# Semiclassical solutions for critical Schrödinger–Poisson systems involving
multiple competing potentials

Lingzheng Kong, Haibo Chen
School of Mathematics and Statistics, Central South University,
Changsha, Hunan 410083, P.R. China Corresponding author: E-mail:
<EMAIL_ADDRESS>(L. Kong<EMAIL_ADDRESS>(H. Chen).

###### Abstract

In this paper, a class of Schrödinger–Poisson system involving multiple
competing potentials and critical Sobolev exponent is considered. Such a
problem cannot be studied with the same argument of the nonlinear term with
only a positive potential, because the weight potentials set
$\\{Q_{i}(x)|1\leq i\leq m\\}$ contains nonpositive, sign-changing, and
nonnegative elements. By introducing the ground energy function and subtle
analysis, we first prove the existence of ground state solution
$v_{\varepsilon}$ in the semiclassical limit via the Nehari manifold and
concentration-compactness principle. Then we show that $v_{\varepsilon}$
converges to the ground state solution of the associated limiting problem and
concentrates at a concrete set characterized by the potentials. At the same
time, some properties for the ground state solution are also studied.
Moreover, a sufficient condition for the nonexistence of the ground state
solution is obtained.

Keywords:  Schrödinger–Poisson system; Semiclassical solutions; Multiple
competing potentials; Variational methods; Critical growth

Mathematics Subject Classification. 35J20; 35J60

##  1 Introduction and statement of results

In this paper, we study the existence and concentration behavior of positive
ground state solutions for the following critical Schrödinger–Poisson system:

$\begin{cases}-\varepsilon^{2}\Delta u+h(x)\phi
u+V(x)u=\sum\limits_{i=1}^{m}Q_{i}(x)|u|^{q_{i}-2}u+K(x)|u|^{4}u,&x\in\mathbb{R}^{3},\\\
-\varepsilon^{2}\Delta\phi=4\pi h(x)u^{2},&x\in\mathbb{R}^{3},\end{cases}$
($SP_{\varepsilon}$)

where $\varepsilon>0$ is a small parameter, $4<q_{1}<q_{2}<\cdots<q_{m}<6$.
$V(x)$ is the potential, $h(x)$ is the electronic potential,
$Q_{i}(x)(i=1,\cdots,m)$ and $K(x)$ are weight potentials satisfying some
competing conditions. Moreover, the nonlinear growth of $|u|^{4}u$ reaches the
Sobolev critical exponent since $2^{*}=6$ for three spatial dimensions, which
is why we call critical Schrödinger–Poisson system in the title.

Replacing the nonlinear term
$\sum\limits_{i=1}^{m}Q_{i}(x)|u|^{q_{i}-2}u+K(x)|u|^{4}u$ by $f(x,u)$,
($SP_{\varepsilon}$) becomes the following system:

$\begin{cases}-\varepsilon^{2}\Delta u+h(x)\phi
u+V(x)u=f(x,u),&x\in\mathbb{R}^{3},\\\ -\varepsilon^{2}\Delta\phi=4\pi
h(x)u^{2},&x\in\mathbb{R}^{3},\end{cases}$ (1.1)

which known as the nonlinear Schrödinger–Maxwell system, arises in an
interesting physical meaning because it appears in quantum mechanics models
[7, 15, 23] and in semiconductor theory [5, 16, 17]. In fact, systems like
(1.1) have been first proposed in [4] as a model describing the interaction
between the electrostatic field and the solitary waves of nonlinear
Schrödinger type equation

$i\hbar\frac{\partial\psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\Delta\psi+W(x)\psi-|\psi|^{p-2}\psi,$ (1.2)

where $i$ is the imaginary unit, $\hbar$ is the Planck constant, $m$ is the
mass of the field $\psi$, $W(x)$ is the time independent potential of the
particle at the position $x\in\mathbb{R}^{3}$. Then, looking for the standing
waves of Eq. (1.2), namely waves of the form $\psi(x,t)=u(x)e^{-i\omega
t/\hbar},\hskip 4.30554ptx\in\mathbb{R}^{3},\hskip 4.30554ptt\in\mathbb{R},$
one is led to the system (1.1) with $\varepsilon^{2}=\frac{\hbar^{2}}{2m}$,
$V(x)=W(x)-\omega$. For more details on physical background, we refer the
reader to [4, 5] and the references therein.

In recent years, the qualitative analysis of positive solutions for problem
(1.1), including the existence, nonexistence, concentration behavior, and
multiplicity, has been widely investigated under various assumptions of the
potentials. For more details, we refer the readers to previous studies [1, 2,
4, 13, 17, 18, 21, 22, 24, 25, 33, 34, 38, 26, 8, 20, 30, 29, 37, 28, 36]. In
particular, if $\varepsilon=V(x)=h(x)=1$, see, for example, Benci and
Fortunato [4] for the case of $f(x,u)=0$, demonstrated the existence of
infinitely many solutions of an eigenvalue problem on bounded domain.
Azzollini et al. [1], when the nonlinearity satisfies Berestycki-Lions type
assumptions, proved the existence of nontrivial solution. [13, 21] for the
pure power nonlinearity $|u|^{p-2}u$ with $p\in(2,6)$ by working in the
subspace of radial functions of $H^{1}(\mathbb{R}^{3})$ i.e.,
$H^{1}_{r}(\mathbb{R}^{3})$, obtained multiple bound state solutions, the
existence and nonexistence results, respectively. Zhao and Zhao [18] for the
case of $f(x,u)=\mu Q(x)|u|^{q-2}u+K(x)|u|^{4}u$ where $q\in[4,6)$ and
$\mu>0$, introduced the assumption on the weight potential $K(x)$:

$(K_{1})$ $|K(x)-K(x_{0})|=o(|x-x_{0}|^{\alpha}),\hskip
4.30554pt\text{where}\hskip 4.30554pt1\leq\alpha<3\hskip
4.30554pt\text{and}\hskip
4.30554ptK(x_{0})=\max\limits_{x\in\mathbb{R}^{3}}K(x),$

to estimate the critical energy level, and proved the existence of a positive
solution based on the methods of Brezis and Nirenberg and Lions’
concentration-compactness principle.

If the electronic potential $h(x)$ is not a constant, Cerami and Vaira [8] for
the case of $f(x,u)=a(x)|u|^{p-1}u$, $p\in(3,5)$ without any symmetry
assumptions, by using the Nehari manifold and establishing compactness lemma,
proved the existence of positive ground state and bound state solutions. Huang
et al. [24] for system (1.1) with $f(x,u)=a(x)|u|^{p-2}u+\mu K(x)u$, where
$p\in(4,6)$, $K(x)\geq 0$ and $a(x)$ can change sign without any symmetry
assumptions, mainly prove the existence of at least two positive solutions via
a comparison between $\mu$ and the first eigenvalue of $-\Delta u+id$. If
$V(x)$ is not a constant, and, then $V(x)$ may not be radial, so one cannot
work in $H^{1}_{r}(\mathbb{R}^{3})$ directly. To overcome this difficulty, by
assuming that $V(x)$ satisfying

$0<V_{0}=\inf\limits_{x\in\mathbb{R}^{3}}V(x)<V_{\infty}=\lim\limits_{|x|\to\infty}V(x),$

which was firstly introduced in [26]. Azzollini et al. [2] for the case of
$f(x,u)=|u|^{p-1}u$, proved the existence of ground state solutions and they
also studied the existence of solutions for the critical growth by
concentration compactness principle. By assuming that $V(x)$, $a(x)$ and
$h(x)$ satisfy some decay rates, Cerami and Molle [20] for the case of
$f(x,u)=a(x)|u|^{p-1}u$, proved the existence of a positive bound state
solution via the Nehari manifold, which complements the results given by [8]
in some sense.

If the parameter $\varepsilon\to 0$, the systems like (1.1) and bound states
are called semiclassical problems and semiclassical states, respectively.
Semiclassical states can be used to describe a kind of transition between
Quantum Mechanics and Newtonian Mechanics. He and Zou [11] proved the
existence of ground state solutions for the critical growth concentrating on
the minima of $V(x)$. Ruiz [22] showed the semiclassical states concentrating
around a sphere when $V(x)$ statisfies some suitable assumptions. J. Wang et
al. [33] for the case of $f(x,u)=b(x)g(u)+|u|^{4}u$, proved that there are two
families of positive solutions concentrating on the maxima of $b(x)$ and the
minima of $V(x)$, respectively. Yang [38] for the case of
$f(x,u)=P(x)g(u)+Q(x)|u|^{4}u$ where $\min\limits_{x\in\mathbb{R}^{3}}V(x)>0$,
$\inf\limits_{x\in\mathbb{R}^{3}}P(x)>0$ and
$\min\limits_{x\in\mathbb{R}^{3}}Q(x)>0$, proved the existence of
semiclassical solutions concentrating at a special set characterized by the
potentials, particularly, if
$\mathcal{V}\cap\mathcal{P}\cap\mathcal{Q}\neq\emptyset$, then the ground
state solusion of $-\Delta v+V_{min}v=P_{max}g(v)+Q_{max}|v|^{4}v$ is
obtained, where
$\mathcal{V}=\\{y\in\mathbb{R}^{3}:V(y)=\min\limits_{x\in\mathbb{R}^{3}}V(x)\\}$,
$\mathcal{P}=\\{y\in\mathbb{R}^{3}:P(y)=\max\limits_{x\in\mathbb{R}^{3}}P(x)\\}$
and
$\mathcal{Q}=\\{y\in\mathbb{R}^{3}:Q(y)=\max\limits_{x\in\mathbb{R}^{3}}Q(x)\\}$.
As one can see, the interaction of several potentials and the nonlinearity
with critical growth have a significant impact on the existence and
concentration behavior of positive solutions for the Schrödinger–Poisson
systems like (1.1). However, the existence and concentration results on them
involving both the critical growth nonlinearity and more than four potentials
have not previously been described. That is the main motivation of this work.

Inspired by the fact mentioned above, the purpose of this paper is to study
how the interaction of more than four potentials will make an impact on the
existence and concentration of the ground state solutions for the critical
Schrödinger–Poisson system ($SP_{\varepsilon}$) when $\varepsilon$ is small.
It is quite natural to ask that: can we obtain a concentration results for the
ground state solutions of ($SP_{\varepsilon}$) ? If so, where? In the present
paper, we shall give some answers for these questions. Furthermore, the aim of
our work is twofold: (i) to study the existence of positive ground state
solution for ($SP_{\varepsilon}$) and its properties, such as concentration,
exponential decay, etc. (ii) to find some sufficient conditions for the
nonexistence of positive ground state solution.

It is well known that system ($SP_{\varepsilon}$) can be easily transformed
into a single Schrödinger equation with a nonlocal term. In fact, as we shall
see in Section 2, for every $u\in H^{1}(\mathbb{R}^{3})$, and any fixed
$\varepsilon>0$, applying the Lax-Milgram theorem, a unique
$\phi_{u/\varepsilon}\in\mathcal{D}^{1,2}(\mathbb{R}^{3})$ is obtained, such
that $-\varepsilon^{2}\Delta\phi=4\pi h(x)u^{2}$ and that, inserted into the
first equation of ($SP_{\varepsilon}$), gives

$-\varepsilon^{2}\Delta
u+h(x)\phi_{u/\varepsilon}u+V(x)u=\sum\limits_{i=1}^{m}Q_{i}(x)|u|^{q_{i}-2}u+K(x)|u|^{4}u.$
($S_{\varepsilon}$)

Hence, $v$ is a solution of ($S_{\varepsilon}$) if and only if
$(v,\phi_{v/\varepsilon})$ is a solution of ($SP_{\varepsilon}$). For
simplicity, in many cases we say $v\in H^{1}(\mathbb{R}^{3})$, instead of
$(v,\phi_{v/\varepsilon})\in
H^{1}(\mathbb{R}^{3})\times\mathcal{D}^{1,2}(\mathbb{R}^{3})$, is a weak
solution of ($SP_{\varepsilon}$). Then in the following we only need to study
the Eq. ($S_{\varepsilon}$).

Before stating the main result of this paper, we introduce the precise
assumptions on $h$, $V$, $K$, and $Q_{i}(i=1,2,\cdots,m)$ :

$(f_{1})$ $V(x),K(x),Q_{i}(x)(1\leq i\leq m)\in
C^{1}(\mathbb{R}^{3},\mathbb{R})$.
$(f_{2})$ $(i-i_{0})Q_{i}(x)\geq 0$ for $i\neq i_{0}$, $Q_{i_{0}}(x)$ is
allowed to change sign.
$(f_{3})$ $0<V_{0}:=\inf\limits_{x\in\mathbb{R}^{3}}V(x)$, $V(x)$ and
$Q_{i}(x)(1\leq i\leq m)$ are bounded in $\mathbb{R}^{3}$.
$(f_{4})$ $0\leq K(x)\leq K^{\infty}:=\limsup\limits_{|x|\to\infty}K(x)$, and
there exists $x_{0}\in\mathbb{R}^{3}$ such that $K(x_{0})=K^{\infty}$.
$(H)$ $h(x)\in
C(\mathbb{R}^{3},\mathbb{R}),0<h_{0}=\inf\limits_{x\in\mathbb{R}^{3}}h(x)\leq
h(x)\leq h_{\infty}:=\lim\limits_{|x|\to\infty}h(x)<\infty.$

For every $s\in\mathbb{R}^{3}$, we consider the following equation with
parameters:

$-\Delta
u+\left[\frac{1}{|x|}*u^{2}\right]h^{2}(s)u+V(s)u=\sum\limits_{i=1}^{m}Q_{i}(s)|u|^{q_{i}-2}u+K(s)|u|^{4}u,\quad
x\in\mathbb{R}^{3},$ (1.3)

where $s\in\mathbb{R}^{3}$ acts as a parameter instead of an independent
variable. The ground energy function $G(s)$, which is defined to be the ground
energy associated with (1.3), and was firstly introduced in [35]. See section
3 for a more detailed. Moreover, let $c_{\infty}$ be the ground energy
associated with “limiting problem” of ($S_{\varepsilon}$), which is given as

$-\Delta
u+\left[\frac{1}{|x|}*u^{2}\right]h^{2}_{\infty}u+V_{\infty}u=\sum\limits_{i=1}^{m}Q_{i}^{\infty}|u|^{q_{i}-2}u+K^{\infty}|u|^{4}u,\quad
x\in\mathbb{R}^{3},$ (1.4)

where $V_{\infty}:=\liminf\limits_{|x|\to\infty}V(x),\hskip
4.30554ptQ_{i}^{\infty}:=\limsup\limits_{|x|\to\infty}Q_{i}(x).$

Then we state our main results of this work as follows.

######  Theorem 1.1.

Suppose that the potentials $V$, $K$, and $Q_{i}(1\leq i\leq m)$ satisfy
conditions $(f_{1})$–$(f_{4})$, the electronic potential $h(x)\equiv 1$ and

$c_{\infty}>c_{0}:=\inf\limits_{s\in\mathbb{R}^{3}}G(s).$ (1.5)

Then, for $\varepsilon>0$ small enough, ($S_{\varepsilon}$) has a positive
ground state solution $v_{\varepsilon}$. Moreover, we have  (i) The positive
ground state solution $v_{\varepsilon}$ possesses a maximum point
$x_{\varepsilon}$ in $\mathbb{R}^{3}$, such that $\lim\limits_{\varepsilon\to
0}dist(x_{\varepsilon},\mathcal{G})=0$. Setting
$\eta_{\varepsilon}(x)=v_{\varepsilon}(\varepsilon x+x_{\varepsilon})$, where
$x_{\varepsilon}\to x_{0},$ as $\varepsilon\to 0$, and $\eta_{\varepsilon}$
converges in $H_{\varepsilon}$ to a positive ground state solution of $-\Delta
u+\phi_{u}u+V(x_{0})u=\sum\limits_{i=1}^{m}Q_{i}(x_{0})|u|^{q_{i}-2}u+K(x_{0})|u|^{4}u,\hskip
4.30554ptx\in\mathbb{R}^{3},$ where
$\mathcal{G}:=\\{s\in\mathbb{R}^{3};G(s)=c_{0}\\}.$ (ii) There exist constants
$C>0$ and $\mu>0$ such that $v_{\varepsilon}(x)\leq
Cexp\left({-\frac{\mu}{\varepsilon}|x-x_{\varepsilon}|}\right)$.

######  Remark 1.1.

The existence of solution can be obtained, if the the electronic potential $h$
satisfies condition $(H)$. However, for concentration, it is complicated since
the nonlocal term

$\int_{\mathbb{R}^{3}}h(\varepsilon x)\left[\frac{1}{|x|}*\left(h(\varepsilon
x)u^{2}\right)\right]u^{2}dx=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}h(\varepsilon
x)u^{2}(x)\frac{h(\varepsilon y)u^{2}(y)}{|x-y|}dydx,$

appears $h(\varepsilon x)$ and $h(\varepsilon y)$. For the sake of simplicity,
we assume that $h(x)\equiv 1$.

######  Remark 1.2.

Conditions like $(f_{1})$–$(f_{4})$ on the nonlinear term was introduced by
[12]. Fan obtained the existence of positive solutions of Kirchhoff-type
problem. Theorem 1.1 extends the main results in [12] to the
Schrödinger–Poisson systems.

Now we give our key idea for the proof of Theorem 1.1. As we deal with the
problem ($S_{\varepsilon}$) in $H^{1}(\mathbb{R}^{3})$, the sobolev embeddings
$H^{1}(\mathbb{R}^{3})\hookrightarrow L^{q}(\mathbb{R}^{3})$, $q\in[2,6)$ are
not compact. The energy functional does not satisfy $(PS)_{c}$ condition at
any energy level $c$. To overcome this obstacle, we try to pull the energy
level down below critical energy level. Different from the assumptions
$(K_{1})$ on $K(x)$ and the methods in [18, 25], we first prove that the upper
bound of the energy level defined in (2.11) no more than the infimum of
$G(s)$. Then, under our conditions, we obtain the estimation of the critical
level for the infimum of $G(s)$ with the help of technique of Brezis and
Nirenberg [27]. On the other hand, one can see that the methods used in [11,
33, 34, 38] to establish a concentrating set of the ground state solutions for
($S_{\varepsilon}$) does not work, owing to the competing relationship among
multiple potentials. We succeed in doing so by introducing a new concentration
set $\mathcal{G}$ for the ground state solutions, which is not relies on the
condition

$\mathcal{M}=\bigcap_{i=1}^{m}\mathcal{Q}_{i}\cap\mathcal{V}\cap\mathcal{K}\neq\emptyset,$
(1.6)

where
$\mathcal{V}=\\{y\in\mathbb{R}^{3}:V(y)=\min\limits_{x\in\mathbb{R}^{3}}V(x)\\}$,
$\mathcal{Q}_{i}=\\{y\in\mathbb{R}^{3}:Q_{i}(y)=\max\limits_{x\in\mathbb{R}^{3}}Q_{i}(x)\\}$
and
$\mathcal{K}=\\{y\in\mathbb{R}^{3}:K(y)=\max\limits_{x\in\mathbb{R}^{3}}K(x)\\}$.
Particularly, if the condition (1.6) holds, it is easy to see that
$\mathcal{M}=\mathcal{G}$.

To overcome the difficulty is mentioned in Remark 1.1, we assume that the
electronic potential $h(x)$ satisfies

$(H_{1})$ $h(x)\in C(\mathbb{R}^{3},\mathbb{R})$, $h(x)\geq 0$,
$\lim\limits_{|x|\to\infty}h(x)=0$ and $h(x)=0$ if $x\in\mathcal{G}.$

Then we have the following result.

######  Theorem 1.2.

Suppose that the potentials $V$, $K$, and $Q_{i}(1\leq i\leq m)$ satisfy
conditions $(f_{1})$–$(f_{4})$, the electronic potential $h$ satisfies
$(H_{1})$ and

$c_{\infty}>c_{0}:=\inf\limits_{s\in\mathbb{R}^{3}}G(s).$

Then, for $\varepsilon>0$ small enough, ($S_{\varepsilon}$) has a positive
ground state solution $v_{\varepsilon}$. Moreover, we have
(i) The positive ground state solution $v_{\varepsilon}$ possesses a maximum
point $x_{\varepsilon}$ in $\mathbb{R}^{3}$, such that
$\lim\limits_{\varepsilon\to 0}dist(x_{\varepsilon},\mathcal{G})=0$. Setting
$\eta_{\varepsilon}(x)=v_{\varepsilon}(\varepsilon x+x_{\varepsilon})$, where
$x_{\varepsilon}\to x_{0},$ as $\varepsilon\to 0$, and $\eta_{\varepsilon}$
converges in $H_{\varepsilon}$ to a positive ground state solution of $-\Delta
u+V(x_{0})u=\sum\limits_{i=1}^{m}Q_{i}(x_{0})|u|^{q_{i}-2}u+K(x_{0})|u|^{4}u,\hskip
4.30554ptx\in\mathbb{R}^{3},$ where
$\mathcal{G}:=\\{s\in\mathbb{R}^{3};G(s)=c_{0}\\}.$ (ii) There exist constants
$C>0$ and $\mu>0$ such that $v_{\varepsilon}(x)\leq
Cexp\left({-\frac{\mu}{\varepsilon}|x-x_{\varepsilon}|}\right).$

######  Remark 1.3.

It is interesting to give some sufficient conditions to guarantee (1.5), in
terms of $V$, $K$ and $Q_{i}(i=1,2,\cdots,m)$. For example, let us consider
the following conditions:
(1) $V_{\infty}=\sup\limits_{x\in\mathbb{R}^{3}}V(x)$,
$Q_{i}^{\infty}=\inf\limits_{x\in\mathbb{R}^{3}}Q_{i}(x)(i=1,2,\cdots,m)$,
$K(x)\equiv C$. (2) There exists $\hat{s}$ such that V_∞≥V(^s),
Q_i^∞≤Q_i(^s)(i=1,2,⋯,m), K(x)≡C with one of the above inequalities being
strict.

If $V$, $K$, and $Q_{i}(i=1,2,\cdots,m)$ are not all constants, then each of
the previous conditions (1) and (2) guarantee $c_{\infty}>c_{0}$ (see[31,
35]).

Finally, to get the nonexistence of ground state solution, we make the
following assumption:

$(f_{5})$ $V(x)\geq V_{\infty}=V_{0},$ and $Q_{i}(x)\leq Q_{i}^{\infty}(1\leq
i\leq m).$ $h(x)\equiv 1$ or $h(x)$ satisfies $(H_{1})$.

The result of this work has the following statement.

######  Theorem 1.3.

Suppose that conditions $(f_{1})$–$(f_{5})$ hold. Then, for any
$\varepsilon>0$, ($S_{\varepsilon}$) has no ground state solution.

The remainder of this paper is organized as follows. In Section 2, we derive a
variational setting for the problem and give some preliminaries. In section 3
and 4, we prove the existence of positive ground state solutions for (2.8)
with some properties, such as concentration, exponential decay etc. The proofs
of Theorems 1.1 and 1.2 will be given in Section 5. Section 6 is dedicated to
the proof of Theorem 1.3.

Hereafter we use the following notations:

$\bullet$ $H^{1}(\mathbb{R}^{3})$ is the usual Sobolev space equipped with the
inner product abd norm

$(u.v)=\int_{\mathbb{R}^{3}}(\nabla u\nabla v)+uv)dx;\hskip
8.61108pt||u||^{2}=\int_{\mathbb{R}^{3}}(|\nabla u|^{2}+u^{2})dx.$

$\bullet$ $\mathcal{D}^{1,2}(\mathbb{R}^{3})$ is the completion of
$C^{\infty}_{0}(\mathbb{R}^{3})$ with respect to the norm

$||u||_{\mathcal{D}^{1,2}}^{2}=\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx.$

$\bullet$ $L^{r}(\Omega)$, $1\leq r\leq\infty$, $\Omega\subset\mathbb{R}^{3}$,
denotes a Lebesgue space, the norm in $L^{r}(\mathbb{R}^{3})$ is denoted by

$|u|_{r,\Omega}$, where $\Omega$ is a proper subset of $\mathbb{R}^{3}$, by
$|u|_{r}$ where $\Omega=\mathbb{R}^{3}$.

$\bullet$ Denote the best constants for the embeddings of
$H^{1}(\mathbb{R}^{3})\hookrightarrow L^{p}(\mathbb{R}^{3})(2\leq p\leq 6)$
and

$\mathcal{D}^{1,2}(\mathbb{R}^{3})\hookrightarrow$ $L^{6}(\mathbb{R}^{3})$ by
$S_{p}$ and $S$, respectively. Then

$|u|_{p}\leq S_{p}^{-\frac{1}{2}}||u||\quad\forall u\in
H^{1}(\mathbb{R}^{3}),$ (1.7)

and

$|u|_{6}\leq S^{-\frac{1}{2}}||u||_{\mathcal{D}^{1,2}}\quad\forall
u\in\mathcal{D}^{1,2}(\mathbb{R}^{3}).$ (1.8)

$\bullet$ For any $R>0$ and for any $z\in\mathbb{R}^{3}$, $B_{R}(z)$ denotes
the ball of rasius $R$ centered at $z$.

$\bullet$ $C$, $C_{k}(k=0,1,\cdots,n+1)$ denote various positive constants may
different from line to line.

$\bullet$ $\rightarrow$ and $\rightharpoonup$ denote the strong and weak
convergence in the related function space respectively.

$\bullet$ $o_{n}(1)$ denotes any quantity which tends to zero when
$n\to\infty$.

##  2 Preliminary results and variational framework

In this section, we outline the variational framework of problem
($SP_{\varepsilon}$) and give some preliminary lemmas. From this section to
the section 5.1, we assume that the electronic potential $h\equiv 1$.

For every $u\in H^{1}(\mathbb{R}^{3})$, the linear functional $L_{u}$ defined
in $\mathcal{D}^{1,2}(\mathbb{R}^{3})$ by

$\displaystyle L_{u}(v)=\int_{\mathbb{R}^{3}}u^{2}vdx.$ (2.1)

Then it follows from Lax-Milgram theorem that there exists a unique
$\phi_{u}\in\mathcal{D}^{1,2}(\mathbb{R}^{3})$ such that

$\displaystyle\int_{\mathbb{R}^{3}}\nabla\phi_{u}\nabla
vdx=\int_{\mathbb{R}^{3}}u^{2}vdx,\hskip 4.30554pt\forall
v\in\mathcal{D}^{1,2}(\mathbb{R}^{3}),$ (2.2)

which is a weak solution of $-\Delta\phi=4\pi u^{2}$ and the following
representation formula holds

$\phi_{u}(x)=\int_{\mathbb{R}^{3}}\frac{u^{2}(y)}{|x-y|}dy=\frac{1}{|x|}*u^{2}.$

Therefore, for any $\varepsilon>0$, we have

$\displaystyle\phi_{u/\varepsilon}=\frac{1}{\varepsilon^{2}}\int_{\mathbb{R}^{3}}\frac{u^{2}(y)}{|x-y|}dy=\varepsilon^{-2}\phi_{u}.$
(2.3)

Let us define the operator
$\Phi$:$H^{1}(\mathbb{R}^{3})\to\mathcal{D}^{1,2}(\mathbb{R}^{3})$ as

$\Phi[u]=\phi_{u}.$

We next state some properties of $\Phi$, which will be useful in the
following.

######  Lemma 2.1.

(i) |  $\Phi$ is continuous;
---|---
(ii) |  $\Phi$ maps bounded sets into bounded sets;
(iii) |  $\Phi(tu)=t^{2}\Phi(u)$;
(iv) |  $||\phi_{u}||_{\mathcal{D}^{1,2}}\leq S^{-\frac{1}{2}}|u|_{12/5}^{2}$.

######  Proof.

The proof of (i) and (ii) can be found in [8]. (iii) and (iv) are clear from
the definition of $\phi_{u}$. Replacing $v$ by $\phi_{u}$ in (2.2) and using
Hölder inequality, we have

$||\phi_{u}||_{\mathcal{D}^{1,2}}^{2}=\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx\leq|\phi_{u}|_{6}|u|_{12/5}^{2}\leq
S^{-\frac{1}{2}}||\phi_{u}||_{\mathcal{D}^{1,2}}|u|_{12/5}^{2},$

and then (iv) holds. $\Box$

######  Lemma 2.2.

Assume $u_{n}\rightharpoonup u\in H^{1}(\mathbb{R}^{3})$.Then
(i) $\Phi(u_{n})\to\Phi(u)$ in $\mathcal{D}^{1,2}(\mathbb{R}^{3})$; (ii)
$\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx\to\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx$;
(iii) $\int_{{\mathbb{R}^{3}}}\phi_{u_{n}}u_{n}\varphi
dx\to\int_{\mathbb{R}^{3}}\phi_{u}u\varphi dx$, $\forall\varphi\in
H^{1}(\mathbb{R}^{3})$.

######  Proof.

We borrow an idea from Cerami et al.[20] to prove this lemma.

(i) By definition of $\Phi$ and $L_{u}$, we have

$||\Phi[u]||_{\mathcal{D}^{1,2}}=||\phi_{u}||_{\mathcal{D}^{1,2}}=||L_{u}||_{\mathcal{L}(\mathcal{D}^{1,2},\mathbb{R})},$

to prove (i), it is enough to show that

$||L_{u_{n}}-L_{u}||_{\mathcal{L}(\mathcal{D}^{1,2},\mathbb{R})}\to 0,\quad
n\to\infty.$

For all $v\in\mathcal{D}^{1,2}(\mathbb{R}^{3})$, then $v\in
L^{6}(\mathbb{R}^{3})$, thus for any $\sigma>0$, there exists $R>0$ large
enough, such that $|v|_{6,B_{R}^{C}(0)}<\sigma$. Thus, for all
$v\in\mathcal{D}^{1,2}(\mathbb{R}^{3})$, we obtain

$\displaystyle 0\leq|L_{u_{n}}(v)-L_{u}(v)|$
$\displaystyle=\left|\int_{\mathbb{R}^{3}}(u_{n}^{2}-u^{2})vdx\right|$ (2.4)
$\displaystyle\leq\int_{B_{R}^{C}(0)}|u_{n}^{2}-u^{2}||v|dx+\int_{B_{R}(0)}|u_{n}^{2}-u^{2}||v|dx$
$\displaystyle\leq|v|_{6,B_{R}^{C}(0)}|u_{n}-u|_{\frac{12}{5}}|u_{n}+u|_{\frac{12}{5}}+\int_{B_{R}(0)}|u_{n}^{2}-u^{2}||v|dx$
$\displaystyle\leq C\sigma+\int_{B_{R}(0)}|u_{n}^{2}-u^{2}||v|dx.$

Furthermore, we may assume, going if necessary to a subsequence, $u_{n}\to u$
in $L_{loc}^{\frac{12}{5}}(\mathbb{R}^{3})$. Hence, we have

$\int_{B_{R}(0)}|u_{n}^{2}-u^{2}||v|dx\leq\left(\int_{{B_{R}}(0)}|u_{n}-u|^{\frac{12}{5}}dx\right)^{\frac{5}{12}}\left(\int_{B_{R}(0)}|u_{n}+u|^{\frac{12}{5}}dx\right)^{\frac{5}{12}}|v|_{6}=o_{n}(1).$
(2.5)

Combining (2.4) and (2.5), the desired conclusion is obtained.

(ii) At first, replacing $v$ by $\phi_{u_{n}}$ and repeating the argument used
in the prove of (i), we can obtain that
$\int_{\mathbb{R}^{3}}\phi_{u_{n}}(u_{n}^{2}-u^{2})dx\to 0$, as $n\to\infty$
holds. On the other hand, we observe that

$\int_{\mathbb{R}^{3}}(\phi_{u_{n}}-\phi_{u})u^{2}dx\leq|u|_{\frac{12}{5}}^{2}|\phi_{u_{n}}-\phi_{u}|_{6}=o(1).$

Indeed, since the embedding $\mathcal{D}^{1,2}(\mathbb{R}^{3})\hookrightarrow
L^{6}(\mathbb{R}^{3})$ is continuous and $\Phi(u_{n})\to\Phi(u)$ in
$\mathcal{D}^{1,2}(\mathbb{R}^{3})$. Hence, we obtain

$\left|\int_{\mathbb{R}^{3}}(\phi_{u_{n}}u_{n}^{2}-\phi_{u}u^{2})dx\right|\leq\left|\int_{\mathbb{R}^{3}}\phi_{u_{n}}(u_{n}^{2}-u^{2})dx\right|+\left|\int_{\mathbb{R}^{3}}(\phi_{u_{n}}-\phi_{u})u^{2}dx\right|=o_{n}(1),$

as desired.

(iii) To show (iii), we first prove that

$\left|\int_{\mathbb{R}^{3}}\phi_{u}(u_{n}-u)\varphi dx\right|=o_{n}(1).$
(2.6)

In fact, using $\varphi\in H^{1}(\mathbb{R}^{3})$ and
$\phi_{u}\in\mathcal{D}^{1,2}(\mathbb{R}^{3})$, it follows from Hölder
inequality that $\phi_{u}\varphi\in L^{2}(\mathbb{R}^{3})$. Since
$u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{3})$and, then, in
$L^{2}(\mathbb{R}^{3})$. Thus, it is easy to see that (2.6) holds. On the
other hand, we have

$\left|\int_{\mathbb{R}^{3}}(\phi_{u_{n}}-\phi_{u})u_{n}\varphi
dx\right|\leq|u_{n}|_{2}|\phi_{u_{n}}-\phi_{u}|_{6}|\varphi|_{3}=o_{n}(1).$
(2.7)

Combining (2.6) and (2.7), we can get that

$\displaystyle\left|\int_{\mathbb{R}^{3}}(\phi_{u_{n}}u_{n}\varphi-\phi_{u}u\varphi)dx\right|$
$\displaystyle\leq\left|\int_{\mathbb{R}^{3}}\phi_{u}(u_{n}-u)\varphi
dx\right|+\left|\int_{\mathbb{R}^{3}}(\phi_{u_{n}}-\phi_{u})u_{n}\varphi
dx\right|$ $\displaystyle=o_{n}(1),$

as desired. $\Box$

######  Remark 2.1.

From Lemma 2.1 (iv) and [8, Lemma 3.1], we obtain that $F\in
C^{2}(H^{1}(\mathbb{R}^{3}),\mathbb{R})$ is well-defined with

$F(u)=\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx.$

Moreover, the functional $F$ and its derivative $F^{\prime}$ posses BL-
splitting property [19, Lemma 2.2], which is similar to Brezis–Lieb lemma
[32].

Substituting (2.3) into ($SP_{\varepsilon}$) and making the change of variable
$x\mapsto\varepsilon x$ , we can rewrite ($S_{\varepsilon}$) as the following
equivalent equation

$-\Delta u+\phi_{u}u+V(\varepsilon x)u=\sum\limits_{i=1}^{m}Q_{i}(\varepsilon
x)|u|^{q_{i}-2}u+K(\varepsilon x)|u|^{4}u,\hskip 4.30554ptx\in\mathbb{R}^{3}.$
(2.8)

Obviously, $v(x)$ is a solution of ($S_{\varepsilon}$) if and only if
$u(x)=v(\varepsilon x)$ is a solution of (2.8). For any $\varepsilon>0$, we
define the Hilbert space $H_{\varepsilon}=\\{u\in
H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}V(\varepsilon
x)|u|^{2}dx<\infty\\}$ equipped with the norm

$||u||_{\varepsilon}^{2}=\int_{\mathbb{R}^{3}}(|\nabla u|^{2}+V(\varepsilon
x)|u|^{2})dx.$

Since $V(x)$ is positive and bounded for all $x\in\mathbb{R}^{3}$, we have
$H_{\varepsilon}=H^{1}(\mathbb{R}^{3})$ and the norm $||\cdot||_{\varepsilon}$
is equivalent to $||\cdot||$. At this step, under our assumptions it is
standard to see that (2.8) is variational and its solutions are the critical
points of the functional $I_{\varepsilon}:H_{\varepsilon}\to\mathbb{R}$ given
by

$I_{\varepsilon}(u)=\frac{1}{2}||u||^{2}_{\varepsilon}+\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx-\frac{1}{6}\int_{\mathbb{R}^{3}}K(\varepsilon x)|u|^{6}dx.$
(2.9)

Moreover, $I_{\varepsilon}\in C^{2}(H_{\varepsilon},\mathbb{R})$. Next, we
introduce the Nehari manifold associated to $I_{\varepsilon}$ by

$\mathcal{N}_{\varepsilon}=\\{u\in H_{\varepsilon}\backslash\\{0\\}:\langle
I_{\varepsilon}^{\prime}(u),u\rangle=0\\}.$

Clearly, $u\in\mathcal{N}_{\varepsilon}$ if and only if

$||u||^{2}_{\varepsilon}+\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx=\sum\limits_{i=1}^{m}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx+\int_{\mathbb{R}^{3}}K(\varepsilon x)|u|^{6}dx.$ (2.10)

Moreover, $I_{\varepsilon}$ is bounded from below on
$\mathcal{N}_{\varepsilon}$. So we can consider the following problem:

$c_{\varepsilon}:=\inf\limits_{u\in\mathcal{N}_{\varepsilon}}I_{\varepsilon}(u).$
(2.11)

Now, we summarize some properties of $I_{\varepsilon}$ on
$\mathcal{N}_{\varepsilon}$.

######  Lemma 2.3.

For any $u\in H_{\varepsilon}\backslash\\{0\\}$, the following statements hold
true.
(i) There exists a unique $t_{\varepsilon}=t_{\varepsilon}(u)>0$ such that
$t_{\varepsilon}u\in\mathcal{N}_{\varepsilon}$ and
$I_{\varepsilon}(t_{\varepsilon}u):=\max\limits_{t\geq 0}I_{\varepsilon}(tu)$.
(ii) There exist constants $0<\alpha_{1}<\alpha_{2}$ independent of
$\varepsilon>0$, such that $\alpha_{1}\leq t_{\varepsilon}\leq\alpha_{2}$.
(iii) $I_{\varepsilon}$ is coercive and bounded from below on
$\mathcal{N}_{\varepsilon}$. (iv) There exists $\kappa>0$ independent of
$\varepsilon>0$, such that $||u||_{\varepsilon}\geq\kappa$,
$I_{\varepsilon}(u)\geq\frac{q_{i_{0}}-2}{2q_{i_{0}}}\kappa^{2}$, $\forall
u_{\varepsilon}\in\mathcal{N}_{\varepsilon}$.

######  Proof.

(i) For $t>0$, we set

$\displaystyle
h(t):=I_{\varepsilon}(tu)=\frac{t^{2}}{2}||u||^{2}_{\varepsilon}+\frac{t^{4}}{4}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx-\sum\limits_{i=1}^{m}\frac{t^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx-\frac{t^{6}}{6}\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u|^{6}dx.$

From the Sobolev embedding inequalities (1.7), we obtain

$h(t)\geq\frac{t^{2}}{2}||u||^{2}_{\varepsilon}-\sum\limits_{i=1}^{m}\frac{t^{q_{i}}}{q_{i}}C_{i}||u||^{q_{i}}_{\varepsilon}-\frac{t^{6}}{6}C_{m+1}||u||^{6}_{\varepsilon}.$

Since $4<q_{i}<6$, it is easy to see that $h(t)>0$ for small $t>0$. Moreover,
it follows from Lemma 2.1 (iv) that,

$h(t)\leq\frac{t^{2}}{2}||u||^{2}_{\varepsilon}+\frac{t^{4}}{4}C||u||^{4}_{\varepsilon}+\sum\limits_{i=1}^{m}\frac{t^{q_{i}}}{q_{i}}C_{i}||u||^{q_{i}}_{\varepsilon}-\frac{t^{6}}{6}\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u|^{6}dx\to-\infty,$

as $t\to\infty$. Consequently, $\max\limits_{t\geq 0}h(t)$ is achieved at
$t_{\varepsilon}=t_{\varepsilon}(u)>0$, hence $h^{\prime}(t_{\varepsilon})=0$
and $t_{\varepsilon}u\in\mathcal{N}_{\varepsilon}$.

Next, we show the uniqueness of $t_{\varepsilon}$. Arguing indirectly, if
there exist $0<t_{1}<t_{2}$ such that
$t_{1}u,t_{2}u\in\mathcal{N}_{\varepsilon}$. Let,

$f(t):=-{t^{2}}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx+\sum\limits_{i=1}^{m}{t^{q_{i}-2}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx+{t^{4}}\int_{\mathbb{R}^{3}}K(\varepsilon x)|u|^{6}dx.$

Taking account of $4<q_{1}<q_{2}<\cdots<q_{m}<6$ and $(f_{2})$, we have $f(t)$
is a strictly increasing function on any interval where $f(t)>0$. Then, we
deduce from $h^{\prime}(t_{1})=0$ and $h^{\prime}(t_{2})=0$ that

$f(t_{1})=||u||_{\varepsilon}^{2}\hskip 4.30554pt\text{and}\hskip
4.30554ptf(t_{2})=||u||_{\varepsilon}^{2},$

which is a contradiction.

(ii) Since $t_{\varepsilon}u\in\mathcal{N}_{\varepsilon}$, then
$f(t_{\varepsilon})=0$, i.e.,

$-{t_{\varepsilon}^{2}}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx+\sum\limits_{i=1}^{m}{t_{\varepsilon}^{q_{i}-2}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx+{t_{\varepsilon}^{4}}\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u|^{6}dx=||u||_{\varepsilon}^{2}.$ (2.12)

It follows from that

$||u||_{\varepsilon}^{2}\leq\sum\limits_{i=1}^{m}t_{\varepsilon}^{q_{i}-2}C_{i}||u||^{q_{i}}_{\varepsilon}+t_{\varepsilon}^{4}C_{m+1}||u||^{6}_{\varepsilon},$
(2.13)

and

${t_{\varepsilon}^{4}}\int_{\mathbb{R}^{3}}|u|^{6}dx\leq||u||_{\varepsilon}^{2}+{t_{\varepsilon}^{2}}C||u||^{4}_{\varepsilon}+\sum\limits_{i=1}^{m}t^{q_{i}-2}C_{i}||u||^{q_{i}}_{\varepsilon}.$

Then, it is easy to see that there exist constants $0<\alpha_{1}<\alpha_{2}$
independent of $\varepsilon>0$, such that $\alpha_{1}\leq
t_{\varepsilon}\leq\alpha_{2}$.

(iii) For $u\in\mathcal{N}_{\varepsilon}$, we have

$\displaystyle I_{\varepsilon}(u)=$ $\displaystyle\hskip
3.99994ptI_{\varepsilon}(u)-\frac{1}{q_{i_{0}}}\langle
I_{\varepsilon}^{\prime}(u),u\rangle$ (2.14) $\displaystyle=$
$\displaystyle\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)||u||^{2}_{\varepsilon}+\left(\frac{1}{4}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{i_{0}-1}\left(\frac{1}{q_{i}}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx+\sum\limits_{i=i_{0}+1}^{m}\left(\frac{1}{q_{i_{0}}}-\frac{1}{q_{i}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx$
$\displaystyle+\left({\frac{1}{q_{i_{0}}}-\frac{1}{6}}\right)\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u|^{6}dx.$ $\displaystyle\geq$
$\displaystyle\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)||u||^{2}_{\varepsilon}>0,$

which implies that $I_{\varepsilon}$ is coercive and bounded from below on
$\mathcal{N}_{\varepsilon}$.

(iv) The conclusion is immediate by (iii) and taking $t_{\varepsilon}=1$ in
(2.13). $\Box$

######  Lemma 2.4.

$I_{\varepsilon}$ has the mountain pass geometry structure.
(i) There exist $\alpha,\rho>0$ independent of $\varepsilon$, such that
$I_{\varepsilon}(u)\geq\alpha$ for $||u||_{\varepsilon}=\rho$; (ii) There
exists an $e\in H_{\varepsilon}$ satisfying $||e||_{\varepsilon}\geq\rho$ such
that $I_{\varepsilon}(e)<0$.

######  Proof.

(i) For any $u\in H_{\varepsilon}\backslash\\{0\\}$, we deduce from the
Sobolev embedding inequalities (1.7) that

$\displaystyle I_{\varepsilon}(u)$
$\displaystyle=\frac{1}{2}||u||^{2}_{\varepsilon}+\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx-\frac{1}{6}\int_{\mathbb{R}^{3}}K(\varepsilon x)|u|^{6}dx$
$\displaystyle\geq\frac{1}{2}||u||^{2}_{\varepsilon}-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}C_{i}||u||^{q_{i}}_{\varepsilon}-\frac{1}{6}C_{n+1}||u||^{6}_{\varepsilon}.$

Set $||u||_{\varepsilon}=\rho$ small enough, such that
$I_{\varepsilon}(u)\geq\alpha$.

(ii) For any $u\in H_{\varepsilon}\backslash\\{0\\}$.

$\displaystyle I_{\varepsilon}(tu)$
$\displaystyle=\frac{t^{2}}{2}||u||^{2}_{\varepsilon}+\frac{t^{4}}{4}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx-\sum\limits_{i=1}^{m}\frac{t^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx-\frac{t^{6}}{6}\int_{\mathbb{R}^{3}}K(\varepsilon x)|u|^{6}dx$
$\displaystyle\leq\frac{t^{2}}{2}||u||^{2}_{\varepsilon}+\frac{t^{4}}{4}C||u||^{4}_{\varepsilon}-\sum\limits_{i=1}^{m}\frac{t^{q_{i}}}{q_{i}}C_{i}||u||^{q_{i}}_{\varepsilon}-\frac{t^{6}}{6}\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u|^{6}dx.$

Since $4<q_{i}<6$ and $K(x)\geq 0$, there exist $e:=t^{\prime}u$ for some
$t^{\prime}>0$ large enough such that $||e||_{\varepsilon}\geq\rho$ and
$I_{\varepsilon}(e)<0$. $\Box$

######  Lemma 2.5.

For any $\varepsilon>0$, we can define

$c_{\varepsilon}^{*}:=\inf\limits_{u\in
H_{\varepsilon}\backslash\\{0\\}}\max\limits_{t\geq
0}I_{\varepsilon}(tu),\quad
c_{\varepsilon}^{**}:=\inf\limits_{\gamma\in\Gamma}\sup\limits_{t\in[0,1]}I_{\varepsilon}(\gamma(t)),$

where $\Gamma=\\{\gamma\in
C([0,1],H_{\varepsilon});\gamma(0)=0,I_{\varepsilon}(\gamma(1))<0)\\}.$
Then,

$c_{\varepsilon}=c_{\varepsilon}^{*}=c_{\varepsilon}^{**}.$ (2.15)

######  Proof.

By Lemma 2.3 (i), we have

$c_{\varepsilon}=\inf\limits_{u\in\mathcal{N}_{\varepsilon}}I_{\varepsilon}(u)=\inf\limits_{u\in
H_{\varepsilon}\backslash\\{0\\}}I_{\varepsilon}(t_{\varepsilon}u)=\inf\limits_{u\in
H_{\varepsilon}\backslash\\{0\\}}\max\limits_{t\geq
0}I_{\varepsilon}(tu)=c_{\varepsilon}^{*}.$

Moreover, by Lemma 2.4 (ii), for any $u\in H_{\varepsilon}\backslash\\{0\\}$,
there exists $k>0$ large enough, such that $I_{\varepsilon}(ku)<0$, set
$\beta(t)=tku$, $t\in[0,1]$, then $\beta\in\Gamma$. Indeed, $\beta(0)=0$,
$\beta(1)=ku$ and $I_{\varepsilon}(\beta(1))<0$. Thus,

$\max\limits_{t\geq
0}I_{\varepsilon}(tu)=\sup\limits_{t\in[0,1]}I_{\varepsilon}(tku)=\sup\limits_{t\in[0,1]}I_{\varepsilon}(\beta(t))\geq
c_{\varepsilon}^{**}.$

It follows that $c_{\varepsilon}^{*}\geq c_{\varepsilon}^{**}$. On the other
hand, the Nehari manifold $\mathcal{N}_{\varepsilon}$ separates
$H_{\varepsilon}$ into two components

$H_{\varepsilon}^{+}=\\{u\in H_{\varepsilon}:\langle
I_{\varepsilon}^{\prime}(u),u\rangle>0\\}\cup\\{0\\}$

and

$H_{\varepsilon}^{-}=\\{u\in H_{\varepsilon}:\langle
I_{\varepsilon}^{\prime}(u),u\rangle<0\\}.$

It follows from (2.14) that $I_{\varepsilon}(u)\geq 0$ for $u\in
H_{\varepsilon}^{+}$, and $\frac{1}{q_{i_{0}}}\langle
I_{\varepsilon}^{\prime}(\gamma(1)),\gamma(1)\rangle\leq
I_{\varepsilon}(\gamma(1))<0$. Thus any $\gamma\in\Gamma$ has to cross
$\mathcal{N}_{\varepsilon}$, since $\gamma(0)\in H_{\varepsilon}^{+}$ and
$\gamma(1)\in H_{\varepsilon}^{-}$, and so $c_{\varepsilon}^{**}\geq
c_{\varepsilon}$. The proof is complete. $\Box$

In order to study (2.8), we need some results about the autonomous problem of
(2.8). For $a>0$, $b_{i_{0}}>0$, $(j-i_{0})b_{j}>0(j=1,2,\cdots,m+1;j\neq
i_{0})$, consider the autonomous problem:

$-\Delta
u+\phi_{u}u+au=\sum\limits_{i=1}^{m}b_{i}|u|^{q_{i}-2}u+b_{m+1}|u|^{4}u,\quad
x\in\mathbb{R}^{3}.$ (2.16)

The associated energy functional is

$\displaystyle
I_{ab}(u)=\frac{1}{2}||u||^{2}_{a}+\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx-\sum\limits_{i=1}^{m}\frac{b_{i}}{q_{i}}\int_{\mathbb{R}^{3}}|u|^{q_{i}}dx-\frac{b_{m+1}}{6}\int_{\mathbb{R}^{3}}|u|^{6}dx,$

where $||u||^{2}_{a}=\int_{\mathbb{R}^{3}}(|\nabla u|^{2}+a|u|^{2})dx$. By
Lemma 2.5, we have

$c_{ab}:=\inf\limits_{u\in\mathcal{N}_{ab}}I_{ab}(u)=\inf\limits_{u\in
H_{\varepsilon}\backslash\\{0\\}}\max\limits_{t\geq
0}I_{ab}(tu)=\inf\limits_{\gamma\in\Gamma_{ab}}\sup\limits_{t\in[0,1]}I_{ab}(\gamma(t)),$
(2.17)

where $\mathcal{N}_{ab}=\\{u\in H_{\varepsilon}\backslash\\{0\\}:\langle
I_{ab}^{\prime}(u),u\rangle=0\\}$. $\Gamma=\\{\gamma\in
C([0,1],H_{\varepsilon});\gamma(0)=0,I_{\varepsilon}(\gamma(1)<0)\\}.$

######  Lemma 2.6.

Problem (2.16) has at least a positive ground state solution in
$H_{\varepsilon}$.

######  Proof.

The idea of the proofs is sketched as follows. At first, similar to Lemma 2.4,
we can show that $I_{ab}$ satisfies the mountain pass geometry structure.
Moreover, similar to (2.14), we can show the (PS) sequence is bounded in
$H_{\varepsilon}$. Secondly, with the help of technique of Brezis and
Nirenberg [27], we can obtain the estimation of the mountain pass critical
level under our conditions for (2.16), which can recover the compactness of
the (PS) sequence. Finally, it follows from the Lions’ concentration-
compactness principle [9, 10] and some standard arguments that (2.16) has a
positive ground state solution. For the details of the proof, we refer readers
to [11, 25]. $\Box$

The following lemma describes a comparison between the Mountain-Pass values
for different parameters, which will play a very important role in obtaining
the existence results.

######  Lemma 2.7.

For $a,\tilde{a}>0$, $b_{i_{0}},\tilde{b}_{i_{0}}>0$, $(j-i_{0})b_{j}>0$,
$(j-i_{0})\tilde{b}_{j}>0$$(j=1,2,\cdots,m+1)$. If

$min\\{a-\tilde{a},\tilde{b}_{j}-b_{j}\\}\geq 0,$

then $c_{ab}\geq c_{\tilde{a}\tilde{b}}$. In particular, if
$max\\{a-\tilde{a},\tilde{b}_{j}-b_{j}\\}>0$ also holds, then
$c_{ab}>c_{\tilde{a}\tilde{b}}$.

######  Proof.

From Lemma 2.6, we choose $u_{ab}$ be a solution of problem (2.16) such that
$c_{ab}=I_{ab}(u_{ab})$. There holds

$c_{ab}=\max\limits_{t\geq 0}I_{ab}(tu_{ab}).$

By the similar arguments to Lemma 2.3 (i), there exists $t_{1}>0$ such that
$t_{1}u_{ab}\in\mathcal{N}_{\tilde{a}\tilde{b}}$. let
$\tilde{u}_{\tilde{a}\tilde{b}}=t_{1}u_{ab}$ be such that
$I_{\tilde{a}\tilde{b}}(\tilde{u}_{\tilde{a}\tilde{b}})=\max\limits_{t\geq
0}I_{\tilde{a}\tilde{b}}(tu_{ab})$. Then, we see that

$\displaystyle c_{ab}=I_{ab}(u_{ab})\geq$ $\displaystyle
I_{ab}(\tilde{u}_{\tilde{a}\tilde{b}})$ $\displaystyle=$ $\displaystyle
I_{\tilde{a}\tilde{b}}(\tilde{u}_{\tilde{a}\tilde{b}})+\frac{a-\tilde{a}}{2}||u||_{\mathcal{D}^{1,2}}$
$\displaystyle-\sum\limits_{i=1}^{m}\frac{b_{i}-\tilde{b}_{i}}{q_{i}}\int_{\mathbb{R}^{3}}|u|^{q_{i}}dx-\frac{b_{m+1}-\tilde{b}_{m+1}}{6}\int_{\mathbb{R}^{3}}|u|^{6}dx$
$\displaystyle\geq$ $\displaystyle c_{\tilde{a}\tilde{b}}.$

In particular, if $max\\{a-\tilde{a},\tilde{b}_{j}-b_{j}\\}>0$ also holds,
then the above inequality implies that $c_{ab}>c_{\tilde{a}\tilde{b}}$. The
proof is complete. $\Box$

##  3 The existence of ground state solution

In this section, we will establish compactness lemma for $I_{\varepsilon}$,
and prove the existence of ground state solution to (2.8). Firstly, we need to
give the energy functional associated with (1.4) by

$I_{\infty}(u)=\frac{1}{2}||u||^{2}_{\mathcal{D}^{1,2}}+\frac{1}{2}\int_{\mathbb{R}^{3}}V_{\infty}u^{2}dx+\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}^{\infty}|u|^{q_{i}}dx-\frac{1}{6}\int_{\mathbb{R}^{3}}K^{\infty}|u|^{6}dx.$

Considering the following minimization problem:

$c_{\infty}:=\inf\limits_{u\in\mathcal{N}_{\infty}}I_{\infty}(u),$

where $\mathcal{N}_{\infty}=\\{u\in H_{\varepsilon}\backslash\\{0\\}:\langle
I_{\infty}^{\prime}(u),u\rangle=0\\}.$ Moreover, we denote the energy
functional for (1.3) by

$\displaystyle I^{s}(u)=$
$\displaystyle\frac{1}{2}||u||^{2}_{\mathcal{D}^{1,2}}+\frac{1}{2}\int_{\mathbb{R}^{3}}V(s)u^{2}dx+\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx$
(3.1)
$\displaystyle-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(s)|u|^{q_{i}}dx-\frac{1}{6}\int_{\mathbb{R}^{3}}K(s)|u|^{6}dx.$

Finally, we define the ground energy function

$G(s):=\inf\limits_{u\in\mathcal{N}^{s}}I^{s}(u),$

where $\mathcal{N}^{s}=\\{u\in
H_{\varepsilon}\backslash\\{0\\}:\langle(I^{s})^{\prime}(u),u\rangle=0\\}$,
which was mentioned in Section 1 and will play an important role in estimating
the critical energy level.

######  Lemma 3.1.

For $s\in\mathbb{R}^{3}$, the ground energy function $G(s)$ is locally
Lipschitz continuous.

The proof of Lemma 3.1 is essentially similar to that in [35, Lemma 2.3, Lemma
2.4 ], so we omit it here.

######  Lemma 3.2.

([32, Lemma 1.21]) Let $r>0$ and $\\{u_{n}\\}$ is bounded in
$H^{1}(\mathbb{R}^{N})$. If

$\sup\limits_{y\in\mathbb{R}^{N}}\int_{B_{r}(y)}|u_{n}|^{2}\to 0,\quad
n\to\infty,$

then $u_{n}\to 0$ in $L^{s}(\mathbb{R}^{N})$ for $2<s<2^{*}$.

To estimate the critical energy level for the critical Schrödinger–Poisson
system involving more than four potentials, we have an important upper bound
for the least energy $c_{\varepsilon}$ defined in (2.11) via the definition of
the ground energy function $G(s)$.

######  Lemma 3.3.

There exists $C>0$ independent of $\varepsilon$, such that
$c_{\varepsilon}\geq C$. Furthermore,

$\limsup\limits_{\varepsilon\to 0}c_{\varepsilon}\leq
c_{0}:=\inf\limits_{s\in\mathbb{R}^{3}}G(s).$ (3.2)

######  Proof.

Taking $a=\inf\limits_{x\in\mathbb{R}^{3}}V(x)$,
$b_{i}=\sup\limits_{x\in\mathbb{R}^{3}}Q_{i}(x)(1\leq i\leq m)$, and
$b_{m}+1=\sup\limits_{x\in\mathbb{R}^{3}}K(x)$ in (2.17) and using Lemma 2.7,
we can obtain that $c_{\varepsilon}\geq c_{ab}>0$. We only need to show (3.2).
It follows from $c_{0}<c_{\infty}$ and Lemma 3.1 that there exists
$s_{0}\in\mathbb{R}^{3}$ such that $G(s_{0})=c_{0}$. Furthermore, by Lemma
2.6, there exists $u_{s_{0}}\in\mathcal{N}^{s_{0}}$, i.e.,

$\displaystyle||u_{s_{0}}||^{2}_{\mathcal{D}^{1,2}}+\int_{\mathbb{R}^{3}}V(s_{0})|u_{s_{0}}|^{2}dx+\int_{\mathbb{R}^{3}}\phi_{u_{s_{0}}}u_{s_{0}}^{2}dx$
(3.3) $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}\int_{\mathbb{R}^{3}}Q_{i}(s_{0})|u_{s_{0}}|^{q_{i}}dx+\int_{\mathbb{R}^{3}}K(s_{0})|u_{s_{0}}|^{6}dx,$

such that $I^{s_{0}}(u_{s_{0}})=c_{0}$. Set
$\omega_{\varepsilon}(x)=u_{s_{0}}\left(x-\frac{s_{0}}{\varepsilon}\right)$,
from Lemma 2.3(i) and (ii), we know that there exists a unique bounded
$t_{\varepsilon}>0$ such that
$t_{\varepsilon}\omega_{\varepsilon}\in\mathcal{N}_{\varepsilon}$, i.e.,

$t_{\varepsilon}||\omega_{\varepsilon}||^{2}_{\varepsilon}+t_{\varepsilon}^{3}\int_{\mathbb{R}^{3}}\phi_{\omega_{\varepsilon}}\omega_{\varepsilon}^{2}dx=\sum\limits_{i=1}^{m}t_{\varepsilon}^{q_{i}-1}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|\omega_{\varepsilon}|^{q_{i}}dx+t_{\varepsilon}^{6}\int_{\mathbb{R}^{3}}K(\varepsilon
x)|\omega_{\varepsilon}|^{6}dx,$

and so

$\displaystyle
t_{\varepsilon}||u_{s_{0}}||^{2}_{\mathcal{D}^{1,2}}+t_{\varepsilon}\int_{\mathbb{R}^{3}}V(\varepsilon
x+s_{0})|u_{s_{0}}|^{2}dx+t_{\varepsilon}^{3}\int_{\mathbb{R}^{3}}\phi_{u_{s_{0}}}u_{s_{0}}^{2}dx$
(3.4) $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}t_{\varepsilon}^{q_{i}-1}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x+s_{0})|u_{s_{0}}|^{q_{i}}dx+t_{\varepsilon}^{6}\int_{\mathbb{R}^{3}}K(\varepsilon
x+s_{0})|u_{s_{0}}|^{6}dx.$

Since $V(x)$ is bounded, then there exists a constant $M$ such that
$|V(\varepsilon x+s_{0})-V(s_{0})|\leq M$. From (1.7), we have $u_{s_{0}}\in
L^{p}(\mathbb{R}^{3})(2\leq p\leq 6)$, then for any $\sigma>0$, there exists
$R>0$ such that $|u_{s_{0}}|_{2,B_{R}^{C}(0)}<\frac{\sigma}{2M}$. Thus, we
have

$\int_{\mathbb{R}^{3}\backslash B_{R}(0)}(V(\varepsilon
x+s_{0})-V(s_{0})|u_{s_{0}}|^{2}dx<\frac{\sigma}{2},$

and obviously we observe that

$\int_{B_{R}(0)}(V(\varepsilon
x+s_{0})-V(s_{0})|u_{s_{0}}|^{2}dx<\frac{\sigma}{2},$

as $\varepsilon\to 0$. Which implies that

$\int_{\mathbb{R}^{3}}V(\varepsilon
x+s_{0})|u_{s_{0}}|^{2}dx\to\int_{\mathbb{R}^{3}}V(s_{0})|u_{s_{0}}|^{2}dx,$
(3.5)

as $\varepsilon\to 0$. Similarly, we can obtain that

$\int_{\mathbb{R}^{3}}K(\varepsilon
x+s_{0})|u_{s_{0}}|^{6}dx\to\int_{\mathbb{R}^{3}}K(s_{0})|u_{s_{0}}|^{6}dx,$
(3.6)

and

$\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x+s_{0})|u_{s_{0}}|^{2}dx\to\int_{\mathbb{R}^{3}}Q_{i}(s_{0})|u_{s_{0}})|^{2}dx,(1\leq
i\leq m),$ (3.7)

as $\varepsilon\to 0$. From (3.3)–(3.7), we have $t_{\varepsilon}\to 1$, as
$\varepsilon\to 0$. Now observe that

$\displaystyle c_{\varepsilon}=$
$\displaystyle\inf\limits_{u\in\mathcal{N}_{\varepsilon}}I_{\varepsilon}(u)$
(3.8) $\displaystyle\leq$ $\displaystyle
I_{\varepsilon}(t_{\varepsilon}\omega_{\varepsilon})$ $\displaystyle=$
$\displaystyle
I^{s_{0}}(t_{\varepsilon}\omega_{\varepsilon})+\frac{t_{\varepsilon}^{2}}{2}\int_{\mathbb{R}^{3}}(V(\varepsilon
x)-V(s_{0}))|\omega_{\varepsilon}|^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{m}\frac{t_{\varepsilon}^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}(Q_{i}(\varepsilon
x)-Q_{i}(s_{0}))|\omega_{\varepsilon}|^{q_{i}}dx-\frac{t_{\varepsilon}^{6}}{6}\int_{\mathbb{R}^{3}}(K(\varepsilon
x)-K(s_{0})|\omega_{\varepsilon}|^{6}dx$ $\displaystyle=$ $\displaystyle
I^{s_{0}}(t_{\varepsilon}u_{s_{0}})+\frac{t_{\varepsilon}^{2}}{2}\int_{\mathbb{R}^{3}}(V(\varepsilon
x+s_{0})-V(s_{0}))|u_{s_{0}}|^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{m}\frac{t_{\varepsilon}^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}(Q_{i}(\varepsilon
x+s_{0})-Q_{i}(s_{0}))|u_{s_{0}}|^{q_{i}}dx$
$\displaystyle-\frac{t_{\varepsilon}^{6}}{6}\int_{\mathbb{R}^{3}}(K(\varepsilon
x+s_{0})-K(s_{0})|u_{s_{0}}|^{6}dx.$

The property of $t_{\varepsilon}$ discussed above implies that
$I^{s_{0}}(t_{\varepsilon}u_{s_{0}})\to I^{s_{0}}(u_{s_{0}})=c_{0}$ as
$\varepsilon\to 0$. Now the desired conclusion is obtained. $\Box$

For any $\sigma>0$, we consider the function $u_{\sigma,x_{0}}$ defined by

$u_{\sigma,x_{0}}(x)=\frac{(3\sigma^{2})^{\frac{1}{4}}}{(\sigma^{2}+|x-x_{0}|^{2})^{\frac{1}{2}}},$

which is a solution of the critical problem $-\Delta u=u^{5}$ in
$\mathbb{R}^{3}$(see [32, 6]), and $x_{0}$ is given in condition $(f_{4})$.
Let $U_{\sigma,x_{0}}=\xi(x-x_{0})u_{\sigma,x_{0}}$, where $\xi(x)\in
C^{\infty}_{0}(B_{2R}(0),[0,1])$ satisfies $\xi(x)\equiv 1$ on $B_{R}(0)$,
where $R$ is a positive constant. As in [27, 6], the following estimations
hold:

$\int_{\mathbb{R}^{3}}|\nabla
U_{\sigma,x_{0}}|^{2}dx=S^{\frac{3}{2}}+O(\sigma)\hskip
4.30554pt\text{and}\hskip
4.30554pt\int_{\mathbb{R}^{3}}|U_{\sigma,x_{0}}|^{6}dx=S^{\frac{3}{2}}+O(\sigma^{3}),$
(3.9)

and for any $t\in[2,6)$,

$|U_{\sigma,x_{0}}|^{t}_{t}=\begin{cases}O(\sigma^{\frac{t}{2}}),&t\in[2,3),\\\
O(\sigma^{\frac{3}{2}}|ln\sigma|),&t=3,\\\
O(\sigma^{\frac{6-t}{2}}),&t\in(3,6).\end{cases}$ (3.10)

In the following lemma, we compare the minimum level of ground energy function
with a suitable number which involves the best Sobolev embedding constant $S$.

######  Lemma 3.4.

$c_{0}=\inf\limits_{s\in\mathbb{R}^{3}}G(s)<\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}$.

######  Proof.

Using the condition $(f_{4})$, there exists $x_{0}\in\mathbb{R}^{3}$ such that
$K(x_{0})=K^{\infty}$. We consider the following equation

$-\Delta
u+\phi_{u}u+V(x_{0})u=\sum\limits_{i=1}^{m}Q_{i}(x_{0})|u|^{q_{i}-2}u+K(x_{0})|u|^{4}u,\quad
x\in\mathbb{R}^{3},$

with the corresponding energy functional

$\displaystyle I^{x_{0}}(u)=$
$\displaystyle\frac{1}{2}||u||^{2}_{\mathcal{D}^{1,2}}+\frac{1}{2}\int_{\mathbb{R}^{3}}V(x_{0})u^{2}dx+\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx$
(3.11)
$\displaystyle-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(x_{0})|u|^{q_{i}}dx-\frac{1}{6}\int_{\mathbb{R}^{3}}K(x_{0})|u|^{6}dx.$

Moreover, by Lemma 2.6, we can obtain that there exists
$u_{x_{0}}\in\mathcal{N}^{x_{0}}$ such that $G(x_{0})=I^{x_{0}}(u_{x_{0}})$.
Taking account of Lemma 2.3 (i), (ii) and (2.15), we have
$G(x_{0})\leq\max\limits_{t\geq 0}I^{x_{0}}(tU_{\sigma,x_{0}}),$ and there
exists a unique bounded $t_{\sigma}>0$ such that
$I^{x_{0}}(t_{\sigma}U_{\sigma,x_{0}})=\max\limits_{t\geq
0}I^{x_{0}}(tU_{\sigma,x_{0}})$. By the Lemma 2.1 and (3.10), we can obtain
that

$\displaystyle I^{x_{0}}(t_{\sigma}U_{\sigma,x_{0}})=$
$\displaystyle\frac{t_{\sigma}^{2}}{2}||U_{\sigma,x_{0}}||^{2}_{\mathcal{D}^{1,2}}+\frac{t_{\sigma}^{2}}{2}\int_{\mathbb{R}^{3}}V(x_{0})U_{\sigma,x_{0}}^{2}dx+\frac{t_{\sigma}^{4}}{4}\int_{\mathbb{R}^{3}}\phi_{{}_{U_{\sigma,x_{0}}}}U_{\sigma,x_{0}}^{2}dx$
(3.12)
$\displaystyle-\sum\limits_{i=1}^{m}\frac{t_{\sigma}^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(x_{0})|U_{\sigma,x_{0}}|^{q_{i}}dx-\frac{t_{\sigma}^{6}}{6}\int_{\mathbb{R}^{3}}K(x_{0})|U_{\sigma,x_{0}}|^{6}dx$
$\displaystyle\leq$
$\displaystyle\frac{t_{\sigma}^{2}}{2}\int_{\mathbb{R}^{3}}\left(|\nabla
U_{\sigma,x_{0}}|^{2}+V(x_{0})U_{\sigma,x_{0}}^{2}\right)dx+\frac{t_{\sigma}^{4}}{4S}\left(\int_{\mathbb{R}^{3}}|U_{\sigma,x_{0}}|^{\frac{12}{5}}dx\right)^{\frac{5}{3}}$
$\displaystyle-\sum\limits_{i=1}^{m}\frac{t_{\sigma}^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(x_{0})|U_{\sigma,x_{0}}|^{q_{i}}dx-\frac{t_{\sigma}^{6}}{6}\int_{\mathbb{R}^{3}}K(x_{0})|U_{\sigma,x_{0}}|^{6}dx$
$\displaystyle\leq$
$\displaystyle\sup\limits_{t_{\sigma}>0}\left\\{\frac{t_{\sigma}^{2}}{2}\int_{\mathbb{R}^{3}}|\nabla
U_{\sigma,x_{0}}|^{2}dx-\frac{t_{\sigma}^{6}}{6}\int_{\mathbb{R}^{3}}K(x_{0})|U_{\sigma,x_{0}}|^{6}dx\right\\}$
$\displaystyle-\sum\limits_{i=1}^{m}\frac{t_{\sigma}^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(x_{0})|U_{\sigma,x_{0}}|^{q_{i}}dx+CO(\sigma)+CS^{-1}O(\sigma^{2}).$

Using the condition $(f_{2})$ and $4<q_{1}<q_{2}<\cdots<q_{m}<6$, one has

$\displaystyle I^{x_{0}}(t_{\sigma}U_{\sigma,x_{0}})\leq$
$\displaystyle\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}+O(\sigma)+CO(\sigma)+CS^{-1}O(\sigma^{2})$
(3.13)
$\displaystyle+\sum\limits_{i=1}^{i_{0}}C_{i}O(\sigma^{\frac{6-q_{i}}{2}})-\sum\limits_{i=i_{0}+1}^{m}C_{i}O(\sigma^{\frac{6-q_{i}}{2}})$
$\displaystyle<$
$\displaystyle\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}},$

as $\sigma>0$ small enough. Consequently, we deduce from the definition of
$c_{0}$ that

$c_{0}\leq G(x_{0})\leq
I^{x_{0}}(t_{\sigma}U_{\sigma,x_{0}})<\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}.$

The desired conclusion is obtained. $\Box$

######  Remark 3.1.

By Lemma 3.3 and Lemma 3.4, we see that
$c_{\varepsilon}<\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}$ for
$\varepsilon>0$ small enough. Moreover, it follows from Lemma 3.3 and (1.5)
that $c_{\varepsilon}<c_{\infty}$.

######  Lemma 3.5.

If $\\{u_{n}\\}\subset H_{\varepsilon}$ be a $(PS)_{c}$ sequence for
$I_{\varepsilon}$, then $\\{u_{n}\\}$ is bounded in $H_{\varepsilon}$.

######  Proof.

Let $\\{u_{n}\\}\subset H_{\varepsilon}$ be a $(PS)_{c}$ sequence for
$I_{\varepsilon}$, for n large enough, we have

$\displaystyle c+1+||u_{n}||_{\varepsilon}\geq$ $\displaystyle\hskip
3.99994ptI_{\varepsilon}(u_{n})-\frac{1}{q_{i_{0}}}\langle
I_{\varepsilon}^{\prime}(u_{n}),u_{n}\rangle$ (3.14) $\displaystyle=$
$\displaystyle\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)||u_{n}||^{2}_{\varepsilon}+\left(\frac{1}{4}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{i_{0}-1}\left(\frac{1}{q_{i}}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u_{n}|^{q_{i}}dx$
$\displaystyle+\sum\limits_{i=i_{0}+1}^{m}\left(\frac{1}{q_{i_{0}}}-\frac{1}{q_{i}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u_{n}|^{q_{i}}dx$
$\displaystyle+\left({\frac{1}{q_{i_{0}}}-\frac{1}{6}}\right)\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u_{n}|^{6}dx$ $\displaystyle\geq$
$\displaystyle\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)||u_{n}||^{2}_{\varepsilon}.$

It follows that $\\{u_{n}\\}$ is bounded in $H_{\varepsilon}$. $\Box$

######  Lemma 3.6.

Let $\\{u_{n}\\}\subset H_{\varepsilon}$ be a $(PS)_{c}$ sequence for
$I_{\varepsilon}$ with
$0<c<\min\left\\{c_{\infty},\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}\right\\}$
and such that $u_{n}\rightharpoonup 0$, as $n\to\infty$. Then, one of the
following conclusions holds.
(i) $u_{n}\to 0$ in $H_{\varepsilon}$, as $n\to\infty$; (ii) there exists a
sequence $\\{y_{n}\\}\subset\mathbb{R}^{3}$ and constants $r,\beta>0$ such
that

$\liminf\limits_{n\to\infty}\int_{B_{r}(y_{n})}u_{n}^{2}dx\geq\beta>0.$

######  Proof.

Suppose that (ii) does not occur, i.e., for any $r>0$, such that

$\limsup\limits_{n\to\infty}\int_{B_{r}(y)}u_{n}^{2}dx=0.$

By Lemma 3.2, we have $u_{n}\to 0$ in $L^{s}(\mathbb{R}^{3})$ for $2<s<6$, as
$n\to\infty$. Hence, we see that

$\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon x)|u_{n}|^{q_{i}}dx\to 0,$ (3.15)

as $n\to\infty$. Moreover, from Lemma 2.1 (iv), we have

$\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx\leq S^{-1}|u_{n}|^{4}_{12/5}\to
0,$ (3.16)

thus, we have $\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx\to 0$, as
$n\to\infty$. Recalling that $\langle
I_{\varepsilon}^{\prime}(u_{n}),u_{n}\rangle=o_{n}(1)$ , we have

$||u_{n}||^{2}_{\varepsilon}=\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u_{n}|^{6}dx+o_{n}(1).$

It follows from Lemma 3.5 that $\\{u_{n}\\}$ is bounded in $H_{\varepsilon}$,
up to a subsequence, we may assume that there exists $l\geq 0$ such that

$||u_{n}||^{2}_{\varepsilon}\to l,\quad\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u_{n}|^{6}dx\to l,$ (3.17)

as $n\to\infty$. If $l=0$, the proof is complete. If $l>0$, by using
(3.15)–(3.17) and $I_{\varepsilon}(u_{n})=c+o_{n}(1)$, we get

$\displaystyle c+o_{n}(1)=$
$\displaystyle\frac{1}{2}||u_{n}||^{2}_{\varepsilon}+\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx$
(3.18)
$\displaystyle-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u_{n}|^{q_{i}}dx-\frac{1}{6}\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u_{n}|^{6}dx$ $\displaystyle=$
$\displaystyle\frac{1}{2}||u_{n}||^{2}_{\varepsilon}-\frac{1}{6}\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u_{n}|^{6}dx+o_{n}(1)$ $\displaystyle=$
$\displaystyle\frac{1}{3}l+o_{n}(1).$

By the Sobolev inequality and $(f_{4})$, we have that

$||u_{n}||^{2}_{\varepsilon}\geq\int_{\mathbb{R}^{3}}|\nabla u_{n}|^{2}dx\geq
S\left(\int_{\mathbb{R}^{3}}|u_{n}|^{6}dx\right)^{\frac{1}{3}}\geq
S|K|^{-\frac{1}{3}}_{\infty}\left(\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u_{n}|^{6}dx\right)^{\frac{1}{3}}.$

Taking the limit as $n\to\infty$ at the last inequality, we obtain

$l\geq S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}.$ (3.19)

From (3.18) and (3.19), we get a contradiction to the definition of $c$.
Therefore, $l=0$ and the desired conclusion is obtained. $\Box$

######  Lemma 3.7.

Let $\\{u_{n}\\}\subset H_{\varepsilon}$ be a $(PS)_{c}$ sequence for
$I_{\varepsilon}$ with
$0<c<\min\left\\{c_{\infty},\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}\right\\}$
and $u_{n}\rightharpoonup 0$ in $H_{\varepsilon}$, then $u_{n}\to 0$ in
$H_{\varepsilon}.$

######  Proof.

Assume by contradiction that $u_{n}\nrightarrow 0$ in $H_{\varepsilon}$. From
Lemma 2.3 (i) and (ii), there exists a positive bounded sequence $\\{t_{n}\\}$
such that $\\{t_{n}u_{n}\\}\subset\mathcal{N}_{\infty}$. Then we claim that
$\limsup\limits_{n\to\infty}t_{n}\leq 1$. Arguing indirectly, there exists
$\delta>$ and a subsequence still denoted by $\\{t_{n}\\}$, such that
$t_{n}\geq 1+\delta$, for all $n\in\mathbb{N}$. Since $\langle
I_{\varepsilon}^{\prime}(u_{n}),u_{n}\rangle=o_{n}(1)$, we get

$\displaystyle||u_{n}||^{2}_{\mathcal{D}^{1,2}}+\int_{\mathbb{R}^{3}}V(\varepsilon
x)|u_{n}|^{2}dx+\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx$ (3.20)
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u_{n}|^{q_{i}}dx+\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u_{n}|^{6}dx+o_{n}(1).$

From $t_{n}u_{n}\in\mathcal{N}_{\infty}$, we have

$\displaystyle
t_{n}^{2}||u_{n}||^{2}_{\mathcal{D}^{1,2}}+t_{n}^{2}\int_{\mathbb{R}^{3}}V_{\infty}|u_{n}|^{2}dx+t_{n}^{4}\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx$
(3.21) $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}t_{n}^{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}^{\infty}|u_{n}|^{q_{i}}dx+t_{n}^{6}\int_{\mathbb{R}^{3}}K^{\infty}|u_{n}|^{6}dx.$

It follows from (3.20) and (3.21) that

$\displaystyle
o_{n}(1)+\left(t_{n}^{2-q_{i_{0}}}-1\right)||u_{n}||^{2}_{\mathcal{D}^{1,2}}+\int_{\mathbb{R}^{3}}\left(t_{n}^{2-q_{i_{0}}}V_{\infty}-V(\varepsilon
x)\right)|u_{n}|^{2}dx$ (3.22)
$\displaystyle+\left(t_{n}^{4-q_{i_{0}}}-1\right)\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx-\sum\limits_{i=1}^{i_{0}-1}\int_{\mathbb{R}^{3}}\left(t_{n}^{q_{i}-q_{i_{0}}}Q_{i}^{\infty}-Q_{i}(\varepsilon
x)\right)|u_{n}|^{q_{i}}dx$ $\displaystyle=$
$\displaystyle\sum\limits_{i=i_{0}}^{m}\int_{\mathbb{R}^{3}}\left(t_{n}^{q_{i}-q_{i_{0}}}Q_{i}^{\infty}-Q_{i}(\varepsilon
x)\right)|u_{n}|^{q_{i}}dx+\int_{\mathbb{R}^{3}}\left(t_{n}^{6-q_{i_{0}}}K^{\infty}-K(\varepsilon
x)\right)|u_{n}|^{6}dx.$

By using the condition $(f_{2})$, $4<q_{1}<q_{2}<\cdots<q_{m}<6$, $t_{n}>1$,
and the definition of $V_{\infty}$ and $Q_{i}^{\infty}$, for any $\sigma>0$,
there exists $R=R(\sigma)>0$ such that

$V(\varepsilon x)\geq V_{\infty}-\sigma>t_{n}^{2-q_{i_{0}}}V_{\infty}-\sigma$
(3.23)

and

$t_{n}^{q_{i}-q_{i_{0}}}Q_{i}^{\infty}+\sigma>Q_{i}^{\infty}+\sigma\geq
Q_{i}(\varepsilon x),\quad 1\leq i\leq i_{0}-1$ (3.24)

and

$t_{n}^{q_{i}-q_{i_{0}}}Q_{i}^{\infty}+\sigma>Q_{i}^{\infty}+\sigma\geq
Q_{i}(\varepsilon x),\quad i_{0}\leq i\leq n,$ (3.25)

for any $|\varepsilon x|\geq R$. Moreover, it follows from $(f_{4})$ that

$t_{n}^{6-q_{i_{0}}}K^{\infty}\geq K^{\infty}>K(\varepsilon x),$ (3.26)

for any $x\in\mathbb{R}^{3}$. Since $u_{n}\rightharpoonup 0$ in
$H_{\varepsilon}$, we get

$u_{n}\to 0\hskip 4.30554pt\text{in}\hskip
4.30554ptL^{q}_{loc}(\mathbb{R}^{3}),\hskip 4.30554ptq\in[2,6).$ (3.27)

Thus, note that $\\{u_{n}\\}$ is bounded in $L^{p}(\mathbb{R}^{3})(2\leq p\leq
6)$, we deduce from (3.22)–(3.27) that

$\int_{\mathbb{R}^{3}}|u_{n}|^{6}dx<C\sigma.$ (3.28)

On the other hand, Lemma 3.6 shows that there exists a sequence
$\\{y_{n}\\}\subset\mathbb{R}^{3}$ and constants $r,\beta>0$ such that

$\liminf\limits_{n\to\infty}\int_{B_{r}(y_{n})}u_{n}^{2}dx\geq\beta>0.$ (3.29)

If we set $v_{n}(x)=u_{n}(x+y_{n})$, then there exists a non-zero function
$v(x)$ such that, up to a subsequence, $v_{n}\rightharpoonup v$ in
$H_{\varepsilon}$, $v_{n}\to v$ in $L_{loc}^{q}(\mathbb{R}^{3})$, $q\in[2,6)$,
and $v_{n}\to v$ a.e. in $\mathbb{R}^{3}$. Moreover, by (3.29), we have that
there exists a subset $\Lambda\subset B_{r}(0)$ with positive measure such
that $v\neq 0$ a.e. in $\Lambda$. It follows from Fatou’s lemma and Sobolev
inequality that

$\int_{\mathbb{R}^{3}}|u_{n}|^{6}dx=\int_{\mathbb{R}^{3}}|v_{n}|^{6}dx\geq\int_{\Lambda}|v|^{6}dx\geq
C\int_{\Lambda}|v_{n}|^{2}dx>\beta_{0}>0,$

for some $\beta_{0}>0$, which contradicts to (3.28) and the claim is true.

We next divide the proof into two separate cases.

Case 1: $\limsup\limits_{n\to\infty}t_{n}=1.$ In this case, there exists a
subsequence, still denote by $\\{t_{n}\\}$, such that $t_{n}\to 1$ as
$n\to\infty$. Observe that

$\displaystyle I_{\infty}(t_{n}u_{n})-I_{\varepsilon}(u_{n})$ (3.30)
$\displaystyle=$
$\displaystyle\frac{t_{n}^{2}-1}{2}||u_{n}||^{2}_{\mathcal{D}^{1,2}}+\frac{1}{2}\int_{\mathbb{R}^{3}}\left(t_{n}^{2}V_{\infty}-V(\varepsilon
x)\right)u_{n}^{2}dx+\frac{t_{n}^{4}-1}{4}\int_{\mathbb{R}^{3}}\phi_{u_{n}}u_{n}^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}\int_{\mathbb{R}^{3}}\left(t_{n}^{q_{i}}Q_{i}^{\infty}-Q_{i}(\varepsilon
x)\right)|u_{n}|^{q_{i}}dx-\frac{1}{6}\int_{\mathbb{R}^{3}}\left(t_{n}^{6}K^{\infty}-K(\varepsilon
x)\right)|u_{n}|^{6}dx.$

Arguing as in the proof of the claim above, fixed $n$ large enough , for any
$\sigma>0$, there exists $R=R(\sigma)>0$ such that

$V(\varepsilon x)\geq t_{n}^{2}V_{\infty}-\sigma,\hskip
4.30554ptt_{n}^{q_{i}}Q_{i}^{\infty}+\sigma\geq Q_{i}(\varepsilon x)(1\leq
i\leq n)$ (3.31)

for any $|\varepsilon x|\geq R$, and

$t_{n}^{6}K^{\infty}\geq K(\varepsilon x),$ (3.32)

for any $x\in\mathbb{R}^{3}$. Thus, we deduce from(3.27), (3.31) and (3.32)
that

$I_{\infty}(t_{n}u_{n})-I_{\varepsilon}(u_{n})\leq o_{n}(1)+C\sigma.$

Taking limit in the above inequality, we have

$\displaystyle c+o_{n}(1)=I_{\varepsilon}(u_{n})\geq$ $\displaystyle
I_{\varepsilon}(u_{n})+c_{\infty}-I_{\infty}(t_{n}u_{n})$ $\displaystyle\geq$
$\displaystyle c_{\infty}+o_{n}(1),$

which is a contradiction.

Case 2: $\limsup\limits_{n\to\infty}t_{n}<1.$ In this case, we suppose that,
without loss of generality, $t_{n}<1$ for all $n\in\mathbb{N}$. From
(3.23)–(3.27) and $\\{u_{n}\\}$ is bounded in $L^{p}(\mathbb{R}^{3})(2\leq
p\leq 6)$, for any $\sigma>0$, we have

$\int_{\mathbb{R}^{3}}\left(V_{\infty}-V(\varepsilon x)\right)u_{n}^{2}dx\leq
C\sigma+o_{n}(1),$ (3.33)
$\int_{\mathbb{R}^{3}}\left(Q_{i}^{\infty}-Q_{i}(\varepsilon
x)\right)|u_{n}|^{q_{i}}dx\geq-C\sigma+o_{n}(1),$ (3.34)
$\int_{\mathbb{R}^{3}}\left(K^{\infty}-K(\varepsilon
x)\right)|u_{n}|^{6}dx\geq-C\sigma+o_{n}(1).$ (3.35)

Then, (3.33)–(3.35) imply that

$\displaystyle c_{\infty}\leq$ $\displaystyle I_{\infty}(t_{n}u_{n})$
$\displaystyle=$ $\displaystyle
I_{\varepsilon}(t_{n}u_{n})+\frac{t_{n}^{2}}{2}\int_{\mathbb{R}^{3}}\left(V_{\infty}-V(\varepsilon
x)\right)u_{n}^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{m}\frac{t_{n}^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}\left(Q_{i}^{\infty}-Q_{i}(\varepsilon
x)\right)|u_{n}|^{q_{i}}dx-\frac{t_{n}^{6}}{6}\int_{\mathbb{R}^{3}}\left(K^{\infty}-K(\varepsilon
x)\right)|u_{n}|^{6}dx$ $\displaystyle\leq$ $\displaystyle
I_{\varepsilon}(t_{n}u_{n})+C\sigma+o_{n}(1)$ $\displaystyle\leq$
$\displaystyle I_{\varepsilon}(u_{n})+C\sigma+o_{n}(1)$ $\displaystyle=$
$\displaystyle c+C\sigma+o_{n}(1).$

Let $\sigma\to 0$ and $n\to\infty$, we have $c_{\infty}\leq c$, which yields a
contradiction ends the proof. $\Box$

######  Lemma 3.8.

$I_{\varepsilon}$ satisfies the $(PS)_{c}$ conditions in $H_{\varepsilon}$
with
$0<c<\min\left\\{c_{\infty},\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}\right\\}$.

######  Proof.

Let $\\{u_{n}\\}\subset H_{\varepsilon}$ be a $(PS)_{\varepsilon}$ for
$I_{\varepsilon}$ with
$0<c<\min\left\\{c_{\infty},\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}\right\\}$.
By Lemma 3.5, we get that $\\{u_{n}\\}$ is bounded in $H_{\varepsilon}$. Then,
there exists $u\in H_{\varepsilon}$ such that $u_{n}\rightharpoonup u$ in
$H_{\varepsilon}$. By a standard argument, we get
$I_{\varepsilon}^{\prime}(u_{n})\to I_{\varepsilon}^{\prime}(u)=0$, i.e., $u$
is a critical point of $I_{\varepsilon}$. Let $\omega_{n}=u_{n}-u$, by Lemma
2.2 and Brezis–Lieb Lemma, it is not difficult to see that

$I_{\varepsilon}(\omega_{n})=I_{\varepsilon}(u_{n})-I_{\varepsilon}(u)+o_{n}(1)=c-I_{\varepsilon}(u)+o_{n}(1).$
(3.36)

Under our conditions, we see that

$\displaystyle I_{\varepsilon}(u)=$ $\displaystyle\hskip
3.99994ptI_{\varepsilon}(u)-\frac{1}{q_{i_{0}}}\langle
I_{\varepsilon}^{\prime}(u),u\rangle$ (3.37) $\displaystyle=$
$\displaystyle\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)||u||^{2}_{\varepsilon}+\left(\frac{1}{4}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\phi_{u}u^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{i_{0}-1}\left(\frac{1}{q_{i}}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx+\sum\limits_{i=i_{0}+1}^{m}\left(\frac{1}{q_{i_{0}}}-\frac{1}{q_{i}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx$
$\displaystyle+\left({\frac{1}{q_{i_{0}}}-\frac{1}{6}}\right)\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u|^{6}dx$ $\displaystyle\geq$
$\displaystyle\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)||u||^{2}_{\varepsilon}$
$\displaystyle\geq$ $\displaystyle 0.$

Then, (3.36) and (3.37) imply that

$I_{\varepsilon}(\omega_{n})\leq c.$

It follows from Lemma 3.7 that $\omega_{n}\to 0$ in $H_{\varepsilon}$, then
$u_{n}\to u$ in $H_{\varepsilon}$ and the proof is complete. $\Box$

######  Lemma 3.9.

Problem (2.8) has at least a positive ground state solution in
$H_{\varepsilon}$.

######  Proof.

From Lemma 2.4, we have that $I_{\varepsilon}$ satisfies the mountain pass
geometry structure. Moreover, using a version of the mountain pass theorem
without $(PS)$ condition, there exists a $(PS)_{c_{\varepsilon}}$ sequence
$\\{u_{n}\\}\subset H_{\varepsilon}$ for $I_{\varepsilon}$. Lemma 3.3 and
Lemma 3.4 imply that
$0<c_{\varepsilon}<\min\left\\{c_{\infty},\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}\right\\}$
for $\varepsilon>0$ small enough. Therefore, with the aim of Lemma 3.8, we
conclude that there exists $u_{\varepsilon}\in H_{\varepsilon}$ such that

$I_{\varepsilon}(u_{\varepsilon})=c_{\varepsilon},\hskip
4.30554pt\text{and}\hskip
4.30554ptI_{\varepsilon}^{\prime}(u_{\varepsilon})=0.$

It follows from (2.15) that $u_{\varepsilon}$ is a ground state solution of
(2.8). If we denote $u_{\varepsilon}^{\pm}=\max\\{\pm u_{\varepsilon},0\\}$,
and replace $I_{\varepsilon}$ by the following functional

$I^{+}_{\varepsilon}(u_{\varepsilon})=\frac{1}{2}||u_{\varepsilon}||^{2}_{\varepsilon}+\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_{u_{\varepsilon}}u_{\varepsilon}^{2}dx-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u_{\varepsilon}^{+}|^{q_{i}}dx-\frac{1}{6}\int_{\mathbb{R}^{3}}K(\varepsilon
x)|u_{\varepsilon}^{+}|^{6}dx.$

Repeating the above proof and calculations, we have

$0=\langle(I^{+})^{\prime}_{\varepsilon}(u_{\varepsilon}),u_{\varepsilon}^{-}\rangle=||u_{\varepsilon}^{-}||^{2}_{\varepsilon}+\int_{\mathbb{R}^{3}}\phi_{u_{\varepsilon}^{-}}(u_{\varepsilon}^{-})^{2}dx\geq||u_{\varepsilon}^{-}||^{2}_{\varepsilon},$

which implies $u_{\varepsilon}\geq 0$ in $\mathbb{R}^{3}$. The strong maximum
principle implies that $u_{\varepsilon}(x)>0$ for all $x\in\mathbb{R}^{3}$.
The proof is complete. $\Box$

##  4 Concentration of positive ground state solutions

In this section, in order to discuss the concentration behavior of ground
state solutions $v_{\varepsilon}$ of ($S_{\varepsilon}$) with $\varepsilon\to
0$, we will consider the family
$u_{\varepsilon}(x)=v_{\varepsilon}(\varepsilon x)$, which is a family of
positive ground state solutions of (2.8).

######  Lemma 4.1.

There exist $\varepsilon_{*}>0$, $\\{y_{\varepsilon}\\}\subset\mathbb{R}^{3}$
and $r$, $\beta>0$, such that

$\int_{B_{r}(y_{\varepsilon})}|u_{\varepsilon}|^{2}dx\geq\beta,\hskip
4.30554pt\text{for all}\hskip 4.30554pt\varepsilon\in(0,\varepsilon_{*}).$

######  Proof.

Arguing by contradiction, suppose that the lemma does not hold. Then, there
exists a sequence $\varepsilon_{n}\to 0$, such that for all $r>0$,

$\lim\limits_{n\to\infty}\sup\limits_{y\in\mathbb{R}^{3}}\int_{B_{r}(y)}|u_{\varepsilon_{n}}|^{2}dx=0.$

Then, repeating the arguments employed in the proof of Lemma 3.6, we can
obtain that

$c_{\varepsilon_{n}}\geq\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}},$

which is a contradiction with
$c_{\varepsilon}<\frac{1}{3}S^{\frac{3}{2}}|K|_{\infty}^{-\frac{1}{2}}.$ The
proof is complete. $\Box$

######  Lemma 4.2.

The family $\\{\varepsilon y_{\varepsilon}\\}$ is bounded as $\varepsilon\to
0$. Moreover, assume that $\varepsilon_{n}y_{\varepsilon_{n}}\to x_{0}$ under
a choice of subsequence, then we have $x_{0}\in\mathcal{G}$, i.e.,

$G(x_{0})=\inf\limits_{s\in\mathbb{R}^{3}}G(s).$

######  Proof.

For the sake of simplicity, we denote $y_{n}=y_{\varepsilon_{n}}$,
$u_{n}(x)=u_{\varepsilon_{n}}(x)$. Arguing by contradiction, suppose that
there is a sequence $\varepsilon_{n}\to 0$ such that
$|\varepsilon_{n}y_{n}|\to\infty$, as $n\to\infty$. Set
$\widetilde{u}_{n}(x)=u_{n}(x+y_{n})$, where
$\widetilde{u}_{n}(x)=\widetilde{u}_{\varepsilon_{n}}(x)$. It follows from
Lemma 4.1 that

$\int_{B_{r}(0)}|\widetilde{u}_{n}|^{2}dx\geq\beta,\hskip 4.30554pt\text{for
all}\hskip 4.30554ptn\in\mathbb{N}.$ (4.1)

Then, $\widetilde{u}_{n}$ is a ground state solution of

$-\Delta\widetilde{u}_{n}+\phi_{\widetilde{u}_{n}}\widetilde{u}_{n}+V(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}=\sum\limits_{i=1}^{m}Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{q_{i}-2}\widetilde{u}_{n}+K(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{4}\widetilde{u}_{n},$

and $||\widetilde{u}_{n}||_{\varepsilon_{n}}=||u_{n}||_{\varepsilon_{n}}$ is
bounded in $\mathbb{R}$. Moreover, up to a subsequence, we may assume that
$\widetilde{u}_{n}\rightharpoonup\widetilde{u}$ in $H_{\varepsilon}$ with
$\widetilde{u}\neq 0$ and $\widetilde{u}\geq 0$ . It follows that

$\displaystyle\int_{\mathbb{R}^{3}}\nabla\widetilde{u}_{n}\nabla\varphi
dx+\int_{\mathbb{R}^{3}}V(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}\varphi
dx+\int_{\mathbb{R}^{3}}\phi_{\widetilde{u}_{n}}\widetilde{u}_{n}\varphi dx$
(4.2) $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{q_{i}-1}\varphi
dx+\int_{\mathbb{R}^{3}}K(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{5}\varphi
dx,$

for any $\varphi\in H^{1}(\mathbb{R}^{3})$. Without loss of generality, we may
asuume that $V(\varepsilon_{n}y_{n})\to V_{\infty}$ and
$Q_{i}(\varepsilon_{n}y_{n})\to Q_{i}^{\infty}$ $(1\leq i\leq m)$ and
$K(\varepsilon_{n}y_{n})\to K^{\infty}$, as $n\to\infty$. Under our
assumptions on $V$, $K$, and $Q_{i}(1\leq i\leq m)$, we get that $V$, $K$, and
$Q_{i}(1\leq i\leq m)$ are uniformly continuous, which implies that

$V(\varepsilon_{n}x+\varepsilon_{n}y_{n})\to V_{\infty}\quad\text{and}\quad
K(\varepsilon_{n}x+\varepsilon_{n}y_{n})\to K^{\infty},$ (4.3)

and

$Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})\to Q_{i}^{\infty}(1\leq i\leq
m),$ (4.4)

as $n\to\infty$ uniformly on bounded sets of $\mathbb{R}^{3}$. Using the weak
limit of $\widetilde{u}_{n}$ and (4.2)–(4.4), we get

$\displaystyle\int_{\mathbb{R}^{3}}\nabla\widetilde{u}\nabla\varphi
dx+\int_{\mathbb{R}^{3}}V_{\infty}\widetilde{u}\varphi
dx+\int_{\mathbb{R}^{3}}\phi_{\widetilde{u}}\widetilde{u}\varphi dx$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}\int_{\mathbb{R}^{3}}Q_{i}^{\infty}|\widetilde{u}|^{q_{i}-1}\varphi
dx+\int_{\mathbb{R}^{3}}K^{\infty}|\widetilde{u}|^{5}\varphi dx,$

for any $\varphi\in H^{1}(\mathbb{R}^{3})$. Which implies that
$\widetilde{u}\in\mathcal{N}_{\infty}$, i.e., $\langle
I_{\infty}^{\prime}(\widetilde{u}),\widetilde{u}\rangle=0.$ We deduce from
Lemma 3.3 and Fatou’s Lemma that

$\displaystyle c_{\infty}\leq$ $\displaystyle
I_{\infty}(\widetilde{u})-\frac{1}{q_{i_{0}}}\langle
I_{\infty}^{\prime}(\widetilde{u}),\widetilde{u}\rangle$ (4.5)
$\displaystyle=$
$\displaystyle\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\left(|\nabla\widetilde{u}|^{2}+V_{\infty}|\widetilde{u}|^{2}\right)dx+\left(\frac{1}{4}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\phi_{\widetilde{u}}\widetilde{u}^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{i_{0}-1}\left(\frac{1}{q_{i}}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}Q_{i}^{\infty}|\widetilde{u}|^{q_{i}}dx+\sum\limits_{i=i_{0}+1}^{m}\left(\frac{1}{q_{i_{0}}}-\frac{1}{q_{i}}\right)\int_{\mathbb{R}^{3}}Q_{i}^{\infty}|\widetilde{u}|^{q_{i}}dx$
$\displaystyle+\left({\frac{1}{q_{i_{0}}}-\frac{1}{6}}\right)\int_{\mathbb{R}^{3}}K^{\infty}|\widetilde{u}|^{6}dx$
$\displaystyle\leq$
$\displaystyle\liminf\limits_{n\to\infty}\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\left(|\nabla\widetilde{u}_{n}|^{2}+V(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{2}\right)dx+\left(\frac{1}{4}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\phi_{\widetilde{u}_{n}}\widetilde{u}_{n}^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{i_{0}-1}\left(\frac{1}{q_{i}}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{q_{i}}dx$
$\displaystyle+\sum\limits_{i=i_{0}+1}^{m}\left(\frac{1}{q_{i_{0}}}-\frac{1}{q_{i}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{q_{i}}dx$
$\displaystyle+\left({\frac{1}{q_{i_{0}}}-\frac{1}{6}}\right)\int_{\mathbb{R}^{3}}K(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{6}dx$
$\displaystyle\leq$
$\displaystyle\limsup\limits_{n\to\infty}\left(I_{\varepsilon_{n}}(\widetilde{u}_{n})-\frac{1}{q_{i_{0}}}\langle
I_{\varepsilon_{n}}^{\prime}(\widetilde{u}_{n}),\widetilde{u}_{n}\rangle\right)$
$\displaystyle=$
$\displaystyle\limsup\limits_{n\to\infty}c_{\varepsilon_{n}}\leq c_{0},$

which contradicts the fact that $c_{0}>c_{\infty}$. Hence, $\\{\varepsilon
y_{\varepsilon}\\}$ is bounded as $\varepsilon\to 0$. Note that if
$\varepsilon_{n}y_{n}\to x_{0}$ under a choice of subsequence, and $V$,
$Q_{i}(1\leq i\leq m)$, $K$ are uniformly continuous, we have

$V(\varepsilon_{n}x+\varepsilon_{n}y_{n})\to V(x_{0})\quad\text{and}\quad
K(\varepsilon_{n}x+\varepsilon_{n}y_{n})\to K(x_{0}),$ (4.6)

and

$Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})\to Q_{i}(x_{0})(1\leq i\leq m),$
(4.7)

as $n\to\infty$ uniformly on bounded sets of $\mathbb{R}^{3}$. Taking
$\varphi=\widetilde{u}$ and limit as $n\to\infty$ in the above Eq. (4.2), we
get

$\displaystyle||\widetilde{u}||^{2}_{\mathcal{D}^{1,2}}+\int_{\mathbb{R}^{3}}V(x_{0})|\widetilde{u}^{2}dx+\int_{\mathbb{R}^{3}}\phi_{\widetilde{u}}\widetilde{u}^{2}dx=\sum\limits_{i=1}^{m}\int_{\mathbb{R}^{3}}Q_{i}(x_{0})|\widetilde{u}|^{q_{i}}dx+\int_{\mathbb{R}^{3}}K(x_{0})|\widetilde{u}|^{6}dx,$

which implies that $\widetilde{u}\in\mathcal{N}^{x_{0}}$, i.e.,
$\langle(I^{x_{0}})^{\prime}(\widetilde{u}),\widetilde{u}\rangle=0.$ We deduce
from Lemma 3.3 and Fatou’s Lemma that

$\displaystyle c_{0}=$ $\displaystyle\inf\limits_{s\in\mathbb{R}^{3}}G(s)\leq
G(x_{0})$ (4.8) $\displaystyle\leq$ $\displaystyle
I^{x_{0}}(\widetilde{u})-\frac{1}{q_{i_{0}}}\langle(I^{x_{0}})^{\prime}(\widetilde{u}),\widetilde{u}\rangle$
$\displaystyle=$
$\displaystyle\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\left(|\nabla\widetilde{u}|^{2}+V(x_{0})|\widetilde{u}|^{2}\right)dx+\left(\frac{1}{4}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\phi_{\widetilde{u}}\widetilde{u}^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{i_{0}-1}\left(\frac{1}{q_{i}}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}Q_{i}(x_{0})|\widetilde{u}|^{q_{i}}dx+\sum\limits_{i=i_{0}+1}^{m}\left(\frac{1}{q_{i_{0}}}-\frac{1}{q_{i}}\right)\int_{\mathbb{R}^{3}}Q_{i}(x_{0})|\widetilde{u}|^{q_{i}}dx$
$\displaystyle+\left({\frac{1}{q_{i_{0}}}-\frac{1}{6}}\right)\int_{\mathbb{R}^{3}}K(x_{0})|\widetilde{u}|^{6}dx$
$\displaystyle\leq$
$\displaystyle\liminf\limits_{n\to\infty}\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\left(|\nabla\widetilde{u}_{n}|^{2}+V(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{2}\right)dx+\left(\frac{1}{4}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}\phi_{\widetilde{u}_{n}}\widetilde{u}_{n}^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{i_{0}-1}\left(\frac{1}{q_{i}}-\frac{1}{q_{i_{0}}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{q_{i}}dx$
$\displaystyle+\sum\limits_{i=i_{0}+1}^{m}\left(\frac{1}{q_{i_{0}}}-\frac{1}{q_{i}}\right)\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{q_{i}}dx$
$\displaystyle+\left({\frac{1}{q_{i_{0}}}-\frac{1}{6}}\right)\int_{\mathbb{R}^{3}}K(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{6}dx$
$\displaystyle\leq$
$\displaystyle\limsup\limits_{n\to\infty}\left(I_{\varepsilon_{n}}(\widetilde{u}_{n})-\frac{1}{q_{i_{0}}}\langle
I_{\varepsilon_{n}}^{\prime}(\widetilde{u}_{n}),\widetilde{u}_{n}\rangle\right)$
$\displaystyle=$ $\displaystyle\limsup\limits_{n\to\infty}c_{\varepsilon_{n}}$
$\displaystyle\leq$ $\displaystyle c_{0},$

it follows that $G(x_{0})=\inf\limits_{s\in\mathbb{R}^{3}}G(s)$. The proof is
complete. $\Box$

######  Lemma 4.3.

$\widetilde{u}_{n}\to\widetilde{u}$ in $H_{\varepsilon}$ as $n\to\infty$.
Furthermore, there exist $C>0$ and $\varepsilon_{*}>0$ such that
$|\widetilde{u}_{n}|_{\infty}\leq C$ and

$\lim\limits_{|x|\to\infty}\widetilde{u}_{\varepsilon}(x)=0\hskip
4.30554pt\text{uniformly on}\hskip
4.30554pt\varepsilon\in(0,\varepsilon_{*}).$

######  Proof.

Due to the Brezis–Lieb lemma and (4.6)–(4.7), we get

$I_{\varepsilon_{n}}(\widetilde{u}_{n}-\widetilde{u}_{)}=I_{\varepsilon_{n}}(\widetilde{u}_{n})-I^{x_{0}}(\widetilde{u})+o_{n}(1).$

Since $\varepsilon_{n}y_{\varepsilon_{n}}\to x_{0}$ and (4.8), we have

$\lim\limits_{n\to\infty}I_{\varepsilon_{n}}(\widetilde{u}_{n}-\widetilde{u})=0.$

Similarly, we also get

$\lim\limits_{n\to\infty}\langle
I_{\varepsilon_{n}}^{\prime}(\widetilde{u}_{n}-\widetilde{u}),\widetilde{u}_{n}-\widetilde{u}\rangle=0.$

Consequently,

$\left(\frac{1}{2}-\frac{1}{q_{i_{0}}}\right)||\widetilde{u}_{n}-\widetilde{u}||^{2}\leq\lim\limits_{n\to\infty}\left(I_{\varepsilon_{n}}(\widetilde{u}_{n}-\widetilde{u})-\frac{1}{q_{i_{0}}}\langle
I_{\varepsilon_{n}}^{\prime}(\widetilde{u}_{n}-\widetilde{u}),\widetilde{u}_{n}-\widetilde{u}\rangle\right)=0,$

which implies that $\widetilde{u}_{n}\to\widetilde{u}$ in $H_{\varepsilon}$ as
$n\to\infty$. From Lemma 4.2, we know that the sequence $\widetilde{u}_{n}$
satisfies

$-\Delta\widetilde{u}_{n}+\left(\phi_{\widetilde{u}_{n}}+V(\varepsilon_{n}x+\varepsilon_{n}y_{n})-K(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{4}\right)\widetilde{u}_{n}=\sum\limits_{i=1}^{m}Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{q_{i}-2}\widetilde{u}_{n}.$
(4.9)

By Lemma 2.1 (iv) and (4.1), we obtain that $0<\phi_{\widetilde{u}_{n}}<C$,
then $\phi_{\widetilde{u}_{n}}+V(\varepsilon_{n}x+\varepsilon_{n}y_{n})\in
L^{\infty}_{loc}(\mathbb{R}^{3})$. Moreover, since
$K(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{4}\in
L^{\frac{3}{2}}(\mathbb{R}^{3})$ and $4<q_{1}<q_{2}<\cdots<q_{m}<6$, using a
result in [11, Proposition 3.3] or [3], we have $\widetilde{u}_{n}\in
L^{t}(\mathbb{R}^{3})$ for all $t\geq 2$. Furthermore, $\widetilde{u}_{n}$
satisfies

$\displaystyle-\Delta\widetilde{u}_{n}$
$\displaystyle\leq-\Delta\widetilde{u}_{n}+\left(\phi_{\widetilde{u}_{n}}+V(\varepsilon_{n}x+\varepsilon_{n}y_{n})\right)\widetilde{u}_{n}$
(4.10)
$\displaystyle=g_{n}(x):=\sum\limits_{i=1}^{m}Q_{i}(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{q_{i}-2}\widetilde{u}_{n}+K(\varepsilon_{n}x+\varepsilon_{n}y_{n})|\widetilde{u}_{n}|^{4}\widetilde{u}_{n},$

where $g_{n}(x)\in L^{\frac{s}{2}}(\mathbb{R}^{3})$, for some $s>3$. Applying
a result of [14] or [11, Proposition 3.4], we can obtain that

$\sup\limits_{x\in B_{r}(y)}\widetilde{u}_{n}(x)\leq
C\left(|\widetilde{u}_{n}|_{L^{2}(B_{2r}(y))}+|\widetilde{u}_{n}|_{L^{\frac{s}{2}}(B_{2r}(y))}\right),\hskip
4.30554pt\text{for any}\hskip 4.30554pty\in\mathbb{R}^{3},$

which implies that $|\widetilde{u}_{n}|_{\infty}\leq C$ and

$\lim\limits_{|x|\to\infty}\widetilde{u}_{n}(x)=0\hskip
4.30554pt\text{uniformly on}\hskip 4.30554ptn\in\mathbb{N}.$

Consequently, there exists $\varepsilon_{*}>0$ such that

$\lim\limits_{|x|\to\infty}\widetilde{u}_{\varepsilon}(x)=0\hskip
4.30554pt\text{uniformly on}\hskip
4.30554pt\varepsilon\in(0,\varepsilon_{*}).$ (4.11)

The proof is complete. $\Box$

In order to see the exponential decay of solutions $u_{\varepsilon}$, it is
enough to show the following result about $\widetilde{u}_{\varepsilon}$.

######  Lemma 4.4.

There exist constants $C>0$ and $\mu>0$ such that

$\widetilde{u}_{\varepsilon}\leq Ce^{-\mu|x|}\hskip 4.30554pt\text{for
all}\hskip 4.30554ptx\in\mathbb{R}^{3}.$

######  Proof.

We borrow an idea from [11, Lemma 3.11]. It follows from (4.11) that there
exists $R>0$ such that

$\sum\limits_{i=1}^{m}Q_{i}(x_{0})|\widetilde{u}_{\varepsilon}|^{q_{i}-2}+K(x_{0})|\widetilde{u}_{\varepsilon}|^{4}\leq\frac{V(x_{0})}{2}\hskip
4.30554pt\text{for all}\hskip 4.30554pt|x|>R\hskip 4.30554pt\text{and}\hskip
4.30554pt\forall\varepsilon\in(0,\varepsilon_{*}).$ (4.12)

Set $\tau(x)=Ce^{-\mu|x|}$ such that $\mu^{2}<\frac{V(x_{0})}{2}$ and
$\tau(R)\geq\widetilde{u}_{\varepsilon}$, we can obtain that

$\nabla\tau(x)=\mu^{2}\tau(x).$ (4.13)

By Lemma 2.1 (iv) and (4.1), we can get

$\displaystyle-\Delta\widetilde{u}_{\varepsilon}+V(x_{0})\widetilde{u}_{\varepsilon}$
$\displaystyle<-\Delta\widetilde{u}_{\varepsilon}+V(x_{0})\widetilde{u}_{\varepsilon}+\phi_{\widetilde{u}_{\varepsilon}}\widetilde{u}_{\varepsilon}$
(4.14)
$\displaystyle=\sum\limits_{i=1}^{m}Q_{i}(x_{0})|\widetilde{u}_{\varepsilon}|^{q_{i}-2}\widetilde{u}_{\varepsilon}+K(x_{0})|\widetilde{u}_{\varepsilon}|^{4}\widetilde{u}_{\varepsilon}$
$\displaystyle\leq\frac{V(x_{0})}{2}\widetilde{u}_{\varepsilon}\hskip
4.30554pt\text{for}\hskip 4.30554pt|x|>R.$

Let $\tau_{\varepsilon}=\tau-\widetilde{u}_{\varepsilon}$, combining (4.13)
and (4.14), we have

$\begin{cases}-\Delta\tau_{\varepsilon}+\frac{V(x_{0})}{2}\tau_{\varepsilon}>0,&|x|>R,\\\
\tau_{\varepsilon}\geq 0,&|x|=R,\\\
\lim\limits_{|x|\to\infty}\tau_{\varepsilon}(x)=0.\end{cases}$ (4.15)

The strong maximum principle implies that $\tau_{\varepsilon}\geq 0$ in
$|x|\geq R$. It follows that

$\widetilde{u}_{\varepsilon}\leq Ce^{-\mu|x|}\hskip 4.30554pt\text{for
all}\hskip 4.30554pt|x|\geq R\hskip 4.30554pt\text{and}\hskip
4.30554pt\forall\varepsilon\in(0,\varepsilon_{*}).$

The proof is complete. $\Box$

##  5 Proof of Theorem 1.1–1.2

In this section, we will prove the existence, concentration and exponential
decay of $v_{\varepsilon}$ in Theorem 1.1–1.2.

###  5.1 Proof of Theorem 1.1

By Lemma 3.9, $u_{\varepsilon}$ is a positive ground state solution of (2.8).
Then, $v_{\varepsilon}(x)=u_{\varepsilon}(\frac{x}{\varepsilon})$ is a
positive ground state solution of ($S_{\varepsilon}$). Suppose $z_{n}$ denotes
a maximum point of $\widetilde{u}_{n}$, it follows from Lemma 4.3 that $z_{n}$
is a bounded sequence in $\mathbb{R}^{3}$. Then, there exists $R>0$, such that
$z_{n}\in B_{R}(0)$. Thus, the global maximum point of $u_{n}$ is
$z_{n}+y_{n}$. Using the boundedness of $\\{z_{n}\\}$ and Lemma 4.2, we have

$\lim\limits_{n\to\infty}\varepsilon_{n}(z_{n}+y_{n})=x_{0}.$

Then, note the relations of $\widetilde{u}_{\varepsilon}$, $u_{\varepsilon}$
and $v_{\varepsilon}$, we have

$v_{\varepsilon}(\varepsilon x+\varepsilon z_{\varepsilon}+\varepsilon
y_{\varepsilon})=u_{\varepsilon}(x+z_{\varepsilon}+y_{\varepsilon})=\widetilde{u}_{\varepsilon}(x+z_{\varepsilon}).$
(5.1)

Set $\eta_{\varepsilon}(x)=v_{\varepsilon}(\varepsilon x+x_{\varepsilon})$,
where $x_{\varepsilon}\to x_{0},$ as $\varepsilon\to 0$. It follows from Lemma
4.1 and (5.1) that $\eta\neq 0$. By the similar arguments to Lemma 4.2 and
Lemma 4.3, we can obtain that $\eta_{\varepsilon}\to\eta$ in $H_{\varepsilon}$
and $\eta$ is a ground state solution of

$\displaystyle-\Delta
u+\phi_{u}u+V(x_{0})u=\sum\limits_{i=1}^{m}Q_{i}(x_{0})|u|^{q_{i}-2}u+K(x_{0})|u|^{4}u,\hskip
4.30554ptx\in\mathbb{R}^{3}.$

From Lemma 4.4, we have

$v_{\varepsilon}(x)=u_{\varepsilon}(\frac{x}{\varepsilon})=\widetilde{u}_{\varepsilon}(\frac{x-\varepsilon
y_{\varepsilon}}{\varepsilon})\leq
Cexp\left({-\frac{\mu}{\varepsilon}|x-x_{\varepsilon}|}\right),$

for all $x\in\mathbb{R}^{3}.$ The proof is complete. $\Box$

###  5.2 Proof of Theorem 1.2

In this subsection, we will consider that the electronic potential $h(x)$
satisfies $(H_{1})$. By the similar strategy to Lemma 3.9, we can obtain that
problem ($S_{\varepsilon}$) has a positive ground state solution, still denote
by $v_{\varepsilon}$. Then, the following energy functional

$\displaystyle\mathcal{I}_{\varepsilon}(u)=$
$\displaystyle\frac{1}{2}||u||^{2}_{\varepsilon}+\frac{1}{4}\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*h(\varepsilon
x)u^{2}\right]h(\varepsilon x)u^{2}\rm{d}x$ (5.2)
$\displaystyle-\sum\limits_{i=1}^{m}\frac{1}{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon
x)|u|^{q_{i}}dx-\frac{1}{6}\int_{\mathbb{R}^{3}}K(\varepsilon x)|u|^{6}dx,$

has a critical point $u_{\varepsilon}$. Next, we want to show the following
claims.

######  Claim 1.

Suppose that $\varepsilon_{n}y_{n}\to\infty$, as $n\to\infty$, then

$\int_{R^{3}}\left[\frac{1}{|x|}*h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}^{2}\right]h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}\varphi
dx\to 0,$

for any $\varphi\in H^{1}(\mathbb{R}^{3})$, as $n\to\infty.$

######  Proof.

From assumption $(H_{1}),$ we know that $h(x)$ is bounded in $\mathbb{R}^{3}$.
Since $\\{\widetilde{u}_{n}\\}$ is bounded in $H_{\varepsilon}$, by Hölder
inequality, we have

$\displaystyle\left|\frac{1}{|x|}*h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}^{2}\right|$
$\displaystyle=\left|\int_{\mathbb{R}^{3}}h(\varepsilon_{n}y+\varepsilon_{n}y_{n})\frac{\widetilde{u}_{n}^{2}(y)}{|x-y|}dy\right|$
(5.3) $\displaystyle\leq
C_{1}\int_{\mathbb{R}^{3}}\frac{\widetilde{u}_{n}^{2}(y)}{|x-y|}dy$
$\displaystyle=C_{1}\int_{|x-y|\leq
1}\frac{\widetilde{u}_{n}^{2}(y)}{|x-y|}dy+C_{1}\int_{|x-y|\geq
1}\frac{\widetilde{u}_{n}^{2}(y)}{|x-y|}dy.$ $\displaystyle\leq
C_{1}\int_{|x-y|\leq 1}\frac{\widetilde{u}_{n}^{2}(y)}{|x-y|}dy+C_{2}$
$\displaystyle\leq C_{1}\left(\int_{|x-y|\leq
1}\widetilde{u}_{n}^{4}(y)dy\right)^{\frac{1}{2}}\left(\int_{|x-y|\leq
1}\frac{1}{|x-y|^{2}}dy\right)^{\frac{1}{2}}+C_{2}$ $\displaystyle\leq
C_{3}\left(\int_{|r|\leq 1}dr\right)^{\frac{1}{2}}+C_{2}$ $\displaystyle\leq
C.$

Since $\varepsilon_{n}y_{n}\to\infty$, as $n\to\infty$, it follows from
$(H_{1})$ that $h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\to 0$, as
$n\to\infty$. Note that

$\displaystyle\widetilde{u}_{n}\rightharpoonup\widetilde{u}$
$\displaystyle\hskip 8.61108pt\text{in}\hskip 4.30554ptH_{\varepsilon};$ (5.4)
$\displaystyle\widetilde{u}_{n}\rightharpoonup\widetilde{u}$
$\displaystyle\hskip 8.61108pt\text{in}\hskip
4.30554ptL^{p}(\mathbb{R}^{3}),\hskip 4.30554pt2\leq p\leq 6;$
$\displaystyle\widetilde{u}_{n}\to\widetilde{u}$ $\displaystyle\hskip
8.61108pt\text{a.e.}\hskip 4.30554pt\text{on}\hskip 4.30554pt\mathbb{R}^{3}.$

We can obtain that
$h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}\to 0$ a.e. on
$\mathbb{R}^{3}$ and
$h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}$ is bounded in
$L^{2}(\mathbb{R}^{3})$. It follows that
$h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}\rightharpoonup 0$
in $L^{2}(\mathbb{R}^{3})$. Then, we have

$\displaystyle\int_{\mathbb{R}^{3}}h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}\varphi
dx\to 0,$ (5.5)

as $n\to\infty$. Combining (5.3) and (5.5), the desired conclusion is
obtained. $\Box$

######  Claim 2.

Suppose that $\varepsilon_{n}y_{n}\to x_{0}$, as $n\to\infty$, then

$\displaystyle\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}^{2}\right]h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}\varphi
dx$ $\displaystyle=$ $\displaystyle
h(x_{0})^{2}\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*\widetilde{u}^{2}\right]\widetilde{u}\varphi
dx+o_{n}(1)$ $\displaystyle=$ $\displaystyle o_{n}(1),$

for any $\varphi\in H^{1}(\mathbb{R}^{3})$.

######  Proof.

For simplicity, we denote $h(\varepsilon_{n}x+\varepsilon_{n}y_{n})$ by
$\widetilde{h}_{n}(x)$. Then, from assumption $(H_{1})$, we have

$\widetilde{h}_{n}(x)\to h(x_{0}),$ (5.6)

as $n\to\infty$ uniformly on bounded sets of $\mathbb{R}^{3}$. Combining (5.4)
and (5.6), we have $\widetilde{h}_{n}(x)\widetilde{u}_{n}\to
h(x_{0})\widetilde{u}$ a.e. on $\mathbb{R}^{3}$ and
$\widetilde{h}_{n}(x)\widetilde{u}_{n}$ is bounded in $L^{2}(\mathbb{R}^{3})$.
It follows that $\widetilde{h}_{n}(x)\widetilde{u}_{n}\rightharpoonup
h(x_{0})\widetilde{u}$ in $L^{2}(\mathbb{R}^{3})$. Then, we have

$\left|\int_{\mathbb{R}^{3}}\left(\widetilde{h}_{n}(x)\widetilde{u}_{n}-h(x_{0})\widetilde{u}\right)\varphi
dx\right|\to 0,$ (5.7)

for any $\varphi\in H^{1}(\mathbb{R}^{3})$, as $n\to\infty.$ By (5.3) and
(5.7), we have

$\left|\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*\widetilde{h}_{n}(x)\widetilde{u}_{n}^{2}\right]\left(\widetilde{h}_{n}(x)\widetilde{u}_{n}-h(x_{0})\widetilde{u}\right)\varphi
dx\right|\to 0,$ (5.8)

for any $\varphi\in H^{1}(\mathbb{R}^{3})$, as $n\to\infty$. On the other
hand, using (5.4) and (5.6), we can deduce that
$\widetilde{h}_{n}(x)\widetilde{u}_{n}^{2}\rightharpoonup
h(x_{0})\widetilde{u}^{2}$ in $L^{\frac{12}{5}}(\mathbb{R}^{3})$. It follows
from Lemma 2.1 (iv) that

$\frac{1}{|x|}*\widetilde{h}_{n}(x)\widetilde{u}_{n}^{2}\rightharpoonup\frac{1}{|x|}*h(x_{0})\widetilde{u}^{2}\hskip
8.61108pt\text{in}\hskip 4.30554pt\mathcal{D}^{1,2}(\mathbb{R}^{3})$

and, then, in $L^{6}(\mathbb{R}^{3})$. Then, we have

$\left|\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*\left(\widetilde{h}_{n}(x)\widetilde{u}_{n}^{2}-h(x_{0})\widetilde{u}^{2}\right)\right]\widetilde{u}\varphi
dx\right|\to 0,$ (5.9)

for any $\varphi\in H^{1}(\mathbb{R}^{3})$, as $n\to\infty.$ Note that

$\displaystyle\left|\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*\widetilde{h}_{n}(x)\widetilde{u}_{n}^{2}\right]\widetilde{h}_{n}(x)\widetilde{u}_{n}\varphi
dx-h(x_{0})^{2}\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*\widetilde{u}^{2}\right]\widetilde{u}\varphi
dx\right|$ (5.10) $\displaystyle\leq$
$\displaystyle\left|\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*\widetilde{h}_{n}(x)\widetilde{u}_{n}^{2}\right]\left(\widetilde{h}_{n}(x)\widetilde{u}_{n}-h(x_{0})\widetilde{u}\right)\varphi
dx\right|$
$\displaystyle+\left|h(x_{0})\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*\left(\widetilde{h}_{n}(x)\widetilde{u}_{n}^{2}-h(x_{0})\widetilde{u}^{2}\right)\right]\widetilde{u}\varphi
dx\right|.$

Thus, using (5.8)–(5.10) and $(H_{1})$, we have

$\displaystyle\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}^{2}\right]h(\varepsilon_{n}x+\varepsilon_{n}y_{n})\widetilde{u}_{n}\varphi
dx$ $\displaystyle=$ $\displaystyle
h(x_{0})^{2}\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*\widetilde{u}^{2}\right]\widetilde{u}\varphi
dx+o_{n}(1)$ $\displaystyle=$ $\displaystyle o_{n}(1),$

as desired. The rest proof is closely similar to that of Theorem 1.1. $\Box$

##  6 Nonexistence of ground state solution

In this section, we will study the nonexistence of ground state solutions for
the problem ($S_{\varepsilon}$). The proof of the case $h(x)\equiv 1$ is
similar to that of $h(x)$ satisfying condition $(H_{1})$, so we just give the
proof for the latter case.

######  Lemma 6.1.

Assume that $(f_{1})$–$(f_{5})$ hold. Then, $c_{\varepsilon}=c_{\infty}$, for
any $\varepsilon>0$.

######  Proof.

By Lemma 2.6, we see that there exists $u_{\infty}\in\mathcal{N}_{\infty}$
such that $\mathcal{I}_{\infty}(u_{\infty})=c_{\infty}$. Let us define
$u_{\infty}^{\zeta_{n}}=u_{\infty}(x-\zeta_{n})$, where
$\zeta_{n}\in\mathbb{R}^{3}$ and $|\zeta_{n}|\to\infty$ as $n\to\infty$. From
Lemma 2.3 (i) and (ii), there exists a positive bounded sequence $\\{t_{n}\\}$
such that $\\{t_{n}u_{\infty}^{\zeta_{n}}\\}\subset\mathcal{N}_{\varepsilon}$.
Then, we have

$\displaystyle\hskip
4.30554ptt_{n}^{2}\left(||u_{\infty}||_{\mathcal{D}^{1,2}}^{2}+\int_{\mathbb{R}^{3}}V(\varepsilon(x+\zeta_{n}))|u_{\infty}|^{2}dx\right)$
$\displaystyle+t_{n}^{4}\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*h(\varepsilon(x+\zeta_{n})|u_{\infty}|^{2}\right]h(\varepsilon(x+\zeta_{n})|u_{\infty}|^{2}dx$
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}t_{n}^{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon(x+\zeta_{n}))|u_{\infty}|^{q_{i}}dx+t_{n}^{6}\int_{\mathbb{R}^{3}}K(\varepsilon(x+\zeta_{n}))|u_{\infty}|^{6}dx.$

Up to a subsequence, we may assume that $t_{n}\to{t}_{0}>0$, as $n\to\infty$.
Taking the limit as $n\to\infty$ in the above equality and using condition
$(f_{5})$, we get

$\displaystyle
t_{0}^{2}\left(||u_{\infty}||_{\mathcal{D}^{1,2}}^{2}+\int_{\mathbb{R}^{3}}V_{\infty}|u_{\infty}|^{2}dx\right)=\sum\limits_{i=1}^{m}t_{0}^{q_{i}}\int_{\mathbb{R}^{3}}Q_{i}^{\infty}|u_{\infty}|^{q_{i}}dx+t_{0}^{6}\int_{\mathbb{R}^{3}}K^{\infty}|u_{\infty}|^{6}dx.$
(6.1)

Note that $u_{\infty}\in\mathcal{N}_{\infty}$, we have

$\left(||u_{\infty}||_{\mathcal{D}^{1,2}}^{2}+\int_{\mathbb{R}^{3}}V_{\infty}|u_{\infty}|^{2}dx\right)=\sum\limits_{i=1}^{m}\int_{\mathbb{R}^{3}}Q_{i}^{\infty}|u_{\infty}|^{q_{i}}dx+\int_{\mathbb{R}^{3}}K^{\infty}|u_{\infty}|^{6}dx.$
(6.2)

Combining (6.1) and (6.2), we can obtain that
$\lim\limits_{n\to\infty}t_{n}=t_{0}=1$. It follows from $V(x)$ is bounded
that there exists a constant $M$ sucht that
$|V(\varepsilon(x+\zeta_{n}))-V_{\infty}|\leq M$. Since $u_{\infty}\in
H_{\varepsilon}$, then for any $\sigma>0$, there exists $R>0$ such that
$|u_{\infty}|_{2,B_{R}^{C}(0)}<\frac{\sigma}{2M}$. Thus, we have

$\int_{\mathbb{R}^{3}\backslash
B_{R}(0)}|V(\varepsilon(x+\zeta_{n}))-V_{\infty}|u_{\infty}^{2}dx<\frac{\sigma}{2},$

and using the condition $(f_{5})$ and $|\zeta_{n}|\to\infty$, as $n\to\infty$,
we have

$\int_{B_{R}(0)}|V(\varepsilon(x+\zeta_{n}))-V_{\infty}|u_{\infty}^{2}dx<\frac{\sigma}{2},$

as $n\to\infty$. Which implies that

$\int_{\mathbb{R}^{3}}V(\varepsilon(x+\zeta_{n}))u_{\infty}^{2}dx\to\int_{\mathbb{R}^{3}}V_{\infty}u_{\infty}^{2}dx,$
(6.3)

as $n\to\infty$. Similarly, we can obtain that

$\int_{\mathbb{R}^{3}}K(\varepsilon(x+\zeta_{n}))|u_{\infty}|^{6}dx\to\int_{\mathbb{R}^{3}}K^{\infty}|u_{\infty}|^{6}dx,$
(6.4)

and

$\int_{\mathbb{R}^{3}}Q_{i}(\varepsilon(x+\zeta_{n}))|u_{\infty}|^{q_{i}}dx\to\int_{\mathbb{R}^{3}}Q_{i}^{\infty}|u_{\infty}|^{q_{i}}dx,$
(6.5)

as $n\to\infty$. From (6.3)–(6.5) and $\lim\limits_{n\to\infty}t_{n}=1$, we
can deduce that

$\displaystyle c_{\varepsilon}=$
$\displaystyle\inf\limits_{u\in\mathcal{N}_{\varepsilon}}\mathcal{I}_{\varepsilon}(u)$
(6.6) $\displaystyle\leq$
$\displaystyle\mathcal{I}_{\varepsilon}(t_{n}u_{\infty}^{\zeta_{n}})$
$\displaystyle=$
$\displaystyle\mathcal{I}_{\infty}(t_{n}u_{\infty})+\frac{t_{n}^{2}}{2}\int_{\mathbb{R}^{3}}\left(V(\varepsilon(x+\zeta_{n}))-V_{\infty}\right)u_{\infty}^{2}dx$
$\displaystyle+\frac{t_{n}^{4}}{4}\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*h(\varepsilon(x+\zeta_{n})u_{\infty}^{2}\right]h(\varepsilon(x+\zeta_{n})u_{\infty}^{2}dx$
$\displaystyle-\sum\limits_{i=1}^{m}\frac{t_{n}^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}(Q_{i}(\varepsilon(x+\zeta_{n}))-Q_{i}^{\infty})|u_{\infty}|^{q_{i}}dx-\frac{t_{n}^{6}}{6}\int_{\mathbb{R}^{3}}(K(\varepsilon(x+\zeta_{n}))-K^{\infty})|u_{\infty}|^{6}dx$
$\displaystyle\to$ $\displaystyle\mathcal{I}_{\infty}(u_{\infty})=c_{\infty},$

as $n\to\infty$. Thus, we get $c_{\varepsilon}\leq c_{\infty}$. On the other
hand, it follows from condition $(f_{5})$ that
$\mathcal{I}_{\infty}(u)\leq\mathcal{I}_{\varepsilon}(u)$, for any $u\in
H_{\varepsilon}$. Then, for any $u\in\mathcal{N}_{\infty}$, by the similar
arguments to Lemma 2.3 (i) and (ii), we can obtain that there exists a unique
bounded $t_{\varepsilon}>0$ such that
$t_{\varepsilon}u\in\mathcal{N}_{\varepsilon}$ and
$\mathcal{I}_{\varepsilon}(t_{\varepsilon}u):=\max\limits_{t\geq
0}\mathcal{I}_{\varepsilon}(tu)$. So it is easy to see that

$c_{\infty}=\inf\limits_{u\in\mathcal{N}_{\infty}}\mathcal{I}_{\infty}(u)\leq\inf\limits_{u\in\mathcal{N}_{\infty}}\mathcal{I}_{\varepsilon}(u)\leq\inf\limits_{u\in\mathcal{N}_{\infty}}\mathcal{I}_{\varepsilon}(t_{\varepsilon}u)=\inf\limits_{u\in\mathcal{N}_{\varepsilon}}\mathcal{I}_{\varepsilon}(u)=c_{\varepsilon}.$
(6.7)

Combining (6.6) and (6.7), the proof is complete. $\Box$

######  Proof of Theorem 1.3.

Arguing by contradiction, suppose that problem ($S_{\varepsilon}$) has a
positive ground state solutions, i.e., there exist $\varepsilon_{0}>0$ and
$u_{0}\in\mathcal{N}_{\varepsilon_{0}}$ such that
$\mathcal{I}_{\varepsilon_{0}}(u_{0})=c_{\varepsilon_{0}}$. By the similar
arguments to Lemma 2.3 (i), there exists $t_{\varepsilon_{0}}>0$ such that
$t_{\varepsilon_{0}}u_{0}\in\mathcal{N}_{\infty}$. From condition $(f_{5})$,
we get that $\mathcal{I}_{\infty}(u)\leq\mathcal{I}_{\varepsilon_{0}}(u)$, for
any $u\in H_{\varepsilon_{0}}$. Then, we have

$c_{\infty}=\inf\limits_{u\in\mathcal{N}_{\infty}}\mathcal{I}_{\infty}(u)\leq\mathcal{I}_{\infty}(t_{\varepsilon_{0}}u_{0})\leq\mathcal{I}_{\varepsilon_{0}}(t_{\varepsilon_{0}}u_{0})\leq\mathcal{I}_{\varepsilon_{0}}(u_{0})=c_{\varepsilon_{0}},$
(6.8)

it follows from Lemma 6.1 that
$\mathcal{I}_{\varepsilon_{0}}(t_{\varepsilon_{0}}u_{0})=\mathcal{I}_{\infty}(t_{\varepsilon_{0}}u_{0})$.
On the other hand, we have

$\displaystyle\mathcal{I}_{\varepsilon_{0}}(t_{\varepsilon_{0}}u_{0})=$
$\displaystyle\mathcal{I}_{\infty}(t_{\varepsilon_{0}}u_{0})+\frac{t_{\varepsilon_{0}}^{2}}{2}\int_{\mathbb{R}^{3}}\left(V(\varepsilon_{0}x)-V_{\infty}\right)u_{\infty}^{2}dx$
(6.9)
$\displaystyle+\frac{t_{\varepsilon_{0}}^{4}}{4}\int_{\mathbb{R}^{3}}\left[\frac{1}{|x|}*h(\varepsilon_{0}x)u_{\infty}^{2}\right]h(\varepsilon_{0}x)u_{\infty}^{2}dx+\sum\limits_{i=1}^{m}\frac{t_{\varepsilon_{0}}^{q_{i}}}{q_{i}}\int_{\mathbb{R}^{3}}(Q_{i}^{\infty}-Q_{i}(\varepsilon_{0}x))|u_{\infty}|^{q_{i}}dx$
$\displaystyle+\frac{t_{\varepsilon_{0}}^{6}}{6}\int_{\mathbb{R}^{3}}(K^{\infty}-K(\varepsilon_{0}x))|u_{\infty}|^{6}dx,$

which implies that
$\mathcal{I}_{\varepsilon_{0}}(t_{\varepsilon_{0}}u_{0})>\mathcal{I}_{\infty}(t_{\varepsilon_{0}}u_{0})$.
This is a contradiction and ends the proof. $\Box$

Acknowledgments This work is supported by National Natural Science Foundation
of China (No. 11671403).

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