# On orbital stability of the physical ground states of the NLS equations

Yavdat Ilyasov  Institute of Mathematics, Ufa Federal Research Centre, RAS,
Chernyshevsky str. 112, 450008 Ufa, Russia Universidade Federal de Goiás,
Instituto de Matemática
74690-900, Goiânia - GO - Brazil<EMAIL_ADDRESS>
###### Abstract.

We apply the nonlinear generalized Rayleigh quotients method to develop new
tools, which can be deployed in studying the ground states of the nonlinear
Schrödinger (NLS) equations with combined power-type nonlinearities. We
introduce a variational functional, the global minimizer of which corresponds
to the so-called fundamental frequency solutions with prescribed action value.
These solutions, in turn, allow us to find a subset of the physical ground
states, i.e., the ground states of the NLS equation that are preserved under
small perturbations of parameters problem. By this approach, we prove the
orbital stability of the set of physical ground states of the NLS equations in
a generalized sense.

###### Key words and phrases:

Schrödinger equations, ground state, orbital stability, Rayleigh quotient,
prescribed action solution

###### 2020 Mathematics Subject Classification:

Primary 35Q55, 35A15; Secondary 35B35 35A01, 35J61

##  1\. Introduction

We consider the nonlinear Schrödinger (NLS) equation with combined power-type
nonlinearities

(1.1)
$i\psi_{t}=\Delta\psi+\mu|\psi|^{p-2}\psi-|\psi|^{q-2}\psi,~{}~{}(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{N},$

where $\psi$ is a complex-valued function of $(t;x)$,
$p,q\in(2,2^{*}:=2N/(N-2))$, $N\geq 3$ and $\mu\in\mathbb{R}$. The NLS
equation with combined power-type nonlinearities has attracted much attention
in the last decades. For physical background of this type of equations, we
refer the reader to [4, 25, 32, 35] and the references therein. According to
[10, 32], the Cauchy problem for (1.1) with the initial value $\psi_{0}\in
H^{1}:=H^{1}(\mathbb{R}^{N})$ is locally well posed and has the unique local
solution $\psi\in C([0,T(\psi_{0})),H^{1})\cap C^{1}([0,T(\psi_{0})),H^{-1})$
for some $T(\psi_{0})>0$, which satisfies the  energy conservation law:

$E\equiv
H_{\mu}(\psi(t)):=\int\left(\frac{1}{2}|\nabla\psi|^{2}-\frac{\mu}{p}|\psi|^{p}+\frac{1}{q}|\psi|^{q}\right)dx,$

and the mass (charge, particle number) conservation law:

$\alpha\equiv Q(\psi(t)):=\frac{1}{2}\int|\psi|^{2}dx.$

In the present paper, we study the existence and stability of the standing
waves $\psi_{\lambda}=e^{i\lambda t}u$ of (1.1), where the amplitude function
$u$ satisfies

(1.2) $-\Delta u-\lambda
u-\mu|u|^{p-2}u+|u|^{q-2}u=0,~{}~{}x\in\mathbb{R}^{N}.$

Here $\lambda\in\mathbb{R}$ is the frequency of the standing wave. Notice that
the action functional associated with the problem

$S_{\lambda,\mu}(u):=H_{\mu}(u)-\lambda Q(u),$

is also a conserved quantity. For a given $\lambda\in\mathbb{R}$, a solution
$\bar{u}$ of (1.2) is said to be ground state if $S_{\lambda,\mu}(\bar{u})\leq
S_{\lambda,\mu}(w)$, for any $w\in H^{1}\setminus 0$ such that
$DS_{\lambda,\mu}(w)=0$. It is important to note that the set of ground states
$G_{\lambda}(\mu)$ of (1.2) may contain nonphysical solutions of (1.2), for
example, solutions that disappear under a small perturbation of the parameter
$\lambda$. In the present paper, we will focus on a subset of ground states
that are preserved under a small perturbation of parameters of the problem. We
call such a subset the set of physical ground states of (1.2).

We also deal with the so-called prescribed action solution of (1.2), i.e., a
function $u^{S}\in H^{1}$ which for a given action $S\in\mathbb{R}^{+}$
satisfies

(1.3)
$S_{\lambda,\mu}(u^{S})=S~{}~{}\mbox{and}~{}~{}DS_{\lambda,\mu}(u^{S})=0,$

with some $\lambda\leq 0$. Note that the standard approach for problem (1.2)
considers solutions with prescribed frequency $\lambda$, and unknowns action
$S$ and mass $\alpha$ (see, e.g, [6, 10]). An alternative formulation which
has also been actively investigated over the last decades consists of finding
the solution $u$ to (1.2) having  prescribed mass $\alpha=Q(u)$, while
$\lambda$ and $S$ are unknown (see, e.g., [5, 11, 18, 29, 28] ). Note that the
frequency $\lambda$ can be also considered as a value of the following
conserved quantity

(1.4) $\lambda=\Lambda^{S}_{\mu}(\psi(t)):=\frac{H_{\mu}(\psi)-S}{Q(\psi)}.$

We call the frequency $\lambda$, action $S$ and mass $\alpha$ the main
parameters of problem (1.1). Thus, from the mathematical point of view, all of
these three approaches, namely, with prescribed frequency, with prescribed
action, and with prescribed mass are equally acceptable. Moreover, all of
these approaches evidently are relevant from the physical point of view. In
particular, the approach with prescribed action arises in the study of inverse
problems and the so-called spectral and scattering control problems (see,
e.g., [13, 17, 34] and the references therein).

We shall pay a special attention to the standing wave $\psi_{\lambda}$ of
(1.1) with $\lambda=0$ which we call the zero frequency solution. The
corresponding equation (1.2) with $\lambda=0$ for the amplitude function $u$
we shall call the zero frequency problem. In the literature, such an equation
is sometimes called the ”zero mass” case problem (cf. [6, 9]). Besides the
fact that this type of problems is interesting from a purely mathematical
point of view (see, e.g. [3, 6, 9, 14, 12, 31]), they often arise in physical
problems, such as in the Euclidean Yang-Mills theory, equations of the
filtration through porous media, the study of solutions with compact supports
of the reaction-diffusion systems, plasma physics, among others (see, e.g.,
[2, 15, 23, 27] and the references therein).

The orbital stability of the ground states of many equations can be often
investigated using the Lyapunov function, determined by the action functional
$S_{\lambda,\mu}(u)$ restricted to the manifold of functions $u$ with fixed
mass integral $\alpha=Q(u)$. The general theorem on orbital stability of
solutions of nonlinear problems based on this idea was proved in the famous
work by T. Cazenave & P.L. Lions [11] in 1983. They showed that if the
following prescribed mass minimization problem

(1.5) $\hat{H}^{\alpha}_{\mu}:=\min\\{H_{\mu}(u):Q(u)=\alpha,~{}u\in
H^{1}\setminus 0\\},~{}\alpha>0$

has a solution, and all of its minimizing sequences are relatively compact,
then the set

$\mathcal{M}_{\mu}(\alpha):=\\{u\in
H^{1}:~{}\hat{H}^{\alpha}_{\mu}:=H_{\mu}(u),~{}Q(u)=\alpha\\}$

is orbital stable in the sense that for any $\varepsilon>0$ there exists
$\delta>0$ such that for any solution (1.1) with initial data $u(0)$ such that
$\inf_{\phi\in\mathcal{M}_{\mu}(\alpha)}\|u(0)-\phi\|_{H^{1}}<\delta$ there
holds

$\inf_{\phi\in\mathcal{M}_{\mu}(\alpha)}\|u(t)-\phi\|_{H^{1}}<\varepsilon,~{}~{}\forall
t\geq 0.$

Nevertheless, it should be noted that this result does not always entail the
orbital stability of the set of ground states of (1.2). Indeed, for any
$u_{\alpha}\in\mathcal{M}_{\mu}(\alpha)$ by the Lagrange multipliers rule
there exists a constant $\nu(u_{\alpha})\in\mathbb{R}$ such that $u_{\alpha}$
solves (1.2) with $\lambda=\nu(u_{\alpha})$. Thus to obtain the orbital
stability of the set of ground states we need to know:

1) Whether the values of the Lagrange multipliers $\nu(u_{\alpha})$ for all
functions $u_{\alpha}$ from the set of solutions of (1.5) are identical, i.e.,
$\nu(u_{\alpha})\equiv\lambda_{\alpha}$, $\forall
u_{\alpha}\in\mathcal{M}_{\mu}(\alpha)$ for some
$\lambda_{\alpha}\in\mathbb{R}$?

2) Whether the set of solutions $\mathcal{M}_{\mu}(\alpha)$ of problem (1.5)
coincides with the set of ground states $G_{\lambda_{\alpha}}$ of (1.2)?

Affirmative answers to these questions can be obtained only for some special
cases of the NLS equation. In particular, this is true for the NLS equation
with a monomial nonlinearity, i.e., for
$i\psi_{t}=\Delta\psi+|\psi|^{q-2}\psi$, where the Lagrange multiplier may be
eliminated by scale changes (see, e.g, [21]). This is also true in the case of
dimension $N=1$, which is characterized by the uniqueness of positive solution
(see, e.g., [19, 22, 24]).

The purpose of this article is to provide some answers to these questions. In
this respect, we present a new type of variational functional associated with
problem (1.1), which makes it possible to find the ground state $\bar{u}$ of
the problem, as well as allows to uniquely determine the corresponding values
of the mass $\alpha$, the frequency $\lambda$ and the action level
$S_{\lambda,\mu}$ of this state.

Our approach is based on the application of the nonlinear generalized Rayleigh
quotient method [16] (the NG-Rayleigh quotient method for short) to (1.2) and
using $\Lambda^{S}_{\mu}(u)$ as the Rayleigh quotient. A distinctive feature
of the NG-Rayleigh method is that it allows to find the critical values of the
problems’ parameters, and at the same time to convert the original variational
functionals into functionals with simpler geometry (see [8, 16]). Notice that
$u^{S}\in H^{1}\setminus 0$ is a prescribed action solution of (1.2) with
action $S>0$, namely it satisfies to (1.3), if and only if $u^{S}$ is a
critical point of $\Lambda^{S}_{\mu}(u)$ with a critical value $\lambda$,
i.e.,

$D\Lambda^{S}_{\mu}(u^{S})=0~{}~{}\mbox{and}~{}~{}\lambda=\Lambda^{S}_{\mu}(u^{S}).$

For a given $S>0$, we call a solution $\hat{u}$ of (1.2) the fundamental
frequency solution (respectively, $e^{i\hat{\lambda}t}\hat{u}$ is said to be
fundamental frequency standing wave of (1.1)) with a fundamental frequency
$\hat{\lambda}^{S}_{\mu}$ if

$\hat{\lambda}^{S}_{\mu}=\Lambda^{S}_{\mu}(\hat{u})\leq\Lambda^{S}_{\mu}(w)$

for any $w\in H^{1}\setminus 0$ such that $D\Lambda^{S}_{\mu}(w)=0$. For
$S>0$, we denote by

(1.6) $G^{S}(\mu):=\\{u\in H^{1}\setminus
0:\Lambda_{\mu}^{S}(u)=\hat{\lambda}_{\mu}^{S},~{}D\Lambda^{S}_{\mu}(u^{S})=0\\}$

the set of fundamental frequency solutions of (1.2) with the fundamental
frequency $\hat{\lambda}_{\mu}^{S}$. Below we show that the existence of the
fundamental frequency solution entails the existence of the ground state and
that the converse is also true (see below Lemma 5.7).

According to the NG-Rayleigh quotient method [16], the functional
$\Lambda^{S}_{\mu}(u)$ corresponds to the following  NG-Rayleigh quotient:

(1.7) $\lambda^{S}_{\mu}(u):=\frac{c_{N}^{S}(\int|\nabla
u|^{2})^{\frac{N}{(N-2)}}-\mu\frac{2}{p}\int|u|^{p}+\frac{2}{q}\int|u|^{q}}{\int|u|^{2}},\,\,u\in
H^{1}\setminus\\{0\\},~{}~{}S>0,$

where $c_{N}^{S}=(N-2)/(N^{\frac{N}{(N-2)}}S^{\frac{2}{(N-2)}})$, for
$p,q\in(2,2^{*})$ and $\mu>0$. Below we will see that any critical point of
$\lambda^{S}_{\mu}(u)$ in $H^{1}\setminus 0$ corresponds, possibly after some
scaling, to a critical point of $\Lambda^{S}_{\mu}(u)$, and thus it gives a
solution of equation (1.2) with prescribed action $S>0$. Moreover, the NG-
Rayleigh quotient $\lambda^{S}_{\mu}(u)$ is characterized by properties
similar to those that has the usual Rayleigh quotient of linear theory [26].
In particular, similar to the spectral theory the following critical value

(1.8) $\displaystyle\hat{\lambda}^{S}_{\mu}:=\min_{u\in H^{1}\setminus
0}\lambda^{S}_{\mu}(u)$

plays a principal role in the investigation of problem (1.2). In the case
$2<q<p<2^{*}$, using the NG-Rayleigh quotient method [16] as well, we
introduce the following principal critical value

(1.9) $\hat{\mu}^{S}=\inf_{u\in\mathcal{D}\setminus 0}\mu^{S}(u).$

Here, $\mathcal{D}:=\mathcal{D}^{1,2}(\mathbb{R}^{N})\cap
L^{q}(\mathbb{R}^{N})$,

(1.10)
$\mu^{S}(u):=\left(\frac{c(p,q,N)}{S^{\frac{2(p-q)}{(2^{*}-q)(N-2)}}}\right)\frac{(\int|u|^{q})^{\frac{2^{*}-p}{2^{*}-q}}(\int|\nabla
u|^{2})^{\frac{2^{*}(p-q)}{2(2^{*}-q)}}}{\int|u|^{p}},~{}~{}u\in H^{1},$

where the constant $c(p,q,N)$ does not depends on $S$ (see below (2.4)).

The NG-Rayleigh quotient $\mu^{S}(u)$ is characterized by the fact that its
critical points correspond to the zero frequency solutions of (1.2). Below we
show that $\hat{\mu}^{S}>0$.

Our first result addresses the question of the existence of ground state and
fundamental frequency solution of (1.2) in nonzero frequency case $\lambda<0$.

######  Theorem 1.1.

Let $S>0$.

$(1^{o})$ If $2<p<q<2^{*}$, then for any $\mu>0$, (1.2) possesses a
fundamental frequency solution $\hat{u}^{S}_{\mu}$ with prescribe action $S$
and frequency
$\hat{\lambda}^{S}_{\mu}=\Lambda^{S}_{\mu}(\hat{u}^{S}_{\mu})<0$.

$(2^{o})$ If $2<q<p<2^{*}$, then for any $\mu>\hat{\mu}^{S}$, (1.2) possesses
a fundamental frequency solution $\hat{u}^{S}_{\mu}$ with prescribe action $S$
and frequency
$\hat{\lambda}^{S}_{\mu}=\Lambda^{S}_{\mu}(\hat{u}^{S}_{\mu})<0$.

$(3^{o})$ If $2<q<p<2^{*}$ and $0\leq\mu<\hat{\mu}^{S}$, then (1.2) has no
weak solutions in $H^{1}$ with any action such that $\tilde{S}\leq S$ and
$\lambda<0$.

Furthermore, $\hat{u}^{S}_{\mu}$ in $(1^{o})$ and $(2^{o})$ is a ground state
of (1.2) and a global minimum point of $\lambda^{S}_{\mu}(u)$ in $H^{1}$,
$\hat{u}^{S}_{\mu}>0$ in $\mathbb{R}^{N}$, $\hat{u}_{\mu}^{S}\in
C^{2}(\mathbb{R}^{N})$.

The existence of the spherically symmetric, decreases with respect to $r:=|x|$
ground state of (1.2) under the assumptions of assertions $(1^{o})$, $(2^{o})$
of Theorem 1.1 follows from the works by H. Berestycki & P.-L. Lions [6] and
W. Strauss [30]. The main novelty of the results in Theorem 1.1 consists of
that the ground state $\hat{u}^{S}_{\mu}$ is obtained as a global minimum of
$\lambda^{S}_{\mu}(u)$ in $H^{1}$, which helps in further investigation of
(1.2).

######  Remark 1.2.

Kato’s Theorem [20] implies that (1.2) can not has a weak, spherically
symmetric, decreases with respect to $r:=|x|$ positive solutions if
$\lambda>0$, see [6].

######  Remark 1.3.

For another type of a threshold value $\hat{\mu}^{*}(\lambda)>0$, which
depends on $\lambda<0$, and divides the intervals for parameter $\mu$ where
equation (1.2) may or may not have solutions, see [6, 30].

For the existence and non existence results for the zero frequency solution of
(1.2) we have the following

######  Theorem 1.4.

Let $S>0$.

$(1^{o})$ If $2<q<p<2^{*}$, then for $\mu=\hat{\mu}^{S}$, the zero frequency
problem (1.2) possesses a fundamental frequency solution
$\hat{u}^{S}_{\hat{\mu}^{S}}\in\mathcal{D}$ with prescribe action $S$.
Furthermore, $\hat{u}^{S}_{\hat{\mu}^{S}}$ is a ground state of (1.2) with
$\lambda=0$ and a global minimum point of $\mu^{S}(u)$ in $\mathcal{D}$;
$\hat{u}^{S}_{\hat{\mu}^{S}}>0$ in $\mathbb{R}^{N}$,
$\hat{u}^{S}_{\hat{\mu}^{S}}\in C^{2}(\mathbb{R}^{N})$.

$(2^{o})$ If $2<p<q<2^{*}$, then the zero frequency problem (1.2) has no weak
solutions in $\mathcal{D}$ for any $\mu>0$.

The assumption in assertion $(1^{o})$ of Theorem 1.4 corresponds to the
sufficient conditions introduced in [6] for the existence of spherically
symmetric ground states from $\mathcal{D}$ of the ”zero-mass” case problem.
However, it appears that the result on the absence of solutions of problems
with zero frequency, as in $(2^{o})$ of Theorem 1.4, has not been known
before. In this article, it is shown that this result has a fairly simple
proof. It should be emphasized that this simplicity is achieved owing to the
application of the NG-Rayleigh quotient method.

######  Remark 1.5.

In the case of $\lambda=0$ in (1.2), the dependence of the problem on $\mu$
can be neglected, since the change of variables
$u=(1/\mu)^{1/(p-q)}v(x/\mu^{(q-2)/2(p-q)})$ transforms (1.2) into $-\Delta
u-|u|^{p-2}u+|u|^{q-2}u=0,~{}~{}x\in\mathbb{R}^{N}$. However for NLS equation
(1.1) the dependence on the parameter $\mu$ can not be neglected.

######  Definition 1.6.

We call $u^{S}\in{G}^{S}(\mu)$ for $S>0$ the physical ground state of (1.2)
with respect to the action value if there exists a sequence $u^{S_{m}}\in
G^{S_{m}}(\mu)$, $m=1,\ldots,$ such that $\lim_{m\to+\infty}S_{m}=S$ and
$u^{S_{m}}\to u^{S}$ in $H^{1}$ as $m\to+\infty$. The set of physical ground
states of (1.2) with respect to the action value we denote by
$\hat{G}^{S}(\mu)$.

It is easily seen that any ground state $v_{\lambda}$ of (1.2) from the
residual set $\hat{G}^{S,c}(\mu):=G^{S}(\mu)\setminus\hat{G}^{S}(\mu)$
disappears after a small perturbation of the parameter $S$. It can also makes
sense to refer to the physical ground states $\hat{G}^{S}(\mu)$ as the ground
states of the branches of solutions of (1.2).

######  Theorem 1.7.

Let $S>0$.

$(1^{o})$ If $2<p<q<2^{*}$, then for any $\mu>0$, there exists a physical
ground state of (1.2), i.e., $\hat{G}^{S}(\mu)\neq\emptyset$.

$(2^{o})$ If $2<q<p<2^{*}$, then for any $\mu>\hat{\mu}^{S}$, there exists a
physical ground state of (1.2), i.e., $\hat{G}^{S}(\mu)\neq\emptyset$.

Furthermore, there exists $\alpha^{S}$ such that $\alpha^{S}=Q(u)$, $\forall
u\in\hat{G}^{S}(\mu)$, for $\mu>0$ and $\mu>\hat{\mu}^{S}$, respectively.

Let us recall some results obtained by M. Shibata in [28]. Consider

$\alpha_{0}(\mu):=\inf\\{\alpha>0:\hat{H}^{\alpha}_{\mu}<0\\}.$

We need the following Shibata Theorem (see Theorems 1.1, 1.3, Lemma 2.3 in
[28]):

######  Theorem 1.8.

(M. Shibata, 2014) Let $\mu>0$ and $2<q<p<2+\frac{4}{N}$. Then
$\alpha_{0}(\mu)>0$ and $\hat{H}^{\alpha}_{\mu}\leq 0$, for any $\alpha>0$.
For any $\alpha>\alpha_{0}(\mu)$, the prescribed mass minimization problem
(1.5) possesses a solution $v_{\alpha}\in H^{1}\setminus 0$, and thus
$\mathcal{M}_{\mu}(\alpha)\neq\emptyset$; if $\alpha\in(0,\alpha_{0}(\mu)]$,
then $\hat{H}^{\alpha}_{\mu}=0$ and (1.5) has no solution. Furthermore,
$\mathcal{M}_{\mu}(\alpha)$ is orbitally stable set of (1.1), and the function
$\alpha\mapsto\hat{H}^{\alpha}_{\mu}$ is nonincreasing.

Our main result on the orbital stability of the set of physical ground states
of (1.2) in a generalized sense is as follows:

######  Theorem 1.9.

Assume that $2<q<p<2+\frac{4}{N}$, $\mu>0$, $N\geq 3$. For any
$\alpha\in(\alpha_{0}(\mu),+\infty)$, the orbtally stable set of solutions
$\mathcal{M}_{\mu}(\alpha)$ of (1.2) with prescribed mass $\alpha$ contains a
set of physical ground states $\hat{G}^{S}(\mu)$ with some action level $S>0$.
Furthermore, $\alpha=Q(u)$, $\hat{\lambda}^{S}_{\mu}=\Lambda^{S}_{\mu}(u)$,
$\forall u\in\hat{G}^{S}(\mu)$.

The article is organized as follows. In Section 2, we give some preliminary
information and introduce the nonlinear generalized Rayleigh quotients. In
Section 3, we prove that the functional $\lambda^{S}_{\mu}(u)$ possesses a
global minimizer. In Section 4, we prove that $\mu^{S}(u)$ attains its global
minimum in $\mathcal{D}$. Section 5 is devoted to the investigation of the
behavior of solutions depend on the main parameters of problem $\lambda$, $S$
and $\alpha$. In Section 6, we conclude the proofs of Theorems 1.1, 1.4 and
1.7. Finally, in Section 7, we prove Theorem 1.9.

##  2\. The nonlinear generalized Rayleigh quotients

We denote by $H^{1}:=H^{1}(\mathbb{R}^{N})$ the Sobolev space of functions
with norm

$\|u\|_{1}=(\int(|u|^{2}+|\nabla u|^{2})^{1/2},$

and use the space

$\mathcal{D}^{1,2}:=\mathcal{D}^{1,2}(\mathbb{R}^{N}):=\\{u\in
L^{2^{*}}(\mathbb{R}^{N}):\nabla u\in L^{2}(\mathbb{R}^{N})\\}$

with inner product $(u,v):=\int\nabla u\cdot\nabla v\,dx$ and norm
$\|u\|_{\mathcal{D}^{1,2}}:=\int|\nabla u|^{2}\,dx$. For abbreviation, we
denote $\int\cdots\,:=\int_{\mathbb{R}^{N}}\cdots\,dx$. For Gateaux
differentiable functional $F:H^{1}(\mathcal{D}^{1,2})\to\mathbb{R}$, its
derivative at $u\in H^{1}(\mathcal{D}^{1,2})$ we denote by $DF(u)$. For $u\in
H^{1}$, we denote $u_{\sigma}:=u(x/\sigma)$, $x\in\mathbb{R}^{N}$, $\sigma>0$,
and

$T(u):=\int|\nabla
u|^{2},~{}~{}Q(u):=\int|u|^{2},~{}~{}A(u):=\int|u|^{p},~{}~{}B(u):=\int|u|^{q}.$

With these notations we have

$S_{\lambda,\mu}(u):=\frac{1}{2}T(u)-\lambda\frac{1}{2}Q(u)-\mu\frac{1}{p}A(u)+\frac{1}{q}B(u).$

For $S\geq 0$, introduce the so-called action-level Rayleigh quotient

(2.1)
$\displaystyle\Lambda^{S}_{\mu}(u):=\frac{\frac{1}{2}T(u)-\mu\frac{1}{p}A(u)+\frac{1}{q}B(u)-S}{\frac{1}{2}Q(u)}.$

Notice that for any $S\in\mathbb{R}$, and $\lambda\in\mathbb{R}$,

  * •

$\Lambda^{S}_{\mu}(u)=\lambda~{}\Leftrightarrow~{}S_{\lambda,\mu}(u)=S$,

  * •

$D\Lambda^{S}_{\mu}(u)=0$ with
$\Lambda^{S}_{\mu}(u)=\lambda~{}\Leftrightarrow~{}DS_{\lambda,\mu}(u)=0$.

Let $u\in H^{1}\setminus 0$, $S>0$, $\sigma>0$, consider

$\Lambda^{S}_{\mu}(u_{\sigma})=\frac{\sigma^{-2}\frac{1}{2}T(u)-\mu\frac{1}{p}A(u)+\frac{1}{q}B(u)-\sigma^{-N}S}{\frac{1}{2}Q(u)}.$

Then

$\frac{d}{d\sigma}(\Lambda^{S}_{\mu})(u_{\sigma})=0~{}\Leftrightarrow~{}\frac{2}{Q(u)}\left(-\sigma^{-3}T(u)+\frac{NS}{\sigma^{N+1}}\right)=0~{}\Leftrightarrow~{}\sigma=\sigma^{S}(u):=\left(\frac{NS}{T(u)}\right)^{\frac{1}{N-2}}.$

Hence, in accordance with the NG-Rayleigh quotient method [16], we are able to
introduce the following NG-Rayleigh quotient (cf. (1.7))

(2.2)
$\lambda^{S}_{\mu}(u):=\lambda^{S}_{\mu}(u_{\sigma^{S}(u)})=\frac{2}{Q(u)}\left(\frac{c_{N}^{S}}{2}T^{\frac{N}{(N-2)}}(u)-\mu\frac{1}{p}A(u)+\frac{1}{q}B(u)\right),$

where

$c_{N}^{S}=\frac{(N-2)}{N^{\frac{N}{(N-2)}}S^{\frac{2}{(N-2)}}}.$

Observe that $\lambda^{S}_{\mu}(u)$ is a $0$-homogeneous functional with
respect to the scale change $\sigma\mapsto u_{\sigma}$, i.e.,
$\lambda^{S}_{\mu}(u_{\sigma})=\lambda^{S}_{\mu}(u)$, $\forall\sigma>0$.

######  Lemma 2.1.

For $S>0$, $u\in H^{1}\setminus 0$, the following $D\lambda^{S}_{\mu}(u)=0$,
$\lambda^{S}_{\mu}(u)=\lambda$, $\sigma^{S}(u)=1$ is satisfied if and only if
$u$ is a weak solution of (1.2) with prescribed action $S$.

###### Proof.

The proof easily follows by direct calculations of derivative
$D\lambda^{S}_{\mu}(u)$. ∎

######  Remark 2.2.

In view of the homogeneity of $\lambda^{S}_{\mu}(u)$, we may always assume
that any critical point $u$ of $\lambda^{S}_{\mu}(u)$ satisfy
$\sigma^{S}(u)=1$.

Consider

$\lambda^{S}_{\mu}(tu)=\frac{1}{Q(u)}\left(t^{\frac{4}{N-2}}c_{N}^{S}T^{\frac{N}{(N-2)}}(u)-\mu\frac{2}{p}t^{p-2}A(u)+\frac{2}{q}t^{q-2}B(u)\right).$

A point $t_{0}>0$ is said to be fibering critical point of
$\lambda^{S}_{\mu}(tu)$ if $(d\lambda^{S}_{\mu}(tu)/dt)|_{t=t_{0}}=0$. Observe
that if $p,q\in(2,2^{*})$, then $\frac{4}{N-2}>\max\\{p-2,q-2\\}$. From this
it is easily seen that

  * •

if $2<p<q<2^{*}$, then for any $u\in H^{1}\setminus 0$ and $\mu>0$, the
fibering function $\lambda^{S}_{\mu}(tu)$ has a unique fibering critical point
$t=t(u)$. Furthermore,
$\lambda^{S}_{\mu}(t(u)u)=\min_{t>0}\lambda^{S}_{\mu}(tu)<0$, $\forall u\in
H^{1}\setminus 0$, $\forall\mu>0$.

  * •

If $2<q<p<2^{*}$, then for $u\in H^{1}\setminus 0$, the fibering function
$\lambda^{S}_{\mu}(tu)$ may has at most two nonzero critical points
$t^{0}_{\mu}(u),t^{1}_{\mu}(u)$ s.t. $0<t^{0}_{\mu}(u)\leq t^{1}_{\mu}(u)$.

We thus have

######  Corollary 2.3.

Assume that $2<p<q<2^{*}$. Then for any $\mu>0$,
$\hat{\lambda}_{\mu}^{S}\equiv\inf_{u\in H^{1}\setminus
0}\lambda^{S}_{\mu}(u)<0$

In the case $2<q<p<2^{*}$, we need to know the value of $\mu$ when the
fibering function $\lambda^{S}_{\mu}(tu)$ has two distinct critical points
$t^{0}_{\mu}(u),t^{1}_{\mu}(u)$, i.e., when $0<t^{0}_{\mu}(u)<t^{1}_{\mu}(u)$.
In order to find such values, we consider the following Rayleigh quotient

$M^{S}(u):=\frac{\frac{c_{N}^{S}}{2}T^{\frac{N}{(N-2)}}(u)+\frac{1}{q}B(u)}{\frac{1}{p}A(u)}.$

Notice that $M^{S}(u)=\mu~{}\Leftrightarrow~{}\lambda^{S}_{\mu}(u)=0$.

For every $u\in H^{1}\setminus 0$, consider the corresponding fibering
function

$M^{S}(su):=\frac{\frac{c_{N}^{S}}{2}s^{2^{*}-p}T^{\frac{N}{(N-2)}}(u)+\frac{1}{q}s^{q-p}B(u)}{\frac{1}{p}A(u)},~{}~{}s>0.$

It is easily seen that the function $s\mapsto M^{S}(su)$ has an unique global
minimum point $s^{S}(u)>0$ such that $M^{S}(su)$ is monotone decreasing in
$(0,s^{S}(u))$ and monotone increasing in $(s^{S}(u),+\infty)$. To find
$s^{S}(u)$, we calculate

$\frac{d}{ds}M^{S}(su)=0~{}\Leftrightarrow~{}s^{2^{*}-q}c_{p,q,N,S}T^{\frac{N}{(N-2)}}(u)=B(u),$

where $c_{p,q,N,S}=c_{N}^{S}q(2^{*}-p)/2(p-q)$. Thus for every $u\in
H^{1}\setminus 0$ the function $M^{S}(su)$ attains its global minimum at the
unique point

$s^{S}(u)=\left(\frac{B(u)}{c_{p,q,N,S}T^{\frac{N}{(N-2)}}(u)}\right)^{1/(2^{*}-q)}.$

Hence, we are able to introduce the following NG-Rayleigh quotient

(2.3) $\mu^{S}(u):=M^{S}(s_{m}(u)u)=\min_{s\geq
0}M^{S}(su)=C_{p,q,N,S}\frac{B^{\frac{2^{*}-p}{2^{*}-q}}(u)T^{\frac{2^{*}(p-q)}{2(2^{*}-q)}}(u)}{A(u)},$

where

$C_{p,q,N,S}=\frac{c(p,q,N)}{S^{\frac{2(p-q)}{(2^{*}-q)(N-2)}}},$ (2.4)
$c(p,q,N)=\left(\frac{(N-2)}{N^{\frac{N}{(N-2)}}}\frac{q(2^{*}-p)}{2(p-q)}\right)^{\frac{(p-q)}{(2^{*}-q)}}\frac{p(2^{*}-q)}{q(2^{*}-p)}.$

It is easily seen that $\mu^{S}(u)$ is $0$-homogeneous with respect to the
both actions: $t\mapsto tu$ and $\sigma\mapsto u_{\sigma}\equiv
u(\cdot/\sigma)$, i.e.,

(2.5)
$\mu^{S}(u_{\sigma})=\mu^{S}(u),~{}~{}\mu^{S}(su)=\mu^{S}(u),~{}~{}\forall\sigma>0,~{}\forall
s>0,~{}\forall u\in H^{1}\setminus 0.$

######  Lemma 2.4.

Assume that $D\mu^{S}(u_{0})=0$, $\mu^{S}(u_{0})=\mu_{0}$ s. t.
$\sigma(u_{0})=1$, $t^{1}_{\mu_{0}}(u_{0})=1$, then
$DS_{\lambda,\mu}(u_{0})=0$, $S_{\lambda,\mu}(u_{0})=S$ with $\lambda=0$,
$\mu=\mu_{0}\equiv\mu^{S}(u_{0})$.

###### Proof.

The proof follows by direct calculations of derivative $D\mu^{S}(u)$. ∎

Observe if $2<q<p<2^{*}$, then by the Gagliardo–Nirenberg interpolation
inequality we have

(2.6) $\displaystyle\int|u|^{p}\leq C_{gn}(\int|\nabla
u|^{2})^{\frac{2^{*}(p-q)}{2(2^{*}-q)}}$
$\displaystyle(\int|u|^{q})^{\frac{2^{*}-p}{2^{*}-q}}~{}\Leftrightarrow$
$\displaystyle A(u)\leq
C_{gn}(T(u))^{\frac{2^{*}(p-q)}{2(2^{*}-q)}}(B(u))^{\frac{2^{*}-p}{2^{*}-q}},$

where a constant $C_{gn}$ does not depend on $u\in\mathcal{D}$. Thus,
$\mu^{S}(u)$ can be extended to the space $\mathcal{D}\setminus 0$.

Consider the principal critical value (1.9), i.e.,

$\hat{\mu}^{S}=\inf_{u\in\mathcal{D}\setminus 0}\mu^{S}(u)\equiv
C_{p,q,N,S}\inf_{u\in\mathcal{D}\setminus
0}\frac{B^{\frac{2^{*}-p}{2^{*}-q}}(u)T^{\frac{2^{*}(p-q)}{2(2^{*}-q)}}(u)}{A(u)}.$

Notice that by (2.3) we have

(2.7) $\hat{\mu}^{S}=\inf_{u\in\mathcal{D}\setminus 0}M^{S}(u).$

Moreover, (2.6) implies

$\hat{\mu}^{S}>0.$

######  Proposition 2.5.

Assume $2<q<p<2^{*}$.

(i):

If $0<\mu\leq\hat{\mu}^{S}$, then $\lambda^{S}_{\mu}(t^{i}_{\mu}(u)u)\geq 0$,
$\forall u\in H^{1}\setminus 0$, $i=0,1$.

(ii):

If $\mu>\hat{\mu}^{S}$, then there exists $u\in H^{1}\setminus 0$ such that
the function $\lambda^{S}_{\mu}(tu)$ has two distinct nonzero critical points
$t^{0}_{\mu}(u),t^{1}_{\mu}(u)$, $0<t^{0}_{\mu}(u)<t^{1}_{\mu}(u)<+\infty$.
Moreover, $\lambda^{S}_{\mu}(t^{0}_{\mu}(u)u)>0$,
$\lambda^{S}_{\mu}(t^{1}_{\mu}(u)u)<0$.

###### Proof.

The proof of (i) follows immediately from the definition of $\hat{\mu}^{S}$ in
(1.9).

Let us prove (ii). Assume that $\mu>\hat{\mu}^{S}$. Then from (1.9) it follows
that there exists $u\in\mathcal{D}\setminus 0$ such that
$\hat{\mu}^{S}<\mu^{S}(u)<\mu$. Since $\mu^{S}(u)$ is a global minimum value
of the function $M^{S}(su)$ and $M^{S}(su)\to+\infty$ as $s\downarrow 0$ and
$s\to+\infty$, we infer that the equation $M^{S}(su)=\mu$ has two distinct
solutions $s^{0}(u)<s^{1}_{\mu}(u)$. Hence
$\lambda^{S}_{\mu}(s^{0}_{\mu}(u)u)=\lambda^{S}_{\mu}(s^{1}_{\mu}(u)u)=0$.
Since $\lambda^{S}_{\mu}(su)<0$ for $s\in(s^{0}_{\mu}(u),s^{1}_{\mu}(u))$,
$\lambda^{S}_{\mu}(su)$ attains its minimum value at a point $t^{1}_{\mu}(u)$
belonging to the interval $(s^{0}_{\mu}(u),s^{1}_{\mu}(u))$, whereas its local
maximum value point $t^{0}_{\mu}(u)\in(0,s^{0}_{\mu}(u))$. ∎

From this we have

######  Corollary 2.6.

Assume that $2<q<p<2^{*}$.

(i) If $\mu>\hat{\mu}^{S}$, then $\hat{\lambda}_{\mu}^{S}=\inf_{u\in
H^{1}\setminus 0}\lambda^{S}_{\mu}(u)<0$.

(ii) If $\mu=\hat{\mu}^{S}$, then
$\hat{\lambda}_{\hat{\mu}^{S}}^{S}=\inf_{u\in H^{1}\setminus
0}\lambda^{S}_{\hat{\mu}^{S}}(u)\geq 0$,

(iii) If $\mu<\hat{\mu}^{S}$, then $\hat{\lambda}_{\mu}^{S}=\inf_{u\in
H^{1}\setminus 0}\lambda^{S}_{\mu}(u)>0$.

###### Proof.

The proofs of (i), (iii) follow directly from Proposition 2.5. Let us prove
(ii). Suppose, contrary to our claim, that
$\hat{\lambda}_{\hat{\mu}^{S}}^{S}<0$. Then by (1.8), there exists $u\in
H^{1}\setminus 0$ such that
$\hat{\lambda}_{\hat{\mu}^{S}}^{S}<\lambda_{\hat{\mu}^{S}}^{S}(u)<0$. This
implies $M^{S}(u)<\hat{\mu}^{S}$, and thus $\mu^{S}(u)=\min_{s\geq
0}M^{S}(su)<\hat{\mu}^{S}$ which contradicts the definition of
$\hat{\mu}^{S}$. ∎

##  3\. Existence of a global minimizer of $\lambda^{S}_{\mu}(u)$

Consider minimization problem (1.8), i.e.,

$\displaystyle\hat{\lambda}_{\mu}^{S}:=\min_{u\in H^{1}\setminus
0}\lambda^{S}_{\mu}(u).$

######  Lemma 3.1.

Assume that $S>0$ and $2<p<q<2^{*}$, $\mu>0$, or $2<q<p<2^{*}$,
$\mu>\hat{\mu}^{S}$, then

(1):

$\hat{\lambda}_{\mu}^{S}<0$ and there exists a minimizer $\hat{u}_{\mu}^{S}$
of (1.8), i.e.,
$\hat{\lambda}_{\mu}^{S}=\lambda^{S}_{\mu}(\hat{u}_{\mu}^{S})$;

(2):

$\hat{u}_{\mu}^{S}$ is a fundamental frequency solution of (1.2) with
prescribe action $S$. Moreover, $\hat{u}^{S}_{\mu}>0$ in $\mathbb{R}^{N}$ and
$\hat{u}_{\mu}\in C^{2}(\mathbb{R}^{N})$.

###### Proof.

We will prove the assertions for the both cases: $2<p<q<2^{*}$, $\mu>0$, and
$2<q<p<2^{*}$, $\mu\geq\hat{\mu}^{S}$, in parallel.

Under the assumptions of the lemma, Corollaries 2.3, 2.6 imply that
$\hat{\lambda}_{\mu}^{S}<0$. Consider a minimizing sequence $(u_{n})$ of
(1.8), i.e., $\lambda^{S}_{\mu}(u_{n})\to\hat{\lambda}_{\mu}^{S}$ as
$n\to+\infty$. Let us show that $(u_{n})$ is bounded in $H^{1}$. In view of
that functional $\lambda^{S}_{\mu}(u)$ is $0$-homogenous, we may assume that
$\|u_{n}\|_{L^{2}}=1$, $n=1,2,\ldots$. Suppose that $\|\nabla
u_{n}\|_{L^{2}}\to+\infty$. By the Hölder and Sobolev inequalities we have

(3.1) $\int|u|^{p}\leq C\|u\|_{L^{2}}^{\kappa}\|\nabla
u\|_{L^{2}}^{\frac{2^{*}(2-\kappa)}{2}}=C\|\nabla
u\|_{L^{2}}^{\frac{2^{*}(2-\kappa)}{2}},u\in H^{1},$

where $\kappa=\frac{2(2^{*}-p)}{2^{*}-2}$, and $0<C<+\infty$ does not depend
on $u\in H^{1}$. Hence and since $2^{*}>\frac{2^{*}(2-\kappa)}{2}$, we get

(3.2) $\displaystyle\lambda^{S}_{\mu}(u_{n})\geq$
$\displaystyle\frac{c_{N}^{S}}{2}\|\nabla
u_{n}\|_{L^{2}}^{2^{*}}-\mu\frac{1}{p}\int|u_{n}|^{p}\geq$
$\displaystyle\frac{c_{N}^{S}}{2}\|\nabla u_{n}\|_{L^{2}}^{2^{*}}-\mu
C\frac{1}{p}\|\nabla
u_{n}\|_{L^{2}}^{\frac{2^{*}(2-\kappa)}{2}}\to+\infty,~{}~{}\mbox{as}~{}~{}\|\nabla
u_{n}\|_{L^{2}}\to+\infty,$

which is a contradiction. Thus, $(u_{n})$ is bounded in $H^{1}$, and therefore
by the Sobolev inequality, the norms $\|u_{n}\|_{L^{p}}$ and
$\|u_{n}\|_{L^{q}}$ are also bounded. This implies by the Banach–Alaoglu and
Sobolev embedding theorems that there exists a subsequence, which we again
denote by $(u_{n})$, such that

$\displaystyle
u_{n}\rightharpoonup\hat{u}_{\mu}^{S}~{}~{}\mbox{in}~{}H^{1}(\mathbb{R}^{N}),$
$\displaystyle
u_{n}\to\hat{u}_{\mu}^{S}~{}\mbox{in}~{}~{}L^{\gamma}_{loc}(\mathbb{R}^{N}),~{}~{}1\leq\gamma<2^{*},$
$\displaystyle u_{n}\to\hat{u}_{\mu}^{S}~{}~{}\mbox{a.e.
on}~{}\mathbb{R}^{N},$

for some $\hat{u}_{\mu}^{S}\in H^{1}$. Let us show that $\hat{u}_{\mu}^{S}\neq
0$. Observe that the sequence $\|u_{n}\|_{L^{p}}^{p}\equiv A(u_{n})$ is
separated from zero. Indeed, if $A(u_{n})\to 0$ as $n\to+\infty$, then

$\hat{\lambda}_{\mu}^{S}=\lim_{n\to+\infty}\frac{2}{Q(u_{n})}\left(\frac{c_{N}^{S}}{2}\|\nabla
u_{n}\|_{L^{2}}^{2^{*}}-\mu\frac{1}{p}\int|u_{n}|^{p}+\frac{1}{q}\int|u_{n}|^{q}\right)\geq
0.$

However, $\hat{\lambda}_{\mu}^{S}<0$, and we thus get a contradiction.

We need the following P.-L. Lions lemma (see Lemma I.1 p.231, in [21])

######  Lemma 3.2.

Let $r>0$ and $1\leq\gamma<2^{*}$. Assume $(u_{n})$ is a bounded in
$L^{\gamma}(\mathbb{R}^{N})$, $|\nabla u_{n}|$ is bounded in
$L^{2}(\mathbb{R}^{N})$ and

$\sup_{y\in\mathbb{R}^{N}}\int_{B(y;r)}|u_{n}|^{\gamma}\to 0,~{}n\to\infty,$

then $u_{n}\to 0$ in $L^{l}(\mathbb{R}^{N})$ for any $l\in(\gamma,2^{*})$.

Let $r>0$. Observe that

$\delta:=\liminf_{n\to\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B(y;r)}|u_{n}|^{\gamma}>0,$

for $1\leq\gamma<p$. Indeed, if this is not true, then by Lemma 3.2, $u_{n}\to
0$ in $L^{p}(\mathbb{R}^{N})$. But by the above this is impossible. Thus, we
infer, choosing a subsequence if necessary, that there exists
$(y_{n})\subset\mathbb{R}^{N}$ such that there holds
$\int_{B(y_{n};r)}|u_{n}|^{\gamma}>\delta/2$, $\forall n=1,\ldots$. Hence we
may assume, redefining $u_{n}:=u_{n}(\cdot+y_{n})$ if necessary, that

$\int_{B(0;r)}|u_{n}|^{\gamma}>\delta/2,~{}n=1,\ldots,~{}~{}1\leq\gamma<p,$

and thus indeed, $\hat{u}_{\mu}^{S}\neq 0$.

Recall the Brezis-Lieb lemma (see [21])

######  Lemma 3.3.

Assume $(u_{n})$ is bounded in $L^{\gamma}(\mathbb{R}^{N})$,
$1\leq\gamma<+\infty$ and $u_{n}\to u$ a. e. on $\mathbb{R}^{N}$, then

(3.3)
$\lim_{n\to+\infty}\|u_{n}\|_{L^{\gamma}}^{\gamma}=\|u\|_{L^{\gamma}}^{\gamma}+\lim_{n\to+\infty}\|u_{n}-u\|_{L^{\gamma}}^{\gamma}.$

From this we have

$\displaystyle\lim_{n\to+\infty}\|\nabla
u_{n}\|^{2}_{L^{2}}=\|\nabla\hat{u}_{\mu}^{S}\|^{2}_{L^{2}}+\lim_{n\to+\infty}\|\nabla(u_{n}-\hat{u}_{\mu}^{S})\|^{2}_{L^{2}},$
$\displaystyle\lim_{n\to+\infty}\|u_{n}\|_{L^{p}}^{p}=\|\hat{u}_{\mu}^{S}\|_{L^{p}}^{p}+\lim_{n\to+\infty}\|u_{n}-\hat{u}_{\mu}^{S}\|_{L^{p}}^{p},$
$\displaystyle\lim_{n\to+\infty}\|u_{n}\|_{L^{q}}^{q}=\|\hat{u}_{\mu}^{S}\|_{L^{q}}^{q}+\lim_{n\to+\infty}\|u_{n}-\hat{u}_{\mu}^{S}\|_{L^{q}}^{q},$
$\displaystyle\bar{Q}:=\lim_{n\to+\infty}\|u_{n}\|^{2}_{L^{2}}=\|\hat{u}_{\mu}^{S}\|^{2}_{L^{2}}+\lim_{n\to+\infty}\|u_{n}-\hat{u}_{\mu}^{S}\|^{2}.$

Hence and since

$(\|\nabla\hat{u}_{\mu}^{S}\|^{2}_{L^{2}}+\|\nabla(u_{n}-\hat{u}_{\mu}^{S})\|^{2}_{L^{2}})^{\frac{N}{(N-2)}}\geq\|\nabla\hat{u}_{\mu}^{S}\|_{L^{2}}^{2^{*}}+\|\nabla(u_{n}-\hat{u}_{\mu}^{S})\|_{L^{2}}^{2^{*}},$

we have

$\displaystyle\hat{\lambda}_{\mu}^{S}=\lim_{n\to+\infty}\lambda^{S}_{\mu}(u_{n})\geq\frac{c_{N}^{S}\|\nabla\hat{u}_{\mu}^{S}\|_{L^{2}}^{2^{*}}-\mu\frac{2}{p}\|\hat{u}_{\mu}^{S}\|_{L^{p}}^{p}+\frac{2}{q}\|\hat{u}_{\mu}^{S}\|_{L^{q}}^{q}}{\bar{Q}}+$
$\displaystyle\frac{c_{N}^{S}\lim_{n\to+\infty}\|\nabla(u_{n}-\hat{u}_{\mu}^{S})\|_{L^{2}}^{2^{*}}-\mu\frac{2}{p}\lim_{n\to+\infty}\|u_{n}-\hat{u}_{\mu}^{S}\|_{L^{p}}^{p}+\frac{2}{q}\lim_{n\to+\infty}\|u_{n}-\hat{u}_{\mu}^{S}\|_{L^{q}}^{q}}{\bar{Q}}.$

In view of that $\hat{\lambda}_{\mu}^{S}\neq 0$, this implies

$\displaystyle\hat{\lambda}_{\mu}^{S}$
$\displaystyle\geq\frac{1}{\bar{Q}}\hat{\lambda}_{\mu}^{S}\cdot\|\hat{u}_{\mu}^{S}\|_{L^{2}}^{2}+$
$\displaystyle\frac{1}{\bar{Q}}\lim_{n\to\infty}\left(c_{N}^{S}\|\nabla(u_{n}-\hat{u}_{\mu}^{S})\|_{L^{2}}^{2^{*}}-\mu\frac{2}{p}\|u_{n}-\hat{u}_{\mu}^{S}\|_{L^{p}}^{p}+\frac{2}{q}\|u_{n}-\hat{u}_{\mu}^{S}\|_{L^{q}}^{q}\right)\geq$
$\displaystyle\frac{1}{\bar{Q}}(\hat{\lambda}_{\mu}^{S}\cdot\|\hat{u}_{\mu}^{S}\|^{2}_{L^{2}}+\hat{\lambda}_{\mu}^{S}\lim_{n\to\infty}\|u_{n}-\hat{u}_{\mu}^{S}\|^{2}_{L^{2}})=\hat{\lambda}_{\mu}^{S}\frac{\lim_{n\to+\infty}\|u_{n}\|^{2}_{L^{2}}}{\bar{Q}}=\hat{\lambda}_{\mu}^{S}.$

However this is possible only if the equality holds. Consequently,

$\lim_{n\to\infty}\left(c_{N}^{S}\|\nabla(u_{n}-\hat{u}_{\mu}^{S})\|_{L^{2}}^{2^{*}}+\|u_{n}-\hat{u}_{\mu}^{S}\|_{L^{p}}^{p}+\|u_{n}-\hat{u}_{\mu}^{S}\|_{L^{q}}^{q}\right)=0.$

Hence it follows easily that $u_{n}\to\hat{u}_{\mu}^{S}$ strongly in $H^{1}$
and $\hat{u}_{\mu}^{S}$ is a minimizer of (1.8).

Due to the homogeneity of $\lambda_{\mu}^{S}(u)$ we may assume that
$\sigma^{S}(\hat{u}_{\mu}^{S})=1$. Hence by Lemma 2.1, we have
$DS_{\lambda,\mu}(\hat{u}_{\mu}^{S})=0$,
$S_{\lambda,\mu}(\hat{u}_{\mu}^{S})=S$, where
$\lambda=\hat{\lambda}_{\mu}^{S}$. Since
$\lambda_{\mu}^{S}(|u|)=\lambda_{\mu}^{S}(u)$ for $u\in H^{1}\setminus 0$, we
may assume that $\hat{u}_{\mu}^{S}\geq 0$ in $\mathbb{R}^{N}$.

Since $\hat{u}_{\mu}\in H^{1}(\mathbb{R}^{N})$, the Brezis & Kato Theorem [7]
and the $L^{\gamma}$ estimates for the elliptic problems [1] yield that
$\hat{u}_{\mu}^{S}\in W^{2,\gamma}_{loc}(\mathbb{R}^{N})$, for any
$\gamma\in(1,+\infty)$, and whence by the regularity theory of the solutions
of the elliptic problems $\hat{u}_{\mu}\in C^{2}(\mathbb{R}^{N})$ (see, e.g.,
[1]). Hence by the Harnack inequality [33], we get that $\hat{u}_{\mu}^{S}>0$
in $\mathbb{R}^{N}$. ∎

######  Proposition 3.4.

Let $S>0$.

$(1^{o})$ Assume that $2<p<q<2^{*}$, then $\hat{\lambda}_{\mu}^{S}\to 0$ as
$\mu\to 0$.

$(2^{o})$ Assume that $2<q<p<2^{*}$, then $\hat{\lambda}_{\mu}^{S}\to 0$ as
$\mu\downarrow\hat{\mu}^{S}$.

###### Proof.

First we prove $(2^{o})$. Fix $S>0$. By Lemma 3.1, for any $\mu>\hat{\mu}^{S}$
there exists a minimizer $\hat{u}_{\mu}^{S}$ of (1.8) so that
${\lambda}_{\mu}^{S}(\hat{u}^{S}_{\mu})<0$. Due to the homogeneity of
${\lambda}_{\mu}^{S}(u)$, we may assume that $Q(\hat{u}^{S}_{\mu})=1$,
$\forall\mu>\hat{\mu}^{S}$. Then by (2.2),

${\lambda}_{\mu}^{S}(\hat{u}^{S}_{\mu})-{\lambda}_{\hat{\mu}^{S}}^{S}(\hat{u}^{S}_{\mu})=-2(\mu-\hat{\mu}^{S})\frac{1}{p}A(\hat{u}^{S}_{\mu})$

Analysis similar to that in the proof of Lemma 3.1 shows that the set
$(\hat{u}^{S}_{\mu})$ for $\mu$ sufficiency close to $\hat{\mu}^{S}$ is
bounded in $H^{1}$, and thus by (3.1), the sequence $A(\hat{u}^{S}_{\mu})$ is
also bounded. Hence,
${\lambda}_{\mu}^{S}(\hat{u}^{S}_{\mu})-{\lambda}_{\hat{\mu}^{S}}^{S}(\hat{u}^{S}_{\mu})\to
0$ as $\mu\downarrow\hat{\mu}^{S}$. Notice that by Corollary 2.6,
${\lambda}_{\hat{\mu}^{S}}^{S}(\hat{u}^{S}_{\mu})\geq 0$, whereas
${\lambda}_{\mu}^{S}(\hat{u}^{S}_{\mu})<0$. This implies that
$\hat{\lambda}_{\mu}^{S}\to 0$ as $\mu\downarrow\hat{\mu}^{S}$.

The proof of $(1^{o})$ is similar. We only note that in this case,
${\lambda}_{\hat{\mu}^{S}}^{S}(u)$ is replaced by
${\lambda}_{0}^{S}(u):={\lambda}_{\mu}^{S}(u)|_{\mu=0}$. Notice that by (2.2),
$\inf_{u\in H^{1}\setminus 0}{\lambda}_{0}^{S}(u)=0$.

∎

##  4\. Existence of a solution of zero frequency problem in $\mathcal{D}$

In this section, we prove the existence of a fundamental frequency solution of
the zero frequency problem (1.2) using minimization problem (1.9).

Denote $\beta:=\frac{2q(2^{*}-p)}{2^{*}(p-q)}$,
$\rho:=\frac{2p(2^{*}-q)}{2^{*}(p-q)}$,

$\mu(u):=\frac{\|u\|_{L^{q}}^{\beta}\|\nabla
u\|_{L^{2}}^{2}}{\|u\|_{L^{p}}^{\rho}(u)}\equiv(S^{\frac{2(p-q)}{(2^{*}-q)(N-2)}}\mu^{S}(u)/c(p,q,N))^{\rho/p},~{}~{}u\in
H^{1}\setminus 0.$

Consider

(4.1) $\bar{\mu}=\inf_{u\in\mathcal{D}\setminus 0}\mu(u).$

Then

(4.2)
$\hat{\mu}^{S}:=c(p,q,N)\frac{1}{S^{\frac{2(p-q)}{(2^{*}-q)(N-2)}}}\bar{\mu}^{p/\rho},~{}~{}\forall
S>0.$

######  Lemma 4.1.

Let $2<q<p<2^{*}$. There exists a minimizer
$\hat{u}^{S}_{\hat{\mu}^{S}}\in\mathcal{D}$ of (1.9) such that
$\hat{u}^{S}_{\hat{\mu}^{S}}$ weakly satisfies to (1.2) with $\lambda=0$,
$\mu=\hat{\mu}^{S}$. Moreover, $\hat{u}^{S}_{\mu}>0$ in $\mathbb{R}^{N}$,
$\hat{u}_{\mu}\in C^{2}(\mathbb{R}^{N})$ and
$S_{0,\hat{\mu}^{S}}(\hat{u}^{S}_{\hat{\mu}^{S}})=S$.

###### Proof.

Let $(v_{i})$ be a minimizing sequence of (4.1), i.e.,
$\mu(v_{i})\to\bar{\mu}$ as $i\to\infty$. Set
$u_{i}=t_{i}(v_{i})_{\sigma_{i}}$, $i=1,2,\ldots$, where

$t_{i}=(\|v_{i}\|_{L^{q}}^{q}/\|v_{i}\|_{L^{p}}^{p})^{1/(p-q)},~{}~{}\sigma_{i}=(\|v_{i}\|_{L^{q}}^{pq}/\|v_{i}\|_{L^{p}}^{qp})^{1/N(p-q)}.$

Then $\|u_{i}\|_{L^{p}}=1$ and $\|u_{i}\|_{L^{q}}=1$, $i=1,\ldots$, and by the
homogeneity of $\mu(u)$, $(u_{i})$ is a minimizing sequence of (4.1). Since
$\bar{\mu}<+\infty$, $(\|\nabla u_{i}\|_{L^{2}})$ is bounded, and thus
$(u_{i})$ is bounded in $\mathcal{D}$ and in $H^{1}_{loc}$. Thus, by the
Banach–Alaoglu and Sobolev embedding theorems, there exists a subsequence,
still denoted by $(u_{i})$, such that

$\displaystyle u_{i}\rightharpoonup\hat{u}^{*}~{}~{}\mbox{in}~{}\mathcal{D},$
$\displaystyle
u_{i}\to\hat{u}^{*}~{}\mbox{in}~{}~{}L^{\gamma}_{loc},~{}~{}1\leq\gamma<2^{*},$
$\displaystyle u_{i}\to\hat{u}^{*}~{}~{}\mbox{a.e. on}~{}\mathbb{R}^{N},$

for some $\hat{u}^{*}\in\mathcal{D}$. Since the sequence
$\|u_{i}\|_{L^{q}}=B(u_{i})$ is bounded, we may apply Lemma 3.2. Hence using
the similar arguments as in the proof of Lemma 3.1 we obtain that for any fix
$r>0$, there exists $(y_{i})\subset\mathbb{R}^{N}$ such that there holds
$\int_{B(y_{i};r)}|u_{i}|^{q}>\delta/2$, $\forall i=1,\ldots$. Hence,
redefining $u_{i}:=u_{i}(\cdot+y_{n})$ if necessary, we have

$\int_{B(0;r)}|u_{i}|^{q}>\delta/2,~{}i=1,\ldots,$

which implies that $\hat{u}^{*}\neq 0$.

Due to the Brezis-Lieb lemma we have

(4.3)
$\displaystyle\|\nabla\hat{u}^{*}\|_{L^{2}}^{2}=\lim_{i\to\infty}\|\nabla{u_{i}}\|_{L^{2}}^{2}-\lim_{i\to\infty}\|\nabla({u_{i}}-\hat{u}^{*})\|_{L^{2}}^{2},$
(4.4)
$\displaystyle\|\hat{u}^{*}\|_{L^{p}}^{p}=\lim_{i\to\infty}\|u_{i}\|_{L^{p}}^{p}-\lim_{i\to\infty}\|u_{i}-\hat{u}^{*}\|_{L^{p}}^{p},$
(4.5)
$\displaystyle\|\hat{u}^{*}\|_{L^{q}}^{q}=\lim_{i\to\infty}\|u_{i}\|_{L^{q}}^{q}-\lim_{i\to\infty}\|u_{i}-\hat{u}^{*}\|_{L^{q}}^{q}.$

Suppose that $\lim_{i\to\infty}\|\nabla({u_{i}}-\hat{u}^{*})\|_{L^{2}}^{2}>0$.
Then, without loss of generality, we may assume that there holds
$\lim_{i\to\infty}\|u_{i}-\hat{u}^{*}\|_{L^{q}}^{q}>0$,
$\lim_{i\to\infty}\|u_{i}-\hat{u}^{*}\|_{L^{p}}^{p}>0$ as well. Hence,

$\displaystyle\bar{\mu}=$
$\displaystyle\lim_{i\to\infty}\|\nabla{u_{i}}\|_{L^{2}}^{2}=\|\nabla\hat{u}^{*}\|_{L^{2}}^{2}+\lim_{i\to\infty}\|\nabla({u_{i}}-\hat{u}^{*})\|_{L^{2}}^{2}\geq$
(4.6)
$\displaystyle\bar{\mu}\left(\frac{\|\hat{u}^{*}\|_{L^{p}}^{\rho}}{\|\hat{u}^{*}\|_{L^{q}}^{\beta}}+\lim_{i\to\infty}\frac{\|u_{i}-\hat{u}^{*}\|_{L^{p}}^{\rho}}{\|u_{i}-\hat{u}^{*}\|_{L^{q}}^{\beta}}\right)=\bar{\mu}\left(\frac{\|\hat{u}^{*}\|_{L^{p}}^{\rho}}{\|\hat{u}^{*}\|_{L^{q}}^{\beta}}+\frac{(1-\|\hat{u}^{*}\|_{L^{p}}^{\rho})}{(1-\|\hat{u}^{*}\|_{L^{q}}^{\beta})}\right)>\bar{\mu},$

which implies a contradiction. Hence
$\|\nabla\hat{u}^{*}\|_{L^{2}}^{2}=\lim_{i\to\infty}\|\nabla
u_{i}\|_{L^{2}}^{2}=\bar{\mu}$,
$\|\hat{u}^{*}\|_{L^{p}}^{p}=\lim_{i\to\infty}\|u_{i}\|_{L^{p}}^{p}$,
$\|\hat{u}^{*}\|_{L^{q}}^{q}=\lim_{i\to\infty}\|u_{i}\|_{L^{q}}^{q}$, and thus
$\hat{u}^{*}$ is a minimizer of (4.1), and whence also of (1.9).

Due to the homogeneity of $\mu^{S}(u)$, any function $s\hat{u}^{*}_{\sigma}$
with $s>0$, $\sigma>0$ is also a minimizer of (1.9). Hence, we can find a
minimizer $\hat{u}^{S}_{\hat{\mu}^{S}}\in\mathcal{D}$ of (1.9) which satisfies
$\sigma^{S}(\hat{u}^{S}_{\hat{\mu}^{S}})=1$ and
$s^{S}(\hat{u}^{S}_{\hat{\mu}^{S}})=1$. From this and since
$D\mu^{S}(\hat{u}^{S}_{\hat{\mu}^{S}})=0$, we have
$DS_{0,\mu}(\hat{u}^{S}_{\hat{\mu}^{S}})=0$. The rest of the proof runs as in
the proof of Lemma 3.1.

∎

##  5\. Further properties of the solutions

In this section, we investigate the behavior of solutions depend on the main
parameters $\lambda$, $S$ and $\alpha$. Observe that the solutions of (1.2)
with prescribed action $S$ can be studied in two ways: first, when for every
given value of the action $S$, equation (1.2) is investigated with respect to
the parameter $\mu>0$, and the second, when for every given parameter $\mu>0$,
(1.2) is investigated with respect to the value $S$. The above results in
Sections 2-4 have been carried out according to the first way. Our next
results are convenient to consider by the second way.

Observe that the function $S\mapsto\hat{\mu}^{S}$ is invertible so that for
any $\mu>0$, we can introduce

(5.1)
$S(\mu)=\bar{\mu}^{\frac{N}{2}}\left(\frac{c(p,q,N)}{\mu}\right)^{\frac{(N-2)(2^{*}-q)}{2(p-q)}},$

such that $\hat{\mu}^{S(\mu)}=\mu$ for any $\mu>0$.

From Corollary 2.6 and Lemma 3.1 it follows

######  Corollary 5.1.

$(1^{o})$:

Suppose $2<p<q<2^{*}$, $\mu>0$, then $G^{S}(\mu)\neq\emptyset$ for any $S>0$.

$(2^{o})$:

Suppose $2<q<p<2^{*}$, $\mu>0$, then

(i):

$G^{S}(\mu)\neq\emptyset$ for any $S\geq S(\mu)$;

(ii):

(1.2) has no solution with $\lambda\leq 0$ and $S\in(0,S(\mu))$.

###### Proof.

The proofs of $(1^{o})$ and (i) of $(2^{o})$ follow immediately from Lemma
3.1. Let us show (ii) of $(2^{o})$. Indeed, if there exists a solution $u_{0}$
of (1.2) with $S_{\lambda,\mu}(u_{0})=S\in(0,S(\mu))$, then by (5.1),
$\mu<\hat{\mu}^{S}$, and thus by Corollary 2.6,
$\lambda=\Lambda^{S}(u_{0})=\lambda^{S}(u_{0})>0$, which contradicts
assumption $\lambda\leq 0$. ∎

Let us prove

######  Proposition 5.2.

Assume that $p,q\in(2,2^{*})$. If $\mu>0$, $S_{2}>S_{1}>0$ and
$G^{S_{j}}(\mu)\neq\emptyset$, $j=1,2$, then

(5.2)
$-2\frac{(S_{2}-S_{1})(S_{2}/S_{1})^{N/(N-2)}}{Q(\hat{u}^{S_{2}}_{\mu})}<\hat{\lambda}_{\mu}^{S_{2}}-\hat{\lambda}_{\mu}^{S_{1}}<-2\frac{(S_{2}-S_{1})(S_{1}/S_{2})^{N/(N-2)}}{Q(\hat{u}^{S_{1}}_{\mu})},$

$\forall\hat{u}^{S_{j}}_{\mu}\in G^{S_{j}}(\mu),~{}j=1,2$.

###### Proof.

Notice that for $\hat{u}^{S_{j}}_{\mu}\in G^{S_{j}}(\mu)$, we have
$\sigma^{S_{j}}(\hat{u}^{S_{j}}_{\mu})=1$, $j=1,2$. Hence,

$\displaystyle\hat{\lambda}_{\mu}^{S_{2}}=\lambda^{S_{2}}_{\mu}(\hat{u}^{S_{2}}_{\mu})\leq\lambda^{S_{2}}_{\mu}(\hat{u}^{S_{1}}_{\mu})$
$\displaystyle=\Lambda^{S_{2}}_{\mu}((\hat{u}^{S_{1}}_{\mu})_{\sigma^{S_{2}}(\hat{u}^{S_{1}}_{\mu})})=$
$\displaystyle\Lambda^{S_{1}}_{\mu}((\hat{u}^{S_{1}}_{\mu})_{\sigma^{S_{2}}(\hat{u}^{S_{1}}_{\mu})})-2\frac{S_{2}-S_{1}}{Q((\hat{u}^{S_{1}}_{\mu})_{\sigma^{S_{2}}((\hat{u}^{S_{1}}_{\mu})})}.$

Since $\sigma^{S_{1}}(\hat{u}^{S_{1}}_{\mu})$ is a global maximum point of the
function
$\mathbb{R}^{+}\ni\sigma\mapsto\Lambda^{S_{1}}((\hat{u}^{S_{1}}_{\mu})_{\sigma})$,

$\hat{\lambda}_{\mu}^{S_{1}}=\Lambda^{S_{1}}_{\mu}((\hat{u}^{S_{1}}_{\mu})_{\sigma^{S_{1}}(\hat{u}^{S_{1}}_{\mu})})>\Lambda^{S_{1}}_{\mu}((\hat{u}^{S_{1}}_{\mu})_{\sigma^{S_{2}}(\hat{u}^{S_{1}}_{\mu})}).$

Hence

$\hat{\lambda}_{\mu}^{S_{2}}-\hat{\lambda}_{\mu}^{S_{1}}<-2\frac{S_{2}-S_{1}}{Q((\hat{u}^{S_{1}}_{\mu})_{\sigma^{S_{2}}(\hat{u}^{S_{1}}_{\mu})})}.$

Now taking into account that

$\frac{1}{Q((\hat{u}^{S_{1}}_{\mu})_{\sigma^{S_{2}}(\hat{u}^{S_{1}}_{\mu})})}=\frac{T(\hat{u}^{S_{1}}_{\mu})^{\frac{N}{(N-2)}}}{(NS_{2})^{N/(N-2)}Q(\hat{u}^{S_{1}}_{\mu})}~{}~{}\mbox{and}~{}~{}\sigma^{S_{1}}(\hat{u}^{S_{1}})=\left(\frac{NS_{1}}{T(\hat{u}^{S_{1}}_{\mu})}\right)^{\frac{1}{N-2}}=1,$

we obtain the second inequality in (5.2). The proof of the first one may be
handled in much the same way. ∎

######  Corollary 5.3.

Assume that $\mu>0$ and $2<p<q<2^{*}$ $(2<q<p<2^{*})$. The function
$S\mapsto\hat{\lambda}_{\mu}^{S}$ is continuous and monotone decreasing on
$(0,\infty)$ $((S(\mu),+\infty))$. Furthermore, $\hat{\lambda}_{\mu}^{S}\to 0$
as $S\to 0$ ($S\to S(\mu)$).

###### Proof.

Let $S_{0}\in(0,\infty)$ $((S(\mu),+\infty))$. By (5.2),
$\frac{S_{2}^{2^{*}}}{Q(\hat{u}^{S_{2}}_{\mu})}>\frac{S_{1}^{2^{*}}}{Q(\hat{u}^{S_{1}}_{\mu})}$
for any $S_{2}>S_{1}>0$ and $\hat{u}^{S_{j}}_{\mu}\in
G^{S_{j}}(\mu),~{}j=1,2$. This implies that the set
$(Q(\hat{u}^{S}_{\mu}))_{S\in(S_{0}-\varepsilon,S_{0}+\varepsilon)}$ is
bounded and separated from zero for any $S_{0}>0$ and $\varepsilon>0$ such
that $S_{0}>\varepsilon$ ($S_{0}>S(\mu)+\varepsilon$). Hence by (5.2), we
infer that $\hat{\lambda}_{\mu}^{S}\to\hat{\lambda}_{\mu}^{S_{0}}$ as $S\to
S_{0}$. In view of (5.2), the function $\hat{\lambda}_{\mu}^{S}$ is monotone
decreasing on $(0,\infty)$ $((S(\mu),+\infty))$. The rest of the proof follows
from Proposition 3.4. ∎

######  Lemma 5.4.

Assume that $\mu>0$ and $2<p<q<2^{*}$ $(2<q<p<2^{*})$. Then
$\hat{G}^{S}(\mu)\neq\emptyset$, $\forall S\in(0,+\infty)$ $(\forall
S\in(S(\mu),+\infty))$.

###### Proof.

Notice that if $2<p<q<2^{*}$, then Corollary 5.1 implies that
$G^{S}(\mu)\neq\emptyset$ for any $\mu>0$ and $S\in(0,+\infty)$. If
$2<q<p<2^{*}$, then by (5.1), the inequality $S>S(\mu)$ implies
$\hat{\mu}^{S}<\mu$, and thus Lemma 3.1 yields $G^{S}(\mu)\neq\emptyset$ .

Let $S\in(0,+\infty)$ ($S\in(S(\mu),+\infty)$). Assume that $(S_{m})$ is a
sequence such that $S_{m}\to S$. Let us fix an arbitrary
$\hat{u}^{S_{m}}_{\mu}\in G^{S_{m}}(\mu)$, $\forall m=1,\ldots$. Due to the
homogeneity of ${\lambda}_{\mu}^{S}(u)$, we may assume that
$Q(\hat{u}^{S_{m}}_{\mu})=1$, $\forall m=1,\ldots$. Analysis similar to that
in the proof of Lemma 3.1 shows that the sequence $(\hat{u}^{S_{m}}_{\mu})$ is
bounded in $H^{1}$. By (2.2),

${\lambda}_{\mu}^{S_{m}}(\hat{u}^{S_{m}}_{\mu})-{\lambda}_{\mu}^{S}(\hat{u}^{S_{m}}_{\mu})=(c_{N}^{S_{m}}-c_{N}^{S})T^{\frac{N}{(N-2)}}(\hat{u}^{S_{m}}_{\mu}),~{}~{}\forall
m=1,\ldots.$

This and the continuity of the function $\hat{\lambda}_{\mu}^{(\cdot)}$
implies that
$\lambda_{\mu}^{S}(\hat{u}^{S_{m}}_{\mu})\to\hat{\lambda}_{\mu}^{S}$, that is
$(\hat{u}^{S_{m}}_{\mu})$ is a minimizing sequence for $\lambda_{\mu}^{S}(u)$.
Arguing as before in the proof of Lemma 3.1 we infer that there exists a
subsequence, which we again denote by $(\hat{u}^{S_{m}}_{\mu})$, such that
$\hat{u}^{S_{m}}_{\mu}\to\hat{u}^{S}_{\mu}$ strongly in $H^{1}$ for some
$\hat{u}^{S}_{\mu}\in G_{\mu}^{S}$. By Definition 1.6, this means that
$\hat{u}^{S}_{\mu}\in\hat{G}^{S}(\mu)$, and thus
$\hat{G}^{S}(\mu)\neq\emptyset$. ∎

######  Lemma 5.5.

Let $\mu>0$.

$(1^{o})$ If $2<p<q<2^{*}$, then the function $\hat{\lambda}_{\mu}^{S}$ is
differentiable at every $S\in(0,+\infty)$, and there holds

(5.3) $\frac{d}{dS}\hat{\lambda}_{\mu}^{S}=-2\frac{1}{Q(\hat{u}^{S}_{\mu})},$

$\forall S\in(0,+\infty),~{}\forall\hat{u}^{S}_{\mu}\in\hat{G}^{S}(\mu)$.
Furthermore, for every $S\in(0,+\infty)$ corresponds a constant $\alpha^{S}>0$
such that

(5.4)
$\alpha^{S}:=\alpha^{S}_{\mu}=Q(\hat{u}^{S}_{\mu}),~{}~{}~{}\forall\hat{u}^{S}_{\mu}\in\hat{G}^{S}(\mu),$

moreover the function $S\mapsto\alpha^{S}$ is continuous on $(0,\infty)$.

$(2^{o})$ If $2<q<p<2^{*}$, then the function $\hat{\lambda}_{\mu}^{S}$ is
differentiable at every $S\in(S(\mu),+\infty)$, and (5.3) holds
$\forall\hat{u}^{S}_{\mu}\in\hat{G}^{S}(\mu)$, $\forall S\in(S(\mu),+\infty)$.
Furthermore, for every $S\in(S(\mu),+\infty)$ corresponds a constant
$\alpha^{S}>0$ such that (5.4) holds, and the function $S\mapsto\alpha^{S}$ is
continuous on $(0,\infty)$.

###### Proof.

We prove $(1^{o})$ and $(2^{o})$, in parallel. Let $\mu>0$ and $2<p<q<2^{*}$
($2<q<p<2^{*}$). Fix $S\in(0,+\infty)$ ($S\in(S(\mu),+\infty)$). Take an
arbitrary sequence $(S_{m})$ such that $S_{m}\to S$ as $m\to+\infty$. By the
proof of Corollary 5.3 we know that the sequence $(Q(\hat{u}^{S_{m}}_{\mu}))$
is bounded and separated from zero. Since
$\sigma^{S_{m}}(\hat{u}^{S_{m}}_{\mu})=1$, i.e.,
$\|\nabla\hat{u}^{S_{m}}_{\mu}\|^{2}_{L^{2}}=NS_{m}$, $m=1,\ldots,$ we
conclude that the set $(\hat{u}^{S_{m}}_{\mu})$ is bounded in $H^{1}$. Hence,
analysis similar to that in the proof of Lemma 3.1 shows that there exists a
limit point $\hat{u}^{S}_{\mu}\in\hat{G}^{S}(\mu)$ such that
$\hat{u}^{S_{m_{k}}}_{\mu}\to\hat{u}^{S}_{\mu}$ in $H^{1}$ for some
subsequence $(m_{k})$ such that $m_{k}\to+\infty$ as $k\to+\infty$. From this
and (5.2) it follows easily that there exists a derivative
$\frac{d}{dS}\hat{\lambda}_{\mu}^{S}$ which satisfies (5.3).

Since $\hat{\lambda}_{\mu}^{(\cdot)}$ is well-defined, the map
$S\mapsto\frac{d}{dS}\hat{\lambda}_{\mu}^{S}$ is unambiguously defined
function, and thus the right hand side of (5.3) is also unambiguously defined.
Thus for every $S\in(0,+\infty)$ ($S\in(S(\mu),+\infty)$), there exists a
unique constant $\alpha^{S}>0$ such that
$\alpha^{S}_{\mu}=Q(\hat{u}^{S}_{\mu})$ for every
$\hat{u}^{S}_{\mu}\in\hat{G}^{S}(\mu)$. From the above convergence
$\hat{u}^{S_{m_{k}}}_{\mu}\to\hat{u}^{S}_{\mu}$ in $H^{1}$ it follows that the
function $S\mapsto\alpha^{S}$ is continuous on $(0,+\infty)$
($(S(\mu),+\infty)$). ∎

Furthermore, we have

######  Corollary 5.6.

Assume that $\mu>0$ and $2<p<q<2^{*}$ $(2<q<p<2^{*})$. Then
$\lim_{S\to+\infty}\hat{\lambda}_{\mu}^{S}=-\infty$. Moreover, there exists an
inverse function $\lambda\mapsto S_{\lambda}$ of $\hat{\lambda}_{\mu}^{S}$
such that $\hat{\lambda}_{\mu}^{S_{\lambda}}=\lambda$,
$\forall\lambda\in(-\infty,0)$. Moreover, $S_{\lambda}$ is a continuous and
monotone increasing function on $(-\infty,0)$.

###### Proof.

Since the function $S\mapsto\hat{\lambda}_{\mu}^{S}$ is continuous and
monotone decreasing on $(0,\infty)$ $((S(\mu),+\infty))$, there exists a limit
$\lim_{S\to+\infty}\hat{\lambda}_{\mu}^{S}=\bar{\lambda}_{\mu}\geq-\infty$.
Suppose, contrary to our claim, that $\bar{\lambda}_{\mu}>-\infty$. Take
$\lambda<\bar{\lambda}_{\mu}$. Then by the Berestycki & Lions Theorem [6]
there exists a solution $\hat{u}_{\lambda}\in H^{1}\setminus 0$ of (1.2).
Denote $S=S_{\lambda,\mu}(\hat{u}_{\lambda})$. Then
$\Lambda^{S}_{\mu}(\hat{u}_{\lambda})=\lambda$, and since
$\sigma^{S}(\hat{u}_{\lambda})=1$, we have
$\Lambda^{S}_{\mu}(\hat{u}_{\lambda})=\lambda^{S}_{\mu}(\hat{u}_{\lambda})=\lambda$.
Hence, $\lambda\geq\hat{\lambda}^{S}_{\mu}>\bar{\lambda}_{\mu}$, which
contradicts the assumption $\lambda<\bar{\lambda}_{\mu}$. The rest of the
proof follows immediately from Corollary 5.3. ∎

Now we are able to prove that the existence of the fundamental frequency
solution entails the existence of the ground state and that the converse is
also true.

######  Lemma 5.7.

Assume that $p,q\in(2,2^{*})$ and $\mu>0$.

$(1^{o})$ Suppose $S>0$ such that there exists a fundamental frequency
solution $\hat{u}^{S}_{\mu}$ of (1.2) with
$\lambda:=\hat{\lambda}_{\mu}^{S}<0$, then $\hat{u}^{S}_{\mu}$ is a ground
state of (1.2) with ground level $S$.

$(2^{o})$ Suppose $\lambda\in(-\infty,0)$ and $\hat{u}_{\lambda}$ is a ground
state of (1.2) with some $S=S_{\lambda,\mu}(\hat{u}_{\lambda})$, then
$\hat{u}_{\lambda}$ is a fundamental frequency solution of (1.2) with
frequency $\lambda=\hat{\lambda}_{\mu}^{S}$.

###### Proof.

Suppose assertion $(1^{o})$ of the lemma is false. Then there exists a
solution $w$ of (1.2) with $\lambda=\hat{\lambda}_{\mu}^{S}$ such that

$S_{1}:=S_{\hat{\lambda}_{\mu}^{S},\mu}(w)<S_{\hat{\lambda}_{\mu}^{S},\mu}(\hat{u}^{S})=S.$

Observe $\Lambda_{\mu}^{S_{1}}(w)=\hat{\lambda}_{\mu}^{S}$ and
$D\Lambda_{\mu}^{S_{1}}(w)=0$. Hence $\sigma^{S_{1}}(w)=1$, and therefore
$\lambda_{\mu}^{S_{1}}(w)=\Lambda_{\mu}^{S_{1}}(w)$. Hence,

$\hat{\lambda}_{\mu}^{S_{1}}=\min_{u\in H^{1}(\mathbb{R}^{N})\setminus
0}\lambda_{\mu}^{S_{1}}(u)\leq\lambda_{\mu}^{S_{1}}(w)=\Lambda_{\mu}^{S_{1}}(w)=\hat{\lambda}_{\mu}^{S}.$

This contradicts the fact that by Lemma 5.5 the function
$\hat{\lambda}_{\mu}^{S}$ is monotone decreasing and $S_{1}<S$.

Let us prove $(2^{o})$. Suppose $2<q<p<2^{*}$, $\mu>0$. Suppose that
$\lambda\in(-\infty,0)$ and $\hat{u}_{\lambda}$ is a ground state of (1.2)
with some $S=S_{\lambda,\mu}(\hat{u}_{\lambda})$. Since
$\sigma^{S}(\hat{u}_{\lambda})=1$ we infer that
$\Lambda^{S}_{\mu}(\hat{u}_{\lambda})=\lambda^{S}_{\mu}(\hat{u}_{\lambda})=\lambda$,
and thus $\lambda\geq\hat{\lambda}^{S}_{\mu}$. By Corollary 5.6, there exists
$S_{\lambda}>S(\mu)$ and the fundamental frequency solution
$\hat{u}^{S_{\lambda}}_{\mu}$ such that
$\lambda=\hat{\lambda}^{S_{\lambda}}_{\mu}$ and
$S_{\lambda,\mu}(\hat{u}^{S_{\lambda}}_{\mu})=S_{\lambda}$. Then
$S_{\lambda}\geq S=S_{\lambda,\mu}(\hat{u}_{\lambda})$ since
$\hat{u}_{\lambda}$ is a ground state of (1.2). Consequently, by Proposition
5.2, $\hat{\lambda}^{S}_{\mu}\geq\hat{\lambda}^{S_{\lambda}}_{\mu}=\lambda$.
At the same time, by the above $\lambda\geq\hat{\lambda}^{S}_{\mu}$, and thus
$\lambda=\hat{\lambda}^{S}_{\mu}$, i.e., $\hat{u}_{\lambda}$ is a fundamental
frequency solution of (1.2) with the fundamental frequency $\lambda$. The
proof of $(2^{o})$, in the case $2<p<q<2^{*}$, is similar.

∎

######  Corollary 5.8.

Let $p,q\in(2,2^{*})$ and $\mu>0$. For any $\lambda\in(-\infty,0)$ there
exists a ground state $\hat{u}_{\lambda}$ of (1.2) such that
$\hat{u}_{\lambda}\in\hat{G}^{S}(\mu)$ with
$S=S_{\lambda,\mu}(\hat{u}_{\lambda})$.

###### Proof.

Let $\lambda\in(-\infty,0)$. Then Corollary 5.3 entails that there exists
uniquely determines $S:=S_{\lambda}\in(0,+\infty)$ $(S\in(S(\mu),+\infty)$
such that $\lambda=\hat{\lambda}^{S}_{\mu}$. Theorem 1.7 implies that there
exists a fundamental frequency solution $\hat{u}^{S}_{\mu}\in\hat{G}^{S}(\mu)$
of (1.2). Applying Lemma 5.7 we conclude that
$\hat{u}_{\lambda}:=\hat{u}^{S}_{\mu}$ is the ground state of (1.2) with
frequency $\lambda$ and ground level $S$. ∎

##  6\. Proofs of Theorems 1.1, 1.4, 1.7

Proof Theorem 1.1: Suppose $2<p<q<2^{*}$, $\mu>0$, or $2<q<p<2^{*}$,
$\mu>\hat{\mu}^{S}$. Then by Lemma 3.1 there exists is a fundamental frequency
solution $\hat{u}^{S}_{\mu}$ of (1.2) with prescribe action $S$ and frequency
$\lambda=\hat{\lambda}^{S}_{\mu}<0$. Moreover, $\hat{u}^{S}_{\mu}>0$ in
$\mathbb{R}^{N}$ and $\hat{u}_{\mu}\in C^{2}(\mathbb{R}^{N})$. By Lemma 5.7 it
follows that $\hat{u}^{S}_{\mu}$ is a ground state of (1.2).

To conclude the proof of the theorem, it remains to show $(3^{o})$. Assume
that $2<q<p<2^{*}$, $0\leq\mu<\hat{\mu}^{S}$. Suppose, contrary to our claim,
that there exists a weak solutions $\tilde{u}\in H^{1}(\mathbb{R}^{N})$ of
(1.2) such that $\Lambda^{\tilde{S}}_{\mu}(\tilde{u})=:\lambda<0$ with
$\tilde{S}\leq S$. Then
$0>\Lambda^{\tilde{S}}_{\mu}(\tilde{u})=\lambda^{\tilde{S}}_{\mu}(\tilde{u})\geq\lambda^{S}_{\mu}(\tilde{u})$,
and thus $M^{S}(\tilde{u})<\mu<\hat{\mu}^{S}$ which contradicts (2.3).

Proof Theorem 1.4: Let us prove $(1^{o})$. Assume that $2<q<p<2^{*}$, $S>0$.
By Lemma 4.1 there exists a minimizer
$\hat{u}^{S}_{\hat{\mu}^{S}}\in\mathcal{D}$ of (1.9) such that
$\hat{u}^{S}_{\hat{\mu}^{S}}$ weakly satisfies to (1.2) with $\lambda=0$,
$\mu=\hat{\mu}^{S}$. Moreover, $\hat{u}^{S}_{\mu}>0$ in $\mathbb{R}^{N}$,
$\hat{u}^{S}_{\mu}\in C^{2}(\mathbb{R}^{N})$ and
$S_{0,\hat{\mu}^{S}}(\hat{u}^{S}_{\hat{\mu}^{S}})=S$.

Let us show that $\hat{u}^{S}_{\hat{\mu}^{S}}$ is a ground state of (1.2).
Conversely, suppose that there exists a weak solution
$v\in\mathcal{D}\setminus 0$ of (1.2) such that $DS_{0,\hat{\mu}^{S}}(v)=0$
and $\tilde{S}:=S_{0,\hat{\mu}^{S}}(v)<S$. Then
$M^{\tilde{S}}(v)=\hat{\mu}^{S}$, and since $\tilde{S}<S$, we have
$M^{S}(v)<M^{\tilde{S}}(v)=\hat{\mu}^{S}$ which contradicts (2.3).

To show that $\hat{u}^{S}_{\hat{\mu}^{S}}$ is a fundamental frequency
solution, it is sufficient to note that by Corollary 2.6 equation (1.2) with
$\mu=\hat{\mu}^{S}$ can not have a solution with frequency $\lambda<0$.

Let us prove $(2^{o})$. Assume $2<p<q<2^{*}$ and $\mu>0$. Suppose, contrary to
our claim, that problem (1.2) with $\lambda=0$ has a weak solution
$\bar{u}_{\mu}\in\mathcal{D}$. Then $0<S_{0,\mu}(\bar{u}_{\mu})<+\infty$, and
for $S:=S_{0,\mu}(\bar{u}_{\mu})$ we have $M^{S}(\bar{u}_{\mu})=\mu$,
$DM^{S}(\bar{u}_{\mu})=0$. Hence
${\displaystyle\frac{d}{ds}M^{S}(s\bar{u}_{\mu})|_{s=1}=0}$. However, in the
case $2<p<q<2^{*}$, the function $s\mapsto M^{S}(s\bar{u}_{\mu})$ can not have
nonzero critical points.

Proof of Theorems 1.7 follows from Lemma 5.4 .

##  7\. Proof of Theorem 1.9

We call a function $w(x)$, $x\in\mathbb{R}^{N}$ radial if it is spherically
symmetric: $w(x)=w(r)$, where $r=|x|$, and decreases with respect to $r$.

######  Proposition 7.1.

If $S>0$ and $2<q<p<2^{*}$, then for any $\mu\geq\hat{\mu}^{S}$, (1.2)
possesses a physical, positive, radial fundamental frequency solution
$\hat{u}^{*,S}_{\mu}\in\hat{G}^{S}(\mu)$ with prescribe action $S$ and
frequency $\lambda=\hat{\lambda}^{S}_{\mu}<0$. Moreover
$\hat{u}^{*,S}_{\mu}\in C^{2}(\mathbb{R}^{N})$.

###### Proof.

If $\mu\geq\hat{\mu}^{S}$, Lemma 3.1 implies the existence of a minimizer
$\hat{u}^{S}_{\mu}$ of $\lambda^{S}_{\mu}(u)$ in $H^{1}\setminus 0$. Let
$\hat{u}^{*,S}_{\mu}$ denotes the Schwarz spherical rearrangement of
$|\hat{u}^{S}_{\mu}|$ [6]. Then one has $\hat{u}^{*,S}_{\mu}\in H^{1}$,
$A(\hat{u}^{*,S}_{\mu})=A(\hat{u}^{S}_{\mu})$,
$B(\hat{u}^{*,S}_{\mu})=B(\hat{u}^{S}_{\mu})$,
$Q(\hat{u}^{*,S}_{\mu})=Q(\hat{u}^{S}_{\mu})$ and $T(\hat{u}^{*,S}_{\mu})\leq
T(\hat{u}^{S}_{\mu})$ (see [6]). Hence
$\lambda^{S}_{\mu}(\hat{u}^{*,S}_{\mu})\leq\hat{\lambda}^{S}_{\mu}(\hat{u}^{S}_{\mu})=\hat{\lambda}^{S}_{\mu}$,
which implies that $\hat{u}^{*,S}_{\mu}$ is also a minimizer of
$\lambda^{S}_{\mu}(u)$ in $H^{1}\setminus 0$, i.e.,
$\hat{u}^{*,S}_{\mu}\in{G}^{S}(\mu)$. Similar arguments as in the proof of
Lemma 3.1 gives that $\hat{u}^{*,S}_{\mu}\in C^{2}(\mathbb{R}^{N})$.

Notice that the strong in $H^{1}$ limit point $u$ of a sequence of positive,
radial function $u_{n}$ is also a positive, radial function. Hence, the same
arguments that have been used in the proof of Lemma 5.4 apply to the positive,
radial ground states of (1.2), yields the existence of the physical, positive,
radial ground state $\hat{u}^{*,S}_{\mu}\in\hat{G}^{S}(\mu)$,
$\forall\mu>\hat{\mu}^{S}$, $\forall S>0$. ∎

The same conclusion can be drawn for the prescribed mass minimization problem
(1.5), namely

######  Proposition 7.2.

For any $\alpha>\alpha_{0}(\mu)$, the set of minimizers
$\mathcal{M}_{\mu}(\alpha)$ of (1.5) contains a positive, radial minimizer
$\check{u}^{*,\alpha}_{\mu}\in C^{2}(\mathbb{R}^{N})\cap H^{1}$.

It is worth pointing out that by the McLeod Theorem (see Theorem 2 in [22]),
under assumptions $2<q<p<2^{*}$, $\mu>0$, $\lambda\leq 0$ equation (1.2) has
at most one positive, radial solution.

We need also the following consequences of the Shibata Theorem

######  Corollary 7.3.

Assume that $\alpha>\alpha_{0}(\mu)$. Any minimizer $w\in H^{1}$ of problem
(1.5) satisfies

(7.1) $DH_{\mu}(w)+\tau DQ(w)=0,$

where $\tau$ is a Lagrange multiplier such that $\tau>0$.

###### Proof.

We follows an idea from [19]. Assume that $w\in H^{1}$ is a minimizer of
(1.5). Then by the Lagrange multipliers rule there exists $\tau\in\mathbb{R}$
such that (7.1) is satisfied. Hence

$-\tau=\frac{DH_{\mu}(w)(w)}{2Q(w)}.$

Suppose that $\tau<0$. Then
$H_{\mu}^{\prime}(w):=DH_{\mu}(w)(w)\equiv\frac{d}{dt}H_{\mu}(tw)|_{t=1}>0$.
Since $H_{\mu}((1-t)w)=\hat{H}^{\alpha}_{\mu}-t(H_{\mu}^{\prime}(w)+o(1))$ as
$t\to 0$, we can fix a small $t_{b}>0$ such that $w_{b}=(1-t_{b})w$ satisfies
$H_{\mu}(w_{b})<\hat{H}^{\alpha}_{\mu}$. Thus, we have

$\hat{H}^{\alpha_{b}}_{\mu}:=\min\\{H_{\mu}(u):Q(u)=\alpha_{b},~{}u\in
H^{1}\setminus 0\\}\leq H_{\mu}(w_{b})<\hat{H}^{\alpha}_{\mu}.$

Observe $\alpha_{b}:=Q(w_{b})=(1-t_{0})^{2}Q(w)<\alpha$. Hence by the Shibata
Theorem $\hat{H}^{\alpha_{b}}_{\mu}\geq\hat{H}^{\alpha}_{\mu}$. We get a
contradiction. ∎

Let $\mu>0$. Define

$\alpha_{1}(\mu):=\inf_{S>S(\mu)}\alpha^{S},~{}~{}\bar{\alpha}(\mu):=\sup_{S>S(\mu)}\alpha^{S}.$

Then $0\leq\alpha_{1}(\mu)<\bar{\alpha}(\mu)\leq+\infty$.

######  Proposition 7.4.

(i) $\bar{\alpha}(\mu)=+\infty$; (ii) $\alpha_{1}(\mu)\leq\alpha_{0}(\mu)$.

###### Proof.

Let us prove (i). Assume by contradiction that $\bar{\alpha}(\mu)<+\infty$.
Fix $\alpha>\max\\{\alpha_{0}(\mu),\bar{\alpha}(\mu)\\}$. By Proposition 7.2
there exists a solution $v_{\alpha}$ of (1.5), i.e.,
$DS_{\tilde{\lambda},\mu}(v_{\alpha})=0$ with some $\tilde{\lambda}$ which by
Corollary 7.3 satisfies $-\tau:=\tilde{\lambda}<0$. It follows from Corollary
5.8 and Proposition 7.1 that (1.2) possesses a physical, positive, radial
ground state $\hat{u}_{\tilde{\lambda}}\in\hat{G}^{\tilde{S}}(\mu)$ such that
$\tilde{S}:=S_{\tilde{\lambda},\mu}(\hat{u}_{\tilde{\lambda}})\in(S(\mu),+\infty)$
and $DS_{\tilde{\lambda},\mu}(\hat{u}_{\tilde{\lambda}})=0$. The McLeod
Theorem [22] yields that $v_{\alpha}=\hat{u}^{\tilde{S}}_{\mu}$, and
consequently $\alpha=Q(v_{\alpha})=Q(\hat{u}^{\tilde{S}}_{\mu})$. But
$Q(\hat{u}^{\tilde{S}}_{\mu})=\alpha^{\tilde{S}}<\bar{\alpha}(\mu)$ for
$\tilde{S}\in(S(\mu),+\infty)$, which contradicts the assumption. The proof of
(ii) is similar. ∎

Let $S>S(\mu)$ such that $\alpha^{S}\in(\alpha_{0}(\mu),+\infty)$. Introduce

(7.2)
$\check{\lambda}_{\mu}^{S}:=\min\\{\Lambda^{S}_{\mu}(u)\equiv\frac{H_{\mu}(u)-S}{Q(u)}:~{}Q(u)=\alpha^{S},~{}u\in
H^{1}\setminus 0\\}.$

Notice that
$\\{S>S(\mu):\alpha^{S}\in(\alpha_{0}(\mu),+\infty)\\}\neq\emptyset$ since
$\bar{\alpha}(\mu)=+\infty$ and $\alpha_{1}(\mu)\leq\alpha_{0}(\mu)$. Observe
that problem (7.2) is equivalent to the mass prescribed minimization problem
(1.5) so that the set of solutions of (7.2) coincides with
$\mathcal{M}_{\mu}(\alpha^{S})$, i.e.,

$\mathcal{M}_{\mu}(\alpha^{S})=\\{u\in
H^{1}:\Lambda_{\mu}^{S}(u)=\check{\lambda}_{\mu}^{S},~{}Q(u)=\alpha^{S}\\},$

and $\check{\lambda}_{\mu}^{S}=(\hat{H}^{\alpha^{S}}_{\mu}-S)/\alpha^{S}$.
Furthermore, since $\Lambda^{S}_{\mu}(u)\leq\lambda^{S}_{\mu}(u)$, $\forall
u\in H^{1}\setminus 0$, we have

(7.3)
$\check{\lambda}_{\mu}^{S}\leq\hat{\lambda}_{\mu}^{S},\,\,\,\forall\alpha^{S}\in(\alpha_{0}(\mu),+\infty).$

######  Lemma 7.5.

For any $\alpha\in(\alpha_{0}(\mu),+\infty)$, there exists $S>S(\mu)$ such
that $\alpha=\alpha^{S}$ and
$\check{\lambda}_{\mu}^{S}=\hat{\lambda}_{\mu}^{S}$.

###### Proof.

Assume that $\alpha>\alpha_{0}(\mu)$. Consider

(7.4) $\kappa:=\min\\{\frac{H_{\mu}(u)}{Q(u)}:~{}Q(u)=\alpha,~{}u\in
H^{1}\setminus 0\\}.$

Evidently if $\alpha=\alpha^{S}$ for $\alpha^{S}\in(\alpha_{0}(\mu),+\infty)$,
then (7.4) is equivalent to (7.2). The Shibata Theorem and Proposition 7.2
imply that there exists a positive, radial minimizer $v_{\alpha}$ of (7.4).
The Lagrange multipliers rule yields

(7.5)
$D\frac{H_{\mu}(v_{\alpha})}{Q(v_{\alpha})}+\nu^{\alpha}DQ(v_{\alpha})=0,$

where $\nu^{\alpha}\in\mathbb{R}$ is a Lagrange multiplier. Thus,
$DS_{\lambda,\mu}(v_{\alpha})=0$, where
$\lambda:=\kappa-\nu^{\alpha}\alpha^{2}$, and $\lambda<0$ due to Corollary
7.3. By the McLeod Theorem [22], $v_{\alpha}$ is a unique, positive, radial
solution of the equation $DS_{\lambda,\mu}(v_{\alpha})=0$. On the other hand,
Proposition 7.1 and Corollary 5.8 implies that this equation has a physical,
positive, radial ground state
$\hat{u}_{\lambda}=\hat{u}_{\mu}^{S}\in\hat{G}^{S}(\mu)$ with
$S=S_{\lambda,\mu}(\hat{u}_{\lambda})$. Hence,
$v_{\alpha}=\hat{u}_{\mu}^{S}\in\hat{G}^{S}(\mu)$ and by (5.4),
$\alpha=Q(v_{\alpha})=Q(\hat{u}_{\mu}^{S})=\alpha^{S}$. Thus,
$\hat{u}_{\mu}^{S}$ is a minimizer of (7.2) and
$\check{\lambda}_{\mu}^{S}=\hat{\lambda}_{\mu}^{S}$, $\alpha=\alpha^{S}$. ∎

Proof Theorem 1.9: For any $\alpha\in(\alpha_{0}(\mu),+\infty)$, there exists
$S>S(\mu)$ such that $\hat{G}^{S}(\mu)\subseteq\mathcal{M}_{\mu}(\alpha)$.
Indeed, fix $\alpha>\alpha_{0}(\mu)$, then by Lemma 7.5 there exists
$S>S(\mu)$ such that $\alpha=\alpha^{S}$ and
$\check{\lambda}_{\mu}^{S}=\hat{\lambda}_{\mu}^{S}$. Hence by (5.4) and (7.2),
we have

$\displaystyle\hat{G}^{S}(\mu)\subseteq\\{u\in H^{1}:$
$\displaystyle\Lambda_{\mu}^{S}(u)=\hat{\lambda}_{\mu}^{S},~{}\sigma^{S}(u)=1,~{}Q(u)=\alpha^{S}\\}\subseteq$
$\displaystyle\\{u\in
H^{1}:\Lambda_{\mu}^{S}(u)=\check{\lambda}_{\mu}^{S},~{}Q(u)=\alpha\\}=\mathcal{M}_{\mu}(\alpha).$

Now taking into account that by the Shibata Theorem
$\mathcal{M}_{\mu}(\alpha)$ is an orbtally stable set of solutions of (1.2),
we obtain the proof.

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