ground state of (1.1) is unique and non-degenerate.

Proof.  Step 1: $\limsup_{\mu\rightarrow
0^{-}}\widetilde{h}(\mu)\leq\widetilde{h}(0)$.

Consider
$\widetilde{t}(\mu,v^{\ast})v^{\ast}\in\mathcal{\widetilde{N}}_{\mu}$. By
$(B_{1})$,
$\widetilde{t}(\mu,v^{\ast})v^{\ast}\rightarrow\widetilde{t}(0,v^{\ast})v^{\ast}=v^{\ast}$
in $W$ as $\mu\rightarrow 0^{-}$. The latter, together with $(N_{2})$, imply
that:

(4.6) $\displaystyle\widetilde{h}(\mu)$ $\displaystyle\leq$
$\displaystyle\widetilde{\Phi}_{\mu}(\widetilde{t}(\mu,v^{\ast})v^{\ast})$
$\displaystyle=$
$\displaystyle\widetilde{\Phi}_{0}(\widetilde{t}(\mu,v^{\ast})v^{\ast})+m_{2}F_{0,q}(\widetilde{t}(\mu,v^{\ast})v^{\ast})-|\mu|^{\frac{q}{q-2}}F(|\mu|^{-\frac{1}{q-2}}\widetilde{t}(\mu,v^{\ast})v^{\ast})$
$\displaystyle=$ $\displaystyle\widetilde{\Phi}_{0}(v^{\ast})+o(1)$
$\displaystyle=$ $\displaystyle\widetilde{h}(0)+o(1).$

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

Step 2: Let $u_{\lambda_{n}}=u_{n}$ be a ground state of (1.1) as
$\lambda_{n}\rightarrow-\infty$ and $v_{n}=v_{\mu_{n}}$ is given by (4.4).
Then $v_{n}\rightarrow v^{\ast}$ in $W$ up to a subsequence.

Since $\Phi_{\lambda_{n}}(u_{n})=h(\lambda_{n})$, we know that
$\widetilde{\Phi}_{\mu_{n}}(v_{n})=\widetilde{h}(\mu_{n})$. By $(B_{2})$,
$v_{n}$ is bounded in $W$. Then by $(N_{2})$,

$|\mu_{n}|^{\frac{q}{q-2}}F(|\mu_{n}|^{-\frac{1}{q-2}}v_{n})-m_{2}F_{0,q}(v_{n})\rightarrow
0.$

Noticing that

$\widetilde{\Phi}_{0}(v_{n})=\widetilde{\Phi}_{\mu_{n}}(v_{n})+|\mu_{n}|^{\frac{q}{q-2}}F(|\mu_{n}|^{-\frac{1}{q-2}}v_{n})-m_{2}F_{0,q}(v_{n})=\widetilde{h}(\mu_{n})+o(1),$

we derive that
$\limsup_{n\rightarrow\infty}\widetilde{\Phi}_{0}(v_{n})\leq\widetilde{h}(0)$.
This, together with $(B_{5})$, enable as to assume that
$\widetilde{\Phi}_{0}(v_{n})\rightarrow c\in(0,\widetilde{h}(0)]$ passing to a
subsequence if necessary. Then by $(N_{2})$, as $n\rightarrow\infty$,

$D_{v}\widetilde{\Phi}_{0}(v_{n})=D_{v}\widetilde{\Phi}_{\mu_{n}}(v_{n})+|\mu_{n}|^{\frac{q-1}{q-2}}D_{v}F(|\mu_{n}|^{-\frac{1}{q-2}}v_{n})-m_{2}D_{v}F_{0,q}(v_{n})\rightarrow
0.$

By $(B_{4})$, up to a subsequence, $v_{n}\rightarrow v_{0}$ in $W$. Since
$\widetilde{\Phi}_{0}\in C^{1}$, $D_{v}\widetilde{\Phi}_{0}(v_{0})=0$.
Noticing that:

$\widetilde{\Phi}_{0}(v_{0})=\lim_{n\rightarrow+\infty}\widetilde{\Phi}_{0}(v_{n})>0,$

we derive that $v_{0}\neq 0$. Thus $v_{0}\in\widetilde{\mathcal{N}}_{0}$.

Next, we will show that $v_{0}\in\widetilde{\mathcal{K}}_{0}$.
$\forall\epsilon>0$, $\exists X>0$, $\forall n>X$, there holds that

(4.7) $\displaystyle\widetilde{h}(0)$ $\displaystyle\leq$
$\displaystyle\widetilde{\Phi}_{0}(v_{0})$ $\displaystyle\leq$
$\displaystyle\widetilde{\Phi}_{0}(v_{n})+\frac{\epsilon}{2}$ $\displaystyle=$
$\displaystyle\widetilde{\Phi}_{\mu_{n}}(v_{n})+|\mu_{n}|^{\frac{q}{q-2}}F(|\mu_{n}|^{-\frac{1}{q-2}}v_{n})-m_{2}F_{0,q}(v_{n})+\frac{\epsilon}{2}$
$\displaystyle\leq$ $\displaystyle\widetilde{\Phi}_{\mu_{n}}(v_{n})+\epsilon.$

Letting $n\rightarrow\infty,$ we get that
$\widetilde{h}(0)\leq\limsup_{\mu\rightarrow 0^{-}}\widetilde{h}(\mu)$.
Therefore,

$\limsup_{\mu\rightarrow 0^{-}}\widetilde{h}(\mu)=\widetilde{h}(0).$

We can derive that $\widetilde{\Phi}_{0}(v_{0})=\widetilde{h}(0)$ by (4.7)
immediately. Thus $v_{0}\in\widetilde{\mathcal{K}}_{0}$. By $(B_{3})$,
$v_{0}=v^{\ast}$.

Step 3: There exists $\Lambda_{2}<0$ such that when $\lambda<\Lambda_{2}$, the
positive ground state of (1.1) is unique and non-degenerate.

Since $v^{\ast}$ is non-degenerate, we can use the implicit function theorem.
Consider

$K(v,\mu)=D_{v}\widetilde{\Phi}_{\mu}(v)=0.$

Then $K(v^{\ast},0)=0$,
$K_{v}(v^{\ast},0)=D_{vv}\widetilde{\Phi}_{0}(v^{\ast})$ is invertible. By
applying the implicit function theorem to the map $K$ at $(v^{\ast},0)$, we
derive that there exists $\delta>0$ small enough such that $v_{\mu}$ is the
unique and non-degenerate solution of $K(v,\mu)=0$ in the neighborhood
$\\{(v,\mu)\in W\times(-\infty,0]:\|v-v^{\ast}\|+|\mu|<\delta\\}$. In
particular, $v_{0}=v^{\ast}$. Therefore, by Step 1, for $\mu<0$ with $|\mu|$
small enough, $v_{\mu}$ is the unique and non-degenerate ground state of
$D_{v}\widetilde{\Phi}_{\mu}(v)=0$. Note that the uniqueness and non-
degeneracy of $v_{\mu}$ is equivalent to the uniqueness and non-degeneracy of
$u_{\lambda}$. Therefore, there exists $\Lambda_{2}<0$ with $|\Lambda_{2}|$
large such that when $\lambda<\Lambda_{2}$, the positive ground state of (1.1)
is unique and non-degenerate. ∎

######  Corollary 4.5.

Assume that $(S_{1})$, $(S_{2})$, $(Q_{1})$, $(N_{2})$, $(B_{1})$ \- $(B_{5})$
hold true. Let $u_{\lambda}$ be the unique ground state of (1.1). Then

$(i)$ $\lim_{\lambda\rightarrow-\infty}Q(u_{\lambda})=0$ if
$\gamma-2/(q-2)>0$;

$(ii)$ $\lim_{\lambda\rightarrow-\infty}Q(u_{\lambda})=Q(v^{\ast})$ if
$\gamma-2/(q-2)=0$;

$(iii)$ $\lim_{\lambda\rightarrow-\infty}Q(u_{\lambda})=+\infty$ if
$\gamma-2/(q-2)<0$.

Proof.  The proof is similar to the one of Corollary 4.2. ∎

######  Corollary 4.6.

Assume that $(S_{1})$, $(S_{2})$, $(Q_{1})$, $(N_{2})$, $(B_{1})$ \- $(B_{5})$
hold true. Then

$(i)$ if $\gamma-2/(q-2)>0$, there exists $c_{3}>0$ small such that for any
$c<c_{3}$, (1.1) admits a ground state $u_{\lambda}$ with $\lambda<0$ and
$|\lambda|$ large such that $Q(u_{\lambda})=c$;

$(ii)$ if $\gamma-2/(q-2)<0$, there exists $c_{4}>0$ large such that for any
$c>c_{4}$, (1.1) admits a ground state $u_{\lambda}$ with $\lambda<0$ and
$|\lambda|$ large such that $Q(u_{\lambda})=c$.

When $D_{u}G\neq 0$, we also assume $(G_{1})$, quite similarly, we can prove
the following results.

######  Theorem 4.7.

Assume that $(S_{1})$, $(S_{2})$, $(G_{1})$, $(N_{2})$, $(B_{1})$ \- $(B_{5})$
hold true. Then there exist $\Lambda_{3}<0$ such that when
$\lambda<\Lambda_{3}$, the positive ground state of (1.1) is unique and non-
degenerate.

######  Corollary 4.8.

Assume that $(S_{1})$, $(S_{2})$, $(G_{1})$, $(Q_{1})$, $(N_{2})$, $(B_{1})$
\- $(B_{5})$ hold true. Let $u_{\lambda}$ be the unique ground state of (1.1).
Then

$(i)$ $\lim_{\lambda\rightarrow-\infty}Q(u_{\lambda})=0$ if
$\gamma-2/(q-2)>0$;

$(ii)$ $\lim_{\lambda\rightarrow-\infty}Q(u_{\lambda})=Q(v^{\ast})$ if
$\gamma-2/(q-2)=0$;

$(iii)$ $\lim_{\lambda\rightarrow-\infty}Q(u_{\lambda})=+\infty$ if
$\gamma-2/(q-2)<0$.

######  Corollary 4.9.

Assume that $(S_{1})$, $(S_{2})$, $(G_{1})$, $(Q_{1})$, $(N_{2})$, $(B_{1})$
\- $(B_{5})$ hold true. Then

$(i)$ if $\gamma-2/(q-2)>0$, there exists $c_{5}>0$ small such that for any
$c<c_{5}$, (1.1) admits a ground state $u_{\lambda}$ with $\lambda<0$ and
$|\lambda|$ large such that $Q(u_{\lambda})=c$;

$(ii)$ if $\gamma-2/(q-2)<0$, there exists $c_{6}>0$ large such that for any
$c>c_{6}$, (1.1) admits a ground state $u_{\lambda}$ with $\lambda<0$ and
$|\lambda|$ large such that $Q(u_{\lambda})=c$.

##  5\. Scalar Field Equations Involving the Fractional Laplacian

In this section and the next ones; we provide some applications of our
abstract framework. In this section, we study scalar field equations involving
the fractional Laplacian. After showing the existence of normalized solutions,
we also prove the orbital stability of standing waves for almost every $L^{2}$
mass in the existence range.

###  5.1. Scalar field equations involving a fractional Laplacian in the
entire space

Consider the scalar field equations involving a fractional Laplacian of the
form

(5.1) $\left\\{\begin{array}[]{cc}(-\Delta)^{s}u=\lambda u+f(|x|,u)\ in\
\mathbb{R}^{N},0<s\leq 1,N\geq 2,\lambda<0,\\\ u\geq 0\ in\
\mathbb{R}^{N},u(x)\rightarrow 0\ as\
|x|\rightarrow+\infty.\end{array}\right.$

When $s=1$, $(-\Delta)^{1}$ is the usual Laplacian; when $0<s<1$, the
fractional Laplacian $(-\Delta)^{s}$ is defined via its multiplier
$|\xi|^{2s}$ in Fourier space. If $u$ is smooth enough, it can also be
computed by the following singular integral:

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

where $P.V.$ is the principal value, and $c_{N,s}$ is a normalization
constant.

As an application of our abstract framework 2, we will show the existence of
normalized solutions. When $s=1$ and $f(|x|,u)=|u|^{q-2}u$. Recall that the
nonlinearity is called $L^{2}$-supercritical case or the mass supercritical
case when $q>2+N/4$, $L^{2}$-critical case when $q=2+N/4$, the
$L^{2}$-subcritical case when $q<2+N/4$. When $s=1$, C.A. Stuart proved the
existence of normalized solutions for (5.1) in a series of works [56, 57, 58]
in the $L^{2}$-subcritical case. In the non-autonomous case, H. Hajaiej and C.
A. Stuart provided a detailed study including a wide range of nonlinearities.
They obtained the existence of the normalized solutions and they proved the
orbital stability of standing waves [20]. In the mass supercritical case,
there are many results showing the existence of normalized solutions when $f$
is autonomous, see, e.g., [2, 3, 25, 27]. As for non-autonomous
nonlinearities, the second author has obtained some results in a recent work
[51]. Here we extend these results to the case of $0<s<1$.

We may assume that $f(r,t)=0$ whenever $t<0$ since we aim to show the
existence of positive solutions. Furthermore, we suppose that $f(r,t)\in
C^{1}([0,+\infty)\times[0,+\infty),\mathbb{R})$:

$(f_{1})$ $f(r,0)=0$ and there exist $2<\alpha\leq\beta<2_{s}^{\ast}$ such
that

(5.2) $0<(\alpha-1)f(r,t)\leq f_{t}(r,t)t\leq(\beta-1)f(r,t),\forall t>0,r>0,$

where $2_{s}^{\ast}=2N/(N-2s)$ if $N>2s$ and $2_{s}^{\ast}=+\infty$ if $N\leq
2s$.

$(f_{2})$ $f(r,1)$ is bounded, i.e. there exists $M>0$ such that $|f(r,1)|\leq
M$, $\forall r>0$.

$(f_{3})$ $F_{r}(r,t)r\leq C(|t|^{q_{1}}+|t|^{q_{2}})$ for some $C>0$,
$q_{1}$, $q_{2}\in(2,2_{s}^{\ast})$ and there exists $\theta\leq 0$ such that

$F_{r}(r,t)r\geq\theta F(r,t),\forall t>0,r>0,$

where $F(r,t)=\int_{0}^{t}f(r,s)ds$.

$(f_{4})$ $F_{r}(r,t)r\geq-C(|t|^{q_{1}}+|t|^{q_{2}})$ for some $C>0$,
$q_{1}$, $q_{2}\in(2,2_{s}^{\ast})$ and there exists $\tau\geq 0$ such that

$F_{r}(r,t)r\leq\tau F(r,t),\forall t>0,r>0.$

In this case, $W=H_{rad}^{s}(\mathbb{R}^{N})$, the radially symmetric
fractional order Sobolev space,
equipped with the norm

$\|u\|=\sqrt{\int_{\mathbb{R}^{N}}(|\xi|^{2s}\hat{u}^{2}+\hat{u}^{2})d\xi},\hat{u}=\mathcal{F}(u),$
$\Phi_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u|^{2}-\lambda
u^{2})dx-\int_{\mathbb{R}^{N}}F(|x|,u)dx,$
$S(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx,G(u)=0,$
$F(u)=\int_{\mathbb{R}^{N}}F(|x|,u)dx,Q(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx,$
$\mathcal{N}_{\lambda}=\\{u\in
H_{rad}^{s}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u|^{2}-\lambda
u^{2})dx-\int_{\mathbb{R}^{N}}f(|x|,u)udx=0\\}.$

######  Remark 5.1.

$H_{rad}^{s}(\mathbb{R}^{N})$ can be replaced by $H^{s}(\mathbb{R}^{N})$
throughout this section if we assume additional assumptions on the integrant
ensuring that all ground states are radial and radially decreasing (Schwarz
symmetric). The reader can refer to the following works for a more detailed
account [21, 22, 23, 24].

All assumptions given in the abstract framework 2 are satisfied and our main
result in this subsection reads:

######  Theorem 5.2.

Assume $(f_{1})$ \- $(f_{4})$ with $(N-2s)\beta<2(N+\theta)$. Furthermore, we
also assume that (5.1) admits at most one positive ground state in
$H_{rad}^{s}(\mathbb{R}^{N})$ for any fixed $\lambda<0$. If

$\alpha>\frac{2(N+\tau)+4s}{N}$

or

$2+\frac{2\tau}{N}<\alpha\leq\beta<\frac{2(N+\theta)+4s}{N},$

then for any $\rho>0$, (5.1) admits at least one solution
$(\lambda,u_{\lambda})\in(-\infty,0)\times H^{s}_{rad}(\mathbb{R}^{N})$ such
that $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=\rho$.

######  Remark 5.3.

Using our approach, it seems challenging to show the uniqueness of positive
GSS for a general function $f$. In the next section, 5.2, when $f$ takes some
special form, namely $f(x,u)=h(|x|)u^{q-2},$ we will prove the uniqueness of
GSS. Our result generalizes the breakthrough papers [13, 14] by using a
totally different method.

######  Lemma 5.4.

$(i)$ Assume $(f_{1})$ and $(f_{2})$, then $(F_{1})$ holds, i.e. for any
$(\lambda,u)\in(-\infty,0)\times
H_{rad}^{s}(\mathbb{R}^{N})\backslash\\{0\\}$, there exists a unique function
$t:(-\infty,0)\times
H_{rad}^{s}(\mathbb{R}^{N})\backslash\\{0\\}\rightarrow(0,+\infty)$ such that

$when\ t>0,\ then\ (\lambda,tu)\in\mathcal{N}\Leftrightarrow t=t(\lambda,u),$

and $\Phi_{\lambda}(t(\lambda,u)u)=\max_{t>0}\Phi_{\lambda}(tu)$, $t\in
C((-\infty,0)\times
H_{rad}^{s}(\mathbb{R}^{N})\backslash\\{0\\},(0,+\infty))$, where
$\mathcal{N}:=\cup_{\lambda<0}\mathcal{N}_{\lambda}$.

Proof.  Consider

$\displaystyle\psi:(0,\infty)\times(-\infty,0)\times
H_{rad}^{s}(\mathbb{R}^{N})\backslash\\{0\\}\rightarrow\mathbb{R}$
$\displaystyle(t,\lambda,u)\mapsto\Phi_{\lambda}(tu).$

Then

$\psi_{t}(t,\lambda,u)=t\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u|^{2}-\lambda
u^{2})dx-\int_{\mathbb{R}^{N}}f(|x|,tu)udx.$

Thus $(\lambda,tu)\in\mathcal{N}$ if and only if $\psi_{t}(t,\lambda,u)=0$.
Note that

$\psi_{tt}(t,\lambda,u)=\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u|^{2}-\lambda
u^{2})dx-\int_{\mathbb{R}^{N}}f_{t}(|x|,tu)u^{2}dx.$

When $\psi_{t}(t,\lambda,u)=0$, then

$\psi_{tt}(t,\lambda,u)=\int_{\mathbb{R}^{N}}(\frac{f(|x|,tu)}{t}u-f_{t}(|x|,tu)u^{2})dx<0.$

By the fact that

$\Phi_{\lambda}(u)\geq\min\\{\frac{1}{2},-\frac{\lambda}{2}\\}\|u\|^{2}-C(\|u\|^{\alpha}+\|u\|^{\beta}),$

it is easy to verify that $\psi(t,\lambda,u)\geq\delta>0$ for suitable $t$
(since there exist $\rho>0$, $\delta>0$ such that
$\inf_{\|v\|=\rho}\Phi_{\lambda}(v)\geq\delta$). Furthermore,
$\psi(t,\lambda,u)\rightarrow-\infty$ as $t\rightarrow+\infty$ for fixed
$\lambda$ and $u$, implying the existence and uniqueness of $t(\lambda,u)$.

We see $\psi_{t}(t,\lambda,u)$ as $\psi_{t}(t,(\lambda,u))$, and

$\psi_{t(\lambda,u)}(t,(\lambda,u))=(\psi_{t\lambda}(t,\lambda,u),\psi_{tu}(t,\lambda,u))$

where

$\psi_{t\lambda}(t,\lambda,u)=-t\int_{\mathbb{R}^{N}}u^{2}dx,$
$\displaystyle\psi_{tu}(t,\lambda,u)(\varphi)$ $\displaystyle=$ $\displaystyle
2t\int_{\mathbb{R}^{N}}((-\Delta)^{\frac{s}{2}}u(-\Delta)^{\frac{s}{2}}\varphi-\lambda
u\varphi)dx$ $\displaystyle-$
$\displaystyle\int_{\mathbb{R}^{N}}(f(|x|,tu)+tf_{t}(|x|,tu)u)\varphi
dx,\forall\varphi\in H_{rad}^{s}(\mathbb{R}^{N}).$

Since $\psi_{tt}(t(\lambda,u),\lambda,u)<0$, it is an isomorphism of
$\mathbb{R}$ onto $\mathbb{R}$. By the Implicit Function Theorem, $t\in
C^{1}((-\infty,0)\times
H_{rad}^{s}(\mathbb{R}^{N})\backslash\\{0\\},(0,+\infty))$. ∎

######  Lemma 5.5.

Assume that $(f_{1})$ and $(f_{2})$ hold true, then $(F_{2})$ holds.

Proof.  By Lemma 5.4, there exists $\delta_{1}=\delta_{1}(\lambda_{0})>0$ such
that $h(\lambda)<h(\lambda_{0})+1$ for all $|\lambda-\lambda_{0}|<\delta_{1}$
and $\lambda<0$. This, together with $u_{\lambda}\in\mathcal{K}_{\lambda}$, we
have

(5.3)
$\left\\{\begin{array}[]{cc}\Phi_{\lambda}(u_{\lambda})=\int_{\mathbb{R}^{N}}(\frac{1}{2}(|(-\Delta)^{\frac{s}{2}}u_{\lambda}|^{2}-\lambda
u_{\lambda}^{2})-F(|x|,u_{\lambda}))dx<h(\lambda_{0})+1,\\\
\langle\Phi^{\prime}_{\lambda}(u_{\lambda}),u_{\lambda}\rangle=\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u_{\lambda}|^{2}-\lambda
u_{\lambda}^{2}-f(|x|,u_{\lambda})u_{\lambda})dx=0.\end{array}\right.$

Thus we have that

$\frac{1}{2}\int_{\mathbb{R}^{N}}f(|x|,u_{\lambda})u_{\lambda}dx-\int_{\mathbb{R}^{N}}F(|x|,u_{\lambda})dx<h(\lambda_{0})+1.$

Since $\alpha F(|x|,t)\leq f(|x|,t)t\leq\beta F(|x|,t)$, it follows that

$\int_{\mathbb{R}^{N}}f(|x|,u_{\lambda})u_{\lambda}dx<\frac{2\alpha}{\alpha-2}(h(\lambda_{0})+1),$

i.e.

$\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u_{\lambda}|^{2}-\lambda
u_{\lambda}^{2})dx<\frac{2\alpha}{\alpha-2}(h(\lambda_{0})+1).$

Therefore, if
$|\lambda-\lambda_{0}|<\min\\{\delta_{1},-\frac{\lambda_{0}}{2}\\}$, which
implies that

(5.4)
$\displaystyle\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u_{\lambda}|^{2}+u_{\lambda}^{2})dx$
$\displaystyle<$
$\displaystyle(1-\frac{1}{\lambda})\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u_{\lambda}|^{2}-\lambda
u_{\lambda}^{2})dx$ $\displaystyle<$
$\displaystyle\frac{2\alpha(\lambda_{0}-2)}{\lambda_{0}(\alpha-2)}(h(\lambda_{0})+1).$

Take $\delta=\min\\{\delta_{1},-\frac{\lambda_{0}}{2}\\}$. For all $\lambda$
with $|\lambda-\lambda_{0}|<\delta$, we have that:

$\|u_{\lambda}\|<\sqrt{\frac{2\alpha(\lambda_{0}-2)}{\lambda_{0}(\alpha-2)}(h(\lambda_{0})+1)}.$

Therefore, $(F_{2})$ holds and the proof is completed. ∎

######  Lemma 5.6.

(Pohozaev identity) Suppose that $(f_{1})$, $(f_{2})$, $(f_{4})$ or $(f_{1})$
\- $(f_{3})$ hold. Let $u\in H_{rad}^{s}(\mathbb{R}^{N})$ be a weak solution
of (5.1), then the following integral identity holds true:

(5.5)
$\displaystyle(N-2s)\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx$
$\displaystyle=$ $\displaystyle
N\lambda\int_{\mathbb{R}^{N}}u^{2}dx+2\int_{\mathbb{R}^{N}}[NF(|x|,u)dx+|x|F_{r}(|x|,u)]dx.$

Proof.  We know that $u\in H_{rad}^{2s+1}(\mathbb{R}^{N})$ if $u\in
H_{rad}^{s}(\mathbb{R}^{N})$ solves (5.1). In fact, [13, Lemma B.2] gives the
proof when $N=1$; for $N\geq 2$, the modifications of the proof given there
are straightforward (the proof of $u\in L^{\infty}$ can be found in [14,
AppendixB.2] and then the cases $s\geq 1/2$ and $0<s<1/2$ can be treated
separately as the proof of [13, Lemma B.2]). We integrate (5.1) after a
multiplication by $x\cdot\nabla u$. Integration by parts implies that

$\langle x\cdot\nabla u,(-\Delta)^{s}u\rangle_{L^{2}}=\frac{2s-N}{2}\langle
u,(-\Delta)^{s}u\rangle_{L^{2}},\langle x\cdot\nabla u,\lambda
u\rangle_{L^{2}}=\frac{-N\lambda}{2}\langle u,u\rangle_{L^{2}},$ $\langle
x\cdot\nabla
u,f(|x|,u)\rangle_{L^{2}}=-\int_{\mathbb{R}^{N}}[NF(|x|,u)+|x|F_{r}(|x|,u)]dx,$

where for the first identity we also use that $[\nabla\cdot
x,(-\Delta)^{s}]=-2s(-\Delta)^{s}$, which is easily verified in Fourier space.
Hence, we deduce (5.5). ∎

######  Lemma 5.7.

Assume $(f_{1})$ \- $(f_{4})$ with $(N-2s)\beta<2(N+\theta)$. If

$\alpha>\frac{2(N+\tau)+4s}{N},$

then $(F_{5})$ and $(F_{6})$ hold with $l\geq k>1$. If

$2+\frac{2\tau}{N}<\alpha\leq\beta<\frac{2(N+\theta)+4s}{N},$

then $(F_{5})$ and $(F_{6})$ hold with $0<k\leq l<1$.

Proof.  By integrating (5.1) after a multiplication by $u$, we obtain

(5.6)
$\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx=\lambda\int_{\mathbb{R}^{N}}u^{2}dx+\int_{\mathbb{R}^{N}}f(|x|,u)udx.$

$(f_{3})$, (5.5) and (5.6) imply that:

(5.7) $\displaystyle-2s\lambda\int_{\mathbb{R}^{N}}u^{2}dx$ $\displaystyle=$
$\displaystyle
2\int_{\mathbb{R}^{N}}[NF(|x|,u)+|x|F_{r}(|x|,u)]dx-(N-2s)\int_{\mathbb{R}^{N}}f(|x|,u)udx$
$\displaystyle\geq$
$\displaystyle[2(N+\theta)-(N-2s)\beta]\int_{\mathbb{R}^{N}}F(|x|,u)dx,$ (5.8)
$\displaystyle 2s\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx$
$\displaystyle=$ $\displaystyle
N\int_{\mathbb{R}^{N}}f(|x|,u)udx-2\int_{\mathbb{R}^{N}}[NF(|x|,u)+|x|F_{r}(|x|,u)]dx$
$\displaystyle\leq$
$\displaystyle[N\beta-2(N+\theta)]\int_{\mathbb{R}^{N}}F(|x|,u)dx.$

$(f_{4})$, (5.5) and (5.6) imply that:

(5.9) $\displaystyle-2s\lambda\int_{\mathbb{R}^{N}}u^{2}dx$ $\displaystyle=$
$\displaystyle
2\int_{\mathbb{R}^{N}}[NF(|x|,u)+|x|F_{r}(|x|,u)]dx-(N-2s)\int_{\mathbb{R}^{N}}f(|x|,u)udx$
$\displaystyle\leq$
$\displaystyle[2(N+\tau)-(N-2s)\alpha]\int_{\mathbb{R}^{N}}F(|x|,u)dx,$ (5.10)
$\displaystyle 2s\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx$
$\displaystyle=$ $\displaystyle
N\int_{\mathbb{R}^{N}}f(|x|,u)udx-2\int_{\mathbb{R}^{N}}[NF(|x|,u)+|x|F_{r}(|x|,u)]dx$
$\displaystyle\geq$
$\displaystyle[N\alpha-2(N+\tau)]\int_{\mathbb{R}^{N}}F(|x|,u)dx.$

Therefore, if $\beta>\frac{2(N+\theta)+4s}{N}$ and
$2(N+\theta)-\beta(N-2s)>0$, then

(5.11) $\displaystyle\Phi_{\lambda}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u|^{2}-\lambda
u^{2})dx-\int_{\mathbb{R}^{N}}F(|x|,u)dx$ $\displaystyle\leq$
$\displaystyle[\frac{N\beta-2(N+\theta)}{4s}-1]\int_{\mathbb{R}^{N}}F(|x|,u)dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}u^{2}dx$
$\displaystyle\leq$
$\displaystyle\\{[\frac{N\beta-2(N+\theta)}{4s}-1]\frac{-2s\lambda}{2(N+\theta)-\beta(N-2s)}-\frac{\lambda}{2}\\}\int_{\mathbb{R}^{N}}u^{2}dx,$

showing that:

$\Phi_{\lambda}(u)\leq l\frac{-\lambda}{2}\int_{\mathbb{R}^{N}}u^{2}dx,$

where $l=\frac{N\beta-2(N+\theta)-4s}{2(N+\theta)-\beta(N-2s)}+1>1$. Thus,
$(F_{5})$ holds with $l>1$.
Similarly, if $2<\beta<\frac{2(N+\theta)+4s}{N}$ and
$2(N+\tau)-\alpha(N-2s)>0$, then

(5.12) $\displaystyle\Phi_{\lambda}(u)$ $\displaystyle\leq$
$\displaystyle[\frac{N\beta-2(N+\theta)}{4s}-1]\int_{\mathbb{R}^{N}}F(|x|,u)dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}u^{2}dx$
$\displaystyle\leq$
$\displaystyle\\{[\frac{N\beta-2(N+\theta)}{4s}-1]\frac{-2s\lambda}{2(N+\tau)-\alpha(N-2s)}-\frac{\lambda}{2}\\}\int_{\mathbb{R}^{N}}u^{2}dx,$

showing that:

$\Phi_{\lambda}(u)\leq l\frac{-\lambda}{2}\int_{\mathbb{R}^{N}}u^{2}dx,$

where $l=\frac{N\beta-2(N+\theta)-4s}{2(N+\tau)-\alpha(N-2s)}+1\in(0,1)$.
Thus, $(F_{5})$ holds with $l\in(0,1)$.
If $\alpha>\frac{2(N+\tau)+4s}{N}$ and $2(N+\tau)-\alpha(N-2s)>0$, then

(5.13) $\displaystyle\Phi_{\lambda}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u|^{2}-\lambda
u^{2})dx-\int_{\mathbb{R}^{N}}F(|x|,u)dx$ $\displaystyle\geq$
$\displaystyle[\frac{N\alpha-2(N+\tau)}{4s}-1]\int_{\mathbb{R}^{N}}F(|x|,u)dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}u^{2}dx$
$\displaystyle\geq$
$\displaystyle\\{[\frac{N\alpha-2(N+\tau)}{4s}-1]\frac{-2s\lambda}{2(N+\tau)-\alpha(N-2s)}-\frac{\lambda}{2}\\}\int_{\mathbb{R}^{N}}u^{2}dx,$

proving that

$\Phi_{\lambda}(u)\geq k\frac{-\lambda}{2}\int_{\mathbb{R}^{N}}u^{2}dx,$

where $k=\frac{N\alpha-2(N+\tau)-4s}{2(N+\tau)-\alpha(N-2s)}+1>1$. Hence,
$(F_{6})$ holds with $k>1$.
If $2+\frac{2\tau}{N}<\alpha<\frac{2(N+\tau)+4s}{N}$ and
$2(N+\theta)-\beta(N-2s)>0$, then

(5.14) $\displaystyle\Phi_{\lambda}(u)$ $\displaystyle\geq$
$\displaystyle[\frac{N\alpha-2(N+\tau)}{4s}-1]\int_{\mathbb{R}^{N}}F(|x|,u)dx-\frac{\lambda}{2}\int_{\mathbb{R}^{N}}u^{2}dx$
$\displaystyle\geq$
$\displaystyle\\{[\frac{N\alpha-2(N+\tau)}{4s}-1]\frac{-2s\lambda}{2(N+\theta)-\beta(N-2s)}-\frac{\lambda}{2}\\}\int_{\mathbb{R}^{N}}u^{2}dx,$

which proves that:

$\Phi_{\lambda}(u)\leq k\frac{-\lambda}{2}\int_{\mathbb{R}^{N}}u^{2}dx,$

where $k=\frac{N\alpha-2(N+\tau)-4s}{2(N+\theta)-\beta(N-2s)}+1\in(0,1)$.
Thus, $(F_{6})$ holds with $k\in(0,1)$.

The proof is complete. ∎

Proof of Theorem 5.2.  Lemmas 5.4, 5.5 and 5.7 yield that $(F_{1})$,
$(F_{2})$, $(F_{5})$ and $(F_{6})$ hold. Standard methods yield that $(F_{3})$
and $(F_{4})$ holds. Since we assume that $f(r,t)\geq 0$ for all $r>0$,
$t\in\mathbb{R}$, nontrivial solutions of $(\ref{eq4.1})$ are strictly
positive by the maximum principle, see e.g. [10, Lemma 4.13 and Remark 4.14]
when $0<s<1$. We also assume that (5.1) admits one positive ground state in
$H_{rad}^{s}(\mathbb{R}^{N})$. Therefore, $\mathcal{K}_{\lambda}$ has a unique
element $u_{\lambda}$ for any $\lambda<0$, i.e.
$\mathcal{K}_{\lambda}=\\{u_{\lambda}\\}$. Thus $d\equiv\hat{d}$ in
$(-\infty,0)$. Then, apply Corollary 3.4 $(iv)$ and the proof is complete. ∎

######  Remark 5.8.

In Theorem 5.2, we have $0\leq(N-2s)\tau<N\theta+4s^{2}\leq 4$ when
$\alpha>\frac{2(N+\tau)+4s}{N}$ and $(N-2s)\beta<2(N+\theta)$ since

$2(N+\theta)>(N-2s)\beta\geq(N-2s)\alpha>(N-2s)\frac{2(N+\tau)+4s}{N};$

and $0\leq\tau<\theta+2s\leq 2s$ when
$2+\frac{2\tau}{N}<\alpha\leq\beta<\frac{2(N+\theta)+4s}{N}$.

######  Remark 5.9.

When $f_{r}(r,t)\leq 0$ for all $r>0$ and $t>0$, applying the moving plane
arguments developed in [38] directly, we can show that any positive solution
of (5.1) is radially symmetric and monotonically decreasing with respect to
$r=|x|$. Since we discuss in $H_{rad}^{s}(\mathbb{R}^{N})$, we do not need the
radially symmetric result here, while the monotonic result will be used in the
proof of non-degeneracy, see Lemma A.3 below.

######  Remark 5.10.

In the special case when

$f(|r|,u)=|u|^{q-2}u,q\in(2,2_{s}^{\ast}),$

we can show the existence of normalized solutions without the hypothesis of
uniqueness for the positive ground state solution, see Corollary 3.4 $(v)$.

In fact, when $q\neq 2+4s/N$, then for any $\rho>0$, (5.1) admits a solution
$(\lambda,u_{\lambda})\in(-\infty,0)\times H^{s}_{rad}(\mathbb{R}^{N})$ such
that $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$. Furthermore,

$\rho=C\frac{2N-(N-2s)q}{2s(q-2)}(-\lambda)^{\frac{4s-N(q-2)}{2s(q-2)}},$

which can be found in Theorem 3.2 $(iv)$. When $q=2+4s/N$,
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx\equiv Const.$ for all
$\lambda\in(-\infty,0)$ and $u_{\lambda}\in\mathcal{K}_{\lambda}$.

###  5.2. A non-autonomous case without the hypothesis of uniqueness of the
ground state solution

Here we come back to the uniqueness assumption that we made in Theorem 5.2,
where we assumed that (5.1) admits a unique positive ground state in
$H_{rad}^{s}(\mathbb{R}^{N})$ for any fixed $\lambda<0$. When $s=1$ and $f$ is
autonomous, there are many results about the uniqueness of positive solution.
When $f$ is non-autonomous, there are quite few results. When $0<s<1$ and
$f(|x|,u)=|u|^{q-2}u$, the uniqueness of positive ground state has been shown
in [13, 14]. However, to the best of our knowledge, there are no results when
$f$ is non-autonomous. Hence, in this subsection, we will apply our abstract
framework 1 to a class of non-autonomous $f$ without the uniqueness
assumption. Then as a corollary, we will show the uniqueness of positive
ground state can be proved when $0<s\leq 1$. This uniqueness result is new.
More precisely, consider

(5.15) $\left\\{\begin{array}[]{cc}(-\Delta)^{s}u=\lambda u+h(|x|)|u|^{q-2}u\
in\ \mathbb{R}^{N},\\\ u(x)\rightarrow 0\ as\
|x|\rightarrow+\infty,\end{array}\right.$

where $0<s\leq 1$, $N\geq 2$, $\lambda<0$, $2<q<2_{s}^{\ast}$, and we assume
on $h(r)$:

$(h_{1})$ $h(r)\in C^{1}[0,+\infty)\cap L^{\infty}[0,+\infty)$, $h(r)>0$ in
$[0,+\infty)$, $h(r)$ and $\frac{rh^{\prime}(r)}{h(r)}$ are non-increasing in
$(0,+\infty)$,
$\theta=\lim_{r\rightarrow+\infty}\frac{rh^{\prime}(r)}{h(r)}>-\infty$.

Our main result in this subsection is the following theorem:

######  Theorem 5.11.

Assume that $(h_{1})$ holds with $(N-2s)q<2(N+\theta)$. Then for any $\rho>0$,
(5.15) admits at least one solution $(\lambda,u_{\lambda})\in(-\infty,0)\times
H^{s}_{rad}(\mathbb{R}^{N})$ such that
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$ if $q>2+4s/N$ or
$2<q<2+(2\theta+4s)/N$.

######  Remark 5.12.

Let $h(r)=1/(1+r^{k})^{l}$, $k\geq 1$, $l>0$. Then $(h_{1})$ holds with
$\theta=-kl$.

We would like to use the abstract framework 1. Let
$E=L^{2}(\mathbb{R}^{N}),W=H^{s}(\mathbb{R}^{N}),\widehat{W}=H^{2s}(\mathbb{R}^{N}),G=O(N)$.
It is not difficult to verify $(H_{1})$ \- $(H_{4})$, $(H_{6})$, $(H_{7})$.
Moreover, a $G$-ground state is indeed a ground state here. To show $(H_{5})$,
we prove the following lemma.

######  Lemma 5.13.

Let $u_{\lambda}\in H_{rad}^{s}(\mathbb{R}^{N})$ be a positive solution of
(5.15) and $\mu_{G}(u_{\lambda})=1$. Suppose that $(h_{1})$ holds true, then
$\ker D_{uu}\Phi_{\lambda}(u_{\lambda})|_{L^{2}_{rad}}=\\{0\\}$.

Proof.  Note that
$\inf\sigma_{ess}(D_{uu}\Phi_{\lambda}(u_{\lambda}))=-\lambda>0$. Arguing by
contradiction, we assume that
$0\in\sigma_{dis}(D_{uu}\Phi_{\lambda}(u_{\lambda})|L^{2}_{rad})$. Since
$\mu_{G}(u_{\lambda})=1$, $0$ is the second eigenvalue. Assume that $v\in
H^{2s}_{rad}$ is an unit eigenfunction corresponding to $0$, then $v=v(|x|)$
changes its sign exactly once for $|x|=r\in(0,+\infty)$, which is a standard
result when $s=1$ and can be found in [14, Theorem 2] when $0<s<1$. We assume
that $v(r^{\ast})=0$, $v(r)\geq 0$ in $(0,r^{\ast})$ and $v(r)\leq 0$ in
$(r^{\ast},+\infty)$.

Direct calculation yields to

(5.16)
$D_{uu}\Phi_{\lambda}(u_{\lambda})(u_{\lambda})=(2-q)h(r)u_{\lambda}^{q-1},$
(5.17) $D_{uu}\Phi_{\lambda}(u_{\lambda})(x\cdot\nabla
u_{\lambda})=(rh^{\prime}(r)+2sh(r))u_{\lambda}^{q-1}+2s\lambda u_{\lambda}.$

Let

$\displaystyle z(r)$ $\displaystyle=$
$\displaystyle(rh^{\prime}(r)+2sh(r))u_{\lambda}^{q-1}+\lambda
u_{\lambda}+kh(r)u_{\lambda}^{q-1}$ $\displaystyle=$ $\displaystyle
h(r)u_{\lambda}^{q-1}[k+2s+\frac{rh^{\prime}(r)}{h(r)}+\frac{2s\lambda}{h(r)u_{\lambda}^{q-2}}],$

where

$k=-(2s+\frac{r^{\ast}h^{\prime}(r^{\ast})}{h(r^{\ast})}+\frac{2s\lambda}{h(r^{\ast})u_{\lambda}(r^{\ast})^{q-2}}),$

ensuring that $z(r^{\ast})=0$. By Remark 5.9, $u_{\lambda}(r)$ is
monotonically decreasing. By $(h_{1})$, $h(r)$ and $rh^{\prime}(r)/h(r)$ are
non-increasing. Thus $z(r)>0$ in $(0,r^{\ast})$ and $z(r)<0$ in
$(r^{\ast},+\infty)$, yielding $\langle z,v\rangle_{L^{2}}>0$. However, since
$z=D_{uu}\Phi_{\lambda}(u_{\lambda})(\frac{k}{2-q}u_{\lambda}+x\cdot\nabla
u_{\lambda})$, we obtain

(5.18) $\displaystyle\langle z,v\rangle_{L^{2}}$ $\displaystyle=$
$\displaystyle\langle
D_{uu}\Phi_{\lambda}(u_{\lambda})(\frac{k}{2-q}u_{\lambda}+x\cdot\nabla
u_{\lambda}),v\rangle_{L^{2}}$ $\displaystyle=$
$\displaystyle\langle\frac{k}{2-q}u_{\lambda}+x\cdot\nabla
u_{\lambda},D_{uu}\Phi_{\lambda}(u_{\lambda})(v)\rangle_{L^{2}}=0,$

which is a contradiction, and this completes the proof. ∎

We assume the positivity of $u_{\lambda}$ in Lemma 5.13. Thus, to apply our
abstract framework 1, we also need the following lemma.

######  Lemma 5.14.

There exists

$\lambda\mapsto u_{\lambda}\in
C^{1}([\lambda^{\ast},\lambda_{\ast}],W\setminus\\{0\\})$

for some $\lambda^{\ast}<\lambda_{\ast}<0$ where $u_{\lambda}$ is a solution
of (5.15).

$(i)$ If $u_{\lambda_{\ast}}>0$, then $u_{\lambda}>0$ for $\lambda$ close
enough to $\lambda_{\ast}$.

$(ii)$ Let $\\{\lambda_{n}\\}\subset(\lambda^{\ast},\lambda_{\ast})$ be a
sequence such that $\lambda_{n}\rightarrow\lambda^{\ast}$, and
$u_{\lambda_{n}}>0$. Then $u_{\lambda^{\ast}}>0$.

Proof. Along the lines of Lemma A.6 $(i)$, we can prove $(i)$, $(ii)$. ∎

Proof of Theorem 5.11. Thanks to Lemmas 5.13 and 5.14, Theorem 2.2 shows that
there exists

$\lambda\mapsto u_{\lambda}\in
C^{1}((-\infty,0),H^{s}_{rad}(\mathbb{R}^{N})\setminus\\{0\\}),$

such that $u_{\lambda}$ solves (5.15), and $\mu_{G}(u_{\lambda})=1$.

Then by Remark 3.5, following the proof of Theorem 5.2, we can use abstract
framework 2 to obtain the asymptotical behaviors of
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx$ as $\lambda\rightarrow-\infty$ and
$\lambda\rightarrow 0^{-}$ to complete the proof. ∎

Finally in this subsection, we show a uniqueness result. The key step is to
show a local uniqueness result when $\lambda\rightarrow-\infty$.

######  Lemma 5.15.

Assume $(h_{1})$ holds with $(N-2s)q<2(N+\theta)$. Then there exists
$\Lambda_{1}<0$ such that the positive solution of $(\ref{eq4.14})$ with
$G$-Morse index $1$ is unique in $H_{rad}^{s}(\mathbb{R}^{N})$ for all fixed
$\lambda<\Lambda_{1}$.

Proof. Assume that for some $\lambda_{0}<0$, there is a positive solution
$u_{\lambda_{0}}$ of $(\ref{eq4.14})$ with $\mu_{G}(u_{\lambda_{0}})=1$.
Similar to the proof of Theorem 5.11, we know that there exists

$\lambda\mapsto u_{\lambda}\in
C^{1}((-\infty,\lambda_{0}],H^{s}_{rad}(\mathbb{R}^{N})\setminus\\{0\\}),$

such that $u_{\lambda}$ solves (5.15), and $\mu_{G}(u_{\lambda})=1$. Using
ideas in Section 4.2, let $\mu=1/\lambda$ and consider

(5.19) $v_{\mu}=|\mu|^{\frac{1}{p-2}}u_{\lambda}(|\mu|^{\frac{1}{2s}}x),$

which satisfies the following equation

(5.20)
$(-\Delta)^{s}v_{\mu}+v_{\mu}=h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q-2}v_{\mu}.$

Moreover, there exists

$\mu\mapsto v_{\mu}\in
C^{1}([\frac{1}{\lambda_{0}},0),H^{s}_{rad}(\mathbb{R}^{N})\setminus\\{0\\}),$

where $\mu_{G}(v_{\mu})=1$. Then along the lines of the proof for Lemma A.5,
we have the following power estimates

$\int_{\mathbb{R}^{N}}h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q}dx\thicksim\int_{\mathbb{R}^{N}}|v_{\mu}|^{2}dx\thicksim\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}v_{\mu}|^{2}dx\gtrsim
1,$

Next, we show that
$\int_{\mathbb{R}^{N}}h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q}dx\lesssim 1$.
Differentiating (5.20) with respect to $\mu$, we get

(5.21)
$L_{\mu}\partial_{\mu}u_{\mu}=\frac{1}{2s\mu}h^{\prime}(|\mu|^{\frac{1}{2s}}|x|)|\mu|^{\frac{1}{2s}}|x||v_{\mu}|^{q-2}v_{\mu},$

showing that

$\partial_{\mu}v_{\mu}=L_{\mu}^{-1}(\frac{1}{2s\mu}h^{\prime}(|\mu|^{\frac{1}{2s}}|x|)|\mu|^{\frac{1}{2s}}|x||v_{\mu}|^{q-2}v_{\mu}),$

where

$L_{\mu}=(-\Delta)^{s}+1-(q-1)h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q-2}.$

Since $L_{\mu}v_{\mu}=(2-q)h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q-2}v_{\mu}$,

(5.22) $\displaystyle\langle
h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q-2}v_{\mu},\partial_{\mu}v_{\mu}\rangle_{L^{2}}$
$\displaystyle=$ $\displaystyle\langle
L_{\mu}^{-1}(h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q-2}v_{\mu}),\frac{1}{2s\mu}h^{\prime}(|\mu|^{\frac{1}{2s}}|x|)|\mu|^{\frac{1}{2s}}|x||v_{\mu}|^{q-2}v_{\mu}\rangle_{L^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{(2-q)2s\mu}\int_{\mathbb{R}^{N}}h^{\prime}(|\mu|^{\frac{1}{2s}}|x|)|\mu|^{\frac{1}{2s}}|x||v_{\mu}|^{q}dx.$

We write
$\phi(\mu)=\int_{\mathbb{R}^{N}}h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q}dx$.
Then

(5.23) $\displaystyle\phi^{\prime}(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{2s\mu}\int_{\mathbb{R}^{N}}h^{\prime}(|\mu|^{\frac{1}{2s}}|x|)|\mu|^{\frac{1}{2s}}|x||v_{\mu}|^{q}dx$
$\displaystyle+q\langle
h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q-2}v_{\mu},\partial_{\mu}v_{\mu}\rangle_{L^{2}}$
$\displaystyle=$
$\displaystyle-\frac{\theta}{(q-2)s\mu}\int_{\mathbb{R}^{N}}h^{\prime}(|\mu|^{\frac{1}{2s}}|x|)|\mu|^{\frac{1}{2s}}|x||v_{\mu}|^{q}dx$
$\displaystyle\leq$ $\displaystyle 0.$

Thus $\phi(\mu)\leq\phi(1/\lambda_{0})$ for all $\mu\in[1/\lambda_{0},0)$. Now
we show that

$\int_{\mathbb{R}^{N}}h(|\mu|^{\frac{1}{2s}}|x|)|v_{\mu}|^{q}dx\thicksim\int_{\mathbb{R}^{N}}|v_{\mu}|^{2}dx\thicksim\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}v_{\mu}|^{2}dx\thicksim
1,$

implying that $v_{\mu}$ is uniformly bounded in $H^{s}(\mathbb{R}^{N})$. Then
along the lines of the proof for Theorem 4.4, we can complete the proof. ∎

######  Theorem 5.16.

Assume $(h_{1})$ holds with $(N-2s)q<2(N+\theta)$. Then the positive ground
state solution of $(\ref{eq4.14})$ is unique in $H^{s}(\mathbb{R}^{N})$ for
any fixed $\lambda<0$.

Proof. By Remark 5.9, any positive solution of (5.15) is radially symmetric if
$(h_{1})$ holds. Then the proof can be completed by the combination of
Corollary 2.6 and Lemma 5.15. ∎

######  Remark 5.17.

It seems that the results in Theorem 5.11 can be extended. We further assume
that $h(r)=r^{\theta},r>R$ for some $R>0$. Similar to the proof of Lemma 5.15,
our abstract framework 3 is helpful to study the asymptotical behavior of
$u_{\lambda}$ as $\lambda\rightarrow 0^{-}$. Then we believe that the
following results hold:

$(i)$ if $q>2+4s/N$ or $2<q<2+(2\theta+4s)/N$, for any $\rho>0$, (5.15) admits
at least one solution $(\lambda,u_{\lambda})\in(-\infty,0)\times
H^{s}_{rad}(\mathbb{R}^{N})$ such that
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$;

$(ii)$ if $q=2+4s/N$, there exists $\rho_{1}>0$ such that
\- for any $\rho>\rho_{1}$, (5.15) admits at least one solution
$(\lambda,u_{\lambda})\in(-\infty,0)\times H^{s}_{rad}(\mathbb{R}^{N})$ such
that $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$,
\- for any $\rho<\rho_{1}$, (5.15) admits no ground state solution
$u_{\lambda}$ such that $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$;

$(iii)$ if $2+(2\theta+4s)/N<q<2+4s/N$, there exists $\rho_{2}>0$ such that
\- (5.15) admits at least one solution
$(\lambda,u_{\lambda})\in(-\infty,0)\times H^{s}_{rad}(\mathbb{R}^{N})$ such
that $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho_{2}$,
\- for any $\rho>\rho_{2}$, (5.15) admits at least two solutions
$(\lambda,u_{\lambda}),(\widehat{\lambda},u_{\widehat{\lambda}})\in(-\infty,0)\times
H^{s}_{rad}(\mathbb{R}^{N})$ such that
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=\int_{\mathbb{R}^{N}}u_{\widehat{\lambda}}^{2}dx=2\rho$,
\- for any $\rho<\rho_{2}$, (5.15) admits no ground state solution
$u_{\lambda}$ such that $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$;

$(iv)$ if $q=2+(2\theta+4s)/N$, there exists $\rho_{3}>0$ such that
\- for any $\rho>\rho_{3}$, (5.15) admits at least one solution
$(\lambda,u_{\lambda})\in(-\infty,0)\times H^{s}_{rad}(\mathbb{R}^{N})$ such
that $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$,
\- for any $\rho<\rho_{3}$, (5.15) admits no ground state solution
$u_{\lambda}$ such that $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$.

Since this is a purely computational aspect, we are not going to prove in this
quite long paper.

######  Remark 5.18.

In Appendix A, we will provide another method to prove Theorem 5.16. The
uniqueness problem of fractional nonlinear Schrödinger equations is a long
standing question. The breakthrough work in [13, 14], gave an affirmative
answer for pure nonlinearity $|u|^{q-2}u$. The discussions in Appendix A are
inspired by [13] and make important extensions.

###  5.3. The orbital stability/instability

If $(\lambda,u_{\lambda})$ is a solution of $(\ref{eq4.1})$, then
$e^{-i\lambda t}u_{\lambda}$ is a standing wave solution of the fractional
nonlinear Schrödinger equation (fNLS)

(5.24)
$\left\\{\begin{array}[]{cc}i\partial_{t}\psi-(-\Delta)^{s}\psi+\tilde{f}(x,\psi)=0,(t,x)\in[0,+\infty)\times\mathbb{R}^{N},\\\
\psi(0,x)=\psi_{0}(x),\end{array}\right.$

where $\tilde{f}:\mathbb{R}^{N}\times\mathbb{C}\rightarrow\mathbb{C}$ is
defined by $\tilde{f}(x,e^{i\theta}u)=e^{i\theta}f(x,u)$, $u\in\mathbb{R}$. In
this subsection, we study the orbital stability or instability of the ground
state solutions for (5.1) which are shown in Theorem 5.2. Recall that such
solutions are called orbitally stable if for each $\epsilon>0$ there exists
$\delta>0$ such that, whenever $\psi_{0}\in
H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$ is such that
$\|\psi_{0}-u_{\lambda}\|_{H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})}<\delta$ and
$\psi(t,x)$ is the solution of (5.24) with $\psi(0,\cdot)=\psi_{0}$ in some
interval $[0,t_{0})$, then $\psi(t,\cdot)$ can be continued to a solution in
$0\leq t<\infty$ and

$\sup_{0<t<\infty}\inf_{w\in\mathbb{R}}\|\psi(t,x)-e^{-i\lambda
w}u_{\lambda}(x)\|_{H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})}<\epsilon;$

otherwise, they are called unstable. To do this, we assume that (5.24) is
locally well-posed in $H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$:

$(LWP)$ For each $\psi_{0}\in H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$, there
exist $t_{0}>0$, only depending on
$\|\psi_{0}\|_{H^{s}(\mathbb{R}^{N},\mathbb{C})}$, and a unique solution
$\psi(t,x)$ of (5.24) with initial datum $\psi_{0}$ in the interval
$I=[0,t_{0})$. A detailed account on this aspect is given in [25].

Our main result in this subsection is as follows.

######  Theorem 5.19.

Under the hypotheses of Theorem 5.2 and $(LWP)$ holds and any positive ground
state of (5.1) is non-degenerate in $H_{rad}^{s}(\mathbb{R}^{N})$ for all
$\lambda\in(-\infty,0)$. $(\lambda,u_{\lambda})$ is given by Theorem 5.2. Then
for a.e. $\rho\in(0,\infty)$, $e^{-i\lambda t}u_{\lambda}$ is orbitally
unstable in $H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$ when
$\alpha>2+(2\tau+4s)/N$ while $e^{-i\lambda t}u_{\lambda}$ is orbitally stable
in $H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$ when
$2+2\tau/N<\alpha\leq\beta<2+(2\theta+4s)/N$ where
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$.

######  Lemma 5.20.

Assume that $(f_{1})$, $(f_{2})$ hold, any positive ground state of (5.1) is
non-degenerate in $H_{rad}^{s}(\mathbb{R}^{N})$ and $\mathcal{K}_{\lambda}$
has only one element $u_{\lambda}$ for any $\lambda<0$, then $\mathcal{K}$ is
a $C^{1}$ curve in $\mathbb{R}\times H_{rad}^{s}(\mathbb{R}^{N})$ where
$\mathcal{K}=\\{(\lambda,u):u\in\mathcal{K}_{\lambda}\\}$.

Proof.  With the help of the result shown in the proof of Step 2 of Theorem
3.1, it is not difficult to verify that $u_{\lambda}\rightarrow
u_{\lambda_{0}}$ in $H^{s}(\mathbb{R}^{N})$ as
$\lambda\rightarrow\lambda_{0}$. In other words, $\mathcal{K}$ is a continuous
curve in $\mathbb{R}\times H_{rad}^{s}(\mathbb{R}^{N})$. Then similar to [59,
Theorem 18], we can prove that $\mathcal{K}$ is a $C^{1}$ curve in
$\mathbb{R}\times H_{rad}^{s}(\mathbb{R}^{N})$ by the implicit function
theorem since $u_{\lambda}$ is non-degenerate in
$H_{rad}^{s}(\mathbb{R}^{N})$. ∎

To study the orbitally stability, we lean on the following result, which
expresses in our context the abstract theory developed in [17]:

######  Proposition 5.21.

Under the hypotheses of Lemma 5.20 and assume that $(LWP)$ holds. Then if
$d^{\prime}(\lambda)<0$ (respectively $>0$), the standing wave $e^{-i\lambda
t}u_{\lambda}(x)$ is orbitally stable (respectively unstable), where
$d(\lambda)=\frac{1}{2}\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx$ here.

Proof of Theorem 5.19:  Case 1: $\alpha>\frac{2(N+\tau)+4s}{N}$.

In this case, Theorem 3.2 yields that
$\lim_{\lambda\rightarrow-\infty}d(\lambda)=0$ and $\lim_{\lambda\rightarrow
0^{-}}d(\lambda)=+\infty$. Thus $d^{-1}(\rho)\subset[e_{1},e_{2}]$ for any
fixed $\rho\in(0,+\infty)$, where $-\infty<e_{1}<e_{2}<0$ and $e_{1}$, $e_{2}$
depend on $\rho$. We assume that $\rho$ is a regular value. Then
$d^{-1}(\rho)$ consist of finite points, i.e.
$d^{-1}(\rho)=\\{\lambda_{\rho}^{(1)},\cdots,\lambda_{\rho}^{(k)}\\}$ and
$d^{\prime}(\lambda_{\rho}^{(i)})\neq 0$, $i=1,2,\cdots,k$. Without loss of
generality, assume $\lambda_{\rho}^{(1)}<\cdots<\lambda_{\rho}^{(k)}$. Then
$d(\lambda)<d(\lambda_{\rho}^{(1)})$ for any $\lambda<\lambda_{\rho}^{(1)}$ by
the continuity of $d(\lambda)$, implying $d^{\prime}(\lambda_{\rho}^{(1)})\geq
0$. Considering $d^{\prime}(\lambda_{\rho}^{(1)})\neq 0$, we conclude that
$d^{\prime}(\lambda_{\rho}^{(1)})>0$. According to Proposition 5.21, the
standing wave $e^{-i\lambda_{\rho}^{(1)}t}u_{\lambda_{\rho}^{(1)}}(x)$ is
orbitally unstable. By Sard’s theorem, we know such $\rho$ is almost
everywhere in $(-\infty,0)$.

Case 2: $2+\frac{2\tau}{N}<\alpha\leq\beta<\frac{2(N+\theta)+4s}{N}$.

In this case, Theorem 3.2 yields that
$\lim_{\lambda\rightarrow-\infty}d(\lambda)=+\infty$ and
$\lim_{\lambda\rightarrow 0^{-}}d(\lambda)=0$. Then we can complete the proof
after quiet similar arguments to case 1. ∎

######  Remark 5.22.

When $f(|x|,u)=|u|^{q-2}u$, $2<q<2+4s/N$ or $2+4s/N<q<2_{s}^{\ast}$, we can
derive the stability or instability for all $\rho\in(-\infty,0)$ rather than
almost everywhere $\rho\in(-\infty,0)$ in Theorem 5.19 above.

Considering the case when $f(|x|,u)=h(|x|)|u|^{q-2}u$, $2<q<2_{s}^{\ast}$, we
have the following corollary:

######  Corollary 5.23.

Under the hypotheses of Theorem 5.11, and $(LWP)$, $(\lambda,u_{\lambda})$ is
given by Theorem 5.11. Then for a.e. $\rho\in(0,\infty)$, $e^{-i\lambda
t}u_{\lambda}$ is orbitally unstable in
$H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$ when $q>2+4s/N$ while $e^{-i\lambda
t}u_{\lambda}$ is orbitally stable in $H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$
when $2<q<2+(2\theta+4s)/N$ where
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$.

###  5.4. Fractional nonlinear Schrödinger equations with a potential I

In this subsection, we consider the fractional nonlinear Schrödinger equation
with a potential

(5.25) $\left\\{\begin{array}[]{cc}(-\Delta)^{s}u+V(|x|)u=\lambda
u+h(|x|)|u|^{q-2}u\ in\ \mathbb{R}^{N},\\\ u(x)\rightarrow 0\ as\
|x|\rightarrow+\infty,\end{array}\right.$

where $0<s\leq 1$, $N\geq 2$, $2<q<2_{s}^{\ast}$, and $h(r)$ satisfies
$(h_{1})$ introduced in Subsection 4.2. Recall that

$(h_{1})$ $h(r)\in C^{1}[0,+\infty)\cap L^{\infty}[0,+\infty)$, $h(r)>0$ in
$[0,+\infty)$, $h(r)$ and $\frac{rh^{\prime}(r)}{h(r)}$ are non-increasing in
$(0,+\infty)$,
$\theta=\lim_{r\rightarrow+\infty}\frac{rh^{\prime}(r)}{h(r)}>-\infty$.
For linear potential $V$, we assume

$(V_{1})$ $V\in C^{1}([0,+\infty))\cap L^{\infty}$, $V(0)=0$, $V(r^{\ast})>0$
for some $r^{\ast}>0$, $\gamma=\sup_{r>0}\frac{rV^{\prime}(r)}{V}<+\infty$,
$V$ and $V+rV^{\prime}(r)$ are non-decreasing on $[0,+\infty)$.

When $(V_{1})$ holds, $(-\Delta)^{s}+V$ acting on
$L^{2}_{rad}(\mathbb{R}^{N})$ is a self-adjoint operator with domain
$H^{2s}_{rad}$ and form domain $H^{s}_{rad}$. Let
$\lambda_{1}=\inf\sigma((-\Delta)^{s}+V)$. By min-max principle,
$0\leq\lambda_{1}\leq\lim_{r\rightarrow+\infty}V(r)$. Furthermore,
$\lambda_{1}>0$. Since
$\sigma_{ess}((-\Delta)^{s}+V)=[\lim_{r\rightarrow+\infty}V(r),+\infty)$, $0$
is an eigenvalue if $\lambda_{1}=0$. Thus there exists $\psi$ with
$\|\psi\|_{L_{2}}=1$ such that
$\int_{\mathbb{R}^{N}}[|(-\Delta)^{\frac{s}{2}}\psi|^{2}+V\psi^{2}]dx=0$,
implying $\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}\psi|^{2}dx=0$. We can
derive $\psi\equiv 0$, contradicting $\|\psi\|_{L_{2}}=1$.

Similar to Lemma 5.13, we have:

######  Lemma 5.24.

Let $u\in H_{rad}^{s}$ be a positive solution of $(\ref{eq4.36})$ and
$\mu_{G}(u)=1$. Suppose that $(h_{1})$ and $(V_{1})$ hold true, then $\ker
D_{uu}\Phi_{\lambda}(u_{\lambda})|_{L^{2}_{rad}}=\\{0\\}$.

Proof.  Let $L_{+,V}=(-\Delta)^{s}+V-\lambda-h(|x|)|u|^{q-2}$. Direct
calculation imply that:

(5.26) $L_{+,V}(u)=(2-q)h(r)u^{q-1},$ (5.27) $L_{+,V}(x\cdot\nabla
u)=(rh^{\prime}(r)+2sh(r))u^{q-1}-2s(V(r)-\lambda+rV^{\prime}(r))u.$

Similarly to the proof of Lemma 5.13, we just need to show that
$-\frac{V(r)-\lambda+rV^{\prime}(r)}{h(r)u(r)^{q-2}}$ is non-increasing in
$(0,+\infty)$. When $\lambda\leq 0$, by $(V_{1})$,
$V(r)-\lambda+rV^{\prime}(r)\geq 0$ is non-decreasing. Since $h(r)u(r)^{q-2}$
is non-increasing, we derive that
$-\frac{V(r)-\lambda+rV^{\prime}(r)}{h(r)u(r)^{q-2}}$ is non-increasing. ∎

######  Theorem 5.25.

Assume $(h_{1})$ and $(V_{1})$ hold with $(N-2s)q<2(N+\theta)$. Then any
positive ground state of $(\ref{eq4.36})$ is unique in $H^{s}$ for any fixed
$\lambda\leq 0$.

Proof.  The proof is quiet similar to the one of Theorem 5.16, using abstract
framework 1. Hence we omit the details. We further provide an alternative
proof in Appendix B. ∎

Next we use the abstract framework 2 to study the existence of normalized
solutions. In this case, $W=H_{rad}^{s}(\mathbb{R}^{N})$,

$\Phi_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}[|(-\Delta)^{\frac{s}{2}}u|^{2}+(V-\lambda)u^{2}]dx-\frac{1}{q}\int_{\mathbb{R}^{N}}h(|x|)|u|^{q}dx,$
$S(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx,G(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}Vu^{2}dx,$
$F(u)=\frac{1}{q}\int_{\mathbb{R}^{N}}h(|x|)|u|^{q}dx,Q(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx.$

Similar to Lemmas 5.4 and 5.5, $(F_{1})$ and $(F_{2})$ hold when $(h_{1})$ and
$(V_{1})$ hold. The following lemma shows $(F_{5})$ holds in mass subcritical
case while $(F_{6})$ holds in mass supercritical case.

After a similar discussion to the proof of Lemma 5.6, we can derive the
Pohozaev identity for (5.25).

######  Lemma 5.26.

(Pohozaev identity) Suppose $(h_{1})$ and $(V_{1})$ hold. Let $u\in
H_{rad}^{s}$ be a weak solution of (5.25), then we have the following integral
identity

$\displaystyle(N-2s)\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx+N\int_{\mathbb{R}^{N}}Vu^{2}dx+\int_{\mathbb{R}^{N}}|x|V^{\prime}(|x|)u^{2}dx$
(5.28)
$\displaystyle=N\lambda\int_{\mathbb{R}^{N}}u^{2}dx+\int_{\mathbb{R}^{N}}\frac{2Nh(|x|)+|x|h^{\prime}(|x|)}{q}|u|^{q}dx.$

######  Lemma 5.27.

Assume that $(h_{1})$ and $(V_{1})$ hold with $(N-2s)\beta<2(N+\theta)$. If
$q>\frac{2N+4s}{N}$, then $(F_{4})$ holds with $k>1$. If
$q<\frac{2(N+\theta)+4s}{N}$, then $(F_{3})$ holds with $0<l<1$ for
$\lambda\leq\lambda^{\ast}<-\frac{(2s+\gamma)\|V\|_{L^{\infty}}}{2s}\frac{2s(q-2)-2\theta}{2(N+\theta)+4s-qN}$.

Proof.  By Lemma 5.26 and

(5.29)
$\int_{\mathbb{R}^{N}}(|(-\Delta)^{\frac{s}{2}}u|^{2}+Vu^{2})dx=\lambda\int_{\mathbb{R}^{N}}u^{2}dx+\int_{\mathbb{R}^{N}}h(|x|)|u|^{q})dx.$

we have

(5.30) $\displaystyle\Phi_{\lambda}(u)$ $\displaystyle\geq$
$\displaystyle-\frac{\lambda}{2}[1+\frac{qN-2N-4s}{2N-q(N-2s)}]\int_{\mathbb{R}^{N}}u^{2}dx$
$\displaystyle+$
$\displaystyle\frac{1}{4s}[1+\frac{qN-2N-4s}{2N-q(N-2s)}]\int_{\mathbb{R}^{N}}(2sV+rV^{\prime})u^{2}dx$
$\displaystyle\geq$ $\displaystyle-k\lambda Q(u),$

where $k=1+\frac{qN-2N-4s}{2N-q(N-2s)}>1$, if $q>\frac{2N+4s}{N}$; and

(5.31) $\displaystyle\Phi_{\lambda}(u)$ $\displaystyle\leq$
$\displaystyle-\frac{\lambda}{2}[1+\frac{qN-2(N+\theta)-4s}{2N-q(N-2s)}]\int_{\mathbb{R}^{N}}u^{2}dx$
$\displaystyle+$
$\displaystyle\frac{1}{4s}[1+\frac{qN-2(N+\theta)-4s}{2N-q(N-2s)}]\int_{\mathbb{R}^{N}}(2sV+rV^{\prime})u^{2}dx$
$\displaystyle\leq$
$\displaystyle-\lambda[1+\frac{qN-2(N+\theta)-4s}{2N-q(N-2s)}](1-\frac{(2s+\gamma)\|V\|_{L^{\infty}}}{2s\lambda})Q(u)$
$\displaystyle\leq$ $\displaystyle-l\lambda Q(u),$

where
$l=[1+\frac{qN-2(N+\theta)-4s}{2N-q(N-2s)}](1-\frac{(2s+\gamma)\|V\|_{L^{\infty}}}{2s\lambda^{\ast}})\in(0,1)$,
if $q<\frac{2(N+\theta)+4s}{N}$ and
$\lambda\leq\lambda^{\ast}<-\frac{(2s+\gamma)\|V\|_{L^{\infty}}}{2s}\frac{2s(q-2)-2\theta}{2(N+\theta)+4s-qN}$.
∎

Theorem 5.25 shows that any positive ground state of (5.25) is unique in
$H^{s}_{rad}$ for any fixed $\lambda\leq 0$. Then applying Corollary 3.4 $(i)$
and $(ii)$, we have

######  Theorem 5.28.

Assume $(h_{1})$ and $(V_{1})$ hold with $(N-2s)q<2(N+\theta)$. Then there
exists $\rho_{1}>0$ such that for any $\rho\geq\rho_{1}$, (5.25) admits at
least one solution $(\lambda,u_{\lambda})\in(-\infty,0]\times H_{rad}^{s}$
such that $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$ if
$2<q<\frac{2(N+\theta)+4s}{N}$; there exists $\rho_{2}>0$ such that for any
$0<\rho\leq\rho_{2}$, (5.25) admits at least one solution
$(\lambda,u_{\lambda})\in(-\infty,0]\times H_{rad}^{s}$ such that
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$ if $q>\frac{2N+4s}{N}$.

######  Remark 5.29.

It is worth noting that results in Theorems 5.25, 5.28 can be extended to the
case $\lim_{r\rightarrow+\infty}V(r)=+\infty$. The method in Appendix B is
invalid since we can not prove that $L_{+,\eta}\rightarrow L_{+,\tilde{\eta}}$
(the definition will be found in Appendix B) in the norm-resolvent sense as
$\eta\rightarrow\tilde{\eta}$. However, the abstract framework 1 also works,
and we can show the uniqueness of the ground state solution of (5.25), and
then use the abstract framework 3 to get the existence of normalized
solutions. For unbounded potentials, the local uniqueness when
$\lambda\rightarrow-\infty$ needs more discussions ( see some ideas used in
[54]).

Similar to the discussions in Subsection 4.3, if $(\lambda,u_{\lambda})$ is a
solution of $(\ref{eq4.36})$, then $e^{-i\lambda t}u_{\lambda}$ is a standing
wave solution of the following fractional nonlinear Schrödinger equation

(5.32)
$\left\\{\begin{array}[]{cc}i\partial_{t}\psi-(-\Delta)^{s}\psi-V(x)\psi+h(|x|)|\psi|^{q-2}\psi=0,(t,x)\in[0,+\infty)\times\mathbb{R}^{N},\\\
\psi(0,x)=\psi_{0}(x),\end{array}\right.$

where $0<s\leq 1$, $N\geq 2$, $2<q<2_{s}^{\ast}$.

Similar arguments to the proof of Theorem 5.19 yield

######  Theorem 5.30.

Assume $(h_{1})$, $(V_{1})$ and $(LWP)$ hold with $(N-2s)q<2(N+\theta)$. When
$2<q<\frac{2(N+\theta)+4s}{N}$, for a.e. $\rho\geq\rho_{1}$, $e^{-i\lambda
t}u_{\lambda}$ is orbitally stable in $H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$
where $\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$. When
$q>\frac{2N+4s}{N}$, for a.e. $\rho\in(0,\rho_{2})$, $e^{-i\lambda
t}u_{\lambda}$ is orbitally unstable in
$H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$ where
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$. $\rho_{1}$, $\rho_{2}$ and
$(\lambda,u_{\lambda})$ are given by Theorem 5.28.

######  Remark 5.31.

We can get more information if the non-degeneracy of positive ground state
hold when $0<\lambda<\lambda_{1}$, when $\lambda_{1}$ is the first eigenvalue
of $(-\Delta)^{s}+V$. Indeed, we may derive that when
$2<q<\frac{2(N+\theta)+4s}{N}$, (5.25) admits at least one solution
$(\lambda,u_{\lambda})\in(-\infty,\lambda_{1})\times H_{rad}^{s}$ such that
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$ for any $\rho>0$, and for a.e.
$\rho>0$, $e^{-i\lambda t}u_{\lambda}$ is orbitally stable in
$H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$. When $q>\frac{2N+4s}{N}$, there
exists $\rho_{3}>0$ such that
\- (5.25) admits at least two solutions $(\lambda,u_{\lambda})$,
$(\tilde{\lambda},u_{\tilde{\lambda}})\in(-\infty,\lambda_{1})\times
H_{rad}^{s}$ such that
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=\int_{\mathbb{R}^{N}}u_{\tilde{\lambda}}^{2}dx=2\rho$
and $\lambda<\tilde{\lambda}$ for any $0<\rho<\rho_{3}$, and for a.e.
$0<\rho\leq\rho_{3}$, $e^{-i\lambda t}u_{\lambda}$ is orbitally unstable while
$e^{-i\tilde{\lambda}t}u_{\tilde{\lambda}}$ is orbitally stable in
$H_{rad}^{s}(\mathbb{R}^{N},\mathbb{C})$;
\- (5.25) admits at least one solution
$(\lambda,u_{\lambda})\in(-\infty,\lambda_{1})\times H_{rad}^{s}$ such that
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho_{3}$;
\- (5.25) admits no ground state
$(\lambda,u_{\lambda})\in(-\infty,\lambda_{1})\times H_{rad}^{s}$ such that
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$ for any $\rho>\rho_{3}$.

######  Remark 5.32.

Let $V(r)=1-1/(1+r^{k})^{l}$, $k\geq 1$, $kl\geq 1$. Then $(V_{1})$ holds.

###  5.5. Nonlinear Schrödinger equations with a potential II

$2sV+rV^{\prime}(r)$ plays an important role when verifying $(F_{5})$ and
$(F_{6})$, see Lemma 5.27. When $V=r^{-2s}$, $2sV+rV^{\prime}(r)=0$. For
simplicity, we just study the case when $s=1$ in this subsection and similar
results can be extended to the case when $0<s<1$. Consider

(5.33) $\left\\{\begin{array}[]{cc}-\Delta u+\frac{1}{|x|^{2}}u=\lambda
u+|u|^{q-2}u\ in\ \mathbb{R}^{N}\backslash\\{0\\},\\\ u(x)\rightarrow 0\ as\
|x|\rightarrow+\infty,\end{array}\right.$

where $N\geq 3$, $2<q<\frac{2N}{N-2}$. [48, Theorem X.28] yields that
$-\Delta+\frac{1}{|x|^{2}}$ is essential self-adjoint on
$C_{0}^{\infty}(\mathbb{R}^{N})$. Furthermore, the form domain is

$H^{1}_{V}:=\\{u\in
H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\frac{1}{|x|^{2}}u^{2}dx<+\infty\\},$

with inner product

$\langle u,v\rangle_{H^{1}_{V}}=\int_{\mathbb{R}^{N}}[\nabla u\nabla
v+(\frac{1}{|x|^{2}}+1)uv]dx$

and norm

$\|u\|_{H^{1}_{V}}=\sqrt{\int_{\mathbb{R}^{N}}[|\nabla
u|^{2}+(\frac{1}{|x|^{2}}+1)u^{2}]dx}.$

Apply the abstract framework 2 where

$W=H^{1}_{V,rad}=\\{u\in H^{1}_{V}:u(x)=u(|x|)\\},$
$\Phi_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}[|\nabla
u|^{2}+(\frac{1}{|x|^{2}}-\lambda)u^{2}]dx-\frac{1}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx,$
$S(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx,G(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}\frac{1}{|x|^{2}}u^{2}dx,$
$F(u)=\frac{1}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx,Q(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx.$

We don’t know whether the positive ground state of (5.33) is unique or not.
However, we can show the existence of normalized solution. Similar to Lemmas
5.4 and 5.5, $(F_{1})$ and $(F_{2})$ hold. Moreover, we will see that both
$(F_{5})$ and $(F_{6})$ hold with $k=l$. To show this, we introduce the
following Pohozaev identity:

######  Lemma 5.33.

([34, Proposition 2.3]) Let $u\in H^{1}_{V}$ be a weak solution of (5.33),
then there holds the following integral identity

(5.34) $(N-2)\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+\frac{1}{|x|^{2}}u^{2})dx=N\lambda\int_{\mathbb{R}^{N}}u^{2}dx+\frac{2N}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx.$

######  Lemma 5.34.

Let $u\in H^{1}_{V}$ be a weak solution of (5.33), then
$\Phi_{\lambda}(u)=(1+\frac{(q-2)N-4}{2N-(N-2)q})Q(u)$, i.e. $(F_{5})$ and
$(F_{6})$ hold with $k=l=1+\frac{(q-2)N-4}{2N-(N-2)q}$.

Proof.  By integrating (5.33) with respect to $u$, we obtain

(5.35) $\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+\frac{1}{|x|^{2}}u^{2})dx=\lambda\int_{\mathbb{R}^{N}}u^{2}dx+\int_{\mathbb{R}^{N}}|u|^{q}dx.$

By (5.34) and (5.35), we get
$\Phi_{\lambda}(u)=(1+\frac{(q-2)N-4}{2N-(N-2)q})Q(u)$. ∎

######  Theorem 5.35.

$(i)$ If $q\neq\frac{2N+4}{N}$, then for any $\rho>0$, (5.33) a solution
$(\lambda,u_{\lambda})\in(-\infty,0)\times H_{V,rad}^{1}$ with
$\frac{1}{2}\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=\rho$ and
$\rho=\frac{C}{k}(-\lambda)^{\frac{1}{k}-1}$ for some $C>0$,
$k=1+\frac{(q-2)N-4}{2N-(N-2)q}$.

$(ii)$ If $q=\frac{2N+4}{N}$, then there exists some $\rho^{\ast}>0$ such that
for any fixed $\lambda<0$, (5.33) admits a solution $u_{\lambda}\in
H_{V,rad}^{1}$ with
$\frac{1}{2}\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=\rho^{\ast}$ (implying that
(5.33) admits infinite solutions $(\lambda,u_{\lambda})\in(-\infty,0)\times
H_{V,rad}^{1}$ with
$\frac{1}{2}\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=\rho^{\ast}$ when $\lambda$
changes) and (5.33) admits no solution $u_{\lambda}\in H_{V,rad}^{1}$ with
$\frac{1}{2}\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=\rho$ for
$\rho\neq\rho^{\ast}$.

Proof.  $(i)$ and $(ii)$ in Theorem 5.35 above are direct corollary of Theorem
3.2 $(iv)$ and Corollary 3.4 $(v)$. ∎

######  Remark 5.36.

By Theorem 3.2 $(iv)$, we also know that all the ground states of (5.33) in
$H_{V,rad}^{1}$ have the same $L^{2}$ norm. For this reason, we conjecture
that the positive ground state of (5.33) is unique in $H_{V,rad}^{1}$ for a
fixed $\lambda<0$.

Similar to the discussions in Subsection 4.3, if $(\lambda,u_{\lambda})$ is a
solution of $(\ref{eq4.55})$, then $e^{-i\lambda t}u_{\lambda}$ is a standing
wave solution of the following nonlinear Schrödinger equation

(5.36)
$\left\\{\begin{array}[]{cc}i\partial_{t}\psi+\Delta\psi-\frac{1}{|x|^{2}}\psi+|\psi|^{q-2}\psi=0,(t,x)\in[0,+\infty)\times\mathbb{R}^{N}\backslash\\{0\\},\\\
\psi(0,x)=\psi_{0}(x),\end{array}\right.$

where $N\geq 3$, $2<q<\frac{2N}{N-2}$.

Since we don’t know the non-degeneracy of ground state for (5.33), we assume
it in the following theorem. In this case, we can get the orbital stability or
instability for all $\rho>0$ rather than almost every $\rho>0$.

######  Theorem 5.37.

Assume $(LWP)$ holds and that any ground state of (5.33) is non-degenerate in
$H_{V,rad}^{1}$. When $2<q<\frac{2N+4}{N}$, for any $\rho>0$, $e^{-i\lambda
t}u_{\lambda}$ is orbitally stable in
$H_{V,rad}^{1}(\mathbb{R}^{N},\mathbb{C})$ where
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$. When
$\frac{2N+4}{N}<q<\frac{2N}{N-2}$, for any $\rho>0$, $e^{-i\lambda
t}u_{\lambda}$ is orbitally unstable in
$H_{V,rad}^{1}(\mathbb{R}^{N},\mathbb{C})$ where
$\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx=2\rho$. $(\lambda,u_{\lambda})$ is the
solution given by Theorem 5.35 and

$H_{V,rad}^{1}(\mathbb{R}^{N},\mathbb{C}):=\\{u\in
H^{1}(\mathbb{R}^{N},\mathbb{C}):u(x)=u(|x|),\int_{\mathbb{R}^{N}}\frac{1}{|x|^{2}}u\bar{u}dx<+\infty\\}.$

Proof.  Let $(\lambda,u_{\lambda})$ be the solution given in Theorem 5.35.
Then $Q(u_{\lambda})=\frac{C}{k}(-\lambda)^{\frac{1}{k}-1}$. Thus
$\partial_{\lambda}Q(u_{\lambda})=-(\frac{1}{k}-1)\frac{C}{k}(-\lambda)^{\frac{1}{k}-2}$.
When $2<q<\frac{2N+4}{N}$, $0<k<1$, implying
$\partial_{\lambda}Q(u_{\lambda})<0$. When $\frac{2N+4}{N}<q<\frac{2N}{N-2}$,
$k>1$, implying $\frac{\partial}{\partial\lambda}Q(u_{\lambda})>0$. Then the
results in Theorem 5.37 follow Proposition 5.21. ∎

##  6\. Fractional Laplacian with mixed nonlinearities

###  6.1. The existence of normalized solutions

Consider

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

where $N\geq 2$, $2<p<2+4s/N<q<2_{s}^{\ast}$. Apply the abstract framework 3
where

$W=H^{s}_{rad}(\mathbb{R}^{N})=\\{u\in H^{s}(\mathbb{R}^{N}):u(x)=u(|x|)\\},$
$\Phi_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}[|(-\Delta)^{\frac{s}{2}}u|^{2}-\lambda
u^{2}]dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx-\frac{1}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx,$
$S(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx,G(u)=0,$
$F(u)=\frac{1}{p}\int_{\mathbb{R}^{N}}|u|^{p}dx+\frac{1}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx,Q(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx.$

Note that $(S_{1})$, $(S_{2})$, $(Q_{1})$, $(N_{1})$ and $(N_{1})$ hold with
$\gamma=N/2s$ and

$T(\lambda)u(x)=u(\frac{x}{|\lambda|^{1/2s}}).$

When $\lambda\rightarrow 0^{-}$,

(6.2) $\displaystyle\widehat{\Phi}_{\lambda}(w)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}w|^{2}dx$
$\displaystyle+$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|w|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|w|^{p}dx-\frac{1}{q}|\lambda|^{\frac{q-p}{p-2}}\int_{\mathbb{R}^{N}}|w|^{q}dx,$
(6.3)
$\displaystyle\widehat{\Phi}_{0}(w)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}w|^{2}dx+\frac{1}{2}\int_{\mathbb{R}^{N}}|w|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|w|^{p}dx.$

We have

######  Lemma 6.1.

$(A_{1})$ \- $(A_{5})$ hold.

Proof.  $(A_{1})$ and $(A_{2})$ can be derived in the same way as in the
proofs of Lemmas 5.4 and 5.5. $(A_{3})$ is standard when $s=1$ and has been
shown in [14, Theorems 3, 4] when $0<s<1$. $(A_{4})$ can be proved by a
standard process, c.f. [65, Lemma 1.20]. Finally, we prove $(A_{5})$. For
$\lambda\in(-1,0)$ and $r>0$ small enough, we have

$\inf_{\|w\|=r}\widehat{\Phi}_{\lambda}(w)\geq\frac{1}{2}r^{2}-C(r^{p}+r^{q})>0.$

Therefore,

$\liminf_{\lambda\rightarrow
0^{-}}\widehat{h}(\lambda)\geq\liminf_{\lambda\rightarrow
0^{-}}\inf_{\|w\|=r}\widehat{\Phi}_{\lambda}(w)>0.$

∎

When $\mu=1/\lambda\rightarrow 0^{-}$,

(6.4) $\displaystyle\widetilde{\Phi}_{\mu}(v)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}v|^{2}dx$
$\displaystyle+$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}|v|^{2}dx-\frac{1}{p}|\mu|^{\frac{q-p}{q-2}}\int_{\mathbb{R}^{N}}|v|^{p}dx-\frac{1}{q}\int_{\mathbb{R}^{N}}|v|^{q}dx,$
(6.5)
$\displaystyle\widetilde{\Phi}_{0}(v)=\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}v|^{2}dx+\frac{1}{2}\int_{\mathbb{R}^{N}}|v|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{N}}|v|^{q}dx.$

We have

######  Lemma 6.2.

$(B_{1})$ \- $(B_{5})$ hold.