# Normalized solutions for the Schrödinger equation with combined Hartree type
and power nonlinearities

Shuai Yaoa,b, Juntao Suna, Tsung-fang Wuc
a _School of Mathematics and Statistics, Shandong University of Technology,
Zibo 255049, PR China_
b _School of Mathematics and Statistics, Central South University, Changsha
410083, PR China_
c _Department of Applied Mathematics, National University of Kaohsiung,
Kaohsiung 811, Taiwan_ E-mail address: shyao2019@163.com(S. Yao)E-mail
address: jtsun@sdut.edu.cn(J. Sun)E-mail address<EMAIL_ADDRESS>(T.-F. Wu)

###### Abstract

We investigate normalized solutions for the Schrödinger equation with combined
Hartree type and power nonlinearities, namely

$\left\\{\begin{array}[]{ll}-\Delta u+\lambda
u=\gamma(I_{\alpha}\ast\left|u\right|^{p})|u|^{p-2}u+\mu|u|^{q-2}u&\quad\text{in}\quad\mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}|u|^{2}dx=c,&\end{array}\right.$

where $N\geq 2$ and $c>0$ is a given real number. Under different assumptions
on $\gamma,\mu,p$ and $q$, we prove several nonexistence, existence and
multiplicity results. In particular, we are more interested in the cases when
the competing effect of Hartree type and power nonlinearities happens, i.e.
$\gamma\mu<0,$ including the cases $\gamma<0,\mu>0$ and $\gamma>0,\mu<0.$ Due
to the different ”strength” of two types of nonlinearities, we find some
differences in results and in the geometry of the corresponding functionals
between these two cases.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
  2. 2 Preliminary
    1. 2.1 Several important inequalities and a property
    2. 2.2 Decomposition of Pohozaev manifold $\mathcal{M}(c)$
    3. 2.3 A general minimax theorem
    4. 2.4 Compactness lemma for Palais-Smale sequence
  3. 3 The case of $\gamma>0$ and $\mu>0$
    1. 3.1 Bounded from below on $S(c)$
      1. 3.1.1 The proof of Theorem 1.2
    2. 3.2 Unbounded from below on $S(c)$
      1. 3.2.1 Mass subcritical and mass supercritical case
      2. 3.2.2 Mass critical and mass supercritical case
      3. 3.2.3 Double mass supercritical case
      4. 3.2.4 The proof of Theorem 1.3
  4. 4 The case of $\gamma<0$ and $\mu>0$
    1. 4.1 Bounded from below on $S(c)$
      1. 4.1.1 The proof of Theorem 1.4
    2. 4.2 Unbounded from below on $S(c)$
      1. 4.2.1 Bounded from below on $\mathcal{M}(c)$
      2. 4.2.2 Unbounded from below on $\mathcal{M}(c)$
      3. 4.2.3 The proof of Theorem 1.5
  5. 5 The case of $\gamma>0$ and $\mu<0$
    1. 5.1 Bounded from below on $S\left(c\right)$
    2. 5.2 The proof of Theorem 1.6
    3. 5.3 Unbounded from below on $S(c)$
      1. 5.3.1 Bounded from below on $\mathcal{M}(c)$
      2. 5.3.2 Unbounded from below on $\mathcal{M}(c)$
    4. 5.4 The proof of Theorem 1.7

##  1 Introduction

In this paper we consider the nonlinear Schrödinger (NLS) equation with
combined Hartree type and power nonlinearities:

$i\partial_{t}\psi-\Delta\psi=\gamma(I_{\alpha}\ast|\psi|^{p})|\psi|^{p-2}\psi+\mu|\psi|^{q-2}\psi,\text{
}(t,x)\in\mathbb{R\times R}^{N},$ (1)

where $N\geq 2,\frac{N+\alpha}{N}<p<2_{\alpha}^{\ast}$
$(2_{\alpha}^{\ast}=\frac{N+\alpha}{N-2}$ if $N\geq 3$ and
$2_{\alpha}^{\ast}=\infty$ if $N=2),2<q<2^{\ast}$ $(2^{\ast}=\frac{2N}{N-2}$
if $N\geq 3$ and $2^{\ast}=\infty$ if $N=2)$, the parameters
$\gamma,\mu\in\mathbb{R}$ and $I_{\alpha}$ is the Riesz potential of order
$\alpha\in(0,N)$ defined by

$I_{\alpha}=\frac{A(N,\alpha)}{|x|^{N-\alpha}}\quad\text{with}\quad
A(N,\alpha)=\frac{\Gamma(\frac{N-\alpha}{2})}{\pi^{N/2}2^{\alpha}\Gamma(\frac{\alpha}{2})}\text{
for each }x\in\mathbb{R}^{N}\backslash\\{0\\},$ (2)

and $\ast$ the convolution product on $\mathbb{R}^{N}.$

An important topic concerning nonlinear evolution equations such as Eq. (1) is
to study their standing wave solutions. A standing wave solution of Eq. (1) is
a solution of the form $\psi(t,x)=e^{-i\lambda t}u(x)$, where
$\lambda\in\mathbb{R}$ and $u$ satisfies the stationary equation

$-\Delta u+\lambda
u=\gamma(I_{\alpha}\ast\left|u\right|^{p})|u|^{p-2}u+\mu|u|^{q-2}u\quad\text{in}\quad\mathbb{R}^{N}.$
(3)

At this point, there are two different ways to deal with Eq. (3) according to
the role of $\lambda$:
$(i)$ the frequency $\lambda$ is a fixed and assigned parameter;
$(ii)$ the frequency $\lambda$ is an unknown of the problem.

For case $(i),$ one can see that solutions of Eq. (3) correspond to critical
points of the functional defined in $H^{1}(\mathbb{R}^{N})$

$I(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx+\frac{\lambda}{2}\int_{\mathbb{R}^{N}}|u|^{2}dx-\frac{\gamma}{2p}\int_{\mathbb{R}^{N}}(I_{\alpha}\ast|u|^{p})|u|^{p}dx-\frac{\mu}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx.$

This case has been extensively studied by many authors in the last years,
depending on $\gamma,\mu,p$ and $q,$ see for example [2, 14, 16, 20, 29, 32,
38, 39, 40] and the references therein.

Alternatively one can look for solutions of Eq. (3) with $\lambda$ unknown. In
this case $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier and
$L^{2}$-norms of solutions are prescribed, which are usually called normalized
solutions. This study seems particularly meaningful from the physical point of
view, since solutions of Eq. (1) conserve their mass along time.

Here we focus on this issue. For $c>0$ given, we consider the problem of
finding solutions to

$\left\\{\begin{array}[]{ll}-\Delta u+\lambda
u=\gamma(I_{\alpha}\ast\left|u\right|^{p})|u|^{p-2}u+\mu|u|^{q-2}u&\quad\text{in}\quad\mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}|u|^{2}dx=c.&\end{array}\right.$ ($P_{c}$)

Solutions of Problem $(P_{c})$ can be obtained as critical points of the
energy functional $E_{\gamma,\mu}:H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}$
given by

$E_{\gamma,\mu}(u):=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{\gamma}{2p}\int_{\mathbb{R}^{N}}(I_{\alpha}\ast|u|^{p})|u|^{p}dx-\frac{\mu}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx$
(4)

on the constraint

$S(c):=\left\\{u\in H^{1}(\mathbb{R}^{N})\text{ }|\text{
}\int_{\mathbb{R}^{N}}|u|^{2}dx=c\right\\}.$ (5)

Note that $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier. It is easy
to show that $E_{\gamma,\mu}$ is a well-defined and $C^{1}$ functional on
$S(c)$ for $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}$ and $2<q<2^{\ast}.$

As we will see, the number and properties of normalized solutions to Problem
$(P_{c})$ are strongly affected by further assumptions on the exponents $p,q$,
on the parameters $\gamma,\mu$ and on the mass $c>0.$ When the problem
involves purely power nonlinearity, i.e. $\gamma=0$ and $\mu>0,$ from the
variational point of view, $E_{0,\mu}$ is bounded from below on
$S\left(c\right)$ for $2<q<2+4/N,$ here $2+4/N$ is called the $L^{2}$-critical
exponent for the power nonlinearity. Thus, for each $c>0$ a ground state can
be found as a global minimizer of $E_{0,\mu}$ on $S\left(c\right)$, see [27,
33, 34]. Moreover, the set of ground states is orbitally stable [11, 33]. For
$2+4/N<q<2^{\ast}$, on the contrary, $E_{0,\mu}$ is unbounded from below on
$S\left(c\right)$. However, Jeanjean [18] showed that a normalized ground
state does exist for each $c>0$ also in this case. The associated standing
wave is strongly unstable [9, 22].

When the problem involves purely Hartree nonlinearity, i.e. $\gamma>0$ and
$\mu=0,$ Eq. (3) is usually called the nonlinear Choquard or Choquard–Pekar
equation. It has several physical origins. In the physical case of $N=3$ and
$\alpha=p=2,$ the problem

$-\Delta u+\lambda u=\gamma\left(I_{2}\ast
u^{2}\right)u\quad\text{in}\quad\mathbb{R}^{3}$ (6)

appears as a model in quantum theory of a polaron at rest [30]. The time-
dependent form of Eq. (6) also describes the self-gravitational collapse of a
quantum mechanical wave-function (see [31]), in which context it is usually
called the Schrödinger–Newton equation. Lieb [24] proved the existence and
uniqueness of normalized solutions for Eq. (6) on $S\left(c\right)$ by using
the symmetrization techniques, and Lions [26] studied the existence and
stability issues of normalized solutions for Eq. (6) on $S\left(c\right)$.
Recently, for the general Choquard equation, i.e. Eq. (3) with $\gamma>0$ and
$\mu=0,$ Li and Ye [23] concluded that if $N\geq 3$ and
$\frac{N+\alpha+2}{N}<p<\frac{N+\alpha}{N-2},$ then $E_{\gamma,0}$ has a MP
geometry on $S(c)$ and there exists a normalized solution for each $c>0.$
Usually, $\frac{N+\alpha+2}{N}$ is called the $L^{2}$-critical exponent for
Hartree nonlinearity.

In the specific case of $N=3,\gamma<0,\mu>0$ and $\alpha=p=2,$ Eq. (1) reduces
to

$i\partial_{t}\psi-\Delta\psi+|\gamma|(I_{2}\ast|\psi|^{2})\psi=\mu|\psi|^{q-2}\psi,\text{
}(t,x)\in\mathbb{R\times R}^{3},$ (7)

and its stationary counterpart

$-\Delta u+\lambda u+|\gamma|(I_{2}\ast
u^{2})u=\mu|u|^{q-2}u\quad\text{in}\quad\mathbb{R}^{3},$ (8)

appearing in the physical literature as an approximation of the Hartree–Fock
model of a quantum many–body system of electrons, see [21] for a mathematical
introduction into Hartree–Fock method and further references therein. Within
this context Eqs. (7) and (8) are known as Schrödinger–Poisson equations.
Clearly, they can be viewed as the NLS equations with combined Hartree type
and power nonlinearities, being the $L^{2}$-subcritical case for Hartree type
nonlinearity, since $2=p<\frac{N+\alpha+2}{N}=7/3.$ Bellazzini and Siciliano
[7, 8] and Kikuchi [19] proved that Eq. (8) has a ground state as a global
minimizer of $E_{\gamma,\mu}$ on $S(c)$ for $2<q<3$ and $c>0$ sufficiently
small, and for $3\leq q<10/3$ and $c>0$ sufficiently large. Bellazzini et al.
[6] concluded that for $10/3<q<6,$ $E_{\gamma,\mu}$ is unbounded from below on
$S(c)$ and critical points exist provided that $c>0$ is sufficiently small by
a mountain pass argument developed on $S(c).$ Moreover, the strong instability
of standing waves of Eq. (7) at the mountain pass energy level is proved. Very
recently, Feng et al. [13] proved that standing waves of Eq. (7) are strongly
unstable by blow up for $q=10/3.$

Concerning on the combined nonlinearities, we would like to introduce some
other known results. Soave [35, 36] studied existence and properties of ground
states for the NLS equation with combined power nonlinearities

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

on the constraint (5), where $q<p\leq 2^{\ast},c>0$ and $\mu\in\mathbb{R}.$
Corresponding results are extended to the fractional NLS equation with same
combined power nonlinearities by Luo and Zhang [28]. Bhattarai [10] considered
the fractional NLS equation with combined Hartree type and power
nonlinearities

$\left\\{\begin{array}[]{ll}(-\triangle u)^{\beta}+\lambda
u=\gamma(I_{\alpha}\ast\left|u\right|^{p})|u|^{p-2}u+\mu|u|^{q-2}u&\text{
in}\quad\mathbb{R}^{N},\\\
\int_{\mathbb{R}^{N}}|u|^{2}dx=c,&\end{array}\right.$

where $N\geq 2,0<\beta<1,I_{\alpha}$ is as in (2) and $\gamma,\mu\geq 0$
satisfying $\gamma+\mu\neq 0.$ When $2\leq p<\frac{N+\alpha+2\beta}{N}$ and
$2<q<2+\frac{4\beta}{N},$ a normalized ground state can be found for each
$c>0$, and the set of ground states is orbitally stable. Cingolani and
Jeanjean [12] investigated NLS equation involving combined logarithmic
convolution and power nonlinearities

$\left\\{\begin{array}[]{ll}-\triangle u+\lambda
u=\gamma(\log|\cdot|\ast|u|^{2})u+\mu|u|^{q-2}u&\text{
in}\quad\mathbb{R}^{2},\\\
\int_{\mathbb{R}^{2}}|u|^{2}dx=c.&\end{array}\right.$ (9)

Under different assumptions on $\gamma,\mu\in\mathbb{R}$ and $p>2,$ several
existence and multiplicity results of Problem (9) are obtained.

One can see that the study of normalized solutions for the NLS equation with
combined Hartree type and power nonlinearities seems to be far from over. As
far as we know, there are only few papers dealing with such problems besides
the ones already mentioned [6, 7, 8, 13, 19] for the Schrödinger-Poisson
equation and [10] for the fractional NLS equation. In this paper, we shall
fully study the nonexistence, existence and multiplicity of normalized
solutions for Problem $(P_{c})$ under all possible assumptions on $p,q,\gamma$
and $\mu.$ In particular, we are more interested in the cases when the
competing effect of Hartree type and power nonlinearities happens, i.e.
$\gamma\mu<0,$ including the cases $\gamma<0,\mu>0$ and $\gamma>0,\mu<0.$ As
we will see, there are some differences in results and the geometry of the
corresponding functionals $E_{\gamma,\mu}$ between these two competing cases,
since ”the strength” of Hartree type and power nonlinearities are different.
For example, for double $L^{2}$-supercritical case, one normalized solution
admits when $\gamma<0$ and $\mu>0$ while two normalized solutions admits when
$\gamma>0$ and $\mu<0.$

###  1.1 Main results

Before stating our main results, we give the definition of ground state in the
following sense.

###### Definition 1.1

We say that $u$ is a ground state of Problem $(P_{c})$ if it is a solution of
Problem $(P_{c})$ having minimal energy among all the solutions:

$E_{\gamma,\mu}^{\prime}|_{S(c)}(u)=0\text{ and
}E_{\gamma,\mu}(u)=\\{E_{\gamma,\mu}(v)\text{ }|\text{
}E_{\gamma,\mu}^{\prime}|_{S(c)}(v)=0\text{ and }v\in S(c)\\}.$

The first case to analyze is given by $\gamma>0$ and $\mu>0$, which means that
two nonlinearities are both focusing. As a first main result, we provide
explicit conditions on $p,q,\gamma,\mu$ under which functional
$E_{\gamma,\mu}$ is bounded from below on $S(c)$ and the infimum

$\sigma(c):=\inf\limits_{u\in S(c)}E_{\gamma,\mu}(u)$ (10)

is achieved. We prove the following result.

###### Theorem 1.2

Let $N\geq 2,\gamma>0$ and $\mu>0.$ Then the following statements are true.

  * $(a)$

Assume that one of the three following conditions holds:
$(i)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and $2<q<2+\frac{4}{N};$
$(ii)$ $p=\frac{N+\alpha+2}{N},2<q<2+\frac{4}{N}$ and
$\gamma^{N/(\alpha+2)}c<\|Q_{(N+\alpha+2)/N}\|_{2}^{2};$
$(iii)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},q=2+\frac{4}{N}$ and
$\mu^{N/2}c<(\frac{N+2}{\overline{S}N})^{N/2},$
where $\overline{S}$ is the sharp constant of the Gagliardo-Nirenberg
inequality (14) below and $Q_{(N+\alpha+2)/N}$ is a ground state of Eq. (16)
below. Then the infimum $\sigma(c)<0$ defined in (10) is achieved by $u_{0}\in
S(c)$ which is a ground state of Problem $(P_{c}).$

  * $(b)$

If $p=\frac{N+\alpha+2}{N}$ and $q=2+\frac{4}{N}$, then there exists a
constant $c_{0}>0$ satisfying

$\frac{2\mu}{q}\overline{S}c_{0}^{2/N}+\gamma
c_{0}^{(2+\alpha)/N}\|Q_{(N+\alpha+2)/2}\|_{2}^{-2(\alpha+2)/N}=1$

such that $\sigma(c)$ has no minimizer for $0<c<c_{0}.$ Moreover,
$\sigma(c)=0$ for $0<c\leq c_{0}.$

We now consider all the other conditions on $p,q,\gamma,\mu$ such that
functional $E_{\gamma,\mu}$ is unbounded from below on $S\left(c\right).$ Then
it will not be possible to find a global minimizer. In order to seek for
critical points of $E_{\gamma,\mu}$ restricted to $S\left(c\right),$ we shall
use the Pohozaev manifold $\mathcal{M}\left(c\right),$ a natural constraint of
$E_{\gamma,\mu},$ that contains all the critical points of $E_{\gamma,\mu}$
restricted to $S(c).$ It is given by

$\mathcal{M}(c):=\left\\{u\in S(c)\text{ }|\text{ }Q(u)=0\right\\},$ (11)

where $Q(u)=0$ is the Pohozaev type identity corresponding to Eq. (3). For its
more details, we refer the reader to Section 2. As we shall see, functional
$E_{\gamma,\mu}$ restricted to $\mathcal{M}(c)$ is bounded from below. We
expect to find critical points with positive level by applying the minimax
method to $E_{\gamma,\mu}$ restricted to $\mathcal{M}(c),$ introduced by
Bartsch and Soave [4, 5]. Set

$c_{\ast}:=\left(\frac{4q(Np-N-\alpha-2)}{\mu N\overline{S}(q-2)(2Np-
Nq-2\alpha)}\right)^{\frac{2(Np-N-\alpha-2)}{4p+2\alpha-q(2+\alpha)}}\left(\frac{2(4-(q-2)N)\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)(2Np-
Nq-2\alpha)}\right)^{\frac{4-N(q-2)}{4p+2\alpha-q(2+\alpha)}},$ (12)

we have the following result.

###### Theorem 1.3

Let $N\geq 2,\gamma>0$ and $\mu>0.$ Then the following statements are true.

  * $(a)$

Assume that one of the two following conditions holds:
$(i)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2<q<2+\frac{4}{N}$ and
$c<\left(\frac{4}{N(q-2)}\right)^{\frac{N(q-2)-4}{4p+2\alpha-q(2+\alpha)}}c_{\ast};$
$(ii)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},2+\frac{4}{N}<q<2^{\ast}$
and
$c<\left(\frac{Np-N-\alpha}{2}\right)^{\frac{2(Np-N-\alpha-2)}{4p+2\alpha-q(2+\alpha)}}c_{\ast}.$
Then Problem $(P_{c})$ has two positive radial solutions
$(\lambda^{\pm},u^{\pm})\in\mathbb{R}^{+}\times H^{1}(\mathbb{R}^{N})$
satisfying $E_{\gamma,\mu}(u^{+})<0<E_{\gamma,\mu}(u^{-})$, where $u^{+}\in
S(c)$ is a radial ground state of Problem $(P_{c}).$ In particular, it holds
$\|\nabla u^{+}\|_{2}<\|\nabla u^{-}\|_{2}$ for $c>0$ sufficiently small.

  * $(b)$

Assume that one of the three following conditions holds:
$(i)$ $p=\frac{N+\alpha+2}{N},2+\frac{4}{N}<q<2^{\ast}$ and
$\gamma^{N/(\alpha+2)}c<\|Q_{(N+\alpha+2)/N}\|_{2}^{2};$
$(ii)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},q=2+\frac{4}{N}$ and
$\mu^{N/2}c<\left(\frac{N+2}{\overline{S}N}\right)^{N/2};$
$(iii)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and
$2+\frac{4}{N}<q<2^{\ast}.$
Then Problem $(P_{c})$ has a radial ground state
$(\lambda^{-},u^{-})\in\mathbb{R^{+}}\times H^{1}(\mathbb{R}^{N})$ satisfying
$E_{\gamma,\mu}(u^{-})>0.$

In Theorem 1.3 $(a),$ the solution $u^{+}$ with negative level can be obtained
as an interior local minimizer of $E_{\gamma,\mu}$ on the set
$\mathbf{A}_{R}:=\left\\{u\in S(c)\text{ }|\text{ }\|\nabla
u\|_{2}^{2}<R\right\\}$ for some $R>0$ small enough.

The second case to analyze is given by $\gamma<0$ and $\mu>0$, which means
that Hartree type nonlinearity is defocusing while the power one is focusing.
First of all, we give the following result when functional $E_{\gamma,\mu}$ is
bounded from below on $S(c).$

###### Theorem 1.4

Let $N\geq 2,\gamma<0$ and $\mu>0.$ Then the following statements are true.

  * $(a)$

Assume that one of the four following conditions holds:
$(i)$
$\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},2<q<\frac{4p+2\alpha}{\alpha+2}$
and $c<c_{1}$ for some $c_{1}>0;$
$(ii)$
$\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},q=\frac{4p+2\alpha}{\alpha+2}$ and
$\mu>\overline{\mu}$ for some $\overline{\mu}>0;$
$(iii)$
$\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},\frac{4p+2\alpha}{\alpha+2}<q<2+\frac{4}{N}$
and $c>c_{3}$ for some $c_{3}>0;$
$(iv)$ $\frac{N+\alpha+2}{N}\leq p<2_{\alpha}^{\ast},2<q<2+\frac{4}{N}$ and
$c<c_{2}$ for some $c_{2}>0.$
Then the infimum $\sigma(c)<0$ defined in (10) is achieved by $u_{0}\in S(c)$
which is a ground state of Problem $(P_{c}).$

  * $(b)$

If $\frac{N+\alpha}{N}<p<2_{\alpha}^{\ast}$ and $q=2+\frac{4}{N}$, then
$\sigma(c)$ has no minimizer for
$\mu^{N/2}c<(\frac{N+2}{N\overline{S}})^{N/2}.$ Moreover, $\sigma(c)=0$ for
$\mu^{N/2}c\leq(\frac{N+2}{N\overline{S}})^{N/2}.$

From Theorem 1.4 $(a)$ one can see that the existing results on Schrödinger-
Poisson equation (see [6, 7, 8, 13, 19]) are extended.

Next, we focus on all the other conditions on $p,q,\gamma,\mu$ such that
functional $E_{\gamma,\mu}$ is unbounded from below on $S(c).$ Set

$\widehat{c}_{\ast}:=\frac{\left(2N-q\left(N-2\right)\right)\|Q_{p}\|_{2}^{2p-2}}{Nq+\alpha
q-2Np}\left(\frac{2q}{\mu
N\overline{S}\left(q-2\right)}\right)^{\frac{2\left(N+\alpha+2-Np\right)}{\left(q-2\right)N-4}},$

we have the result as follows.

###### Theorem 1.5

Let $N\geq 2,\gamma<0$ and $\mu>0.$ Assume that one of the three following
conditions holds:
$(i)$ $\frac{N+\alpha}{N}<p\leq\frac{N+\alpha+2}{N},2+\frac{4}{N}<q<2^{\ast}$
and $|\gamma|c^{\frac{\alpha q+2q-2\alpha-4p}{(q-2)N-4}}<\widehat{c}_{\ast};$
$(ii)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<2^{\ast}$
with $2Np-Nq-2\alpha\leq 0$ and $|\gamma|c^{\frac{N+\alpha-(N-2)p}{2}}$ is
sufficiently small;
$(iii)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<2^{\ast}$
with $2Np-Nq-2\alpha>0$ and $|\gamma|$ is sufficiently small.
Then Problem $(P_{c})$ has a positive radial solution
$(\lambda^{-},u^{-})\in\mathbb{R}^{+}\times H^{1}(\mathbb{R}^{N})$ satisfying
$E_{\gamma,\mu}(u^{-})>0.$ In particular, such solution is a radial ground
state under condition $(i)$ or $(ii).$

The minimax method to $E_{\gamma,\mu}$ restricted to $\mathcal{M}(c)$ can be
used to prove Theorem 1.5 $(i)-(ii).$ However, in the proof of Theorem 1.5
$(iii),$ due to the complicate competing effect between Hartree type
nonlinearity and the power nonlinearity, it is difficult to verify that
functional $E_{\gamma,\mu}$ is bounded below on $\mathcal{M}(c)$. Furthermore,
we can not guarantee that an arbitrary $u\in S(c)$ can always be projected
onto $\mathcal{M}(c)$ for $c>0$. For these reasons, the minimax argument to
$E_{\gamma,\mu}$ restricted to $\mathcal{M}(c)$ can not be applied directly.
In order to overcome this difficulty, we shall introduce a submanifold of
codimension $1$ in $S(c)$ given by

$\Lambda(c):=V\cap\mathcal{M}(c),$

where $V$ is an open subset of $S(c)$ defined as (4.2.2) below satisfying the
dilation $u_{t}:=t^{N/2}u(tx)\in V$ for any $u\in V$ and $t>0$, and then apply
the minimax method to $E_{\gamma,\mu}$ restricted to $\Lambda(c).$ Since $V$
has a boundary, we need to insure that our deformation arguments take place
inside $V$ by imposing on the parameter $|\gamma|.$ It is unclear if the
solution $u^{-}$ obtained by us is a ground state of Problem $(P_{c})$ under
condition $(iii),$ since $E_{\gamma,\mu}$ is restricted to $\Lambda(c),$ not
to $\mathcal{M}(c).$

We now turn to the third case when $\gamma>0$ and $\mu<0,$ which means that
Hartree type nonlinearity is focusing while the power one is defocusing. First
of all, we give the following result when $E_{\gamma,\mu}$ is bounded from
below on $S(c)$.

###### Theorem 1.6

Let $N\geq 2,\gamma>0$ and $\mu<0.$ Then the following statements are true.

  * $(a)$

Assume that one of the five following conditions holds:
$(i)$
$\max\\{\frac{N+\alpha}{N},\frac{(\alpha+2)q-2\alpha}{4}\\}<p<\frac{N+\alpha+2}{N},2<q<2+\frac{4}{N}$
and $c>c_{4}$ for some $c_{4}>0;$
$(ii)$ $\frac{N+\alpha}{N}<p<\frac{(\alpha+2)q-2\alpha}{4}$ and
$2+\frac{4\alpha}{N(\alpha+2)}<q<2+\frac{4}{N};$
$(iii)$
$p=\frac{(\alpha+2)q-2\alpha}{4},2+\frac{4\alpha}{N(\alpha+2)}<q<2+\frac{4}{N}$
and $\gamma>\overline{\gamma}$ for some $\overline{\gamma}>0;$
$(iv)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and $2+\frac{4}{N}\leq
q<2^{\ast};$
$(v)$ $p=\frac{N+\alpha+2}{N},2+\frac{4}{N}<q<2^{\ast}$ and
$\gamma^{N/\left(\alpha+2\right)}c>\|Q_{(N+\alpha+2)/N}\|_{2}^{2}.$
Then the infimum $\sigma(c)$ defined in (10) is achieved by $u_{0}\in S(c)$
which is a ground state of Problem $(P_{c}).$

  * $(b)$

If $p=\frac{N+\alpha+2}{N}$ and $2<q<2^{\ast}$, then $\sigma(c)$ has no
minimizer for $\gamma^{N/(\alpha+2)}c\leq\|Q_{(N+\alpha+2)/N}\|_{2}^{2}.$
Moreover, $\sigma(c)=0$ for
$\gamma^{N/(\alpha+2)}c<\|Q_{(N+\alpha+2)/N}\|_{2}^{2}.$

By the above theorem we find a threshold value of $c>0$ separating existence
and nonexistence of minimizers when $p=\frac{N+\alpha+2}{N}$ and
$2+\frac{4}{N}<q<2^{\ast}.$

We are now interested in all the other conditions on $p,q,\gamma,\mu$ such
that functional $E_{\gamma,\mu}$ is unbounded from below on $S(c).$ Set

$\widetilde{c}_{\ast}:=\frac{q(N+\alpha-(N-2)p)}{\overline{S}(2Np-Nq-\alpha
q)}\left(\frac{2\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)}\right)^{\frac{4-(q-2)N}{2(Np-N-\alpha-2)}},$

we derive the following result.

###### Theorem 1.7

Let $N\geq 2,\gamma>0$ and $\mu<0.$ Then the following statements are true.

  * $(a)$

Assume that one of the four following conditions holds:
$(i)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2<q\leq 2+\frac{4}{N}$ with
$2Np-Nq-\alpha q>0$ and $|\mu|c^{\frac{4p+2\alpha-\alpha
q-2q}{2(Np-N-\alpha-2)}}<\widetilde{c}_{\ast};$
$(ii)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2<q\leq 2+\frac{4}{N}$ with
$2Np-Nq-\alpha q\leq 0;$
$(iii)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<2^{\ast}$
with $2Np-Nq-\alpha q>0$ and $|\mu|c^{\frac{2N-(N-2)q}{4}}$ is sufficiently
small;
$(iv)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<2^{\ast}$
with $2Np-Nq-\alpha q\leq 0\leq 2Np-Nq-2\alpha.$
Then Problem $(P_{c})$ has a radial ground state
$(\lambda^{-},u^{-})\in\mathbb{R}^{+}\times H^{1}(\mathbb{R}^{N})$ satisfying
$E_{\gamma,\mu}(u^{-})>0.$

  * $(b)$

If $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and $2+\frac{4}{N}<q<2^{\ast}$
with $2Np-Nq-2\alpha<0,$ then for $|\mu|$ sufficiently small, Problem
$(P_{c})$ has two positive radial solutions
$(\lambda^{\pm},u^{\pm})\in\mathbb{R}^{+}\times H^{1}(\mathbb{R}^{N})$
satisfying $E_{\gamma,\mu}(u^{+})<0<E_{\gamma,\mu}(u^{-})$ and $\|\nabla
u^{-}\|_{2}<\|\nabla u^{+}\|_{2}.$

Similar to Theorems 1.3 and 1.5 $(i)-(ii),$ we can apply the minimax method to
$E_{\gamma,\mu}$ restricted to $\mathcal{M}(c)$ to prove Theorem 1.7 $(a)$.
For Theorem 1.7 $(b)$, the solution $u^{-}$ with positive level can be found
by adopting the argument of Theorem 1.5 $(iii)$, just replacing $V$ with
another open subset $U$ of $S(c)$ (see (5.3.2)), and the solution $u^{+}$ with
negative level can be obtained as an interior local minimizer of
$E_{\gamma,\mu}$ on the set $\mathbf{A}_{r_{0}}:=\left\\{u\in U\text{ }|\text{
}\|\nabla u\|_{2}^{2}<r_{0}\right\\}$ for some $r_{0}>0$ large enough. It is
worth noting that we are not sure whether $u^{+}$ is a ground state of Problem
$(P_{c})$, although its energy level is negative.

By comparing Theorem 1.5 $(iii)$ with Theorem 1.7 $(b),$ we find that Problem
$(P_{c})$ has different number of solutions, although they are both competing
and double $L^{2}$-supercritical case. It shows that the power nonlinearity is
in some sense stronger that Hartree type one.

Note that two normalized solutions are found in both Theorems 1.3 $(a)$ and
1.7 $(b)$. However, from the variational point of view, the geometry of the
corresponding functionals $E_{\gamma,\mu}$ are totally different. In fact, the
critical point $u^{+}$ with negative level is closer to the origin than the
critical point $u^{-}$ with positive level in Theorem 1.3 $(a)$ while the
opposite is true in Theorem 1.7 $(b).$

Finally, let us consider the simplest case when $\gamma<0$ and $\mu<0.$ We
observe that if $\gamma<0,\mu<0,\frac{N+\alpha}{N}<p<2_{\alpha}^{\ast}$ and
$2<q<2^{\ast}$, for each $u\in S(c)$ the fibering map
$g_{u}(t):=E_{\gamma,\mu}(u_{t})$ is strictly increasing, and so the following
nonexistence result is proved.

###### Theorem 1.8

Assume that $N\geq 2,\gamma<0,\mu<0,\frac{N+\alpha}{N}<p<2_{\alpha}^{\ast}$
and $2<q<2^{\ast}$. Then functional $E_{\gamma,\mu}$ does not have critical
points on $S(c)$. In other words, Problem $(P_{c})$ does not exist solution.

###### Remark 1.9

$(I)$ By Theorems 1.2 $(a)$, 1.4 $(a)$ and 1.6 $(a),$ we can prove that any
minimizing sequence has, up to translation, a subsequence converging strongly
in $H^{1}(\mathbb{R}^{N})$, which implies that the set of standing waves
associated to the set of minimizers for $E_{\gamma,\mu}$ on $S(c)$ is
orbitally stable by the classical Cazenave-Lions’ stability argument [11],
further developed in [17].
$(II)$ From Theorems 1.2 $(b),$ 1.4 $(b)$ and 1.6, we only obtain that the
infimum $\sigma(c)$ is not achieved under one of the three following
conditions:
$(i)$ $\gamma>0,\mu>0,p=\frac{N+\alpha+2}{N}$ and $q=2+\frac{4}{N};$
$(ii)$ $\gamma<0,\mu>0,\frac{N+\alpha}{N}<p<2_{\alpha}^{\ast}$ and
$q=2+\frac{4}{N};$
$(iii)$ $\gamma>0,\mu<0,p=\frac{N+\alpha+2}{N}$ and $2<q\leq 2+\frac{4}{N}.$
However, it is not clear if Problem $(P_{c})$ has solution. In a future paper
we will study the existence of solution for Problem $(P_{c})$ when either
$\gamma<0,\mu>0,\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and
$q=2+\frac{4}{N},$ or $\gamma>0,\mu<0,p=\frac{N+\alpha+2}{N}$ and
$2<q<2+\frac{4}{N}.$ For the rest cases, they are open.

##  2 Preliminary

###  2.1 Several important inequalities and a property

For convenience, we set

$A(u)=\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx,\text{
}B(u)=\int_{\mathbb{R}^{N}}\left(I_{\alpha}\ast|u|^{p}\right)|u|^{p}dx,\text{
}C(u)=\int_{\mathbb{R}^{N}}|u|^{q}dx\quad\text{and
}D(u)=\int_{\mathbb{R}^{N}}|u|^{2}dx.$

Then functional $E_{\gamma,\mu}$ in $(\ref{e1-4})$ becomes the following

$E_{\gamma,\mu}(u)=\frac{1}{2}A(u)-\frac{\gamma}{2p}B(u)-\frac{\mu}{q}C(u).$
(13)

In what follows, we introduce several important inequalities.

(1) Gagliardo-Nirenberg inequality ([43]): Let $N\geq 1$ and
$r\in(2,2^{\ast})$. Then there exists a sharp constant
$\overline{S}:=\overline{S}(N,r)>0$ such that

$\|u\|_{r}\leq\overline{S}^{1/r}\|\nabla
u\|_{2}^{\frac{N(r-2)}{2r}}\|u\|_{2}^{\frac{2N-r(N-2)}{2r}}.$ (14)

(2) Gagliardo-Nirenberg inequality of Hartree type ([29, 44]): Let $N\geq 1$
and $\frac{N+\alpha}{N}<p<2_{\alpha}^{\ast}$. Then it holds

$\int_{\mathbb{R}^{N}}(I_{\alpha}\ast|u|^{p})|u|^{p}dx\leq\frac{p}{\|Q_{p}\|_{2}^{2p-2}}\|\nabla
u\|_{2}^{Np-N-\alpha}\|u\|_{2}^{N+\alpha-p(N-2)},$ (15)

where $Q_{p}$ is a positive ground state solution of the following equation

$-\frac{Np-N-\alpha}{2}\Delta
u+\frac{N+\alpha-(N-2)p}{2}u=(I_{\alpha}\ast|u|^{p})|u|^{p-2}u\quad\text{in}\quad\mathbb{R}^{N}.$
(16)

(3) Hardy-Littlewood-Sobolev inequality ([25]): Let $s,r>1$ and $0<\alpha<N$
with $\frac{1}{s}+\frac{N-\alpha}{N}+\frac{1}{r}=2$. For $u\in
L^{s}(\mathbb{R}^{N})$ and $v\in L^{r}(\mathbb{R}^{N}),$ there exists a sharp
constant $C(N,\alpha,s)>0$, independent of $u,v,$ such that

$\iint\limits_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{u(x)v(y)}{|x-y|^{N-\alpha}}dydx\leq
C(N,\alpha,s)\|u\|_{s}\|u\|_{r}.$

###### Remark 2.1

For $p\in[\frac{N+\alpha}{N},2_{\alpha}^{\ast}]$, there exists
$\widetilde{C}:=C(N,\alpha,\frac{2N}{N+\alpha})>0$ such that

$\int_{\mathbb{R}^{N}}(I_{\alpha}\ast|u|^{p})|u|^{p}dx\leq\widetilde{C}\|u\|_{\frac{2Np}{N+\alpha}}^{2p}.$
(17)

Next, we show a splitting property for functional $B$ and its derivative
$B^{\prime}$, which is similar to the Brezis-Lieb type Lemma for nonlocal
nonlinearities [1, 2].

###### Lemma 2.2

Let $u_{n}$ be a bounded sequence in $H^{1}(\mathbb{R}^{N})$. If
$u_{n}\rightarrow u$ a.e. in $\mathbb{R}^{N}$, then
$(i)$ $B(u_{n}-u)=B(u_{n})-B(u)+o(1);$
$(ii)$ $B^{\prime}\left(u_{n}-u\right)=B^{\prime}(u_{n})-B^{\prime}(u)+o(1)$
in $H^{-1}(\mathbb{R}^{N}).$

###  2.2 Decomposition of Pohozaev manifold $\mathcal{M}(c)$

We firstly establish the following Pohozaev identity.

###### Lemma 2.3

Let $u$ be a weak solution to the equation:

$-\Delta u+\lambda u=\gamma(I_{\alpha}\ast u^{p})|u|^{p-2}u+\mu|u|^{q-2}u.$

Then $u$ satisfies the Pohozaev identity

$\frac{N-2}{2}A(u)+\frac{\lambda
N}{2}D(u)=\frac{\gamma(N+\alpha)}{2p}B(u)+\frac{\mu N}{q}C(u).$

As a consequence it satisfies

$A(u)-\frac{\gamma(Np-N-\alpha)}{2p}B(u)-\frac{\mu(q-2)N}{2q}C(u)=0.$ (18)

Proof. The proof is similar to that of [29, Proposition 3.1], we omit it here.

Following the idea of Soave [35, 36] and Cingolani and Jeanjean [12], we will
introduce a natural constraint manifold $\mathcal{M}(c)$ that contains all the
critical points of functional $E_{\gamma,\mu}$ restricted to $S(c)$. For each
$u\in H^{1}(\mathbb{R}^{N})\backslash\\{0\\}$ and $t>0$, we set

$u_{t}(x):=t^{\frac{N}{2}}u(tx).$

Then a direct calculation shows that $\|u_{t}\|_{2}^{2}=\|u\|_{2}^{2},$
$A(u_{t})=t^{2}A(u),$ $B(u_{t})=t^{Np-N-\alpha}B(u)\quad$and
$C(u_{t})=t^{N(q-2)/2}C(u).$ Define the fibering map $t\in(0,\infty)\mapsto
g_{u}(t):=E_{\gamma,\mu}(u_{t})$ given by

$g_{u}(t)=\frac{t^{2}}{2}A(u)-\frac{\gamma t^{Np-N-\alpha}}{2p}B(u)-\frac{\mu
t^{N(q-2)/2}}{q}C(u).$ (19)

By calculating the first and second derivatives of $g_{u}(t),$ we have

$g_{u}^{\prime}(t)=tA(u)-\frac{\gamma(Np-N-\alpha)t^{Np-N-\alpha-1}}{2p}B(u)-\frac{\mu(q-2)Nt^{N(q-2)/2}}{2q}C(u)$

and

$\displaystyle g_{u}^{\prime\prime}(t)$ $\displaystyle=$ $\displaystyle
A(u)-\frac{\gamma(Np-N-\alpha)(Np-N-\alpha-1)}{2p}B(u)t^{Np-N-\alpha-2}$
$\displaystyle-\frac{\mu N(q-2)(N(q-2)-2)}{4q}C(u)t^{\frac{(q-2)N-4}{2}}.$

Notice that $\frac{d}{dt}E(u_{t})=g_{u}^{\prime}(t)=\frac{Q(u_{t})}{t},$ where

$Q(u):=\frac{d}{dt}|_{t=1}E(u_{t})=A(u)-\frac{\gamma(Np-N-\alpha)}{2p}B(u)-\frac{\mu
N(q-2)}{2q}C(u).$

Actually $Q(u)=0$ corresponds to the Pohozaev identity (18). Then by (11) one
has

$\mathcal{M}(c)=\left\\{u\in S(c)\text{ }|\text{ }Q(u)=0\right\\}=\left\\{u\in
S(c)\text{ }|\text{ }g_{u}^{\prime}(1)=0\right\\},$ (20)

which appears as a natural constraint. We also recognize that for any $u\in
S(c)$, the dilated function $u_{t}(x)=t^{N/2}u(tx)$ belongs to the constraint
manifold $\mathcal{M}(c)$ if and only if $t\in\mathbb{R}$ is a critical value
of the fibering map $t\in(0,\infty)\mapsto g_{u}(t)$, namely,
$g_{u}^{\prime}(t)=0$. Thus, it is natural split $\mathcal{M}(c)$ into three
parts corresponding to local minima, local maxima and points of inflection.
Following [42], we define

$\displaystyle\mathcal{M}^{+}(c)$ $\displaystyle:$ $\displaystyle=\left\\{u\in
S(c)\text{ }|\text{ }g_{u}^{\prime}(1)=0,g_{u}^{\prime\prime}(1)>0\right\\};$
$\displaystyle\mathcal{M}^{-}(c)$ $\displaystyle:$ $\displaystyle=\left\\{u\in
S(c)\text{ }|\text{ }g_{u}^{\prime}(1)=0,g_{u}^{\prime\prime}(1)<0\right\\};$
$\displaystyle\mathcal{M}^{0}(c)$ $\displaystyle:$ $\displaystyle=\left\\{u\in
S(c)\text{ }|\text{ }g_{u}^{\prime}(1)=0,g_{u}^{\prime\prime}(1)=0\right\\}.$

Then it holds,

$\displaystyle g_{u}^{\prime\prime}(1)$ $\displaystyle=$
$\displaystyle(N+\alpha+2-Np)A(u)-\frac{\mu N(q-2)(Nq-2Np+2\alpha)}{4q}C(u)$
(21) $\displaystyle=$
$\displaystyle\frac{4-N(q-2)}{2}A(u)-\frac{\gamma(Np-N-\alpha)(2Np-
qN-2\alpha)}{4p}B(u)$ $\displaystyle=$
$\displaystyle\frac{\gamma(Np-N-\alpha)(N+\alpha+2-Np)}{2p}B(u)+\frac{\mu
N(q-2)(4-N(q-2))}{4q}C(u).$

Furthermore, following the argument of Soave [35], we have the following
lemma.

###### Lemma 2.4

If $\mathcal{M}^{0}(c)=\emptyset$, then $\mathcal{M}(c)$ is a smooth
submanifold of codimension $2$ of $H^{1}(\mathbb{R}^{N})$ and a submanifold of
codimension $1$ in $S(c).$

###  2.3 A general minimax theorem

In this subsection, we shall give a general minimax theorem to establish the
existence of a Palais–Smale sequence.

###### Definition 2.5

[15, Definition 3.1] Let $\Theta$ be a closed subset of a metric space
$X\subset H^{1}(\mathbb{R}^{N})$. We say that a class $\mathcal{F}$ of compact
subsets of $X$ is a homotopy-stable family with closed boundary $B$ provided

  * $(a)$

every set in $\mathcal{F}$ contains $\Theta$;

  * $(b)$

for any set $H\in\mathcal{F}$ and any $\eta\in C([0,1]\times X,X)$ satisfying
$\eta(s,x)=x$ for all $(s,x)\in(\\{0\\}\times X)\cup([0,1]\times\Theta)$, we
have that $\eta(\\{1\\}\times H)\in\mathcal{F}$.

###### Lemma 2.6

[15, Theorem 3.2] Let $\varphi$ be a $C^{1}$-functional on a complete
connected $C^{1}$-Finsler manifold $X$ (without boundary) and consider a
homotopy stable family $\mathcal{F}$ of compact subsets of $X$ with a closed
boundary $B$. Set

$c=c(\varphi,\mathcal{F})=\inf_{H\in\mathcal{F}}\max_{u\in H}\varphi(u)$

and suppose that $\sup\varphi(\Theta)<c.$ Then for any sequence of sets
$\\{H_{n}\\}$ in $\mathcal{F}$ such that
$\lim_{n\rightarrow\infty}\sup_{H_{n}}\varphi=c$, there exists a sequence
$\\{u_{n}\\}$ in $X$ such that
$(i)$ $\lim_{n\rightarrow\infty}\varphi(u_{n})=c;$ $(ii)$
$\lim_{n\rightarrow\infty}\left\|\varphi^{\prime}(u_{n})\right\|=0;$ $(iii)$
$\lim_{n\rightarrow\infty}\text{dist}(u_{n},H_{n})=0.$
Furthermore, if $d\varphi$ is uniformly continuous, then $u_{n}$ can be chosen
to be in $H_{n}$ for each $n$.

###  2.4 Compactness lemma for Palais-Smale sequence

###### Lemma 2.7

Let $\\{u_{n}\\}\subset\mathcal{M}(c)\cap H_{r}^{1}(\mathbb{R}^{N})$ be a
bounded Palais-Smale sequence for $E(u)|_{S(c)}$ at level
$\overline{\beta}\neq 0$. Then, up to a subsequence, $u_{n}\rightarrow u$
strongly in $H^{1}(\mathbb{R}^{N})$, and $u\in S(c)$ is a solution of Problem
$(P_{c})$ for some $\lambda>0$.

Proof. Since $\left\\{u_{n}\right\\}\subset\mathcal{M}(c)$ is bounded and the
embedding $H_{r}^{1}(\mathbb{R}^{N})\hookrightarrow L^{s}(\mathbb{R}^{N})$ is
compact for $s\in(2,2^{\ast})$, there exists $u\in H_{r}^{1}(\mathbb{R}^{N})$
such that $u_{n}\rightharpoonup u$ weakly in $H_{r}^{1}(\mathbb{R}^{N})$,
$u_{n}\rightarrow u$ strongly in $L^{s}(\mathbb{R}^{N})$ for
$s\in(2,2^{\ast})$, and a.e. in $\mathbb{R}^{N}$. By the Lagrange multipliers
rule, there exists $\lambda_{n}\in\mathbb{R}$ such that for every $\varphi\in
H^{1}(\mathbb{R}^{N})$,

$\int_{\mathbb{R}^{N}}\nabla u_{n}\nabla\varphi
dx+\lambda_{n}\int_{\mathbb{R}^{N}}u_{n}\varphi
dx-\gamma\iint\limits_{\mathbb{R}^{N}\times\mathbb{R}^{N}}\frac{|u_{n}|^{2p-2}u_{n}}{|x-y|^{N-\alpha}}\varphi
dxdy-\mu\int_{\mathbb{R}^{N}}|u_{n}|^{q-2}u_{n}\varphi dx=o(1)\|\varphi\|.$
(23)

In particular, we have

$\lambda_{n}c=-A(u_{n})+\gamma B(u_{n})+\mu C(u_{n})+o(1),$

which implies that $\\{\lambda_{n}\\}$ is bounded, and so we can assume that
$\lambda_{n}\rightarrow\lambda\in\mathbb{R}$. Since $Q(u_{n})=0$, it holds

$\lambda_{n}c=\gamma\left(1-\frac{Np-N-\alpha}{2p}\right)B(u_{n})+\mu\left(1-\frac{N(q-2)}{2q}\right)C(u_{n})+o(1).$
(24)

We now divide the arguments into several cases as follows.

Case $(a):\gamma>0,\mu>0,\frac{N+\alpha}{N}<p<2_{\alpha}^{\ast}$ and
$2<q<2^{\ast}$. It follows from $(\ref{e2-14})$ that $\lambda\geq 0$, with
equality only if $u\equiv 0$. But $u$ cannot be identically $0$, since
$E_{\gamma,\mu}(u_{n})\rightarrow\overline{\beta}\neq 0$. Thus, we obtain that
up to a subsequence $\lambda_{n}\rightarrow\lambda>0$.

Case $(b):\gamma<0,\mu>0,\frac{N+\alpha}{N}<p<2_{\alpha}^{\ast}$ and
$2+\frac{4}{N}<q<2^{\ast}$. Since $Q(u_{n})=0$, we have

$A(u_{n})=\frac{\gamma(Np-N-\alpha)}{2p}B(u_{n})+\frac{\mu
N(q-2)}{2q}C(u_{n})\leq\frac{\mu N(q-2)}{2q}C(u_{n}).$

Then from (14) it follows that

$A(u)\leq\frac{\mu N(q-2)}{2q}C(u)\leq\frac{\mu
N(q-2)\overline{S}}{2q}\|u\|_{2}^{\frac{2N-q(N-2)}{2}}A(u)^{\frac{N(q-2)}{4}}.$

As in Case $(a)$, we have $u\neq 0.$ Hence, using the fact that
$\|u\|_{2}^{2}\leq c$ by weak lower semi-continuity, we deduce that

$A(u)\geq\left(\frac{2q}{\mu
N(q-2)\overline{S}c^{\frac{2N-q(N-2)}{4}}}\right)^{\frac{4}{N(q-2)-4}}.$ (25)

Since $\lambda_{n}\rightarrow\lambda$ and $u_{n}\rightarrow u\neq 0$ weakly in
$H_{r}^{1}(\mathbb{R}^{N})$, and strongly in $L^{q}(\mathbb{R}^{N})$, the
equality (23) implies that $u$ is a weak solution to

$-\Delta u+\lambda
u=\gamma(I_{\alpha}\ast|u|^{p})|u|^{p-2}u+\mu|u|^{q-2}u\quad\text{in}\,\mathbb{R}^{N}.$

By Lemma 2.3, we infer that $Q(u)=0.$ Moreover, we have

$\displaystyle\lambda\|u\|_{2}^{2}$ $\displaystyle\geq$
$\displaystyle\frac{2N-q(N-2)}{N(q-2)}A(u)+\frac{\gamma(Nq+\alpha
q-2Np)}{N(q-2)\|Q_{p}\|_{2}^{2p-2}}c^{\frac{N+\alpha-p(N-2)}{2}}A(u)^{\frac{Np-N-\alpha}{2}}$
(26) $\displaystyle\geq$ $\displaystyle
A(u)^{\frac{Np-N-\alpha}{2}}\left[\frac{2N-q(N-2)}{N(q-2)}\left(\frac{2q}{\mu
N(q-2)\overline{S}c^{\frac{2N-(N-2)q}{4}}}\right)^{\frac{2(N+\alpha+2-Np)}{(q-2)N-4}}\right.$
$\displaystyle\left.+\frac{\gamma(Nq+\alpha
q-2Np)}{N(q-2)\|Q_{p}\|_{2}^{2p-2}c^{-\frac{N+\alpha-p(N-2)}{2}}}\right].$

Case $(b-i):Nq+\alpha q-2Np\leq 0$. It is clear that $\lambda>0,$ by
$(\ref{e2-15})-(\ref{e2-16})$.
Case $(b-ii):Nq+\alpha q-2Np>0$ and
$\frac{N+\alpha}{N}<p\leq\frac{N+\alpha+2}{N}$. For

$|\gamma|c^{{}^{\frac{\alpha
q+2q-2\alpha-4p}{(q-2)N-4}}}<\frac{(2N-q(N-2))\|Q_{p}\|_{2}^{2p-2}}{Nq+\alpha
q-2Np}\left(\frac{2q}{\mu
N\overline{S}(q-2)}\right)^{\frac{2(N+\alpha+2-Np)}{(q-2)N-4}},$

it follows from $(\ref{e2-15})-(\ref{e2-16})$ that $\lambda>0$.
Case $(b-iii):$ $Nq+\alpha q-2Np>0$ and
$\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$. Since $\\{u_{n}\\}$ is a bounded
sequence, there exists a constant $D_{1}>0$ such that
$A(u)\leq\lim_{n\rightarrow\infty}A(u_{n})\leq D_{1}$. Let us consider

$f(s):=\frac{2N-q(N-2)}{N(q-2)}s+\frac{\gamma(Nq+\alpha
q-2Np)}{N(q-2)\|Q_{p}\|_{2}^{2p-2}}c^{\frac{N+\alpha-p(N-2)}{2}}s^{\frac{Np-N-\alpha}{2}}\text{
for }s>0.$

A direct calculation shows that $f(A(u))>0$ for

$0<A(u)<\left(\frac{(2N-q(N-2))\|Q_{p}\|_{2}^{2p-2}}{|\gamma|(Nq+\alpha
q-2Np)c^{\frac{N+\alpha-p(N-2)}{2}}}\right)^{\frac{2}{Np-N-\alpha-2}}.$

Taking $|\gamma|$ small enough such that

$\left(\frac{(2N-(N-2)q)\|Q_{p}\|_{2}^{2p-2}}{|\gamma|(Nq+\alpha
q-2Np)c^{\frac{N+\alpha-p(N-2)}{2}}}\right)^{\frac{2}{Np-N-\alpha-2}}>D_{1}>\left(\frac{2q}{\mu
N(q-2)\overline{S}c^{\frac{2N-q(N-2)}{4}}}\right)^{\frac{4}{N(q-2)-4}}.$

Thus, for $|\gamma|$ small enough, one has

$\lambda\|u\|_{2}^{2}\geq\frac{2N-q(N-2)}{N(q-2)}A(u)+\frac{\gamma(Nq+\alpha
q-2Np)}{N(q-2)\|Q_{p}\|_{2}^{2p-2}}c^{\frac{N+\alpha-p(N-2)}{2}}A(u)^{\frac{Np-N-\alpha}{2}}>0,$

which shows that $\lambda>0$.

Case $(c):\gamma>0,\mu<0,\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and
$2<q<2^{\ast}$. Since $Q(u_{n})=o(1)$, it holds

$\displaystyle A(u_{n})$ $\displaystyle=$
$\displaystyle\frac{\gamma(Np-N-\alpha)}{2p}B(u_{n})+\frac{\mu
N(q-2)}{2q}C(u_{n})+o(1)$ $\displaystyle\leq$
$\displaystyle\frac{\gamma(Np-N-\alpha)}{2p}B(u_{n})+o(1).$

Then by (15), one has

$A(u)\leq\frac{\gamma(Np-N-\alpha)}{2p}B(u)\leq\frac{\gamma(Np-N-\alpha)}{2\|Q_{p}\|_{2}^{2p-2}}\|u\|_{2}^{N+\alpha-(N-2)p}A(u)^{\frac{Np-N-\alpha}{2}}.$

As in Case $(a)$, we have $u\neq 0.$ Thus, using that $\|u\|_{2}^{2}\leq c$ by
weak lower semi-continuity, we deduce that

$A(u)\geq\left(\frac{2\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)c^{\frac{N+\alpha-(N-2)p}{2}}}\right)^{\frac{2}{Np-N-\alpha-2}}.$
(27)

Since $\lambda_{n}\rightarrow\lambda$ and $u_{n}\rightarrow u\neq 0$ weakly in
$H_{r}^{1}(\mathbb{R}^{N})$, and strongly in $L^{q}(\mathbb{R}^{N})$, the
equality (23) indicates that $u$ is a weak solution to

$-\Delta u+\lambda
u=\gamma(I_{\alpha}\ast|u|^{p})|u|^{p-2}u+\mu|u|^{q-2}u\quad\text{in}\,\mathbb{R}^{N}.$

It follows from Lemma 2.3 that $Q(u)=0$ and

$\displaystyle\lambda\|u\|_{2}^{2}$ $\displaystyle\geq$
$\displaystyle\frac{N+\alpha-p(N-2)}{Np-N-\alpha}A(u)+\frac{\mu\overline{S}(2Np-
Nq-\alpha q)}{q(Np-N-\alpha)}c^{\frac{2N-q(N-2)}{4}}A(u)^{\frac{N(q-2)}{4}}$
(28) $\displaystyle\geq$ $\displaystyle
A(u)^{\frac{N(q-2)}{4}}\left[\frac{N+\alpha-p(N-2)}{Np-N-\alpha}\left(\frac{2\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)c^{\frac{N+\alpha-p(N-2)}{2}}}\right)^{\frac{4-N(q-2)}{2(Np-N-\alpha-2)}}\right.$
$\displaystyle\left.+\frac{\mu\overline{S}(2Np-Nq-\alpha
q)}{q(Np-N-\alpha)c^{-\frac{2N-q(N-2)}{4}}}\right].$

Case $(c-i):2Np-Nq-\alpha q\leq 0.$ By $(\ref{e2-17})-(\ref{e2-18})$ it is
easy to get $\lambda>0$.
Case $(c-ii):2Np-Nq-\alpha q>0$ and $2<q\leq 2+\frac{4}{N}$. For

$|\mu|c^{\frac{4p+2\alpha-\alpha
q-2q}{2(Np-N-\alpha-2)}}<\frac{q(N+\alpha-p(N-2))}{\overline{S}(2Np-Nq-\alpha
q)}\left(\frac{2\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)}\right)^{\frac{4-N(q-2)}{2(Np-N-\alpha-2)}},$

it follows from $(\ref{e2-17})-(\ref{e2-18})$ that $\lambda>0$.
Case $(c-iii):2Np-Nq-\alpha q>0$ and $2+\frac{4}{N}<q<2^{\ast}.$ Note that
$\\{u_{n}\\}$ is a bounded sequence. Then there exists a constant $D_{2}>0$
such that $A(u)\leq\lim_{n\rightarrow\infty}A(u_{n})\leq D_{2}$. Now, we
consider the function

$f(s):=\frac{N+\alpha-p(N-2)}{Np-N-\alpha}s+\frac{\mu\overline{S}(2Np-
Nq-\alpha q)}{q(Np-N-\alpha)}c^{\frac{2N-q(N-2)}{4}}s^{\frac{N(q-2)}{4}}\text{
for }s>0.$

It is not difficult to verify that $f(A(u))>0$ for

$0<A(u)<\left(\frac{q(N+\alpha-p(N-2))}{|\mu|\overline{S}(2Np-Nq-\alpha
q)c^{\frac{2N-q(N-2)}{4}}}\right)^{\frac{4}{N(q-2)-4}}.$

Let us take $|\mu|c^{\frac{2N-(N-2)q}{4}}$ small enough such that

$\left(\frac{q(N+\alpha-p(N-2))}{|\mu|\overline{S}(2Np-Nq-\alpha
q)c^{\frac{2N-(N-2)q}{4}}}\right)^{\frac{4}{N(q-2)-4}}>D_{2}>\left(\frac{2\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)c^{\frac{N+\alpha-(N-2)p}{2}}}\right)^{\frac{2}{Np-N-\alpha-2}}.$

Thus, for $|\mu|c^{\frac{2N-(N-2)q}{4}}$ small enough, one has

$\lambda\|u\|_{2}^{2}\geq\frac{N+\alpha-p(N-2)}{Np-N-\alpha}A(u)+\frac{\mu\overline{S}(2Np-
Nq-\alpha
q)}{q(Np-N-\alpha)}c^{\frac{2N-q(N-2)}{4}}A(u)^{\frac{N(q-2)}{4}}>0,$

which indicates that $\lambda>0$.

By (23), we have

$E_{\gamma,\mu}^{\prime}(u)[\varphi]+\lambda\int_{\mathbb{R}^{N}}u\varphi
dx=0$ (29)

for every $\varphi\in H^{1}(\mathbb{R}^{N})$. Taking $\varphi=u_{n}-u$ in (23)
and (29), we have

$(E_{\gamma,\mu}^{\prime}(u_{n})-E_{\gamma,\mu}^{\prime}(u))[u_{n}-u]+\lambda\int_{\mathbb{R}^{N}}|u_{n}-u|^{2}dx=o(1).$

Since $u_{n}\rightarrow u$ in $L^{r}$ for $r\in(2,2^{\ast})$, we infer that
$B(u_{n}-u)=o(1)$ and $C(u_{n}-u)=o(1)$. Then we obtain that

$\int_{\mathbb{R}^{N}}|\nabla(u_{n}-u)|^{2}+\lambda|u_{n}-u|^{2}dx=o(1),$

which implies that $u_{n}\rightarrow u$ in $H^{1}(\mathbb{R}^{N}),$ since
$\lambda>0.$ Consequently, we complete the proof.

##  3 The case of $\gamma>0$ and $\mu>0$

###  3.1 Bounded from below on $S(c)$

###### Lemma 3.1

Assume that $N\geq 2,\gamma>0,\mu>0$ and that one of the four following
conditions holds:
$(i)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and $2<q<2+\frac{4}{N};$
$(ii)$ $p=\frac{N+\alpha+2}{N},2<q<2+\frac{4}{N}$ and
$\gamma^{N/(\alpha+2)}c<\|Q_{(N+\alpha+2)/N}\|_{2}^{2};$
$(iii)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},q=2+\frac{4}{N}$ and
$\mu^{N/2}c<\left(\frac{N+2}{\overline{S}N}\right)^{N/2};$
$(iv)$ $p=\frac{N+\alpha+2}{N},q=2+\frac{4}{N}$ and $c<c_{0}$ for some
$c_{0}>0.$
Then functional $E_{\gamma,\mu}$ is bounded from below and coercive on $S(c)$.

Proof. We consider the following four cases:
Case $(i):\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and $2<q<2+\frac{4}{N}$.
By $(\ref{e2-0})-(\ref{e2-3})$, we have

$E_{\gamma,\mu}(u)\geq\frac{1}{2}A(u)-\frac{\gamma}{2\|Q_{p}\|_{2}^{2p-2}}A(u)^{\frac{Np-N-\alpha}{2}}c^{\frac{N+\alpha-p(N-2)}{2}}-\frac{\mu}{q}\overline{S}A(u)^{\frac{N(q-2)}{4}}c^{\frac{2N-q(N-2)}{4}},$

which implies that $E_{\gamma,\mu}$ is bounded from below and coercive on
$S(c).$
Case $(ii):p=\frac{N+\alpha+2}{N},2<q<2+\frac{4}{N}$ and
$c\gamma^{N/(\alpha+2)}<\|Q_{(N+\alpha+2)/N}\|_{2}^{2}.$ From
$(\ref{e2-0})-(\ref{e2-3})$ it follows that

$E_{\gamma,\mu}(u)\geq\left(\frac{1}{2}-\frac{\gamma}{2\|Q_{(N+\alpha+2)/2}\|_{2}^{2(\alpha+2)/N}}c^{(2+\alpha)/N}\right)A(u)-\frac{\mu}{q}\overline{S}A(u)^{\frac{(q-2)N}{4}}c^{\frac{2N-(N-2)q}{4}},$

which implies that $E_{\gamma,\mu}$ is bounded from below and coercive on
$S(c)$.
Case $(iii):\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},q=2+\frac{4}{N}$ and
$c\mu^{N/2}<\left(\frac{N+2}{\overline{S}N}\right)^{N/2}$. Using
$(\ref{e2-0})-(\ref{e2-3})$ gives

$E_{\gamma,\mu}(u)\geq\left(\frac{1}{2}-\frac{\mu}{q}\overline{S}c^{2/N}\right)A(u)-\frac{\gamma}{2\|Q_{p}\|_{2}^{2p-2}}A(u)^{\frac{Np-N-\alpha}{2}}c^{\frac{N+\alpha-p(N-2)}{2}},$

which implies that $E_{\gamma,\mu}$ is bounded from below and coercive on
$S(c)$.
Case $(iv):p=\frac{N+\alpha+2}{N}$ and $q=2+\frac{4}{N}.$ We use
$(\ref{e2-0})-(\ref{e2-3})$ to get

$E_{\gamma,\mu}(u)\geq\left(\frac{1}{2}-\frac{\mu}{q}\overline{S}c^{2/N}-\frac{\gamma}{2\|Q_{(N+\alpha+2)/2}\|_{2}^{2(\alpha+2)/N}}c^{(2+\alpha)/N}\right)A(u).$

Now, we consider

$\widehat{f}(c)=\frac{1}{2}-\frac{\mu}{q}\overline{S}c^{2/N}-\frac{\gamma}{2\|Q_{(N+\alpha+2)/2}\|_{2}^{2(\alpha+2)/N}}c^{(2+\alpha)/N}.$

Observe that $\widehat{f}(c)$ is decreasing on $c>0,$
$\widehat{f}(0)=\frac{1}{2}$ and $\widehat{f}(c)\rightarrow-\infty$ as
$c\rightarrow+\infty$. Then there exists a constant $c_{0}>0$ such that

$\frac{1}{2}-\frac{\mu}{q}\overline{S}c_{0}^{2/N}-\frac{\gamma}{2\|Q_{(N+\alpha+2)/2}\|_{2}^{2(\alpha+2)/N}}c_{0}^{(2+\alpha)/N}=0.$

Hence, we have $E_{\gamma,\mu}(u)\geq 0$ for $0<c\leq c_{0}$ and
$E_{\gamma,\mu}$ is coercive for $0<c<c_{0}$. Consequently, we complete the
proof.

Next, we study the strict sub-additivity inequality for $\sigma(c).$

###### Lemma 3.2

Assume that $N\geq 2,\gamma>0,\mu>0$ and that one of the three following
conditions holds:
$(i)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and $2<q<2+\frac{4}{N};$
$(ii)$ $p=\frac{N+\alpha+2}{N},2<q<2+\frac{4}{N}$ and
$\gamma^{N/(\alpha+2)}c<\|Q_{(N+\alpha+2)/N}\|_{2}^{2};$
$(iii)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},q=2+\frac{4}{N}$ and
$\mu^{N/2}c<(\frac{N+2}{\overline{S}N})^{N/2}.$
Then we have $-\infty<\sigma(c)<0$ and $\sigma(c)<\sigma(\eta)+\sigma(c-\eta)$
for $0<\eta<c.$

Proof. We introduce the test function $u_{t}(x):=t^{N/2}u(tx)$ with $t>0$.
Clearly, $u_{t}\in S(c).$

Case $(i):\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and $2<q<2+\frac{4}{N}$.
For $t>0$ small enough, we have

$E_{\gamma,\mu}(u_{t})=\frac{t^{2}}{2}A(u)-\frac{\gamma
t^{Np-N-\alpha}}{2p}B(u)-\frac{\mu t^{\frac{N(q-2)}{2}}}{q}C(u)<0,$ (30)

since $Np-N-\alpha<2,$ which implies that $\sigma(c)<0$ for each $c>0$. Let
$\\{u_{n}\\}\subset S(c)$ be a bounded minimizing sequence for $\sigma(c)$.
Then for any $c>0$ and $u\in S(c)$, it follows that $\theta u_{n}\in
S(\theta^{2}c)$ and

$E_{\gamma,\mu}(\theta
u_{n})-\theta^{2}E_{\gamma,\mu}(u_{n})=\frac{\theta^{2}-\theta^{2p}}{2p}\gamma
B(u_{n})+\frac{\theta^{2}-\theta^{q}}{q}\mu C(u_{n}).$

Let $\theta>1.$ Then we have $E_{\gamma,\mu}(\theta
u_{n})<\theta^{2}E_{\gamma,\mu}(u_{n})$, implying that
$\sigma(\theta^{2}c)\leq\theta^{2}\sigma(c)$. The equality holds if and only
if $B(u_{n})+C(u_{n})\rightarrow 0$ as $n\rightarrow\infty$. But this is not
possible, since otherwise we find that

$0>\sigma(c)=\lim_{n\rightarrow\infty}E_{\gamma,\mu}(u_{n})\geq\lim\inf_{n\rightarrow\infty}\frac{1}{2}A(u_{n})\geq
0.$

So, $\sigma(\theta^{2}c)<\theta^{2}\sigma(c)$. Without loss of generality, we
may assume that $0<\eta<c-\eta$. Then it holds

$\sigma(c)<\frac{c}{c-\eta}\sigma(c-\eta)=\sigma(c-\eta)+\frac{\eta}{c-\eta}\sigma(c-\eta)<\sigma(c-\eta)+\sigma(\eta).$

Case $(ii):p=\frac{N+\alpha+2}{N},2<q<2+\frac{4}{N}$ and
$c\gamma^{N/(\alpha+2)}<\|Q_{(N+\alpha+2)/N}\|_{2}^{2}$. For $t>0$ small
enough, it holds

$E_{\gamma,\mu}(u_{t})=\frac{t^{2}}{2}\left(A(u)-\frac{\gamma
N}{N+\alpha+2}B(u)\right)-\frac{\mu t^{\frac{N(q-2)}{2}}}{q}C(u)<0,$

since $A(u)-\frac{\gamma N}{N+\alpha+2}B(u)>0$ and $0<\frac{N(q-2)}{2}<2$. It
indicates that $\sigma(c)<0$ for each
$0<c\gamma^{N/(\alpha+2)}<\|Q_{(N+\alpha+2)/N}\|_{2}^{2}$. We take $\theta>1.$
Then we have

$E_{\gamma,\mu}(\theta
u_{n})-\theta^{2}E_{\gamma,\mu}(u_{n})=\frac{\theta^{2}-\theta^{2p}}{2p}\gamma
B(u_{n})+\frac{\theta^{2}-\theta^{q}}{q}\mu C(u_{n})<0,$

which implies that $\sigma(\theta^{2}c)<\theta^{2}\sigma(c)$ by using the same
argument in Case $(i)$. Without loss of generality, we may assume that
$0<\eta<c-\eta$, then

$\sigma(c)<\frac{c}{c-\eta}\sigma(c-\eta)=\sigma(c-\eta)+\frac{\eta}{c-\eta}\sigma(c-\eta)<\sigma(c-\eta)+\sigma(\eta).$

Case $(iii):\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},q=2+\frac{4}{N}$ and
$\mu^{N/2}c<(\frac{N+2}{\overline{S}N})^{N/2}.$ The proof is very similar to
that of Case $(ii)$, we omit it here. Consequently, we complete the proof.

####  3.1.1 The proof of Theorem 1.2

###### Lemma 3.3

Assume that the assumptions of Theorem 1.2 $(a)$ hold. Let $\\{u_{n}\\}\subset
H^{1}(\mathbb{R}^{N})$ be a sequence such that
$E_{\gamma,\mu}(u_{n})\rightarrow\sigma(c)$ and $\|u_{n}\|_{2}^{2}\rightarrow
c$. Then $\\{u_{n}\\}$ is relatively compact in $H^{1}(\mathbb{R}^{N})$ up to
translations, that is, there exist a subsequence, still denoted by
$\\{u_{n}\\}$, a sequence of points $\\{y_{n}\\}\subset\mathbb{R}^{N}$, and a
function $u_{0}\in S(c)$ such that $u_{n}(\cdot+y_{n})\rightarrow u_{0}$
strongly in $H^{1}(\mathbb{R}^{N}).$

Proof. By Lemma 3.1, we obtain that the sequence $\\{u_{n}\\}$ is bounded in
$H^{1}(\mathbb{R}^{N})$. Then by concentration-compactness principle [26],
there exists a subsequence, still denoted by $\\{u_{n}\\}$ satisfying one of
the following three possibilities:
$(i)$ vanishing:

$\lim_{n\rightarrow\infty}\sup_{y\in\mathbb{R}^{N}}\int_{B_{R}(y)}|u_{n}|^{2}dx=0\text{
for }R>0.$

$(ii)$ dichotomy: there exists $0<\eta<c$ and $\\{u_{n}^{(1)}\\}$,
$\\{u_{n}^{(2)}\\}$ bounded in $H^{1}(\mathbb{R}^{N})$ such that as
$n\rightarrow\infty$, we have

$\|u_{n}-(u_{n}^{(1)}+u_{n}^{(2)})\|_{r}\rightarrow 0\text{ and
}dist(\mathrm{supp}u_{n}^{(1)},\mathrm{supp}u_{n}^{(2)})\rightarrow+\infty;$
$\left|\int_{\mathbb{R}^{N}}|u_{n}^{(1)}|^{2}dx-\eta\right|\rightarrow 0\text{
and }\left|\int_{\mathbb{R}^{N}}|u_{n}^{(2)}|^{2}dx-(c-\eta)\right|\rightarrow
0;$ $\lim\inf_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}(|\nabla
u_{n}|^{2}-|\nabla u_{n}^{(1)}|^{2}-|\nabla u_{n}^{(2)}|^{2})dx\geq 0.$

$(iii)$ compactness: there exists $y_{n}\in\mathbb{R}^{N}$ such that
$\forall\varepsilon>0$, $\exists R>0$, we have

$\int_{B_{R}(y_{n})}|u_{n}|^{2}dx\geq c-\varepsilon.$

Firstly, we verify that the vanishing cannot occur. Suppose that the contrary,
we have $u_{n}\rightarrow 0$ strongly in $L^{s}(\mathbb{R}^{N})$ for
$2<s<2^{\ast}$ and $C(u_{n})\rightarrow 0$ as $n\rightarrow\infty.$ Using (17)
gives $B(u_{n})\rightarrow 0$ as $n\rightarrow\infty$, which implies that
$\sigma(c)\geq 0$. This is a contradiction with $\sigma(c)<0$.

Next, we prove that the dichotomy cannot occur. Suppose on the contrary. Then
it holds

$\sigma(c)=\lim_{n\rightarrow\infty}E_{\gamma,\mu}(u_{n})\geq\lim_{n\rightarrow\infty}E_{\gamma,\mu}(u_{n}^{(1)})+\lim_{n\rightarrow\infty}E_{\gamma,\mu}(u_{n}^{(2)})\geq\sigma(\eta)+\sigma(c-\eta),$

since $B(u_{n})=B(u_{n}^{(1)})+B(u_{n}^{(2)})+o(1)$, which contradicts with
Lemma 3.2. So the compactness holds, namely, there exist subsequence
$\\{u_{n}\\},\\{y_{n}\\}\subset\mathbb{R}^{N}$ and $u_{0}\in S(c)$ such that
$\overline{u}_{n}:=u_{n}(\cdot+y_{n})\rightarrow u_{0}$ in
$L^{s}(\mathbb{R}^{N})$ for $2\leq s<2^{\ast}$, which implies that
$B(\overline{u}_{n})\rightarrow B(u_{0})$ by (17). Therefore, we have

$\sigma(c)\leq
E_{\gamma,\mu}(u_{0})\leq\lim\inf_{n\rightarrow\infty}E_{\gamma,\mu}(\overline{u}_{n})=\lim\inf_{n\rightarrow\infty}E_{\gamma,\mu}(u_{n})=\sigma(c).$

This shows that $u_{0}$ is a minimizer for $\sigma(c)$. Consequently, we
complete the proof.

Now we give the proof of Theorem 1.2: $(a)$ By Lemmas 3.1 and 3.3, there
exists a minimizer $u_{0}$ for $E_{\gamma,\mu}$ on $S(c).$ In other words, the
infimum $\sigma(c)$ is achieved by $u_{0}\in S(c)$ which is a ground state
solution of Problem $(P_{c})$ for some $\lambda\in\mathbb{R}$. After
multiplying the equation in Eq. (3) by $u_{0}$ and integrating, we deduce that

$\lambda c=\gamma B(u_{0})+\mu
C(u_{0})-A(u_{0})=-2\sigma(c)+\frac{\gamma(p+1)}{p}B(u_{0})+\frac{\mu(q+2)}{q}C(u_{0})>0,$

which implies that $\lambda>0$. We complete the proof of Theorem 1.2 $(a).$

$(b)$ When $p=\frac{N+\alpha+2}{N},q=2+\frac{4}{N}$ and $0<c\leq c_{0}$, by
Lemma 3.1 $(iv)$, we have

$E_{\gamma,\mu}(u)\geq\left(\frac{1}{2}-\frac{\mu}{q}\overline{S}c^{2/N}-\frac{\gamma}{2\|Q_{(N+\alpha+2)/2}\|_{2}^{2(\alpha+2)/N}}c^{(2+\alpha)/N}\right)A(u)\geq
0.$

In additional, for all $c>0$, it holds $\sigma(c)\leq
E_{\gamma,\mu}(u_{t})=t^{2}\left(\frac{1}{2}A(u)-\frac{\gamma}{2p}B(u)-\frac{\mu}{q}C(u)\right)\rightarrow
0$ as $t\rightarrow 0^{+}.$ Thus, we have $\sigma(c)=0$ for $0<c\leq c_{0}$.

Suppose that there exists $\widetilde{c}\in(0,c_{0})$ such that
$\sigma(\widetilde{c})$ has a minimizer $\widetilde{u}\in S(\widetilde{c})$.
Then we obtain that $E_{\gamma,\mu}(\widetilde{u})=\sigma(\widetilde{c})=0$
and

$A(\widetilde{u})=\frac{\gamma}{p}B(\widetilde{u})+\frac{2\mu}{q}C(\widetilde{u})\leq\left(\frac{2\mu}{q}\overline{S}c^{2/N}+\frac{\gamma}{\|Q_{(N+\alpha+2)/2}\|_{2}^{2(\alpha+2)/N}}c^{(2+\alpha)/N}\right)A(\widetilde{u})<A(\widetilde{u}),$

which is a contradiction. Hence, $\sigma(c)$ has no minimizer for $0<c<c_{0}$.
Consequently, we complete the proof.

###  3.2 Unbounded from below on $S(c)$

For the other subcases in the case of $\gamma>0$ and $\mu>0$, functional
$E_{\gamma,\mu}$ will be no longer bounded from below on $S(c)$, and so it
will not be possible to obtain a global minimizer. In this subsection, in
order to find critical points of functional $E_{\gamma,\mu},$ we shall
restrict it to a natural constraint manifold $\mathcal{M}(c)$ defined in
Section 2, on which $E_{\gamma,\mu}$ is bounded below.

####  3.2.1 Mass subcritical and mass supercritical case

First of all, we show that $\mathcal{M}^{0}(c)$ is empty, whence
$\mathcal{M}(c)=\mathcal{M}^{+}(c)\cup\mathcal{M}^{-}(c)$.

###### Lemma 3.4

Assume that $N\geq 2,\gamma>0,\mu>0$ and that one of the two following
conditions holds:
$(i)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and $2<q<2+\frac{4}{N};$
$(ii)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and
$2+\frac{4}{N}<q<2^{\ast}.$
Then $\mathcal{M}^{0}(c)=\emptyset$ for all $c<c_{\ast}$, where $c_{\ast}>0$
is as in (12).

Proof. We only consider the case of $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$
and $2<q<2+\frac{4}{N},$ since the other one is similar. Suppose on the
contrary. Let $u\in\mathcal{M}^{0}(c)$. Then it follows from (14) and (21)
that

$\frac{4q(Np-N-\alpha-2)}{\mu N(q-2)(2Np-
Nq-2\alpha)}A(u)=C(u)\leq\overline{S}c^{\frac{2N-q(N-2)}{4}}A(u)^{\frac{N(q-2)}{4}},$

which indicates that

$A(u)\leq\left(\frac{\mu N\overline{S}(q-2)(2Np-
Nq-2\alpha)}{4q(Np-N-\alpha-2)}c^{\frac{2N-q(N-2)}{4}}\right)^{\frac{4}{4-N(q-2)}}.$
(31)

On the other hand, using (15) and (2.2) gives

$\frac{2p(4-(q-2)N)}{\gamma(Np-N-\alpha)(2Np-
Nq-2\alpha)}A(u)=B(u)\leq\frac{p}{\|Q_{p}\|_{2}^{2p-2}}c^{\frac{N+\alpha-p(N-2)}{2}}A(u)^{\frac{Np-N-\alpha}{2}},$

which indicates that

$A(u)\geq\left(\frac{2(4-(q-2)N)\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)(2Np-
Nq-2\alpha)}c^{-\frac{N+\alpha-p(N-2)}{2}}\right)^{\frac{2}{Np-N-\alpha-2}}.$
(32)

Thus, by (31) and (32), one has

$\displaystyle\left(\frac{2(4-(q-2)N)\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)(2Np-
Nq-2\alpha)}c^{-\frac{N+\alpha-p(N-2)}{2}}\right)^{\frac{2}{Np-N-\alpha-2}}$
$\displaystyle\leq$ $\displaystyle A(u)\leq\left(\frac{\mu
N\overline{S}(q-2)(2Np-
Nq-2\alpha)}{4q(Np-N-\alpha-2)}c^{\frac{2N-q(N-2)}{4}}\right)^{\frac{4}{4-N(q-2)}}.$

Hence, if the submanifold $\mathcal{M}^{0}(c)$ is nonempty, we must have
$c\geq c_{\ast},$ which is a contradiction. Consequently, we complete the
proof.

###### Lemma 3.5

Assume that $N\geq 2,\gamma>0,\mu>0$ and that one of the two following
conditions holds:
$(i)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and $2<q<2+\frac{4}{N};$
$(ii)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and
$2+\frac{4}{N}<q<2^{\ast}.$
Then functional $E_{\gamma,\mu}$ is bounded from below and coercive on
$\mathcal{M}(c)$ for all $c>0$. In particular, if
$c<\left(\frac{4}{N(q-2)}\right)^{\frac{N(q-2)-4}{4p+2\alpha-q(2+\alpha)}}c_{\ast}$
for $(i)$ or
$c<\left(\frac{Np-N-\alpha}{2}\right)^{\frac{2(Np-N-\alpha-2)}{4p+2\alpha-q(2+\alpha)}}c_{\ast}$
for $(ii),$ then $E_{\gamma,\mu}$ is bounded from below by a positive constant
on $\mathcal{M}^{-}(c).$

Proof. We only consider the case of $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$
and $2<q<2+\frac{4}{N}.$ For $u\in\mathcal{M}(c)$, we have

$A(u)-\frac{\gamma(Np-N-\alpha)}{2p}B(u)-\frac{\mu N(q-2)}{2q}C(u)=0.$ (33)

Then it follows from (14) and (33) that

$\displaystyle E_{\gamma,\mu}(u)$ $\displaystyle=$
$\displaystyle\frac{Np-N-\alpha-2}{2(Np-N-\alpha)}A(u)-\frac{\mu(2Np-
Nq-2\alpha)}{2q(Np-N-\alpha)}C(u)$ (34) $\displaystyle\geq$
$\displaystyle\frac{Np-N-\alpha-2}{2(Np-N-\alpha)}A(u)-\frac{\mu\overline{S}(2Np-
Nq-2\alpha)}{2q(Np-N-\alpha)}c^{\frac{2N-(N-2)q}{4}}A(u)^{\frac{N(q-2)}{4}},$

where $\frac{Np-N-\alpha-2}{2(Np-N-\alpha)}>0$ and $\frac{\mu(2Np-2\alpha-
qN)}{2q(Np-N-\alpha)}>0$. This implies that $E_{\gamma,\mu}$ is bounded from
below and coercive on $\mathcal{M}(c),$ since $\frac{N(q-2)}{4}<1$.

For $u\in\mathcal{M}^{-}(c)$, it holds

$\displaystyle A(u)$ $\displaystyle<$
$\displaystyle\frac{\gamma(Np-N-\alpha)(2Np-qN-2\alpha)}{2p(4-N(q-2))}B(u)$
$\displaystyle\leq$ $\displaystyle\frac{\gamma(Np-N-\alpha)(2Np-
qN-2\alpha)}{2(4-N(q-2))\|Q_{p}\|_{2}^{2p-2}}c^{\frac{N+\alpha-p(N-2)}{2}}A(u)^{\frac{Np-N-\alpha}{2}},$

which implies that

$A(u)\geq\left(\frac{2(4-N(q-2))\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)(2Np-
Nq-2\alpha)}c^{-\frac{N+\alpha-p(N-2)}{2}}\right)^{\frac{2}{Np-N-\alpha-2}}.$
(35)

Thus, for
$c<\left(\frac{4}{N(q-2)}\right)^{\frac{N(q-2)-4}{4p+2\alpha-q(2+\alpha)}}c_{\ast},$
it follows from (34) and (35) that there exists a constant $D_{0}>0$ such that
$E_{\gamma,\mu}(u)>D_{0}$. Consequently, we complete the proof.

Next, we consider the nonemptiness of submanifolds $\mathcal{M}^{-}(c)$ and
$\mathcal{M}^{+}(c)$. Set

$t_{\max}\left(u\right):=\left[\frac{2p(4-N(q-2))A(u)}{\gamma(Np-N-\alpha)(2Np-2\alpha-
qN)B(u)}\right]^{\frac{1}{Np-(N+\alpha+2)}}\text{ for }u\in S(c).$

###### Lemma 3.6

Assume that $N\geq 2,\gamma>0,\mu>0$ and that one of the two following
conditions holds:
$(i)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and $2<q<2+\frac{4}{N};$
$(ii)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and
$2+\frac{4}{N}<q<2^{\ast}.$
Then for each $u\in S(c)$ and $c<c_{\ast}$, there exist two constants
$t^{+}\left(u\right)<t_{\max}\left(u\right)<t^{-}\left(u\right)$ such that
$u_{t^{\pm}\left(u\right)}\in\mathcal{M}^{\pm}(c).$ Furthermore, we have
$E_{\gamma,\mu}(u_{t^{+}\left(u\right)})=\inf_{0\leq t\leq
t_{\max}\left(u\right)}E_{\gamma,\mu}(u_{t})<0.$

Proof. We only consider the case of $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$
and $2<q<2+\frac{4}{N}.$ Fix $u\in S(c)$. Let

$m(t)=\frac{2q}{\mu N(q-2)}t^{\frac{4-N(q-2)}{2}}A(u)-\frac{\gamma
q(Np-N-\alpha)}{\mu pN(q-2)}t^{\frac{2Np-2\alpha-qN}{2}}B(u).$

Note that $u_{t}\in\mathcal{M}(c)$ if and only if $m(t)=C(u).$ A direct
calculation shows that

$m^{\prime}(t)=\frac{q(4-N(q-2))}{\mu
N(q-2)}t^{\frac{2-N(q-2)}{2}}A(u)-\frac{\gamma q(Np-N-\alpha)(2Np-2\alpha-
qN)}{2\mu Np(q-2)}t^{\frac{2Np-2\alpha-qN-2}{2}}B(u),$

which implies that $m(t)$ is increasing for $t\in(0,t_{\max}\left(u\right))$
and decreasing for $t\in(t_{\max}\left(u\right),\infty)$ with
$\lim_{t\rightarrow\infty}m(t)=-\infty$. Moreover, we have

$\displaystyle m(t_{\max}\left(u\right))$ $\displaystyle\geq$
$\displaystyle\frac{4q(Np-N-\alpha-2)}{\mu N(q-2)(2Np-2\alpha-
qN)}\left[\frac{2p(4-N(q-2))}{\gamma(Np-N-\alpha)(2Np-2\alpha-
qN)}\right]^{\frac{4-N(q-2)}{2(Np-N-\alpha-2)}}$
$\displaystyle\times\left[\frac{\|Q_{p}\|_{2}^{2p-2}}{pc^{\frac{N+\alpha-p(N-2)}{2}}}\right]^{\frac{4-N(q-2)}{2(Np-N-\alpha-2)}}\overline{S}^{-1}c^{\frac{q(N-2)-2N}{4}}C(u)$
$\displaystyle\geq$ $\displaystyle\frac{4q(Np-N-\alpha-2)}{\mu
N(q-2)(2Np-2\alpha-
qN)}\left[\frac{2(4-N(q-2))\|Q_{p}\|_{2}^{2p-2}}{\gamma(Np-N-\alpha)(2Np-2\alpha-
qN)}\right]^{\frac{4-N(q-2)}{2(Np-N-\alpha-2)}}$
$\displaystyle\times\overline{S}^{-1}c^{\frac{q(2+\alpha)-4p-2\alpha}{2(Np-N-\alpha-2)}}C(u)$
$\displaystyle>$ $\displaystyle C(u),$

since $0<c<c_{\ast}$. This implies that there are unique $t^{+}\left(u\right)$
and $t^{-}\left(u\right)$ satisfying
$0<t^{+}\left(u\right)<t_{\max}\left(u\right)<t^{-}\left(u\right)$ such that

$m(t^{\pm}\left(u\right))=C(u)\text{ and
}m^{\prime}(t^{+}\left(u\right))>0>m^{\prime}(t^{-}\left(u\right)).$

In other words, $u_{t^{\pm}\left(u\right)}\in\mathcal{M}(c).$ By calculating
the second derivative of $g_{u}(t)$ one has

$g_{u_{t^{+}\left(u\right)}}^{\prime\prime}(1)=\frac{\mu
N(q-2)}{2q}(t^{+}\left(u\right))^{\frac{N(q-2)+2}{2}}m^{\prime}(t^{+}\left(u\right))>0\text{
}$

and

$g_{u_{t^{-}\left(u\right)}}^{\prime\prime}(1)=\frac{\mu
N(q-2)}{2q}(t^{-}\left(u\right))^{\frac{N(q-2)+2}{2}}m^{\prime}(t^{-}\left(u\right))<0,$

which indicates that $u_{t^{\pm}\left(u\right)}\in\mathcal{M}^{\pm}(c)$.
Furthermore, it holds

$E_{\gamma,\mu}(u_{t^{-}\left(u\right)})\geq E_{\gamma,\mu}(u_{t})\geq
E_{\gamma,\mu}(u_{t^{+}\left(u\right)})\quad\text{for
}t\in[t^{+}\left(u\right),t^{-}\left(u\right)]$

and

$E_{\gamma,\mu}(u_{t^{+}\left(u\right)})\leq E_{\gamma,\mu}(u_{t})\text{ for
}t\in(0,t^{+}\left(u\right)).$

Thus, we have

$E_{\gamma,\mu}(u_{t^{+}\left(u\right)})=\inf_{0\leq t\leq
t_{\max}\left(u\right)}E_{\gamma,\mu}(u_{t})<0\quad\text{and}\quad
E_{\gamma,\mu}(u_{t^{-}\left(u\right)})=\sup_{t\geq
t_{\max}\left(u\right)}E_{\gamma,\mu}(u_{t})>0.$

Consequently, we complete the proof.

According to Lemmas 3.5 and 3.6, it follows that

$-\infty<m^{+}(c)<0\text{ and }m^{-}(c)>0.$

For each $u\in S(c)$, by (14) and (15) one has

$E_{\gamma,\mu}(u)\geq\frac{1}{2}A(u)-\frac{\gamma
c^{\frac{N+\alpha-p(N-2)}{2}}}{2\|Q_{p}\|_{2}^{2p-2}}A(u)^{\frac{Np-N-\alpha}{2}}-\frac{\mu\overline{S}c^{\frac{2N-q(N-2)}{4}}}{q}A(u)^{\frac{N(q-2)}{4}}.$

Define a function $h$ by

$h(s):=\frac{1}{2}s-\frac{\gamma
c^{\frac{N+\alpha-p(N-2)}{2}}}{2\|Q_{p}\|_{2}^{2p-2}}s^{\frac{Np-N-\alpha}{2}}-\frac{\mu\overline{S}c^{\frac{2N-q(N-2)}{4}}}{q}s^{\frac{N(q-2)}{4}}\text{
for }s>0.$

Under the assumptions of Theorem 1.3 $(a),$ there exist two constants
$0<R<\widetilde{R}$, both depending on $c,\gamma$ and $\mu$, such that
$h(R)=0=h(\widetilde{R})$. We note that $R$ and $\widetilde{R}$ both tend to
$0$ as $c\rightarrow 0.$ Then it follows from Lemma 3.6 that the set
$\mathcal{M}^{+}(c)$ is contained in $\mathbf{A}_{R}:=\left\\{u\in S(c)\text{
}|\text{ }A(u)<R\right\\}.$ Furthermore, we have the following lemma.

###### Lemma 3.7

Assume that $N\geq 2,\gamma>0,\mu>0$ and that one of the two following
conditions holds:
$(i)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and $2<q<2+\frac{4}{N};$
$(ii)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and
$2+\frac{4}{N}<q<2^{\ast}.$
Then for each $c<c_{\ast}$, we have
$-\infty<\widehat{m}(c):=\inf_{u\in\mathbf{A}_{R}}E_{\gamma,\mu}(u)<0.$
Furthermore, it holds
$\widehat{m}(c)=m^{+}(c):=\inf_{u\in\mathcal{M}^{+}(c)}E_{\gamma,\mu}(u)$ and
$\widehat{m}(c)<\inf_{\overline{\mathbf{A}}_{R}\backslash\mathbf{A}_{R-\rho}}E_{\gamma,\mu}(u)$
for $\rho>0$ sufficiently small.

Proof. For $u\in\mathbf{A}_{R}$, it holds $E_{\gamma,\mu}(u)\geq
h(A(u))\geq\min_{s\in[0,R]}h(s)>-\infty,$ leading to $\widehat{m}(c)>-\infty$.
Moreover, for any $u\in S(c)$, we have $A(u_{t})<R$ and
$E_{\gamma,\mu}(u_{t})<0$ for $t>0$ small enough, and hence
$\widehat{m}(c)<0.$

Since $\mathcal{M}^{+}(c)\subset\mathbf{A}_{R},$ we have
$\widehat{m}(c)\leq\inf_{u\in\mathcal{M}^{+}(c)}E_{\gamma,\mu}(u)$. On the
other hand, if $u\in\mathbf{A}_{R}$, then
$u_{t^{+}(u)}\in\mathcal{M}^{+}(c)\subset\mathbf{A}_{R}$ and
$E_{\gamma,\mu}(u_{t^{+}(u)})=\min\left\\{E_{\gamma,\mu}(u_{t})\text{ }|\text{
}t>0\text{ and }A(u_{t})<R\right\\}\leq E_{\gamma,\mu}(u),$ which implies that
$\inf_{u\in\mathcal{M}^{+}(c)}E_{\gamma,\mu}(u)\leq\widehat{m}(c)$. Thus, we
get $\widehat{m}(c)=\inf_{u\in\mathcal{M}^{+}(c)}E_{\gamma,\mu}(u)=m^{+}(c).$

According to the continuity of $h,$ there exists $\rho>0$ such that
$h(s)\geq\widehat{m}(c)/2$ for $s\in[R-\rho,R]$. Hence, we have

$E_{\gamma,\mu}(u)\geq h(A(u))\geq\frac{\widehat{m}(c)}{2}>\widehat{m}(c)$

for every $u\in S(c)$ with $R-\rho\leq A(u)\leq R.$ Consequently, we complete
the proof.

We now give the existence of a local minimizer for $E_{\gamma,\mu}.$

###### Theorem 3.8

Assume that $N\geq 2,\gamma>0,\mu>0$ and that one of the two following
conditions holds:
$(i)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and $2<q<2+\frac{4}{N};$
$(ii)$ $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and
$2+\frac{4}{N}<q<2^{\ast}.$
Then for each $c<c_{\ast}$, functional $E_{\gamma,\mu}$ has a local minimizer
$u^{+}$ on $S(c)$ satisfying $E_{\gamma,\mu}(u^{+})<0.$

Proof. Following the idea by Soave [35]. Let us consider a minimizing sequence
$\\{v_{n}\\}$ for $E_{\gamma,\mu}|_{\mathbf{A}_{R}}$. Since
$(v_{n})_{t^{+}(v_{n})}\in\mathcal{M}^{+}(c)$, we have
$A((v_{n})_{t^{+}(v_{n})})<R$ and

$E_{\gamma,\mu}((v_{n})_{t^{+}(v_{n})})=\min\left\\{E_{\gamma,\mu}((v_{n})_{t})\text{
}|\text{ }t>0\text{ and }A((v_{n})_{t^{+}(v_{n})})<R\right\\}\leq
E_{\gamma,\mu}(v_{n}).$

In this way we obtain a new minimizing sequence
$\\{\omega_{n}\\}\subset\mathcal{M}^{+}(c),$ where
$\omega_{n}:=(v_{n})_{t^{+}(v_{n})}$. It follows from Lemma 3.7 that
$A(\omega_{n})<R-\rho$ for every $n\in\mathbb{N}$. Then by applying Ekeland’s
variational principle, we can find a new minimizing sequence
$\\{u_{n}\\}\subset\mathbf{A}_{R}$ for $\widehat{m}(c)$ such that
$\|u_{n}-\omega_{n}\|_{H^{1}}^{2}\rightarrow 0$ as $n\rightarrow\infty,$ which
is also a Palais-Smale sequence for $E_{\gamma,\mu}$ on $S(c)$. Since
$\|u_{n}-\omega_{n}\|_{H^{1}}^{2}\rightarrow 0$ as $n\rightarrow\infty$ and
$\\{\omega_{n}\\}\subset\mathbf{A}_{R}$ is bounded, we have
$Q(u_{n})\rightarrow 0$ as $n\rightarrow\infty.$ Thus, by Lemma 2.7, up to a
subsequence $u_{n}\rightarrow u^{+}$ strongly in $H^{1}(\mathbb{R}^{N})$, and
$u^{+}$ is an interior local minimizer for $E_{\gamma,\mu}|_{\mathbf{A}_{R}}$,
which solves Problem $(P_{c})$ for some $\lambda>0$. Consequently, we complete
the proof.

Next, we seek for a second critical point for $E_{\gamma,\mu}|_{S(c)}$. First
of all, we shall use Lemma 2.6 in Section 2.3 to establish the existence of a
Palais–Smale sequence $\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$ for
$E_{\gamma,\mu}$ restricted to $S(c)$. Our arguments are inspired by [5, 12].

Observe that $\Theta=\emptyset$ is admissible. Now we define the functional
$J^{-}:S(c)\mapsto\mathbb{R}\quad$by

$J^{-}(u):=E_{\gamma,\mu}(u_{t^{-}\left(u\right)}).$

###### Lemma 3.9

The map $u\in S(c)\mapsto t^{-}(u)\in\mathbb{R}$ is of class $C^{1}$.

Proof. We apply the implicit function theorem on the $C^{1}$ function
$F:\mathbb{R\times}S(c)\rightarrow\mathbb{R}$ defined by
$F(t,u)=g_{u}^{\prime}(t)$. Since $F(t^{-}(u),u)=0$ and
$\mathcal{M}^{0}(c)=\emptyset$, we have
$\partial_{t}F(t^{-}(u),u)=g_{u}^{\prime\prime}(t^{-}(u))>0$, and the fact
that it is impossible to pass with continuity from $\mathcal{M}^{-}(c)$ to
$\mathcal{M}^{+}(c)$.

By the above lemma, we obtain that functional $J^{-}$ is of class $C^{1}$.

###### Lemma 3.10

The map $T_{u}S(c)\rightarrow T_{u_{t^{-}(u)}}S\left(c\right)$ defined by
$\psi\rightarrow\psi_{t^{-}(u)}$ is isomorphism.

Proof. Clearly the map is linear. For $\psi\in T_{u}S(c),$ we have

$\int_{\mathbb{R}^{N}}u_{t}(x)\psi_{t}(x)dx=\int_{\mathbb{R}^{N}}t^{N/2}u(tx)t^{N/2}\psi(tx)dx=\int_{\mathbb{R}^{N}}u(y)\psi(y)dy=0,$

which implies that $\psi_{t}\in T_{u_{t}}S(c)$ and the map is well-defined.
The rest of the proof is standard, see for example [5, Lemma 3.6].

###### Lemma 3.11

It holds
$(J^{-})^{\prime}(u)[\psi]=E_{\gamma,\mu}^{\prime}(u_{t^{-}(u)})[\psi_{t^{-}(u)}]$
for any $u\in S(c)$ and $\psi\in T_{u}S(c).$

Proof. The proof is similar to that of [12, Lemma 3.15], we omit it here.

###### Lemma 3.12

Let $\mathcal{F}$ be a homotopy stable family of compact subsets of $S(c)$
with closed boundary $\Theta$ and let

$e_{\mathcal{F}}^{-}:=\inf_{H\in\mathcal{F}}\max_{u\in H}J^{-}(u).$

Assume that $\Theta$ is contained in a connected component of
$\mathcal{M}^{-}(c)$ and $\max\\{\sup
J^{-}(\Theta),0\\}<e_{\mathcal{F}}^{-}<\infty$. Then there exists a
Palais–Smale sequence $\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$ for
$E_{\gamma,\mu}$ restricted to $S(c)$ at level $e_{\mathcal{F}}^{-}$.

Proof. First of all, we take $\\{D_{n}\\}\subset\mathcal{F}$ such that
$\max_{u\in D_{n}}J^{-}(u)<e_{\mathcal{F}}^{-}+\frac{1}{n}$ and define
$\eta:[0,1]\times S(c)\rightarrow S(c)$ by

$\eta(s,u)=u_{1-s+st^{-}(u)}.$

Since $t^{-}(u)=1$ for any $u\in\mathcal{M}^{-}(c)$ and
$\Theta\subset\mathcal{M}^{-}(c)$, we have $\eta(s,u)=u$ for
$(s,u)\in(\\{0\\}\times S(c))\cup([0,1]\times\Theta)$. Also note that $\eta$
is continuous. Then it follows from the definition of $\mathcal{F}$ that
$\mathbf{A}_{n}:=\eta(\\{1\\}\times D_{n})=\\{u_{t^{-}(u)}$ $|$ $u\in
D_{n}\\}\in\mathcal{F}.$ Clearly, $\mathbf{A}_{n}\subset\mathcal{M}^{-}(c)$
for all $n\in\mathbb{N}$. Let $v\in\mathbf{A}_{n}$. Then $v=u_{t^{-}(u)}$ for
some $u\in D_{n},$ and $J^{-}(u)=J^{-}(v)$, which shows that
$\max_{\mathbf{A}_{n}}J^{-}=\max_{D_{n}}J^{-}.$ Thus,
$\\{\mathbf{A}_{n}\\}\subset\mathcal{M}^{-}(c)$ is another minimizing sequence
of $e_{\mathcal{F}}^{-}$. Applying Lemma 2.6 yields a Palais–Smale sequence
$\\{\widehat{u}_{n}\\}$ for $J^{-}$ on $S(c)$ at level $e_{\mathcal{F}}^{-}$
satisfying
$\text{dist}_{H^{1}(\mathbb{R}^{N})}(\widehat{u}_{n},\mathbf{A}_{n})\rightarrow
0$ as $n\rightarrow\infty$. Setting $t_{n}=t^{-}(\widehat{u}_{n})$ and
$u_{n}=(\widehat{u}_{n})_{t_{n}}\in\mathcal{M}^{-}(c)$. We claim that there
exists a constant $C_{0}>0$ such that

$\frac{1}{C_{0}}\leq t_{n}^{2}\leq C_{0}$ (36)

for $n\in\mathbb{N}$ large enough. Indeed, note that

$t_{n}^{2}=\frac{A((\widehat{u}_{n})_{t_{n}})}{A(\widehat{u}_{n})}=\frac{A(u_{n})}{A(\widehat{u}_{n})}.$
(37)

Since $E_{\gamma,\mu}(u_{n})=J^{-}(\widehat{u}_{n})\rightarrow
e_{\mathcal{F}}^{-}$ as $n\rightarrow\infty$, it follows from Lemma 3.5 that
there exists a constant $M_{0}>0$ such that

$\frac{1}{M_{0}}\leq A(u_{n})\leq M_{0}.$ (38)

On the other hand, since $\\{\mathbf{A}_{n}\\}\subset\mathcal{M}^{-}(c)$ is a
minimizing sequence for $e_{\mathcal{F}}^{-}$ and $E_{\gamma,\mu}$ is coercive
on $\mathcal{M}^{-}(c)$, we conclude that $\\{\mathbf{A}_{n}\\}$ is uniformly
bounded in $H^{1}(\mathbb{R}^{N})$, and combining the fact of
$\text{dist}_{H^{1}(\mathbb{R}^{N})}\left(\widehat{u}_{n},\mathbf{A}_{n}\right)\rightarrow
0$ as $n\rightarrow\infty,$ gives $\sup_{n}A(\widehat{u}_{n})<\infty$. Also,
since $\\{\mathbf{A}_{n}\\}$ is compact for every $n\in\mathbb{N}$, there
exists a $v_{n}\in\mathbf{A}_{n}$ such that
dist${}_{H^{1}(\mathbb{R}^{N})}(\widehat{u}_{n},\mathbf{A}_{n})=\|v_{n}-\widehat{u}_{n}\|_{H^{1}(\mathbb{R}^{N})}.$
Then by Lemma 3.5, we obtain that for some $\sigma>0,$

$A(\widehat{u}_{n})\geq
A(v_{n})-A(\widehat{u}_{n}-v_{n})\geq\frac{\sigma}{2}.$ (39)

Thus, by (37)–(39), we proves the claim.

Next, we show that $\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$ is a Palais–Smale
sequence for $E_{\gamma,\mu}$ on $S(c)$ at level $e_{\mathcal{F}}^{-}$. Denote
by $\|\cdot\|_{\ast}$ the dual norm of $(T_{u_{n}}S(c))^{\ast}$. Then we have

$\|E_{\gamma,\mu}^{\prime}(u_{n})\|_{\ast}=\sup_{\psi\in
T_{u_{n}}S(c),\|\psi\|\leq
1}|E_{\gamma,\mu}^{\prime}(u_{n})[\psi]|=\sup_{\psi\in
T_{u_{n}}S(c),\|\psi\|\leq
1}|E_{\gamma,\mu}^{\prime}(u_{n})[(\psi_{-t_{n}})_{t_{n}}]|.$

By Lemma 3.10, we obtain that $T_{\widehat{u}_{n}}S(c)\rightarrow
T_{u_{n}}S(c)$ defined by $\psi\rightarrow\psi_{t_{n}}$ is an isomorphism.
Moreover, it follows from Lemma 3.11 that

$(J^{-})^{\prime}(\widehat{u}_{n})[\psi_{-t_{n}}]=E_{\gamma,\mu}^{\prime}((\widehat{u}_{n})_{t_{n}})[(\psi_{-t_{n}})_{t_{n}}]=E_{\gamma,\mu}^{\prime}(u_{n})[(\psi_{-t_{n}})_{t_{n}}].$

It follows that

$\|E_{\gamma,\mu}^{\prime}(u_{n})\|_{\ast}=\sup_{\psi\in
T_{u_{n}}S(c),\|\psi\|\leq
1}|(J^{-})^{\prime}(\widehat{u}_{n})[\psi_{-t_{n}}]|.$ (40)

Note that $\|\psi_{-t_{n}}\|_{H^{1}}\leq C\|\psi\|_{H^{1}}\leq C$ for some
$C>0$ by (36). Thus, from (40) we deduce that
$\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$ is a Palais–Smale sequence for
$E_{\gamma,\mu}$ on $S(c)$ at level $e_{\mathcal{F}}^{-}$. Consequently, we
complete the proof.

###### Lemma 3.13

Under the assumptions of Theorem 1.3 $(a),$ there exists a Palais–Smale
sequence $\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$ for $E_{\gamma,\mu}$
restricted to $S(c)$ at the level
$m^{-}(c):=\inf_{u\in\mathcal{M}^{-}(c)}E_{\gamma,\mu}(u)$.

Proof. Let $\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$. By Lemma 3.12, we choose
the set $\overline{\mathcal{F}}$ of all singletons belonging to $S(c)$ and
$\Theta=\emptyset$, which is clearly a homotopy stable family of compact
subsets of $S(c)$ (without boundary). Note that
$e_{\overline{\mathcal{F}}}^{-}=\inf_{H\in\overline{\mathcal{F}}}\max_{u\in
H}J^{-}(u)=\inf_{u\in S(c)}J^{-}(u)=m^{-}(c).$ Then the lemma follows directly
from Lemma 3.12. This completes the proof.

####  3.2.2 Mass critical and mass supercritical case

###### Lemma 3.14

Assume that $N\geq 2,\gamma>0,\mu>0$ and that one of the two following
conditions holds:
$(i)$ $p=\frac{N+\alpha+2}{N},2+\frac{4}{N}<q<2^{\ast}$ and
$c\gamma^{N/(\alpha+2)}<\|Q_{(N+\alpha+2)/N}\|_{2}^{2};$
$(ii)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},q=2+\frac{4}{N}$ and
$c\mu^{N/2}<\left(\frac{N+2}{\overline{S}N}\right)^{N/2}.$
Then functional $E_{\gamma,\mu}$ is bounded from below by a positive constant
and coercive on $\mathcal{M}(c)=\mathcal{M}^{-}(c)$.

Proof. Clearly, $\mathcal{M}(c)=\mathcal{M}^{-}(c)$ by (21). Next, we only
prove the conclusion under the condition $(i)$, since the other one is
similar. For $u\in\mathcal{M}(c)=\mathcal{M}^{-}(c)$, we have

$A(u)-\frac{\gamma N}{N+\alpha+2}B(u)-\frac{\mu N(q-2)}{2q}C(u)=0.$ (41)

Using this, together with (14) and (15), gives

$A(u)\leq\frac{1}{\|Q_{p}\|_{2}^{2(\alpha+2)/N}}\gamma
c^{\frac{\alpha+2}{N}}A(u)+\frac{\mu
N(q-2)}{2q}\overline{S}c^{\frac{2N-(N-2)q}{4}}A(u)^{\frac{(q-2)N}{4}},$

which implies that there exists a constant $D_{1}>0,$ depending on
$\gamma,\mu$ and $c,$ such that

$A(u)\geq D_{1},$ (42)

since $c\gamma^{N/(\alpha+2)}<\|Q_{(N+\alpha+2)/N}\|_{2}^{2}$ and
$\frac{N(q-2)}{4}>1.$ Then it follows from (15), (41) and (42) that

$\displaystyle E_{\gamma,\mu}(u)$ $\displaystyle=$
$\displaystyle\frac{N(q-2)-4}{2N(q-2)}A(u)-\frac{\gamma(N(q-2)-4)}{2(N+\alpha+2)(q-2)}B(u)$
$\displaystyle\geq$ $\displaystyle\frac{N(q-2)-4}{2N(q-2)}\left(1-\frac{\gamma
c^{\frac{\alpha+2}{N}}}{\|Q_{(N+\alpha+2)/N}\|_{2}^{(2\alpha+4)/N}}\right)A(u)$
$\displaystyle>$ $\displaystyle\frac{N(q-2)-4}{2N(q-2)}\left(1-\frac{\gamma
c^{\frac{\alpha+2}{N}}}{\|Q_{(N+\alpha+2)/N}\|_{2}^{(2\alpha+4)/N}}\right)D_{1}>0,$

which implies that $E_{\gamma,\mu}$ is bounded from below by a positive
constant and coercive on $\mathcal{M}(c)$, since
$c\gamma^{N/(\alpha+2)}<\|Q_{(N+\alpha+2)/N}\|_{2}^{2}$ and $q>2+\frac{4}{N}.$
This completes the proof.

####  3.2.3 Double mass supercritical case

###### Lemma 3.15

Assume that $N\geq 2,\gamma>0,\mu>0,\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$
and $2+\frac{4}{N}<q<2^{\ast}$. Then functional $E_{\gamma,\mu}$ is bounded
from below by a positive constant and coercive on
$\mathcal{M}(c)=\mathcal{M}^{-}(c)$ for all $c>0$.

Proof. By (21) one has $\mathcal{M}(c)=\mathcal{M}^{-}(c)$. For
$u\in\mathcal{M}(c),$ it follows from (14) and (15) that

$A(u)\leq\frac{\gamma
pNc^{\frac{N+\alpha-(N-2)p}{2}}}{(N+\alpha+2)\|Q_{p}\|_{2}^{2p-2}}A(u)^{\frac{Np-N-\alpha}{2}}+\frac{\mu
N(q-2)\overline{S}c^{\frac{2N-(N-2)q}{4}}}{2q}A(u)^{\frac{(q-2)N}{4}},$

which implies that there exists a constant $D_{2}>0,$ depending on
$\gamma,\mu$ and $c,$ such that

$A(u)\geq D_{2},$ (43)

since $\frac{Np-N-\alpha}{2}>1$ and $\frac{N(q-2)}{4}>1.$ Next, we prove the
conclusion by separating two cases as follows:
Case $(i):2Np-qN-2\alpha\leq 0.$ Note that $\mathcal{M}(c)=\mathcal{M}^{-}(c)$
by (21). It follows from (41) and (43) that

$\displaystyle E_{\gamma,\mu}(u)$ $\displaystyle=$
$\displaystyle\frac{Np-N-\alpha-2}{2(Np-N-\alpha)}A(u)-\frac{\mu(2Np-2\alpha-
qN)}{2q(Np-N-\alpha)}C(u)$ $\displaystyle\geq$
$\displaystyle\frac{Np-N-\alpha-2}{2(Np-N-\alpha)}D_{2}>0.$

Hence, $E_{\gamma,\mu}$ is bounded from below by a positive constant and
coercive on $\mathcal{M}(c)$.
Case $(ii):2Np-qN-2\alpha>0.$ Clearly $\mathcal{M}(c)=\mathcal{M}^{-}(c)$ by
(2.2). Then by (41) and (43) one has

$\displaystyle E_{\gamma,\mu}(u)$ $\displaystyle=$
$\displaystyle\frac{N(q-2)-4}{2N(q-2)}A(u)-\frac{\gamma(Nq-2Np+2\alpha)}{2Np(q-2)}B(u)$
$\displaystyle\geq$ $\displaystyle\frac{N(q-2)-4}{2N(q-2)}D_{2}>0,$

which implies that $E_{\gamma,\mu}$ is bounded from below by a positive
constant and coercive on $\mathcal{M}(c).$ This completes the proof.

####  3.2.4 The proof of Theorem 1.3

First of all, we give the proof of Theorem 1.3 $(a).$ By Lemma 3.12, we find a
Palais–Smale sequence $\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$ for $E$
restricted to $S(c)$ at level $m^{-}(c)>0$, which is bounded in
$H^{1}(\mathbb{R}^{N})$ via Lemma 3.5. Thus, from Lemma 2.7 we obtain that
$u_{n}\rightarrow u^{-}$ strongly in $H^{1}(\mathbb{R}^{N})$, and $u^{-}\in
S(c)$ is a solution to Problem $(P_{c})$ for some $\lambda^{-}>0$. Therefore,
combining Theorem 3.8, Problem $(P_{c})$ has two positive radial solutions
$(\lambda^{\pm},u^{\pm})\in\mathbb{R}^{+}\times H^{1}(\mathbb{R}^{N})$ with
$E_{\gamma,\mu}(u^{-})>0>E_{\gamma,\mu}(u^{+})$, where $u^{+}$ is a ground
state solution. Moreover, by (35), one has $A(u^{-})\rightarrow\infty$ as
$c\rightarrow 0,$ and together with the fact of $R\rightarrow 0$ as
$c\rightarrow 0,$ leading to $\|\nabla u^{+}\|_{2}<\|\nabla u^{-}\|_{2}$ for
$c>0$ sufficiently small.

Next, we prove Theorem 1.3 $(b).$ Similar to Lemma 3.12, we can construct a
Palais–Smale sequence $\\{u_{n}\\}\subset\mathcal{M}(c)=\mathcal{M}^{-}(c)$
for $E$ restricted to $S(c)$ at level $m^{-}(c)>0$, which is bounded in
$H^{1}(\mathbb{R}^{N})$ by Lemmas 3.14 and 3.15. Thus, it follows from Lemma
2.7 that $u_{n}\rightarrow u^{-}$ strongly in $H^{1}(\mathbb{R}^{N})$. Hence,
Problem $(P_{c})$ has a radial ground state solution $u^{-}\in S(c)$
satisfying $E_{\gamma,\mu}(u^{-})=m^{-}(c)>0$ for some $\lambda^{-}>0$. This
completes the proof.

##  4 The case of $\gamma<0$ and $\mu>0$

###  4.1 Bounded from below on $S(c)$

###### Lemma 4.1

Assume that $N\geq 2,\gamma<0,\mu>0,\frac{N+\alpha}{N}<p<2_{\alpha}^{\ast}$
and $2<q<2+\frac{4}{N}$. Then functional $E_{\gamma,\mu}$ is bounded from
below and coercive on $S(c)$.

Proof. Since $\gamma<0$, from (14), we have

$E_{\gamma,\mu}(u)\geq\frac{1}{2}A(u)-\frac{\mu}{q}\overline{S}A(u)^{\frac{N(q-2)}{4}}c^{\frac{2N-q(N-2)}{4}},$

which implies that the conclusion holds. We complete the proof.

Next, following the idea of [7, 8], we discuss the strong subadditivity
properties for $\sigma(c)$ under some cases of $p,q$ when $\gamma<0$ and
$\mu>0.$

###### Lemma 4.2

Assume that $N\geq 2,\gamma<0,\mu>0,\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$
and $\frac{4p+2\alpha}{\alpha+2}<q<2+\frac{4}{N}.$ Then there exists a
constant $c_{3}>0$ such that for $c>c_{3},$

$-\infty<\sigma(c)<0\text{ and }\sigma(c)<\sigma(\eta)+\sigma(c-\eta)\text{
for }0<\eta<c.$

Proof. Define $u_{\theta}(x):=\theta^{1-\frac{N}{2}\beta}u(\theta^{-\beta}x)$.
Then we have $\|u_{\theta}\|_{2}^{2}=\theta^{2}\|u\|_{2}^{2}$ and

$\displaystyle E_{\gamma,\mu}(u_{\theta})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\theta^{2-2\beta}A(u)-\frac{\gamma}{2p}\theta^{2p-\beta(Np-N-\alpha)}B(u)-\frac{\mu}{q}\theta^{q(1-\frac{N}{2}\beta)+N\beta}C(u)$
$\displaystyle=$
$\displaystyle\theta^{2}\left[E_{\gamma,\mu}(u)+f_{\beta}(\theta,u)\right],$

where

$f_{\beta}(\theta,u):=\frac{1}{2}(\theta^{-2\beta}-1)A(u)-\frac{\gamma}{2p}(\theta^{2p-\beta(Np-N-\alpha)-2}-1)B(u)-\frac{\mu}{q}(\theta^{q(1-\frac{N}{2}\beta)+N\beta-2}-1)C(u).$
(44)

From now on, we let $\beta=\frac{2p-2}{Np-N-\alpha-2}<0.$ Then it holds

$E_{\gamma,\mu}(u_{\theta})=\frac{1}{2}\theta^{\frac{2(N+\alpha-p(N-2))}{N+\alpha+2-Np}}A(u)-\frac{\gamma}{2p}\theta^{\frac{2(N+\alpha-p(N-2))}{N+\alpha+2-Np}}B(u)-\frac{\mu}{q}\theta^{\frac{q(\alpha+2)-2N(p-1)}{N+\alpha+2-Np}}C(u)<0$

for $\theta>0$ sufficiently large, since
$\frac{q(\alpha+2)-2N(p-1)}{N+\alpha+2-Np}>\frac{2(N+\alpha-p(N-2))}{N+\alpha+2-Np}$.
This indicates that there exists a constant $c_{3}>0$ such that $\sigma(c)<0$
for $c>c_{3}$.

For $c>c_{0},$ let $\\{u_{n}\\}$ be a minimizing sequence in $S(c)$ with
$\sigma(c)<0$. Then one get

$0<C_{1}<A(u_{n})<C_{2}\quad\text{and }0<C_{3}<C(u_{n})<C_{4}.$

Note that $f_{\beta}(1,u_{n})=0.$ Since $q>\frac{4p+2\alpha}{\alpha+2},$ we
have

$\displaystyle\frac{d}{d\theta}f_{\beta}(\theta,u_{n})|_{\theta=1}$
$\displaystyle=$
$\displaystyle\frac{4(p-1)}{N+\alpha+2-Np}\left[\frac{1}{2}A(u_{n})-\frac{\gamma}{2p}B(u_{n})\right]$
(45) $\displaystyle-\frac{\mu(\alpha+2)(q-2)}{q(N+\alpha+2-Np)}C(u_{n})<0.$

Moreover, for all $\theta>1$, it follows that

$\displaystyle\frac{d^{2}}{d\theta^{2}}f_{\beta}(\theta,u_{n})$
$\displaystyle=$
$\displaystyle\frac{4(p-1)(4(p-1)-N-\alpha-2+Np)}{(N+\alpha+2-Np)^{2}}\theta^{\frac{4p-2N-2\alpha+2Np-8}{N+\alpha+2-Np}}\left[\frac{1}{2}A(u_{n})-\frac{\gamma}{2p}B(u_{n})\right]$
(46)
$\displaystyle-\frac{\mu(\alpha+2)(q-2)((\alpha+2)(q-2)-N-\alpha-2+Np)}{q(N+\alpha+2-Np)^{2}}\theta^{\frac{q(\alpha+2)-2N-4\alpha-8+2Np}{N+\alpha+2-Np}}C(u_{n})$
$\displaystyle<$ $\displaystyle 0.$

Using (45) and (46) gives $f_{\beta}(\theta,u_{n})<0$ for all $\theta>1$. So
we have

$\sigma(\theta^{2}c)<\theta^{2}E_{\gamma,\mu}(u_{n})=\theta^{2}\sigma(c)+o(1)\text{
for all }c>c_{3}\text{ and }\theta>1.$ (47)

Without loss of generality, we assume that $0<\eta<c-\eta$. Then we
distinguish the following three cases:

$(i)\text{ }\eta<c-\eta<c_{3};\text{ }(ii)\text{ }\eta<c_{3}<c-\eta;\text{
}(iii)\text{ }c_{3}<\eta<c-\eta.$

For the first case, we observe that $\sigma(c)<0,\sigma(c-\eta)>0$ and
$\sigma(\eta)>0$. Clearly, $\sigma(c)<\sigma(c-\eta)+\sigma(\eta)$. For the
second one, since $c-\eta>c_{3}$ and $\sigma(c-\eta)<0,$ it follows from (47)
that $\sigma(\theta^{2}(c-\eta))<\theta^{2}\sigma(c-\eta),$ which implies that

$\sigma(c)<\frac{c}{c-\eta}\sigma(c-\eta)=\frac{\eta}{c-\eta}\sigma(c-\eta)+\sigma(c-\eta)<\sigma(\eta)+\sigma(c-\eta),$

where we have used the fact of $\sigma(\eta)>0.$ For the last one, by (47) we
get

$\sigma(c)<\frac{c}{c-\eta}\sigma(c-\eta)=\sigma(c-\eta)+\frac{\eta}{c-\eta}\sigma(c-\eta)<\sigma(c-\eta)+\sigma(\eta).$

We complete the proof.

###### Lemma 4.3

Assume that $N\geq 2,\gamma<0,\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$ and
$q=\frac{4p+2\alpha}{\alpha+2}$. Then there exists a constant
$\overline{\mu}>0$ such that for $\mu>\overline{\mu},$

$-\infty<\sigma(c)<0\text{ and }\sigma(c)<\sigma(\eta)+\sigma(c-\eta)\text{
for }0<\eta<c.$

Proof. Fix $u\in S(1)$, i.e. $u\in H^{1}(\mathbb{R}^{N})$ with
$\|u\|_{2}^{2}=1.$ Then there exists a constant $\overline{\mu}>0$ large
enough such that

$E_{\gamma,\mu}(u)=\frac{1}{2}A(u)-\frac{\gamma}{2p}B(u)-\frac{\mu}{q}C(u)<0\text{
for all }\mu>\overline{\mu}.$

Let $u_{\theta}(x)=\theta u\left(\theta^{\frac{2(p-1)}{\alpha+2}}x\right),$
where $\theta^{\frac{2(N+\alpha+2-Np)}{\alpha+2}}=c>0$. Then a direct
calculation shows that
$\|u_{\theta}\|_{2}^{2}=\theta^{\frac{2(N+\alpha+2-Np)}{\alpha+2}}\|u\|_{2}^{2}=c$
and

$\displaystyle E_{\gamma,\mu}(u_{\theta})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\theta^{\frac{2(N+\alpha-p(N-2))}{\alpha+2}}A(u)-\frac{\gamma}{2p}\theta^{\frac{2(N+\alpha-p(N-2))}{\alpha+2}}B(u)-\frac{\mu}{q}\theta^{\frac{2(N+\alpha-p(N-2))}{\alpha+2}}C(u)$
$\displaystyle=$
$\displaystyle\theta^{\frac{2(N+\alpha-p(N-2))}{\alpha+2}}E_{\gamma,\mu}(u)<0,$

which implies that $\sigma(c)<0$ for all $\mu>\overline{\mu}.$

For $u\in H^{1}(\mathbb{R}^{N})$ with $0<\|u\|_{2}^{2}=\eta<c$, we set
$u_{\theta}(x)=\theta u\left(\theta^{\frac{2(p-1)}{\alpha+2}}x\right)$ and
$\theta^{\frac{2(N+\alpha+2-Np)}{\alpha+2}}=\delta$, where $1<\delta\leq
c/\eta$. Then we have
$\|u_{\theta}\|_{2}^{2}=\theta^{\frac{2(N+\alpha+2-Np)}{\alpha+2}}\|u\|_{2}^{2}=\delta\eta\leq
c$ and

$\sigma(\delta\eta)\leq
E_{\gamma,\mu}(u_{\theta})=\theta^{\frac{2(N+\alpha-p(N-2))}{\alpha+2}}E_{\gamma,\mu}(u).$

Let $\\{u_{n}\\}$ be a minimizing sequence for $\sigma(\eta)<0$. Then it holds

$\sigma(\delta\eta)\leq\theta^{\frac{2(N+\alpha-p(N-2))}{\alpha+2}}\lim_{n\rightarrow\infty}E_{\gamma,\mu}(u_{n})=\theta^{\frac{2(N+\alpha-p(N-2))}{\alpha+2}}\sigma(\eta)<\theta^{\frac{2(N+\alpha+2-Np)}{\alpha+2}}\sigma(\eta)=\delta\sigma(\eta),$

since $\theta>1$ and $\sigma(\eta)<0$. Without loss of generality, we may
assume that $0<\eta<c-\eta$, and so

$\sigma(c)\leq\frac{c}{\eta}\sigma(\eta)=\sigma(\eta)+\frac{c-\eta}{\eta}\sigma(\eta)<\sigma(\eta)+\sigma(c-\eta).$

We complete the proof.

By Lemmas $\ref{L4-0}-\ref{L4-12},$ we have the following compactness result.

###### Lemma 4.4

Assume that $N\geq 2,\gamma<0,\mu>0$ and that one of the two following
conditions holds:
$(i)$
$\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},\frac{4p+2\alpha}{\alpha+2}<q<2+\frac{4}{N}$
and $c>c_{3}$ for some $c_{3}>0;$
$(ii)$
$\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},q=\frac{4p+2\alpha}{\alpha+2}$ and
$\mu>\overline{\mu}$ for some $\overline{\mu}>0.$
Let $\\{u_{n}\\}\subset H^{1}(\mathbb{R}^{N})$ be a sequence such that
$E_{\gamma,\mu}(u_{n})\rightarrow\sigma(c)$ and $\|u_{n}\|_{2}^{2}\rightarrow
c$. Then $\\{u_{n}\\}$ is relatively compact in $H^{1}(\mathbb{R}^{N})$ up to
translations, that is, there exist a subsequence, still denoted by
$\\{u_{n}\\}$, a sequence of points $\\{y_{n}\\}\subset\mathbb{R}^{N}$, and a
function $u_{0}\in S(c)$ such that $u_{n}(\cdot+y_{n})\rightarrow u_{0}$
strongly in $H^{1}(\mathbb{R}^{N}).$

Proof. The proof is very similar to that of Lemma 3.3, we omit it here.

For some other cases of $p,q$ when $\gamma<0$ and $\mu>0,$ it seems to be
difficult to verify the strong subadditivity inequality for $\sigma(c).$ In
view of this, the concentration-compactness principle [26] is not applicable.
We shall use the approach developed by Bellazzini and Siciliano [7] to obtain
the compactness.

First of all, we give two definitions and a lemma. Let
$W:H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}$ be a functional of the
following form

$W(u):=\frac{1}{2}A\left(u\right)+T(u),$

where the nonlinear operator $T\in C^{1}(H^{1}(\mathbb{R}^{N}),\mathbb{R})$
satisfies some suitable assumptions. In particular, we require that $T$ is
invariant for the noncompact group of translations in $\mathbb{R}^{N}.$ We
look at the constrained minimization problem

$\xi(c):=\inf_{u\in S(c)}W(u),$

where $\xi>-\infty$ is assumed.

###### Definition 4.5

[7, Definition 2.1] Let $u\in H^{1}(\mathbb{R}^{N})\backslash\\{0\\}$. A
continuous path
$\hat{g}_{u}:\theta\in\mathbb{R}^{+}\mapsto\hat{g}_{u}(\theta)\in
H^{1}(\mathbb{R}^{N})$ such that $\hat{g}_{u}(1)=u$ is said to be a scaling
path of $u$ if

$\Pi_{\hat{g}_{u}}(\theta):=\|\hat{g}_{u}(\theta)\|_{2}^{2}\|u\|_{2}^{-2}\text{is
differentiable}\quad\text{and}\quad\Pi_{\hat{g}_{u}}^{\prime}(1)\neq 0,$ (48)

where the prime denotes the derivative. We denote with $\mathcal{G}_{u}$ the
set of the scaling paths of $u$.

###### Definition 4.6

[7, Definition 2.2] Let $u\neq 0$ be fixed and
$\hat{g}_{u}\in\mathcal{G}_{u}$. We say that the scaling path $\hat{g}_{u}$ is
admissible for functional $W,$ if $\chi_{{}_{\hat{g}_{u}}}$ is a
differentiable function, where

$\chi_{\hat{g}_{u}}(\theta):=W(\hat{g}_{u}(\theta))-\Pi_{\hat{g}_{u}}(\theta)W(u)\text{
for }\theta\geq 0.$ (49)

###### Lemma 4.7

[7, Theorem 2.1] Let $c>0$ and $\\{u_{n}\\}$ be a minimizing sequence for
$\xi(c)$ weakly convergent, up to translations, to a nonzero function $u_{0}.$
Assume that

$T(u_{n}-u_{0})+T(u_{0})=T(u_{n})+o(1);$ (50)
$T(\alpha_{n}(u_{n}-u_{0}))-T(u_{n}-u_{0})=o(1),\quad\text{where}\quad\alpha_{n}=\frac{c-\|u_{0}\|_{2}^{2}}{\|u_{n}-u_{0}\|_{2}^{2}};$
(51) $\left\langle T^{\prime}(u_{n}),u_{n}\right\rangle=O(1)\text{ and
}\left\langle
T^{\prime}(u_{n})-T^{\prime}(u_{m}),u_{n}-u_{m}\right\rangle=o(1)\quad\text{as}\quad
n,m\rightarrow\infty.$ (52)

In addition, assume that $\xi(c)\leq\xi(\eta)+\xi(c-\eta)$ for all $0<\eta<c$
(weak subadditivity inequality), $-\infty<\xi(s)<0\quad$for all$\quad
s>0\quad(W(0)=0)$ and the following condition

$s\mapsto\xi(s)\text{ is continuous and }\lim_{s\rightarrow
0}\frac{\xi(s)}{s}=0.$ (53)

Then for every $c>0$ the set $\mathfrak{B}(c)=\cup_{\eta\in(0,c]}\\{u\in
S(\eta)$ $|$ $W(u)=\xi(\eta)\\}$ is nonempty. If in addition

$\text{for any }u\in\mathfrak{B}(c),\text{ there exists
}\hat{g}_{u}\in\mathcal{G}_{u}\text{ admissible such that
}\chi_{\hat{g}_{u}}^{\prime}(1)\neq 0,$ (54)

then the function $s\mapsto\frac{\xi(s)}{s}$ is monotone decreasing.
Furthermore, it holds $\|u_{n}-u_{0}\|_{H^{1}}\rightarrow 0$ as
$n\rightarrow\infty$ and $W(u_{0})=\xi(c).$

Using Lemma 4.7, we have the following result.

###### Lemma 4.8

Assume that $N\geq 2,\gamma<0,\mu>0$ and that one of the two following
conditions holds:
$(i)$
$\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N},2<q<\frac{4p+2\alpha}{\alpha+2}$
and $c<c_{1}$ for some $c_{1}>0;$
$(ii)$ $\frac{N+\alpha+2}{N}\leq p<2_{\alpha}^{\ast},2<q<2+\frac{4}{N}$ and
$c<c_{2}$ for some $c_{2}>0.$
Then we have $\sigma(c)<0$. Furthermore, if $\\{u_{n}\\}$ is a minimizing
sequence weakly convergent to a certain nonzero function $u_{0}$, then
$\|u_{n}-u_{0}\|_{H^{1}}\rightarrow 0$ as $n\rightarrow\infty$ and
$E_{\gamma,\mu}(u_{0})=\sigma(c)<0$.

Proof. We verify all the hypotheses of Lemma 4.7 step by step.

Step 1. Since functional $E_{\gamma,\mu}$ is translation-invariant, we always
have

$\sigma(c)\leq\sigma(\eta)+\sigma(c-\eta)\text{ for all}\,0<\eta<c.$

Moreover, functional
$T_{\gamma,\mu}\left(u\right):=-\frac{\gamma}{2p}B(u)-\frac{\mu}{q}C(u)$
satisfies (50)–(52). These are proved in [8, Proposition 3.1] and [37,
Proposition 2.3], respectively.

Step 2. $\sigma(c)<0$ for each $c>0.$

Under condition $(i)$, we define
$u_{\theta}(x):=\theta^{1-\frac{N}{2}\beta}u(\theta^{-\beta}x)$ for
$\theta>0.$ Then a direct calculation shows that
$\|u_{\theta}\|_{2}^{2}=\theta^{2}\|u\|_{2}^{2}$ and

$E_{\gamma,\mu}(u_{\theta})=\frac{1}{2}\theta^{2-2\beta}A(u)-\frac{\gamma}{2p}\theta^{2p-\beta(Np-N-\alpha)}B(u)-\frac{\mu}{q}\theta^{q(1-\frac{N}{2}\beta)+N\beta}C(u).$

Take $\beta=\frac{2p-2}{Np-N-\alpha-2}$, we have

$E_{\gamma,\mu}(u_{\theta})=\frac{1}{2}\theta^{\frac{2(N+\alpha-p(N-2))}{N+\alpha+2-Np}}A(u)-\frac{\gamma}{2p}\theta^{\frac{2(N+\alpha-p(N-2))}{N+\alpha+2-Np}}B(u)-\frac{\mu}{q}\theta^{\frac{q(\alpha+2)-2N(p-1)}{N+\alpha+2-Np}}C(u)<0$

for $\theta$ sufficiently small, since
$(\alpha+2)q-2N(p-1)<2(N+\alpha-p(N-2))$. Then there exists a constant
$c_{1}>0$ such that $\sigma(c)<0$ for $0<c\leq c_{1}$. By iterating this
procedure, we obtain that $\sigma(c)<0$ for all $c>0$. Under condition $(ii)$,
we consider another text function $u_{\theta}(x):=\theta^{N/2}u(\theta x)$ for
$\theta>0.$ Then $\|u_{\theta}\|_{2}^{2}=\|u\|_{2}^{2}$ and for $\theta>0$
small enough, we have

$E_{\gamma,\mu}(u_{\theta})=\frac{1}{2}\theta^{2}A(u)-\frac{\gamma}{2p}\theta^{Np-N-\alpha}B(u)-\frac{\mu}{q}\theta^{\frac{N(q-2)}{2}}C(u)<0,$

since $\frac{N(q-2)}{2}<2\leq Np-N-\alpha$. Thus, $\sigma(c)<0$ for all $c>0.$

Step 3. For every $c>0$, all the minimizing sequences $\\{u_{n}\\}$ for
$\sigma(c)$ have a weak limit, up to translations, different from zero, and
the weak limit is in $\mathfrak{B}(c)$. These are proved in [7].

Step 4. The function $s\mapsto\sigma(s)$ satisfies (53).

First of all, we prove that if $c_{n}\rightarrow c,$ then
$\lim_{n\rightarrow\infty}\sigma(c_{n})=\sigma(c)$. For every
$n\in\mathbb{N}$, let $\tau_{n}\in S(c_{n})$ such that
$E_{\gamma,\mu}(\tau_{n})<\sigma(c_{n})+\frac{1}{n}<\frac{1}{n}.$ Using (14)
gives

$\frac{1}{2}\|\nabla\tau_{n}\|_{2}^{2}-\frac{\mu}{q}\overline{S}c_{n}^{\frac{2N-q(N-2)}{4}}\|\nabla\tau_{n}\|_{2}^{\frac{N(q-2)}{2}}\leq\frac{1}{2}\|\nabla\tau_{n}\|_{2}^{2}-\frac{\mu}{q}\|\tau_{n}\|_{q}^{q}\leq
E(\tau_{n})<\frac{1}{n},$

which implies that $\\{\tau_{n}\\}$ is a bounded sequence in
$H^{1}(\mathbb{R}^{N}),$ since $\frac{N(q-2)}{2}<2$ and $\\{c_{n}\\}$ is
bounded. In particular, $\\{A(\tau_{n})\\},\\{C(\tau_{n})\\}$ and
$\\{B(\tau_{n})\\}$ are all bounded sequences. Thus, we get

$\sigma(c)\leq
E_{\gamma,\mu}\left(\frac{c}{c_{n}}\tau_{n}\right)=E_{\gamma,\mu}(\tau_{n})+o(1)<\sigma(c_{n})+o(1).$
(55)

On the other hand, given a minimizing sequence $\\{\nu_{n}\\}\subset S(c)$ for
$\sigma(c)$, it holds

$\sigma(c_{n})\leq
E_{\gamma,\mu}\left(\frac{c_{n}}{c}\nu_{n}\right)+o(1)=\sigma(c)+o(1).$ (56)

So, by (55) and (56) one has
$\lim_{n\rightarrow\infty}\sigma(c_{n})=\sigma(c)$. Notice that
$\frac{\rho(c)}{c}\leq\frac{\sigma(c)}{c}<0,$ where $\rho(c)=\inf_{u\in
S(c)}\\{\frac{1}{2}A(u)-\frac{\mu}{q}C(u)\\}$. Since
$\frac{\rho(c)}{c}\rightarrow 0$ as $c\rightarrow 0$ (see [7, Lemma A.1]), we
have $\lim_{c\rightarrow 0}\frac{\sigma(c)}{c}=0$.

Step 5. For small $c>0$, functional $E_{\gamma,\mu}$ satisfies (54).

Since $0\notin\mathfrak{B}(c)$, we obtain that $A(u),B(u)$ and $C(u)$ are all
different from zero whenever $u\in\mathfrak{B}(c)$. Let $u\in\mathfrak{B}(c)$
and define $v_{\theta}(u):=\theta^{-\frac{N}{2}}u(\theta^{-1}x)$ for
$\theta>0.$ Then $\|v_{\theta}(u)\|_{2}^{2}=\|u\|_{2}^{2}$. Moreover, we have
$A(v_{\theta}(u))=\theta^{-2}A(u),\quad B(v_{\theta}(u))=\theta^{N+\alpha-
Np}B(u)\quad$and $C(v_{\theta}(u))=\theta^{-\frac{N(q-2)}{2}}C(u).$ Since the
map $\theta\mapsto E_{\gamma,\mu}(v_{\theta}(u))$ is differentiable and $u$
achieves the minimum on $S(\eta)$ for some $0<\eta<c$, we get

$\frac{d}{d\theta}E_{\gamma,\mu}(v_{\theta}(u))|_{\theta=1}=-A(u)+\frac{\gamma(Np-N-\alpha)}{2p}B(u)+\frac{\mu
N(q-2)}{2q}C(u)=0.$ (57)

Define
$\mathcal{G}_{u}^{\beta}:=\\{\hat{g}_{u}(\theta)=\theta^{1-\frac{N}{2}\beta}u(x/\theta^{\beta})\\}\subset\mathcal{G}_{u}.$
We find that
$\Pi_{\hat{g}_{u}}(\theta)=\|\hat{g}_{u}(\theta)\|_{2}^{2}\|u\|_{2}^{-2}=\theta^{2}$
and

$\chi_{\hat{g}_{u}}(\theta)=\frac{1}{2}(\theta^{2-2\beta}-\theta^{2})A(u)-\frac{\gamma}{2p}(\theta^{2p-\beta(Np-N-\alpha)}-\theta^{2})B(u)-\frac{\mu}{q}(\theta^{q(1-\frac{N}{2}\beta)+N\beta}-\theta^{2})C(u),$

where $\Pi_{\hat{g}_{u}}$ and $\chi_{\hat{g}_{u}}$ are given in (48) and (49),
respectively. This shows that the paths in $\mathcal{G}_{u}^{\beta}$ are
admissible, i.e. $\chi_{\hat{g}_{u}}$ is differentiable for every
$\hat{g}_{u}\in\mathcal{G}_{u}^{\beta}$. For
$\hat{g}_{u}\in\mathcal{G}_{u}^{\beta}$, we have

$\chi_{\hat{g}_{u}}^{\prime}(1)=-\beta
A(u)-\frac{\gamma(2p-2-\beta(Np-N-\alpha))}{2p}B(u)-\frac{\mu(q(1-\frac{N}{2}\beta)+N\beta-2)}{q}C(u).$
(58)

Next, we show that the admissible scaling path satisfying
$\chi_{\hat{g}_{u}}^{\prime}(1)\neq 0$ can be chosen in
$\mathcal{G}_{u}^{\beta}$. Assume on the contrary. Then there exists a
sequence $\\{u_{n}\\}\subset\mathfrak{B}(c)$ with
$c\geq\|u_{n}\|_{2}^{2}=c_{n}\rightarrow 0$ such that for all
$\beta\in\mathbb{R}$, $\chi_{\hat{g}_{u_{n}}}^{\prime}(1)=0.$ If we take
$\beta=0,$ then

$\chi_{\hat{g}_{u_{n}}}^{\prime}(1)=-\frac{\gamma(p-1)}{p}B(u_{n})-\frac{\mu(q-2)}{q}C(u_{n})=0,$

and using (57) gives

$B(u_{n})=\frac{2p}{-\gamma\alpha}A(u_{n})\quad\text{and
}C(u_{n})=\frac{2q(p-1)}{\mu\alpha(q-2)}A(u_{n}).$ (59)

So, we have

$E_{\gamma,\mu}(u_{n})=\frac{(\alpha+2)(q-2)-4(p-1)}{2\alpha(q-2)}A(u_{n}).$
(60)

The contradiction is achieved by showing that relations (59) and (60) are
impossible for $c>0$ small. Note that
$E_{\gamma,\mu}(u_{n})=\sigma(c_{n})\rightarrow 0$ by continuity. Then we have
$A(u_{n}),$ $B(u_{n})$ and $C(u_{n})\rightarrow 0$ by (59) and (60).

Firstly, we consider condition $(i)$.
Case $(i-a):2<q<\frac{2Np}{N+\alpha}.$ Since
$q<\frac{2Np}{N+\alpha}<\frac{4p+2\alpha}{\alpha+2},$ using Sobolev
inequality, (17) and (59) leads to

$B(u_{n})\leq\widetilde{C}\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p}\leq\widetilde{C}\|u_{n}\|_{q}^{\frac{2Nq(N+\alpha-p(N-2))}{N(2N-q(N-2))}}\|u_{n}\|_{2^{\ast}}^{\frac{2(2Np-q(N+\alpha))}{2N-q(N-2)}}\leq\widetilde{C}B(u_{n})^{\frac{(2-q)(N+\alpha)+4p}{2N-q(N-2)}},$

which implies that for all $n\in\mathbb{N},$ $B(u_{n})\geq\widetilde{D}$ for
some $\widetilde{D}>0.$ This is a contradiction with $B(u_{n})\rightarrow 0$.
Case $(i-b):q=\frac{2Np}{N+\alpha}$. Thanks to (17) and (59), we get

$\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{\frac{2Np}{N+\alpha}}=\widetilde{C}B(u_{n})\leq\widetilde{C}\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p},$

which contradicts with $C(u_{n})\rightarrow 0$.
Case $(i-c):\frac{2Np}{N+\alpha}<q<\frac{2Np-2\alpha}{N}$. It follows from
(17), (59) and Holder’s inequality that

$\|u_{n}\|_{q}^{q}=\widetilde{C}B(u_{n})\leq\widetilde{C}\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p}\leq\widetilde{C}\|u_{n}\|_{2}^{\frac{2(q(N+\alpha)-2Np)}{N(q-2)}}\|u_{n}\|_{q}^{\frac{2q(Np-N-\alpha)}{N(q-2)}}=\widetilde{C}c_{n}^{\frac{q(N+\alpha)-2Np}{N(q-2)}}\|u_{n}\|_{q}^{\frac{2q(Np-N-\alpha)}{N(q-2)}}.$

Since $\frac{2Np}{N+\alpha}<q<\frac{2Np-2\alpha}{N}$ and $c_{n}\rightarrow 0$,
we get $\|u_{n}\|_{q}\rightarrow\infty$, which contradicts with
$C(u_{n})\rightarrow 0$.
Case $(i-d):q=\frac{2Np-2\alpha}{N}$. Using (17), (59) and Holder’s inequality
yields

$B(u_{n})\leq\widetilde{C}\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p}\leq\widetilde{C}\|u_{n}\|_{2}^{\frac{2N\alpha(p-1)-2\alpha^{2}}{N(Np-N-\alpha)}}\|u_{n}\|_{\frac{2Np-2\alpha}{N}}^{{\frac{2Np-2\alpha}{N}}}\leq\widetilde{C}c_{n}^{\frac{2N\alpha(p-1)-2\alpha^{2}}{N(Np-N-\alpha)}}B(u_{n}),$

which contradicts with $c_{n}\rightarrow 0,$ since
$\frac{2N\alpha(p-1)-2\alpha^{2}}{N(Np-N-\alpha)}>0.$
Case $(i-e):\frac{2Np-2\alpha}{N}<q<\frac{4p+2\alpha}{\alpha+2}$. For $u_{0}$
satisfying (59) and (60) with $\left\|u_{0}\right\|_{2}^{2}=c,$ we have

$\displaystyle\frac{\sigma(\theta^{2}c)}{\theta^{2}c}$ $\displaystyle\leq$
$\displaystyle\frac{E_{\gamma,\mu}(\hat{g}_{u_{0}(\theta)})}{\theta^{2}c}$
$\displaystyle=$
$\displaystyle\frac{1}{c}\left(\frac{1}{2}\theta^{-2\beta}A(u_{0})+\frac{1}{\alpha}\theta^{2p-2-\beta(Np-N-\alpha)}A(u_{0})-\frac{2(p-1)}{(q-2)\alpha}\theta^{q(1-\frac{N}{2}\beta)+N\beta-2}A(u_{0})\right).$

Now let us choose $\beta=\frac{2(2-q)}{4+2N-Nq}$ so that

$0<-2\beta=(1-\frac{N}{2}\beta)q+N\beta-2<2p-2-(Np-N-\alpha)\beta.$

Thus we obtain

$\displaystyle\frac{\sigma(\theta^{2}c)}{\theta^{2}c}$ $\displaystyle\leq$
$\displaystyle\frac{E_{\gamma,\mu}(\hat{g}_{u_{0}(\theta)})}{\theta^{2}c}$
$\displaystyle=$
$\displaystyle\frac{A(u_{0})}{c}\left[\frac{1}{2}\theta^{\frac{4(q-2)}{4+2N-Nq}}-\frac{2(p-1)}{(q-2)\alpha}\theta^{\frac{4(q-2)}{4+2N-Nq}}+\frac{1}{\alpha}\theta^{2p-2-\frac{2(Np-N-\alpha)(2-q)}{4+2N-Nq}}\right]$
$\displaystyle=$
$\displaystyle\frac{2(q-2)\alpha}{(\alpha+2)(q-2)-4(p-1)}\frac{E(u_{0})}{c}$
$\displaystyle\times\left[\frac{(q-2)\alpha-4(p-1)}{2(q-2)\alpha}\theta^{\frac{4(q-2)}{4+2N-Nq}}+\frac{1}{\alpha}\theta^{2p-2-\frac{2(Np-N-\alpha)(2-q)}{4+2N-Nq}}\right].$

Let $\theta^{2}c=s$. Then it holds

$\frac{\sigma(s)}{s}\leq-\widetilde{C}s^{\frac{2(q-2)}{4+2N-Nq}}+o\left(s^{\frac{2(q-2)}{4+2N-Nq}}\right)\text{
for sufficiently small }s>0.$ (61)

On the other hand, for $u_{n}$ satisfying (59) and (60), we have

$\|u_{n}\|_{q}^{q}=\widetilde{C}B(u_{n})\leq\widetilde{C}\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p}\leq\widetilde{C}\|u_{n}\|_{2}^{\frac{2(q(N+\alpha)-2Np)}{N(q-2)}}\|u_{n}\|_{q}^{\frac{2q(Np-N-\alpha)}{N(q-2)}},$

which implies that
$\|u_{n}\|_{q}^{q}\leq\widetilde{C}c_{n}^{\frac{Nq+q\alpha-2Np}{Nq+2\alpha-2Np}}.$
Using this, together with (59) and (60), gives
$E_{\gamma,\mu}(u_{n})=-\widetilde{C}C(u_{n})\geq-\widetilde{C}c_{n}^{\frac{Nq+\alpha
q-2Np}{Nq+2\alpha-2Np}},$ which shows that

$\frac{\sigma(c_{n})}{c_{n}}\geq-\widetilde{C}c_{n}^{\frac{\alpha(q-2)}{Nq+2\alpha-2Np}}.$
(62)

Thus it follows from (61) and (62) that

$-\widetilde{C}c_{n}^{\frac{\alpha(q-2)}{Nq+2\alpha-2Np}}\leq\frac{\sigma(c_{n})}{c_{n}}\leq-\widetilde{C}c_{n}^{\frac{2(q-2)}{4+2N-Nq}}+o\left(c_{n}^{\frac{2\left(q-2\right)}{4+2N-Nq}}\right).$

This leads to a contradiction for $c_{n}\rightarrow 0,$ since
$\frac{\alpha(q-2)}{Nq+2\alpha-2Np}>\frac{2(q-2)}{4+2N-Nq}.$

Next, we consider condition $(ii)$.
When $\frac{2Np}{N+\alpha}>2+\frac{4}{N}$, for
$2<q<2+\frac{4}{N}<\frac{2Np}{N+\alpha}$, it follows from (17), (59) and
Sobolev inequality that

$B(u_{n})\leq\widetilde{C}\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p}\leq\widetilde{C}B(u_{n})^{\frac{2(N+\alpha)-2p(N-2)}{2N-q(N-2)}}B(u_{n})^{\frac{2Np-q(N+\alpha)}{2N-q(N-2)}}.$

Since $q<\frac{2Np}{N+\alpha}<\frac{4p+2\alpha}{\alpha+2}$, we get

$\frac{2((N+\alpha)-p(N-2))}{2N-q(N-2)}+\frac{2Np-q(N+\alpha)}{2N-q(N-2)}>1,$

this implies that there exists $\widetilde{C}_{0}>0$ such that
$B(u_{n})\geq\widetilde{C}_{0},$ this is in contradiction with
$B(u_{n})\rightarrow 0.$

If $\frac{2Np}{N+\alpha}\leq 2+\frac{4}{N}$, then we consider the following
three situations.
Case $(ii-a):2<q<\frac{2Np}{N+\alpha}$. The argument is same as the above one.
Case $(ii-b):q=\frac{2Np}{N+\alpha}$. It follows from (17) and (59) that

$\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{\frac{2Np}{N+\alpha}}=\widetilde{C}B(u_{n})\leq\widetilde{C}\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p},$

which contradicts with $C(u_{n})\rightarrow 0$.
Case $(ii-c):\frac{2Np}{N+\alpha}<q<2+\frac{4}{N}.$ By (17), (59) and Holder’s
inequality, we have

$\|u_{n}\|_{q}^{q}=\widetilde{C}B(u_{n})\leq\widetilde{C}\|u_{n}\|_{\frac{2Np}{N+\alpha}}^{2p}\leq\widetilde{C}c_{n}^{\frac{q(N+\alpha)-2Np}{N(q-2)}}\|u_{n}\|_{q}^{\frac{2q(Np-N-\alpha)}{N(q-2)}}.$

Since $\frac{2Np}{N+\alpha}<q,$ we have $q<\frac{2q(Np-N-\alpha)}{N(q-2)}$,
together with $c_{n}\rightarrow 0,$ leading to a contradiction with
$C(u_{n})\rightarrow 0$. Consequently, we complete the proof.

####  4.1.1 The proof of Theorem 1.4

First of all, we give the proof of Theorem 1.4 $(a).$ Under conditions $(i)$
and $(iv)$ of Theorem 1.4 $(a),$ it follows from Lemma 4.8 that there exists a
minimizer $u_{0}$ for $E_{\gamma,\mu}$ restricted to $S\left(c\right)$. Under
conditions $(ii)$ and $(iii)$ of Theorem 1.4 $(a),$ by Lemmas 4.2 and 4.3 we
can obtain a minimizer $u_{0}$ for $E_{\gamma,\mu}$ restricted to $S(c)$ by
adapting the similar argument of Lemma 3.3. Hence, the infimum
$\sigma\left(c\right)$ is achieved by $u_{0}\in S(c)$ which is a ground state
solution of Problem $(P_{c}).$

Next, we prove Theorem 1.4 $(b).$ Since $\gamma<0,\mu>0$ and
$\mu^{N/2}c\leq(\frac{N+2}{N\overline{S}})^{N/2}$, it follows from (14) that

$E_{\gamma,\mu}(u)\geq\frac{1}{2}A(u)-\frac{\mu}{q}C(u)\geq(\frac{1}{2}-\frac{\mu}{q}\overline{S}c^{2/N})A(u)\geq
0.$

Moreover, it holds $\sigma(c)\leq E_{\gamma,\mu}(u_{t})\rightarrow 0$ as
$t\rightarrow 0^{+}$ for all $c>0,$ where $E_{\gamma,\mu}(u_{t})$ is given in
(30). Thus, we have $\sigma(c)=0$ for all
$\mu^{N/2}c\leq(\frac{N+2}{N\overline{S}})^{N/2}$. Suppose that there exists a
constant $\widetilde{c}>0$ satisfying
$\mu^{N/2}\widetilde{c}<(\frac{N+2}{N\overline{S}})^{N/2}$ such that
$\sigma(\widetilde{c})$ has a minimizer $\widetilde{u}\in S(\widetilde{c})$.
Clearly, $E_{\gamma,\mu}(\widetilde{u})=\sigma(\widetilde{c})=0$. Then we
obtain that

$A(\widetilde{u})=\frac{\gamma}{p}B(\widetilde{u})+\frac{2\mu}{q}C(\widetilde{u})\leq\frac{2\mu}{q}C(\widetilde{u})\leq\frac{2\mu\overline{S}\widetilde{c}^{2/N}}{q}A(\widetilde{u})<A(\widetilde{u}),$

which is a contradiction. Hence, $\sigma(c)$ has no minimizer for
$0<\mu^{N/2}c<(\frac{N+2}{N\overline{S}})^{N/2}$. We complete the proof.

###  4.2 Unbounded from below on $S(c)$

####  4.2.1 Bounded from below on $\mathcal{M}(c)$

###### Lemma 4.9

Assume that $N\geq 2,\gamma<0,\mu>0$ and that one of the two following
conditions holds:
$(i)$ $\frac{N+\alpha}{N}<p\leq\frac{N+\alpha+2}{N}$ and
$2+\frac{4}{N}<q<2^{\ast};$
$(ii)$ $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<2^{\ast}$ and
$2Np-qN-2\alpha\leq 0.$
Then functional $E_{\gamma,\mu}$ is bounded from below by a positive constant
and coercive on $\mathcal{M}(c)=\mathcal{M}^{-}(c)$ for all $c>0$.

Proof. Clearly, $\mathcal{M}(c)=\mathcal{M}^{-}(c)$ by (21). Next, we only
prove the conclusion under the condition $(i)$, since the other one is
similar. Let $u\in\mathcal{M}(c)=\mathcal{M}^{-}(c)$. Using (14) and (20)
yields

$A(u)\leq\frac{\mu
N(q-2)}{2q}\overline{S}c^{\frac{2N-(N-2)q}{4}}A(u)^{\frac{(q-2)N}{4}},$

which implies that there exists a constant $D_{3}>0$ depending on $\mu$ and
$c,$ such that

$A\left(u\right)\geq D_{3},$ (63)

since $\frac{(q-2)N}{4}>1.$ Then by (20) and (63), we get

$\displaystyle E_{\gamma,\mu}(u)$ $\displaystyle=$
$\displaystyle\frac{N(q-2)-4}{2N(q-2)}A(u)-\frac{\gamma(Nq-2Np+2\alpha)}{2Np(q-2)}B(u)$
$\displaystyle>$ $\displaystyle\frac{N(q-2)-4}{2N(q-2)}D_{3}>0,$

Hence, $E_{\gamma,\mu}$ is bounded from below by a positive constant and
coercive on $\mathcal{M}(c)$. We complete the proof.

Similar to the arguments in Section 3.2.2, we also apply Lemma 2.6 to
construct a Palais–Smale sequence $\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$ for
$E_{\gamma,\mu}$ restricted to $S(c)$. Here we only give the conclusion
without proof.

###### Lemma 4.10

Let $\mathcal{F}$ be a homotopy stable family of compact subsets of $S(c)$
with closed boundary $\Theta$ and let

$e_{\mathcal{F}}^{-}:=\inf_{H\in\mathcal{F}}\max_{u\in H}J^{-}(u),$

where $J^{-}(u)=E_{\gamma,\mu}(u_{t^{-}}).$ Assume that $\Theta$ is contained
in a connected component of $\mathcal{M}^{-}(c)$ and $\max\\{\sup
J^{-}(\Theta),0\\}<e_{\mathcal{F}}^{-}<\infty$. Then there exists a
Palais–Smale sequence $\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$ for
$E_{\gamma,\mu}$ restricted to $S(c)$ at level $e_{\mathcal{F}}^{-}.$

According to Lemma 4.10, similar to the argument of Lemma 3.13, we have the
following result.

###### Lemma 4.11

Assume that $N\geq 2,\gamma<0,\mu>0\ $and that one of the three following
conditions holds:
$(i)$ $\frac{N+\alpha}{N}<p\leq\frac{N+\alpha+2}{N},2+\frac{4}{N}<q<2^{\ast}$
and $|\gamma|c^{\frac{\alpha q+2q-2\alpha-4p}{(q-2)N-4}}<\widehat{c}_{\ast};$
$(ii)$
$\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<\frac{2N}{N-2}$ with
$2Np-Nq-2\alpha\leq 0$ and $|\gamma|c^{\frac{N+\alpha-(N-2)p}{2}}$ is
sufficiently small.
Then there exists a Palais–Smale sequence
$\\{u_{n}\\}\subset\mathcal{M}^{-}(c)$ for $E_{\gamma,\mu}$ restricted to
$S(c)$ at the level $m^{-}(c)>0.$

####  4.2.2 Unbounded from below on $\mathcal{M}(c)$

Under the case of $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and
$2+\frac{4}{N}<q<2^{\ast}$ with $2Np-Nq-2\alpha>0$, it is very difficult to
verify that $\mathcal{M}^{0}(c)=\emptyset$ and that $E_{\gamma,\mu}$ is
bounded from below on $\mathcal{M}(c)$. In view of this, we need to introduce
a novel natural constraint manifold.

Set

$\Gamma:=\frac{q(Np-N-\alpha-2)}{2Np-Nq-2\alpha}\left(\frac{|\gamma|(2Np-
Nq-2\alpha)(Np-N-\alpha)c^{\frac{N+\alpha-p(N-2)}{2}}}{2\|Q_{p}\|_{2}^{2p-2}(N(q-2)-4)}\right)^{\frac{N(q-2)-4}{2(Np-N-\alpha-2)}}.$
(64)

Clearly, $\Gamma>0$ and $\Gamma\rightarrow 0$ as $|\gamma|\rightarrow 0,$
since $\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast}$ and
$2+\frac{4}{N}<q<2^{\ast}$ with $2Np-Nq-2\alpha>0.$ We define a set $V$ by

$\displaystyle V$ $\displaystyle:$ $\displaystyle=\left\\{u\in S(c)\text{
}|\text{\ }\mu C(u)>\Gamma A(u)^{\frac{N(q-2)}{4}}\right\\}$ $\displaystyle=$
$\displaystyle\left\\{u\in S(c)\text{ }|\text{\
}(t_{u}^{\ast})^{2}A(u)<K\right\\},$

where

$t_{u}^{\ast}:=\left(\frac{4q(Np-N-\alpha-2)A(u)}{\mu N(q-2)(2Np-
Nq-2\alpha)C(u)}\right)^{\frac{2}{N(q-2)-4}}$ (66)

and

$K:=\left(\frac{4}{(q-2)N}\right)^{\frac{4}{N(q-2)-4}}\left(\frac{2\|Q_{p}\|_{2}^{2p-2}((q-2)N-4)}{|\gamma|(2Np-
Nq-2\alpha)(Np-N-\alpha)c^{\frac{N+\alpha-p(N-2)}{2}}}\right)^{\frac{2}{Np-N-\alpha-2}}.$

To show that the set $V$ is not empty, we consider the following nonlinear
Schrödinger equations:

$\left\\{\begin{array}[]{ll}-\Delta u+\lambda u=\mu|u|^{q-2}u&\text{ in
}\mathbb{R}^{N},\\\ \int_{\mathbb{R}^{N}}|u|^{2}dx=c>0,&\end{array}\right.$
($E_{\mu}^{\infty}$)

where $N\geq 3$ and $2+\frac{4}{N}<q<2^{\ast}$. From [18] we know that Eq.
$(E_{\mu}^{\infty})$ admits a ground state normalized solution $\omega_{0}$
with positive energy $\beta$. Then
$\beta=\inf_{u\in\mathbf{N}_{\mu}(c)}G_{\mu}(u)=G_{\mu}(\omega_{0})>0,$ where
$G_{\mu}$ is the energy functional related to Eq. $(E_{\mu}^{\infty})$ given
by

$G_{\mu}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{\mu}{q}\int_{\mathbb{R}^{N}}|u|^{q}dx$

and

$\mathbf{N}_{\mu}(c)=\left\\{u\in S(c)\text{ }|\text{ }P_{\mu}(u)=0\right\\}$

with $P_{\mu}(u)=\int_{\mathbb{R}^{N}}|\nabla
u|^{2}dx-\frac{\mu(q-2)N}{2q}\int_{\mathbb{R}^{N}}|u|^{q}dx$. Clearly, it
holds

$\beta=\frac{1}{2}A(\omega_{0})-\frac{\mu}{q}C(\omega_{0})=\frac{N(q-2)-4}{2N(q-2)}A(\omega_{0})>0.$

Set

$\displaystyle\gamma_{\ast}$ $\displaystyle:$
$\displaystyle=\left[\left(\frac{N(q-2)-4}{2N\beta(q-2)}\right)^{1/2}\left(\frac{2(2Np-
Nq-2\alpha)}{N(Np-N-\alpha-2)(q-2)}\right)^{\frac{2}{N(q-2)-4}}\right]^{Np-N-\alpha-2}$
$\displaystyle\text{ \
}\times\left(\frac{2(N(q-2)-4)\|Q_{p}\|_{2}^{2p-2}}{(Np-N-\alpha)(2Np-
Nq-2\alpha)c^{\frac{N+\alpha-p(N-2)}{2}}}\right).$

For $0<\left|\gamma\right|<\gamma_{\ast}$, we have $\mu
C(\omega_{0})=\frac{2q}{N(q-2)}A(\omega_{0})=\frac{4q\beta}{N(q-2)-4}>\Gamma
A(\omega_{0})^{\frac{N(q-2)}{4}},$ which implies that $\omega_{0}\in V,$
namely, $V$ is nonempty.

Now we define the set $\Lambda(c):=\left\\{u\in V\text{ }|\text{\
}g_{u}^{\prime}(1)=0\right\\},$ which is a natural constraint. Similar to
Pohozaev manifold $\mathcal{M}(c),$ we can split $\Lambda(c)$ into three parts
corresponding to local minima, local maxima and points of inflection as
follows

$\displaystyle\Lambda^{+}(c)$ $\displaystyle:$ $\displaystyle=\left\\{u\in
V\text{ }|\text{\ }g_{u}^{\prime}(1)=0,g_{u}^{\prime\prime}(1)>0\right\\};$
$\displaystyle\Lambda^{-}(c)$ $\displaystyle:$ $\displaystyle=\left\\{u\in
V\text{ }|\text{\ }g_{u}^{\prime}(1)=0,g_{u}^{\prime\prime}(1)<0\right\\};$
$\displaystyle\Lambda^{0}(c)$ $\displaystyle:$ $\displaystyle=\left\\{u\in
V\text{ }|\text{\ }g_{u}^{\prime}(1)=0,g_{u}^{\prime\prime}(1)=0\right\\}.$

Then we have the following results.

###### Lemma 4.12

Assume that $N\geq
2,\gamma<0,\mu>0,\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<2^{\ast}$
with $2Np-qN-2\alpha>0$. Then $\Lambda^{0}(c)=\emptyset$ for all $c>0$.

Proof. Suppose that $\Lambda^{0}(c)\neq\emptyset$. Then for
$u\in\Lambda^{0}(c)$, it follows from (14) and (21) that

$\displaystyle\frac{N(q-2)-4}{2}A(u)$ $\displaystyle=$
$\displaystyle\frac{|\gamma|(Np-N-\alpha)(2Np-Nq-2\alpha)}{4p}B(u)$
$\displaystyle\leq$ $\displaystyle\frac{|\gamma|(Np-N-\alpha)(2Np-
Nq-2\alpha)}{4\|Q_{p}\|_{2}^{2p-2}}c^{\frac{N+\alpha-p(N-2)}{2}}A(u)^{\frac{Np-N-\alpha}{2}},$

which indicates that

$A(u)\geq\left[\frac{2\|Q_{p}\|_{2}^{2p-2}(N(q-2)-4)}{|\gamma|(Np-N-\alpha)(2Np-
Nq-2\alpha)}c^{-\frac{N+\alpha-p(N-2)}{2}}\right]^{\frac{2}{Np-N-\alpha-2}}.$
(67)

On the other hand, by using (2.2) and the fact of $\mu C(u)>\Gamma
A(u)^{\frac{N(q-2)}{4}},$ we have

$A(u)<\left(\frac{4}{N(q-2)}\right)^{\frac{4}{N(q-2)-4}}\left(\frac{2\|Q_{p}\|_{2}^{2p-2}(N(q-2)-4)}{|\gamma|(2Np-
Nq-2\alpha)(Np-N-\alpha)c^{\frac{N+\alpha-p(N-2)}{2}}}\right)^{\frac{2}{Np-N-\alpha-2}}.$
(68)

Hence, using (67) and (68) leads to $q<2+\frac{4}{N},$ which is a
contradiction. This completes the proof.

For $u\in\Theta$, we define

$\widehat{t}_{u}:=\left(\frac{2qA(u)}{\mu
N(q-2)C(u)}\right)^{\frac{2}{N(q-2)-4}}.$

###### Lemma 4.13

Assume that $N\geq
2,\gamma<0,\mu>0,\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<2^{\ast}$
with $2Np-qN-2\alpha>0$. Then for each $u\in V$ there exist two positive
constants $t_{\ast}^{+}(u)$ and $t_{\ast}^{-}(u)$ satisfying
$\widehat{t}_{u}<t_{\ast}^{-}(u)<t_{u}^{\ast}<t_{\ast}^{+}(u)$ such that
$u_{t_{\ast}^{\pm}(u)}\in\Lambda^{\pm}(c)$, where $t_{u}^{\ast}$ is as in
(66).

Proof. Define

$\widehat{h}(t)=A(u)t^{N+\alpha+2-Np}-\frac{\mu
N(q-2)}{2q}C(u)t^{\frac{Nq+2\alpha-2Np}{2}}\quad\text{for}\quad t>0.$

Clearly, $u_{t}\in\Lambda(c)$ if and only if
$\widehat{h}(t)=\frac{\gamma(Np-N-\alpha)}{2p}B(u)$. A straightforward
evaluation shows that $\widehat{h}(\widehat{t}_{u})=0,$ $\lim_{t\rightarrow
0^{+}}\widehat{h}(t)=\infty$ and $\lim_{t\rightarrow\infty}\widehat{h}(t)=0.$
Note that

$\widehat{h}^{\prime}(t)=t^{N+\alpha+1-Np}\left[(N+\alpha+2-Np)A(u)-\frac{\mu
N(q-2)(Nq+2\alpha-2Np)}{4q}C(u)t^{\frac{N(q-2)-4}{2}}\right].$

Then we obtain that $\widehat{h}(t)$ is decreasing when $0<t<t_{u}^{\ast}$ and
is increasing when $t>t_{u}^{\ast}$, which implies that

$\displaystyle\inf_{t>0}\widehat{h}(t)$ $\displaystyle=$
$\displaystyle\widehat{h}\left(\left(\frac{4q(Np-N-\alpha-2)A(u)}{\mu
N(q-2)(2Np-Nq-2\alpha)C(u)}\right)^{\frac{2}{N(q-2)-4}}\right)$
$\displaystyle=$ $\displaystyle-\frac{N(q-2)-4}{2Np-
Nq-2\alpha}\left(\frac{4q(Np-N-\alpha-2)A(u)}{\mu N(q-2)(2Np-
Nq-2\alpha)C(u)}\right)^{\frac{2(N+\alpha+2-Np)}{N(q-2)-4}}A(u).$

Since $u\in V$, we have $\mu C(u)>\Gamma A(u)^{\frac{N(q-2)}{4}}.$ This
indicates that

$\inf_{t>0}\widehat{h}(t)<\frac{\gamma(Np-N-\alpha)}{2\|Q_{p}\|_{2}^{2p-2}}A(u)^{\frac{Np-N-\alpha}{2}}c^{\frac{N+\alpha-p(N-2)}{2}}\leq\frac{\gamma(Np-N-\alpha)}{2p}B(u),$

which implies that there exist two constants $t_{\ast}^{+}(u)$ and
$t_{\ast}^{-}(u)$ satisfying
$\widehat{t}_{u}<t_{\ast}^{-}(u)<t_{u}^{\ast}<t_{\ast}^{+}(u)$ such that

$\widehat{h}(t_{\ast}^{\pm}(u))=\frac{\gamma(Np-N-\alpha)}{2p}B(u).$

Namely, $u_{t_{\ast}^{\pm}(u)}\in\Lambda(c).$ By a calculation on the second
order derivatives, we find that

$g_{u_{t_{\ast}^{-}(u)}}^{\prime\prime}(1)=(t_{\ast}^{-}(u))^{Np-N-\alpha+1}(\widehat{h})^{\prime}(t_{\ast}^{-}(u))<0\text{
and
}g_{u_{t_{\ast}^{+}\left(u\right)}}^{\prime\prime}\left(1\right)=\left(t_{\ast}^{+}\left(u\right)\right)^{Np-N-\alpha+1}(\widehat{h})^{\prime}\left(t_{\ast}^{+}\left(u\right)\right)>0.$

These imply that
$u_{t_{\ast}^{\pm}\left(u\right)}\in\Lambda^{\pm}\left(c\right)$. This
completes the proof.

###### Lemma 4.14

Assume that $N\geq
2,\gamma<0,\mu>0,\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<2^{\ast}$
with $2Np-qN-2\alpha>0$. Then functional $E_{\gamma,\mu}$ is bounded from
below by a positive constant and coercive on $\Lambda^{-}(c)$ for all $c>0.$

Proof. Let $u\in\Lambda^{-}(c)$. Then we have

$A(u)-\frac{\gamma(Np-N-\alpha)}{2p}B(u)-\frac{\mu N(q-2)}{2q}C(u)=0$ (69)

and

$(Np-N-\alpha-2)A(u)>\frac{\mu N(q-2)(2Np-2\alpha-qN)}{4q}C(u).$ (70)

It follows from (14) and (69) that

$\displaystyle A(u)$ $\displaystyle=$
$\displaystyle\frac{\gamma(Np-N-\alpha)}{2p}B(u)+\frac{\mu N(q-2)}{2q}C(u)$
$\displaystyle\leq$ $\displaystyle\frac{\mu
N(q-2)}{2q}\overline{S}c^{\frac{2N-q(N-2)}{4}}A(u)^{\frac{N(q-2)}{4}},$

which implies that $A(u)\geq D_{3}$, where $D_{3}>0$ is as in (63). Using
this, together with (69) and (70), leads to

$\displaystyle E_{\gamma,\mu}(u)$ $\displaystyle=$
$\displaystyle\frac{Np-N-\alpha-2}{2(Np-N-\alpha)}A(u)-\frac{\mu(2Np-2\alpha-
qN)}{2q(Np-N-\alpha)}C(u)$ $\displaystyle>$
$\displaystyle\frac{(Np-N-\alpha-2)(N(q-2)-4)}{2N(Np-N-\alpha)(q-2)}A(u)$
$\displaystyle>$
$\displaystyle\frac{(Np-N-\alpha-2)(N(q-2)-4)}{2N(Np-N-\alpha)(q-2)}D_{3}>0.$

This shows that $E_{\gamma,\mu}$ is bounded from below by a positive constant
and coercive on $\Lambda^{-}(c)$. We complete the proof.

By the above lemma, we can define

$l^{-}(c):=\inf_{u\in\Lambda^{-}(c)}E_{\gamma,\mu}(u)>0.$

Next, we shall establish the existence of a Palais–Smale sequence
$\\{u_{n}\\}\subset\Lambda^{-}(c)$ by Lemma 2.6 in Section 2.3. Define the
functional $F^{-}:V\mapsto\mathbb{R}\quad$by

$F^{-}(u):=E_{\gamma,\mu}(u_{t_{\ast}^{-}(u)}),$

where $t_{\ast}^{-}(u)$ is given in Lemma 4.13.

As in Section 3.2.2, we have the following results without proof.

###### Lemma 4.15

If $\Lambda^{0}(c)=\emptyset$, then $\Lambda(c)=V\cap\mathcal{M}(c)$ is smooth
submanifold of codimension $2$ of $H^{1}(\mathbb{R}^{N})$ and a submanifold of
codimension $1$ in $S(c)$.

###### Lemma 4.16

The map $u\in V\mapsto t_{\ast}^{-}(u)\in\mathbb{R}$ is of class $C^{1}$.

###### Lemma 4.17

The map $T_{u}V\rightarrow T_{u_{t_{\ast}^{-}(u)}}V$ defined by
$\psi\rightarrow\psi_{t_{\ast}^{-}(u)}$ is isomorphism.

###### Lemma 4.18

We have that
$(F^{-})^{\prime}(u)[\psi]=E^{\prime}_{\gamma,\mu}(u_{t_{\ast}^{-}(u)})[\psi_{t_{\ast}^{-}(u)}]$
for any $u\in V$ and $\psi\in T_{u}V$.

Since $V$ has a boundary, in order to guarantee that our deformation arguments
happen inside $V,$ we need the following lemma.

###### Lemma 4.19

Assume that $0<|\gamma|<\gamma_{\ast}$. If $\\{v_{n}\\}\subset V$ is a
sequence with $v_{n}\rightarrow\bar{v}\in\partial V$ strongly in
$H^{1}(\mathbb{R}^{N})$, then there exist a sequence
$\\{t_{\ast}^{-}(v_{n})\\}$ with $t_{\ast}^{-}(v_{n})\rightarrow t_{\infty}>0$
satisfying $g_{\bar{v}_{t_{\infty}}}^{\prime\prime}(1)\leq 0$ and a constant
$\overline{M}=\overline{M}(\gamma,c)>0$ satisfying
$\overline{M}\rightarrow\infty$ as $|\gamma|\rightarrow 0$ such that
$E_{\gamma,\mu}(\bar{v}_{t_{\infty}})>\overline{M}.$

Proof. Let $\\{v_{n}\\}\subset V$ such that
$v_{n}\rightarrow\bar{v}\in\partial V$ strongly in $H^{1}(\mathbb{R}^{N})$ as
$n\rightarrow\infty$. Then it holds $t_{v_{n}}^{\ast}\rightarrow
t_{\bar{v}}^{\ast}\neq
0\quad$and$\quad\widehat{t}_{v_{n}}\rightarrow\widehat{t}_{\bar{v}}\neq
0\quad$as$\,n\rightarrow\infty,$ where $t_{u}^{\ast}$ and $\widehat{t}_{u}$
are defined in (63) and (66), respectively. Since $v_{n}\in V$ and
$\bar{v}\in\partial V$, we have $(t_{v_{n}}^{\ast})^{2}A(v_{n})<K$ and
$(t_{\bar{v}}^{\ast})^{2}A(\bar{v})=K$. Moreover, it follows from Lemma 4.13
that $\widehat{t}_{v_{n}}<t_{\ast}^{-}(v_{n})\leq t_{v_{n}}^{\ast}\leq
t_{\ast}^{+}(v_{n}),$ which implies that the sequence
$\\{t_{\ast}^{-}(v_{n})\\}$ is bounded. Then we can assume that
$\\{t_{\ast}^{-}(v_{n})\\}$, up to a subsequence, converges to $t_{\infty}\neq
0$. Since $v_{n}\rightarrow\bar{v}$ strongly in $H^{1}(\mathbb{R}^{N})$ and
$Q((v_{n})_{t_{\ast}^{-}(v_{n})})=0$, we infer that
$Q(\bar{v}_{t_{\infty}})=0$, namely,

$A(\bar{v}_{t_{\infty}})-\frac{\gamma(Np-N-\alpha)}{2p}B(\bar{v}_{t_{\infty}})-\frac{\mu
N(q-2)}{2q}C(\bar{v}_{t_{\infty}})=0.$ (71)

Then we have

$A(\bar{v}_{t_{\infty}})\leq\frac{\mu
N(q-2)}{2q}C(\bar{v}_{t_{\infty}})=\frac{N\Gamma(q-2)}{2q}A(\bar{v}_{t_{\infty}})^{\frac{N(q-2)}{4}},$

which implies that

$A(\bar{v}_{t_{\infty}})\geq\left(\frac{2q}{N\Gamma(q-2)}\right)^{\frac{4}{N(q-2)-4}},$
(72)

where $\Gamma$ is as in (64) and $\Gamma\rightarrow 0$ as
$\left|\gamma\right|\rightarrow 0$. Since
$g_{(v_{n})_{t_{\ast}^{-}(v_{n})}}^{\prime\prime}(1)<0$, we have
$g_{\bar{v}_{t_{\infty}}}^{\prime\prime}(1)\leq 0$, namely,

$(Np-N-\alpha-2)A(\bar{v}_{t_{\infty}})\geq\frac{\mu N(q-2)(2Np-
Nq-2\alpha)}{4q}C(\bar{v}_{t_{\infty}}).$ (73)

Then it follows from (71)–(73) that

$\displaystyle E_{\gamma,\mu}(\bar{v}_{t_{\infty}})$ $\displaystyle=$
$\displaystyle\frac{Np-N-\alpha-2}{2(Np-N-\alpha)}A(\bar{v}_{t_{\infty}})-\frac{\mu(2Np-2\alpha-
qN)}{2q(Np-N-\alpha)}C(\bar{v}_{t_{\infty}})$ $\displaystyle>$
$\displaystyle\frac{(Np-N-\alpha-2)(N(q-2)-4)}{2N(Np-N-\alpha)(q-2)}A(\bar{v}_{t_{\infty}})$
$\displaystyle\geq$
$\displaystyle\frac{(Np-N-\alpha-2)(N(q-2)-4)}{2N(Np-N-\alpha)(q-2)}\left(\frac{2q}{N\Gamma(q-2)}\right)^{\frac{4}{N(q-2)-4}}:=\overline{M}.$

Clearly, $\overline{M}\rightarrow\infty$ as $|\gamma|\rightarrow 0$. We
complete the proof.

###### Lemma 4.20

Let $\mathcal{F}$ be a homotopy stable family of compact subsets of $\Theta$
with closed boundary $B$ and let

$e_{\mathcal{F}}^{-}:=\inf_{H\in\mathcal{F}}\max_{u\in H}F^{-}(u).$

Suppose that $\Theta$ is contained in a connected component of
$\Lambda^{-}(c)$ and $\max\\{\sup
J^{-}(\Theta),0\\}<e_{\mathcal{F}}^{-}<\infty$. Then there exists a
Palais–Smale sequence $\\{u_{n}\\}\subset\Lambda^{-}(c)$ for $E_{\gamma,\mu}$
restricted to $V$ at level $e_{\mathcal{F}}^{-}$ for $|\gamma|$ sufficiently
small.

Proof. Take $\\{D_{n}\\}\subset\mathcal{F}$ such that $\max_{u\in
D_{n}}F^{-}(u)<e_{\mathcal{F}}^{-}+\frac{1}{n}$ and $\eta:[0,1]\times
V\rightarrow V,$ $\eta(s,u)=u_{1-s+st_{\ast}^{-}(u)}.$ Since
$t_{\ast}^{-}(u)=1$ for any $u\in\Lambda^{-}(c)$ and
$\Theta\subset\Lambda^{-}(c)$, we obtain $\eta(s,u)=u$ for
$(s,u)\in(\\{0\\}\times V)\cup([0,1]\times\Theta)$. Also note that $\eta$ is
continuous. Then, using the definition of $\mathcal{F}$ gives
$\mathbf{A}_{n}:=\eta(\\{1\\}\times D_{n})=\left\\{u_{t(u)}\text{ }|\text{
}u\in D_{n}\right\\}\in\mathcal{F}.$ Clearly,
$\mathbf{A}_{n}\subset\Lambda^{-}(c)$ for all $n\in\mathbb{N}$. Let
$v\in\mathbf{A}_{n}$, i.e. $v=u_{t_{\ast}^{-}(u)}$ for some $u\in D_{n}$ and
$F^{-}(u)=F^{-}(v)$. Then we have
$\max_{\mathbf{A}_{n}}F^{-}=\min_{D_{n}}F^{-},$ and so
$\\{\mathbf{A}_{n}\\}\subset\Lambda^{-}(c)$ is another minimizing sequence of
$e_{\mathcal{F}}^{-}$. Note that the minimax level $e_{\mathcal{F}}^{-}$
depends on the parameter $\gamma$ and is decreasing as $|\gamma|\searrow
0^{+}$. Then for $|\gamma|$ small enough, by Lemmas 2.6 and 4.19, we obtain a
Palais–Smale sequence $\\{\widetilde{u}_{n}\\}$ for $F^{-}$ on $V$ at level
$e_{\mathcal{F}}^{-}$ such that
dist${}_{H^{1}(\mathbb{R}^{N})}(\widetilde{u}_{n},\mathbf{A}_{n})\rightarrow
0$ as $n\rightarrow\infty.$ Next, by using the standard arguments as in Lemma
3.12, we easily obtain the conclusion.

###### Lemma 4.21

Assume that $N\geq
2,\gamma<0,\mu>0,\frac{N+\alpha+2}{N}<p<2_{\alpha}^{\ast},2+\frac{4}{N}<q<2^{\ast}$
with $2Np-qN-2\alpha>0$. Then there exists a Palais–Smale sequence
$\\{u_{n}\\}\subset\Lambda^{-}(c)$ for $E_{\gamma,\mu}$ restricted to $S(c)$
at the level $l^{-}(c)>0.$

Proof. Let $\\{u_{n}\\}\subset\Lambda^{-}(c)$. By Lemma 4.20, we choose the
set $\overline{\mathcal{F}}$ of all singletons belonging to $V$ and
$\Theta=\emptyset$, which is clearly a homotopy stable family of compact
subsets of $V$ (without boundary). Note that
$e_{\overline{\mathcal{F}}}^{-}=\inf_{H\in\overline{\mathcal{F}}}\max_{u\in
H}F^{-}(u)=\inf_{u\in V}F^{-}(u)=l^{-}(c).$ Then the lemma follows directly
from Lemma 4.20. This completes the proof.

####  4.2.3 The proof of Theorem 1.5

Now we give the proof of Theorem 1.5. Under condition $(i)$ or $(ii)$ of
Theorem 1.5, by Lemma 4.11, there exists a Palais–Smale sequence
$\\{u_{n}\\}\subset\mathcal{M}(c)=\mathcal{M}^{-}(c)$ for $E_{\gamma,\mu}$
restricted to $S(c)$ at level $m^{-}(c)>0$, which is bounded in
$H^{1}(\mathbb{R}^{N})$ via Lemma 4.9. So, according to Lemma 2.7, Problem
$(P_{c})$ admits a radial ground state solution $u^{-}$ satisfying
$E_{\gamma,\mu}(u^{-})=m^{-}(c)>0$ for some $\lambda>0$.

Under condition $(iii)$ of Theorem 1.5, let $\\{u_{n}\\}\subset\Lambda^{-}(c)$
be a Palais–Smale sequence for $E_{\gamma,\mu}$ restricted to $S(c)$ at level
$l^{-}(c)>0$ whose existence is ensured by Lemma 4.21. Lemma 4.14 shows that
$\\{u_{n}\\}$ is bounded in $H^{1}(\mathbb{R}^{N})$. Hence, it follows from
Lemma 2.7 that Problem $(P_{c})$ has a positive radial solution $u^{-}$
satisfying $E_{\gamma,\mu}(u^{-})=l^{-}(c)>0$ for some $\lambda>0$. We
complete the proof.

##  5 The case of $\gamma>0$ and $\mu<0$

###  5.1 Bounded from below on $S\left(c\right)$

###### Lemma 5.1

Assume that $N\geq 2,\gamma>0,\mu<0,\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$
and $2<q<2^{\ast}.$ Then functional $E_{\gamma,\mu}$ is bounded from below and
coercive on $S(c).$

Proof. Since $\mu<0$ and $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$, it
follows from (15) that

$E_{\gamma,\mu}(u)\geq\frac{1}{2}A(u)-\frac{\gamma}{2\|Q_{p}\|_{2}^{2p-2}}c^{\frac{N+\alpha-p(N-2)}{2}}A(u)^{\frac{Np-N-\alpha}{2}},$

which implies that $E_{\gamma,\mu}$ is bounded from below and coercive on
$S(c)$. We complete the proof.

We now study the strong sub-additivity inequalities for $\sigma(c)$ under some
cases of $p,q$ when $\gamma>0$ and $\mu<0.$

###### Lemma 5.2

Assume that $N\geq
2,\gamma>0,\mu<0,\max\\{\frac{N+\alpha}{N},\frac{q(\alpha+2)-2\alpha}{4}\\}<p<\frac{N+\alpha+2}{N}$
and $2<q<2+\frac{4}{N}.$ Then there exists a constant $c_{4}>0$ such that for
all $c>c_{4},$

$\sigma(c)<0\text{ and }\sigma(c)<\sigma(\eta)+\sigma(c-\eta)\text{ for
}0<\eta<c.$

Proof. Define $u_{\theta}(x):=\theta^{1-\frac{N}{2}\beta}u(\theta^{-\beta}x)$.
Then a direct calculation shows that
$\|u_{\theta}\|_{2}^{2}=\theta^{2}\|u\|_{2}^{2}$ and
$E_{\gamma,\mu}(u_{\theta})=\theta^{2}\left[E_{\gamma,\mu}(u)+f_{\beta}(\theta,u)\right],$
where $f_{\beta}(\theta,u)$ is given in (44). Now we always let
$\beta=\frac{2(q-2)}{N(q-2)-4}.$ Then it holds

$E_{\gamma,\mu}(u_{\theta})=\frac{1}{2}\theta^{\frac{2(2N-q(N-2))}{4-N(q-2)}}A(u)-\frac{\gamma}{2p}\theta^{\frac{2(4p-(q-2)(N+\alpha))}{4-N(q-2)}}B(u)-\frac{\mu}{q}\theta^{\frac{2(2N-q(N-2))}{4-N(q-2)}}C(u),$

which implies that $E_{\gamma,\mu}(u_{\theta})<0$ for $\theta>0$ sufficiently
large, since $\frac{4p-(q-2)(N+\alpha)}{4-N(q-2)}>\frac{2N-q(N-2)}{4-N(q-2)}$.
So there exists a constant $c_{4}>0$ such that $\sigma(c)<0$ for $c>c_{4}.$

Let $\\{u_{n}\\}$ be a minimizing sequence in $S(c)$ with $\sigma(c)<0$. Then
we obtain that $A(u_{n})$ and $C(u_{n})$ are both bounded. Note that

$\displaystyle f_{\beta}(\theta,u_{n})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\theta^{\frac{2(2N-q(N-2))}{4-N(q-2)}}-1\right)A(u_{n})-\frac{\gamma}{2p}\left(\theta^{\frac{2(4p-(q-2)(N+\alpha))}{4-N(q-2)}}-1\right)B(u_{n})$