\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 142, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/142\hfil 
 Ground states for fractional Kirchhoff equations]
{Existence of ground states for fractional Kirchhoff equations with
general potentials \\ via Nehari-Pohozaev manifold}

\author[J. Chen, X. Tang, S. Chen \hfil EJDE-2018/142\hfilneg]
{Jing Chen, Xianhua Tang, Sitong Chen}

\address{Jing Chen \newline
School of Mathematics and Computing Sciences,
Hunan University of Science and Technology,
Xiangtan, Hunan 411201, China}
\email{cjhnust@aliyun.com}

\address{Xianhua Tang \newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{tangxh@mail.csu.edu.cn}

\address{Sitong Chen \newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{mathsitongchen@163.com}

\dedicatory{Communicated by Binlin Zhang}

\thanks{Submitted November 15, 2017. Published July 13, 2018.}
\subjclass[2010]{35J20, 35J65}
\keywords{Fractional Kirchhoff equation; Nehari-Pohozaev manifold;
\hfill\break\indent ground state solutions}

\begin{abstract}
 We consider the nonlinear fractional Kirchhoff equation
 \[
 \Big(a+b\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)
 (-\Delta)^\alpha u+V(x)u=f(u) \quad \text{in } \mathbb R^3,
 u\in H^{\alpha}(\mathbb R^3),
 \]
 where $a>0$, $b\ge 0$, $\alpha\in(3/4, 1)$ are three constants, $V(x)$
 is differentiable and $f\in C^1(\mathbb R, \mathbb R)$.
 Our main results show the existence of ground state solutions of
 Nehari-Pohozaev type, and the existence of the least energy solutions
 to the above problem with general superlinear and subcritical nonlinearity.
 These results are proved by applying variational methods and some techniques
 from \cite{TC}. 
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

 In this article, we study the fractional Kirchhoff equation
 \begin{equation}\label{KE}
\begin{gathered}
 \Big(a+b\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)
 (-\Delta)^\alpha u+V(x)u=f(u) \quad \text{in } \mathbb R^3, \\
 u\in H^{\alpha}(\mathbb R^3),
 \end{gathered}
 \end{equation}
where $a>0$, $b\ge 0$, $\alpha\in(3/4,1)$ are three constants, the operator
$(-\Delta)^\alpha$ is the fractional Laplacian defined as
$\mathcal{F}((-\Delta)^\alpha\phi)(\xi)=|\xi|^{2\alpha}\mathcal{F}(\phi)(\xi)$,
where $\mathcal{F}$ is the Fourier transform. $V:\mathbb R^3\to \mathbb{R}$ and
$f:\mathbb{R}\to \mathbb{R}$ satisfy the following assumptions:
\begin{itemize}
\item[(A1)] $V\in C(\mathbb R^3, [0,\infty))$;

\item[(A2)] $V_\infty:=\lim\inf_{|y|\to \infty}V(y)\ge(\not\equiv)V(x)$ for all
$x\in \mathbb R^3$;

\item[(A3)]  $f\in C^1(\mathbb{R}, \mathbb{R})$ and there exist constants $C_0>0$ and
$p\in (2, 2_\alpha^*)$, $ 2^*_\alpha:=\frac{6}{3-2\alpha}$, such that
 $$
 |f(t)|\le C_0(1+|t|^{p-1}),\quad\forall  t\in \mathbb{R};
 $$

\item[(A4)] $f(t)=o(t)$ as $t\to 0$.
\end{itemize}

When $a=1$ and $b=0$, problem \eqref{KE} reduces to the  fractional
 Schr\"odinger equation
 \begin{equation}\label{S}
(-\Delta)^\alpha u+V(x)u=f(u) \quad  \text{in } \mathbb R^3.
 \end{equation}
 As we all known, the fractional Laplacian $(-\Delta)^\alpha$ is now experiencing
impressive applications in different subjects, such as phase transition,
 anomalous diffusion, fractional quantum mechanics and so on, see \cite{Hi}.
Equation \eqref{S}  concerns fractional quantum mechanics, exactly,
in the study of particles on stochastic fields modeled by L\'{e}vy processes.
For this kind of fractional and nonlocal problems, Caffarelli and Silvestre \cite{CS}
 constructed general fractional power of Laplacian from an extension problem
to the upper half space by mapping the Dirichlet condition to Neumann condition
for a specific elliptic equation. Their work devotes efforts to dealing with
 nonlinear variational problems with fractional Laplacian by standard local
perturbation method from variational method. After that, many results on the
existence of ground state and multiplicity for solutions of \eqref{S} have
been obtained. Just to mention a few, we recall, for instance, the
following papers and the references therein \cite{Zh1, TJ, ZTZ, SS, CX}.
Moreover, if $\alpha=1$, \eqref{S} reduces to the  classic nonlinear
Schr\"odinger equation, we refer to \cite{BL, TA2, TLY} and the references
therein for the recent research progress on this field.

If $\alpha=1$, then \eqref{KE} formally reduces to the well-known Kirchhoff equation
 \begin{equation}\label{K}
 -\Big(a+b\int_{\mathbb R^3}|\nabla u|^2\,\mathrm{d}x\Big)\Delta u+V(x)u=f(u)\quad
\text{in } \mathbb R^3.
 \end{equation}
This equation is related to the stationary analogue of the Kirchhoff equation
 \begin{equation}\label{KE1}
 u_{tt}-\Big(a+b\int_{\Omega}|\nabla u|^{2}\,\mathrm{d}x\Big)\Delta u=f(x, u),
 \end{equation}
 where $\Omega\subset\mathbb R^3$ is a bounded domain.
In 1883, Kirchhoff \cite{Kirchhoff} proposed \eqref{KE1} as a generalization
of the well-known D'Alembert wave equation
 $$
\rho\frac{\partial^2 u}{\partial t^2}
-\Big(\frac{r_0}{\lambda}+\frac{K}{2L}\int_0^L|
 \frac{\partial u}{\partial x}|^2\,\mathrm{d}x\Big)
\frac{\partial^2 u}{\partial x^2}=f(x,u)
 $$
for describing the changes in length of the elastic string arising from
transversal oscillations. Here, $L$ is the length of the string, $K$ is the
Young modulus of the material, $\rho$ is the mass density, $r_0$ is the
initial tension and $f$ denotes the area of the cross section,
see \cite{PXZ}. In addition, Kirchhoff-type equation also models several
physical and biological systems for the applications of the nonlocal effect.
For more detail on the background of Kirchhoff type problems, we refer the
readers to \cite{AC,AA} and the reference therein.
With the widespread applications of Kirchhoff type equations, abundant
results on the solvability of this kind of equations have brought out after
the pioneer work \cite{LJL}. Without attempts to provide a complete list
of references, we refer the reader to \cite{CBT, LG, Gu, WM, DY, HZ2, Y, Ta, TC}
for the existence of ground state solutions, nodal solutions, sign-changing
solutions, positive solutions and the concentration phenomena of solutions.

In the context of fractional quantum mechanics, a great interest has been
devoted to fractional Kirchhoff type equations in recent years.
Similar with classical Kirchhoff's model, Fiscella and Valdinoci \cite{FV}
first proposed the  stationary Kirchhoff equation involving
 nonlocal integro-differential operators
 \begin{equation}\label{KE2}
\begin{gathered}
 M\Big(\int_{\mathbb R^N}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)
(-\Delta)^\alpha u=f(x, u), \quad x\in \Omega,\\
 u=0, \quad  x\in \mathbb R^N\setminus\Omega,
 \end{gathered}
\end{equation}
 which takes into account the nonlocal aspect of the tension produced by
nonlocal measurement of the fractional length of the string.
Ambrosio and Isernia \cite{AI} investigated the  fractional Kirchhoff equation
 \begin{equation}\label{KN1}
\Big(a+b\int_{\mathbb R^N}|(-\Delta)^{\alpha/2} u|^2
\,\mathrm{d}x\Big)(-\Delta)^\alpha u=f(u) \quad \text{in }\mathbb R^N,
 \end{equation}
when $f$ is an odd subcritical nonlinearity satisfying the well known
Berestycki-Lions assumptions introduced in \cite{BL}. By minimax arguments,
the authors established a multiplicity result in the fractional radial function
space $H_{\text{rad}}^\alpha(\mathbb R^N)$ while the parameter $b$ is
sufficiently small.  Jin and Liu \cite{JH} considered the  Kirchhoff equation
 with the critical growth
 \begin{equation}\label{KN3}
 \Big(a+b\int_{\mathbb R^N}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)
 (-\Delta)^\alpha u+u=f(u) \quad \text{in } \mathbb R^N,
 \end{equation}
when $N>2\alpha$. By using a perturbation approach, they proved the existence
of positive radial solutions to \eqref{KN3} without the (AR) condition when
the parameter $b$ is small. A natural question now arises on whether \eqref{KE}
has ground state solution if the potential function $V(x)$ is not a constant.
 Recently, Liu, Squassina and Zhang  \cite{LZ} studied the  equation with general
potential
 \begin{equation}\label{KN2}
\Big(a+b\int_{\mathbb R^N}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)
(-\Delta)^\alpha u+V(x)u=f(u) \quad  \text{in } \mathbb R^N
\end{equation}
 in low dimension $2\alpha<N<4\alpha$, that is $N=2, 3$.
When $N=3$ with $\alpha\in (3/4,1)$, the potential $V$ in \eqref{KN2} satisfied
not only the assumptions (A1)-(A2) but also the following assumptions:
\begin{itemize}
\item[(A5)] $V\in C^1(\mathbb R^3, \mathbb{R})$ and
$\|\max\{ \nabla V(x)\cdot x,0\}\|_{L^{\frac{3}{2\alpha}}(\mathbb R^3)}
 <2a\alpha S_\alpha$, where $\cdot$ denotes the inner product in
$\mathbb R^3$ and $S_\alpha$ will be defined in Section 2.
\end{itemize}
 Moreover, the nonlinearity $f$ in \eqref{KN2} satisfies the following assumptions
 in addition to (A4):
\begin{itemize}
\item[(A6)]  $f\in C^1(\mathbb R^+, \mathbb{R})$, $f(t)=0$ for all $t\le 0$;

 \item[(A7)]  $\lim_{t\to \infty}\frac{f(t)}{t^{2^*_\alpha-1}}=1$;

\item[(A8)]  there are $D>0$ and $2<q<2_\alpha^*$ such that
 $f(t)\ge t^{2^*_\alpha-1}+Dt^{q-1}$ for any $t\ge 0$.
\end{itemize}

Clearly, when $N=3$, weak solutions to \eqref{KN2} correspond to critical points of
 the energy functional defined in $H^{\alpha}(\mathbb R^3)$ by
 \begin{equation}\label{IU}
\begin{aligned}
 I(u) & = \frac{1}{2}\int_{\mathbb R^3}[a|(-\Delta)^{\alpha/2} u|^2+V(x)u^2]
 \,\mathrm{d}x+\frac{b}{4}
 \Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)^2 \\
 & \quad -\int_{\mathbb R^3}F(u)\,\mathrm{d}x.
\end{aligned}
 \end{equation}

Here, $H^{\alpha}(\mathbb R^3)$ denotes the usual fractional Sobolev space
and $F(u):=\int_{0}^{u}f(s)\,\mathrm{d}s$. Under above assumptions on $V$ and $f$,
Liu et al.\ proved that \eqref{KN2} has a positive least energy solution when $D$
is large enough. In fact, under the assumptions on $N$ and $\alpha$,
it follows that $2_\alpha^*>4$. Thus $f(x,t)$ is critical growth at $t=\infty$ and
$$
\lim_{t\to \infty}\frac{F(t)}{t^4}=\infty \quad \text{uniformly in } x\in\mathbb R^3.
$$
Then it easily follows that the energy functional $I$ possesses a mountain-pass
geometry. But it is difficult to get a bounded (PS) sequence because of without
(AR) condition. To overcome this difficulty, Liu et al.\ used Jeanjean's
monotonicity  \cite{JL} to show that there exists a bounded (PS) sequence
$\{u_{n}(\lambda)\}$ at the level $c_{\lambda}$ for almost
$\lambda\in [1/2, 1]$. However, by the presence of the Kirchhoff term,
it is not easy to show that $I$ has a critical point since in general,
for any $v\in C_0^\infty(\mathbb R^3)$, we do not know that
\begin{align*}
&\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u_n|^2\,\mathrm{d}x
\int_{\mathbb R^3}(-\Delta)^{\alpha/2} u_n(-\Delta)^{\alpha/2} v\,\mathrm{d}x \\
&\to \int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x
 \int_{\mathbb R^3}(-\Delta)^{\alpha/2} u(-\Delta)^{\alpha/2} v\,\mathrm{d}x
\end{align*}
from $u_n\rightharpoonup u$ in $H^\alpha(\mathbb R^3)$.
Although $I_\lambda'(u)=0$ from $u_n\rightharpoonup u$ in $H^\alpha(\mathbb R^3)$
can not be directly concluded, partly inspired by \cite{LG}, they consider
a family of related functionals $J_\lambda$, whose corresponding problem is
a non-Kirchhoff equation. Through establishing a profile decomposition
of the (PS) sequence $\{u_n(\lambda)\}$ related to $J_\lambda$, they obtained
 a nontrivial critical point $u_\lambda$ of $I_\lambda$ at the level
$c_\lambda$. Subsequently, by choosing a sequence $\{\lambda_n\}\subset [1/2,1]$
with $\lambda_n\to 1$, thanks to the Pohozaev identity they obtain a bounded
$(PS)_{c_{1}}$ sequence of the original functional $I$. We point out that
$f\in C^1$ is very crucial to get the profile decomposition of the (PS)
sequence in \cite{LZ}. In addition, the result is heavily dependent on the
existence of a constant $D>0$ large enough in (A8).

This article is motivated by \cite{CT, TA, TC, TA2, LZ}.
Provided $f \in C^1(\mathbb{R}, \mathbb{R})$  with superlinear growth at infinity, under some
mild assumptions on $V$ and $f$,  we  obtain the existence of a ground solution
of Nehari-Pohozaev type for \eqref{KE}  which is a minimizer of $I$ on the
Nehari-Pohozaev manifold  $\mathcal{M}$.
Moreover, we  prove the existence of the least energy solutions for \eqref{KE}.

 To state our results, in addition to (A1)--(A4), we make the following
assumptions on $V$ and $f$.
\begin{itemize}
\item[(A9)] $V(x)$ is weakly differentiable, and there exists
$\theta\in[0,1)$ such that
 $$
(\nabla V(x),x)\leq \frac{2\alpha\theta a K(\alpha)}{|x|^{2\alpha}}, \quad\text{a.e. }
x\in \mathbb R^3\backslash\{0\},
 $$
where $K(\alpha):=\frac{1}{\pi}\Gamma^2(\frac{1+2\alpha}{2})$ and $\Gamma$
is the Gamma function;


\item[(A10)] $V\in C^1(\mathbb R^3, \mathbb{R})$ and there exists $\theta\in [0, 1)$ such that
\[
4\alpha t^{4\alpha}[V(x)-V(tx)]-(1-t^{4\alpha})(\nabla V(x),x)
\ge -\frac{2\alpha\theta a K(\alpha)(1-t^{2\alpha})^2}{|x|^{2\alpha}},
\]
 for all $t>0$, $x\in \mathbb R^3\backslash\{0\}$;

\item[(A11)]  $\lim_{|t|\to \infty}\frac{F(t)}{|t|^{2}}=\infty$;


\item[(A12)] $\frac{(4\alpha-3)f(t)t+6F(t)}{(4\alpha-3)t|t|}$ is nondecreasing
on $(-\infty, 0)\cup(0, \infty)$.
\end{itemize}

\begin{remark} \label{rmk1.1}\rm
 Note that the  monotonicity condition
\begin{itemize}
\item[(A13)]  $\frac{f(t)}{|t|}$ is nondecreasing on $(-\infty, 0)\cup(0, \infty)$
\end{itemize}
 is the weakened Nehari type condition. When $\alpha\in(3/4,1)$, it is easy
 to prove that (A13)  implies (A12).  In fact, there are many functions
satisfying (A12)  but not (A13).  For example,
 $$
 f(t)=\Big(\frac{t^2}{5}+3|t|+2\sin t+t\cos t\Big)t.
 $$
Furthermore by (A3) and (A4), for any $\varepsilon>0$, there exists
$C_{\varepsilon}>0$  such that
 \begin{equation}\label{Cv}
 |f(t)|\le \varepsilon |t|+C_{\varepsilon}|t|^{p-1}, \quad  \forall  t\in \mathbb{R}.
 \end{equation}
\end{remark}

Before starting our main results, we define a functional on $H^{\alpha}(\mathbb R^3)$
as follows
 \begin{equation}\label{Ju}
\begin{aligned}
 J(u):=&\alpha a\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x
 +\frac{1}{2}\int_{\mathbb R^3}[4\alpha V(x)+(\nabla V(x),x)]u^2\,\mathrm{d}x \\
 &+\alpha b\Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)^2
 -\frac{1}{2}\int_{\mathbb R^3}\left[(4\alpha-3)f(u)u+6F(u)\right]\,\mathrm{d}x.
\end{aligned}
 \end{equation}
In addition, we set
 \begin{equation}\label{Ne}
 \mathcal{M}:= \{u\in H^{\alpha}(\mathbb R^3)\setminus \{0\} : J(u)=0\}
 \end{equation}
as Nehari-Pohozaev manifold.
We are now in a position to state the main results of this article.

\begin{theorem} \label{thm1.2}
 Assume that $V$ and $f$ satisfy {\rm (A1)-(A4), (A10)--(A12)}. Then
 Problem \eqref{KE} has a solution $\bar{u}\in H^{\alpha}(\mathbb R^3)$ such
that $I(\bar{u})=\inf_{\mathcal{M}}I>0$.
\end{theorem}

\begin{theorem} \label{thm1.3}
 Assume that $V$ and $f$ satisfy {\rm (A1)--(A4), (A9),  (A11), (A12)}.
 Then  Problem \eqref{KE} has a least energy solution
$\bar{u}\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$.
\end{theorem}

 To prove Theorem \ref{thm1.2}, we first look for a minimizer of the functional
restricted to Nehari-Pohozaev manifold $\mathcal{M}$ which is defined by a
 condition through combining the Nehari equation with the Pohozaev equality,
for the usual method of Nehari manifold becomes invalid in this case.
Note that such type of manifold was first introduced by Ruiz \cite{RD}
for the study of the Schr\"odinger-Poisson problem.
Then we prove that the minimizer on $\mathcal{M}$ is a critical point.
To show Theorem \ref{thm1.3}, we  use Jeanjeans monotonicity tricks, which is partly
followed by \cite{LZ}. These results will greatly improve the existing
ones on fractional Kirchhoff problems.

 To illustrate conveniently, we introduce some useful notation.
In the sequel, $\|u\|_s  =(\int_{\mathbb R^3}|u|^s\,\mathrm{d}x)^{1/s}$
 denotes the norm of the Lebesgue space $L^s(\mathbb R^3) (1\le s\le \infty)$.
we set $(\cdot)_t:=t^{4\alpha-3}[(\cdot)(\frac{x}{t})]$ for $t>0$, where
$(\cdot)$ denotes any function belongs to $H^{\alpha}(\mathbb R^3)\setminus \{0\}$.
For $x\in \mathbb R^3$ and $r>0$, $B_r(x):=\{y\in \mathbb R^3: |y-x|<r \}$.
Throughout the paper, $C_1, C_2,\dots$ denote various positive constants.

 This article is organized as follows.
In Section 2, we give some preliminaries.
In Section 3, the limited problem is discussed and the proof of 
Theorem \ref{thm1.2} is given.
 Section 4 is devoted to finding a least energy
 solution for \eqref{KE}. Theorem \ref{thm1.3} will be proved in this section.


\section{Preliminaries and variational setting}

 The fractional Sobolev space $H^\alpha(\mathbb R^3)$ is defined by
 $$
H^\alpha(\mathbb R^3):=\Big\{u\in L^2(\mathbb R^3):\frac{|u(x)-u(y)|}
{|x-y|^{\frac{3+2\alpha}{2}}}\in L^2(\mathbb R^3\times\mathbb R^3)\Big\}.
$$
It is known that $$\int_{\mathbb R^3}\int_{\mathbb R^3}
\frac{|u(x)-u(y)|^2}{|x-y|^{3+2\alpha}}\,\mathrm{d}x\,\mathrm{d}y
=2C_\alpha^{-1}\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x,
$$
 where
 $$
C_\alpha=\Big(\int_{\mathbb R^3}\frac{1-\cos\zeta_1}
{|\zeta|^{3+2\alpha}}\,\mathrm{d}\zeta\Big)^{-1}.
 $$
We endow the space $H^\alpha(\mathbb R^3)$ with the norm
$$
\|u\|_{H^\alpha(\mathbb R^3)}
:=\Big(\int_{\mathbb R^3}|u|^2\,\mathrm{d}x
+\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)^{1/2}.
$$
The space $H^\alpha(\mathbb R^3)$ is the completion of 
$C_0^\infty(\mathbb R^3)$ with $\|\cdot\|_{H^\alpha(\mathbb R^3)}$ and 
it is continuously embedded into $L^q(\mathbb R^3)$ for $q\in [2,2^*_\alpha]$
 and compactly embedded into $L_{\text{loc}}^q(\mathbb R^3)$ for any 
$q \in [2,2_\alpha^*)$. Moreover, the best embedding constant is 
 $$
 S_\alpha:=\inf_{u\in D^{\alpha,2}(\mathbb R^3),\, u\neq 0}
\frac{\int_{\mathbb R^3}|(-\Delta)^{\alpha/2}u|^2\,\mathrm{d}x}
{\big(\int_{\mathbb R^3}|u|^{2_\alpha^*}\,\mathrm{d}x\big)^{2/{2^*_\alpha}}}.
$$
Let
 $$ 
\mathcal{H}:=\Big\{u\in H^\alpha(\mathbb R^3): 
\int_{\mathbb R^3}V(x)u^2\,\mathrm{d}x<\infty\Big\}
 $$
be a Hilbert space equipped with the inner product
 $$
\langle u,v\rangle_\mathcal{H}:=a\int_{\mathbb R^3}
(-\Delta)^{\alpha/2}u(-\Delta)^{\alpha/2}v\,\mathrm{d}x
+\int_{\mathbb R^3}V(x)uv\,\mathrm{d}x
 $$
 and the corresponding induced norm
 $$
\|u\|:=\Big(\int_{\mathbb R^3}a|(-\Delta)^{\alpha/2}u|^2\,\mathrm{d}x
+\int_{\mathbb R^3}V(x)u^2\Big)^{1/2}.
 $$
The homogeneous Sobolev space $D^{\alpha,2}(\mathbb R^3)$ is defined by
 $$
D^{\alpha,2}(\mathbb R^3):=\big\{u\in L^{2_\alpha^*}(\mathbb R^3):
\frac{|u(x)-u(y)|}{|x-y|^{\frac{3+2\alpha}{2}}}
\in L^2(\mathbb R^3\times\mathbb R^3)\big\},
 $$
which is also the completion of $C_0^\infty(\mathbb R^3)$ under the norm 
$$
\|u\|_{D^{\alpha,2}(\mathbb R^3)}
:=\Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2}u|^2\,\mathrm{d}x\Big)^{1/2}
=\|(-\Delta)^{\alpha/2}u\|_2.
$$

 A function $u\in \mathcal{H}$ is a weak solution to problem \eqref{KE} if,
 for every $\phi\in \mathcal{H}$, we have
\begin{align*}
&\Big(a+b\int_{\mathbb R^3}|(-\Delta)^{\alpha/2}u|^2\,\mathrm{d}x\Big)
\int_{\mathbb R^3}(-\Delta)^{\alpha/2}u(-\Delta)^{\alpha/2}\phi\,\mathrm{d}x
 +\int_{\mathbb R^3}V(x)u\phi\,\mathrm{d}x \\
&=\int_{\mathbb R^3} f(u)\phi\,\mathrm{d}x.
\end{align*}

 \begin{lemma}[\cite{LZ}] \label{lem2.1} 
 Assume that $V$ satisfies {\rm (A2)}.
 Then for every $\epsilon >0$ there exists $\tau_\epsilon >0$ such that
 $$
\int_{\mathbb R^3} ((a-\epsilon)|(-\Delta)^{\alpha/2}u|^2+V(x)u^2)\,\mathrm{d}x
\ge \tau_\epsilon\int_{\mathbb R^3}u^2\,\mathrm{d}x, \quad 
 u\in H^\alpha(\mathbb R^3).
 $$
\end{lemma}

 From the above lemma, it easily follows that the  norm $\|\cdot\|$ is equivalent 
to $\|\cdot\|_{H^\alpha}$.
 
\begin{lemma}[\cite{SS}] \label{lem2.2}
 Assume that $\{u_n\}_{n\in\mathbb{N}} $ is bounded in $\mathcal{H}$ and
 $$
\lim_{n\to \infty}\sup_{y\in \mathbb R^3}\int_{B_r(y)}|u_n|^2\,\mathrm{d}x=0,
 $$
for some $r>0$. Then $u_n\to 0$ in $L^s(\mathbb R^3)$ for all $s\in (2, 2^*_\alpha)$.
\end{lemma}

\section{Limit problem and  proof of Theorem \ref{thm1.2}}

In this section, through discussing the corresponding limit problem for 
\eqref{KE}, 
we will obtain the proof of Theorem \ref{thm1.2}. To this aim, we define two new functionals on 
$H^\alpha(\mathbb R^3)$ as follows:
 \begin{gather}\label{Iinf}
\begin{aligned}
 I^{\infty}(u)
&=\frac{1}{2}\int_{\mathbb R^3}\big[a|(-\Delta)^{\alpha/2} u|^2+V_{\infty} u^2\big]\,
\mathrm{d}x \\
&\quad +\frac{b}{4}\Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,
  \mathrm{d}x\Big)^2 
 -\int_{\mathbb R^3}F(u)\,\mathrm{d}x,
\end{aligned} \\
\label{Jinf}
\begin{aligned}
J^\infty(u)
&=\alpha a\|(-\Delta)^{\alpha/2} u\|_2^2+2\alpha V_\infty \|u\|_2^2
+\alpha b\|(-\Delta)^{\alpha/2} u\|_2^4  \\
&\quad -\frac{1}{2}\int_{\mathbb R^3}[(4\alpha-3)f(u)u+6F(u)]\,\mathrm{d}x.
\end{aligned}
 \end{gather}
 Set
\begin{gather*}
\mathcal{M}^\infty:=\{u\in H^\alpha(\mathbb R^3)\backslash\{0\}:J^\infty(u)=0\},\\
m_0:=\inf_{u\in \mathcal{M}}I(u), \quad 
m^\infty:=\inf_{u\in\mathcal{M}^\infty}I^\infty(u).
\end{gather*}

 \begin{lemma} \label{lem3.1}
 Assume that {\rm (A3)} and {\rm(A12)} hold and $\alpha\in(3/4, 1)$. Then
 \begin{equation}\label{B20}
\frac{(4\alpha-3)(1-t^{4\alpha})}{8\alpha}f(w)w
 -\frac{4\alpha-3+3t^{4\alpha}}{4\alpha}F(w)
 +t^3F(t^{\frac{4\alpha-3}{2}}w)\ge 0, 
 \end{equation}
for all $t\ge0$ and $w\in \mathbb{R}$.
\end{lemma}


\begin{proof} 
It is clear that \eqref{B20} holds for $w=0$. For $w\ne 0$, let
 \begin{equation}\label{B21}
 h(t)=\frac{4\alpha-3}{8\alpha}(1-t^{4\alpha})f(w)w
-\frac{4\alpha-3+3t^{4\alpha}}{4\alpha}F(w)+t^3F(t^{\frac{4\alpha-3}{2}}w), \quad
 t\ge 0.
 \end{equation}
Then from (A12), one has
 \begin{equation}\label{B22}
\begin{aligned}
 h'(t) 
 &= \frac{t^{4\alpha-1}w^{2}}{2}
\Big[\frac{f(t^{\frac{4\alpha-3}{2}}w)(t^{\frac{4\alpha-3}{2}}w)
 +\frac{6}{4\alpha-3}F(t^{\frac{4\alpha-3}{2}}w)}{(t^{\frac{4\alpha-3}{2}}w)^{2}}\\
&\quad -\frac{f(w)w+\frac{6}{4\alpha-3}F(w)}{w^{2}}\Big] \\
&= \begin{cases} 
 \ge 0,  & t\ge 1,\\
 \le 0,  & 0<t<1.
 \end{cases}
\end{aligned}
\end{equation}
From this and the continuity of $h$ it follows that $h(t)\ge h(1)=0$ for $t\ge0$. 
This implies \eqref{B20} holds.
\end{proof}


 \begin{lemma} \label{lem3.2}
Assume that  {\rm (A1), (A3), (A4), (A10), (A12)} hold. Then
 \begin{equation}\label{B31}
 I(u)\ge  I(u_t)+\frac{1-t^{4\alpha}}{4\alpha}J(u)
 +\frac{a(1-\theta)(1-t^{2\alpha})^2}{4}\|(-\Delta)^{\alpha/2} u\|_2^2, \\
 \end{equation}
for all $u\in H^{\alpha}(\mathbb R^3)$, $t>0$.
\end{lemma}

\begin{proof} According to the fractional Hardy inequality in \cite{BD}, we have
 \begin{equation}\label{FH}
 \|(-\Delta)^{\alpha/2} u\|_2^2\ge\frac{1}{\pi}\Gamma^2\big(\frac{1+2\alpha}{2}\big)
\int_{\mathbb R^3}\frac{u^2}{|x|^{2\alpha}}\,\mathrm{d}x.
 \end{equation}
Note that
 \begin{equation}\label{B33}
\begin{aligned}
 I(u_t)
 & =  \frac{at^{2\alpha}}{2}\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{t^{4\alpha}}{2}\int_{\mathbb R^3}V(tx)u^2\,\mathrm{d}x
 +\frac{bt^{4\alpha}}{4}\|(-\Delta)^{\alpha/2} u\|_2^4 \\
 &\quad -t^3\int_{\mathbb R^3}F\big(t^{\frac{4\alpha-3}{2}}u\big)\,\mathrm{d}x.
\end{aligned}
 \end{equation}
 Thus, in view of (A10), \eqref{IU}, \eqref{B20}, \eqref{FH} and \eqref{B33}, one has
 \begin{align*}
&  I(u)-I(u_t)\\
& =  \frac{a(1-t^{2\alpha})}{2}\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{1}{2}\int_{\mathbb R^3}[V(x)-t^{4\alpha} V(tx)]u^2\,\mathrm{d}x
 \\
& \quad +\frac{b(1-t^{4\alpha})}{4}\|(-\Delta)^{\alpha/2} u\|_2^4
 +\int_{\mathbb R^3}\left[t^3F(t^{\frac{4\alpha-3}{2}}u)-F(u)\right]\,\mathrm{d}x\\
& =  \frac{1-t^{4\alpha}}{4\alpha}
 \Big\{\alpha(a+b\|(-\Delta)^{\alpha/2} u\|_2^2)\|(-\Delta)^{\alpha/2} u\|_2^2\\
&\quad +\frac{1}{2}\int_{\mathbb R^3}\left[4\alpha V(x)+(\nabla V(x),x)\right]
 u^2\,\mathrm{d}x \\
&\quad -\frac{1}{2}\int_{\mathbb R^3}[(4\alpha-3)f(u)u+6F(u)]\,\mathrm{d}x\Big\}
 +\frac{a(1-t^{2\alpha})^2}{4}\|(-\Delta)^{\alpha/2} u\|_2^2\\
 &\quad +\frac{1}{8\alpha}\int_{\mathbb R^3}
 \{4\alpha t^{4\alpha}[V(x)-V(tx)]-(1-t^{4\alpha})(\nabla V(x),x)\}u^2\,\mathrm{d}x \\
 &\quad +\int_{\mathbb R^3}\Big[\frac{(4\alpha-3)(1-t^{4\alpha})}{8\alpha}f(u)u
 -\frac{4\alpha-3+3t^{4\alpha}}{4\alpha}F(u)+t^3F(t^{\frac{4\alpha-3}{2}}u)\Big]
 \,\mathrm{d}x \\
 & \ge  \frac{1-t^{4\alpha}}{4\alpha}J(u)+\frac{a(1-\theta)(1-t^{2\alpha})^2}{4}
  \|(-\Delta)^{\alpha/2} u\|_2^2.
 \end{align*}
 This shows that \eqref{B31} holds.
\end{proof}

 From Lemma \ref{lem3.2}, we have the following two corollaries.

 
 \begin{corollary} \label{coro3.3}
 Assume that {\rm (A3), (A4), (A12)}  hold. Then
\begin{equation}\label{IIn}
 I^\infty(u)\ge I^\infty(u_t)+\frac{1-t^{4\alpha}}{4\alpha}J^\infty(u)
+\frac{a(1-t^{2\alpha})^2}{4}\|(-\Delta)^{\alpha/2} u\|_2^2,
\end{equation}
 for all $u \in H^\alpha(\mathbb R^3)$ and $t>0$.
\end{corollary}

 \begin{corollary} \label{coro3.4} Assume that 
{\rm (A1),  (A3), (A4), (A10), (A12)} hold. Then for $u\in \mathcal{M}$,
 \begin{equation}\label{Imax}
 I(u) = \max_{t>0}I(u_t).
 \end{equation}
\end{corollary}

 \begin{lemma} \label{lem3.5}
 Assume that {\rm (A1), (A2), (A10)} hold. Then there exist two constants
 $\omega_1, \omega_2>0$ such that
\begin{equation}\label{SC1}
\omega_1\|u\|^2\le \alpha a\|(-\Delta)^{\alpha/2} u\|_2^2
+\frac{1}{2}\int_{\mathbb R^3}[4\alpha V(x)+(\nabla V(x), x)]u^2\,\mathrm{d}x
\le \omega_2\|u\|^2,
\end{equation}
for all $u\in H^\alpha(\mathbb R^3)$.
\end{lemma}

\begin{proof} Using (A2) and Lemma \ref{lem2.1}, in a same way as in
\cite[Lemma 2.5]{TC}, we can prove \eqref{SC1} holds.
\end{proof}

 \begin{lemma} \label{lem3.6}
 Assume that Assumptions  {\rm (A1)-(A4), (A10)--(A12)}  hold. Then for any
 $u\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$, there exists a unique $t(u)>0$ 
such that $u_{t(u)}\in \mathcal{M}$.
\end{lemma}

 
\begin{proof}   Let $u\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$ be 
fixed and define a function $g(t):=I(u_t)$
 on $(0, \infty)$. From \eqref{B33}, we have that
 \begin{equation}\label{B41}
\begin{aligned}
&g'(t)=0 \\
& \Leftrightarrow
  \alpha at^{2\alpha}\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{t^{4\alpha}}{2}\int_{\mathbb R^3}[4\alpha V(tx)
 +(\nabla V(tx), tx)]u^2\,\mathrm{d}x  \\
 &\quad +\alpha bt^{4\alpha}\|(-\Delta)^{\alpha/2} u\|_2^4-\frac{t^3}{2}
 \int_{\mathbb R^3}[(4\alpha-3)f(t^{\frac{4\alpha-3}{2}}u)
t^{\frac{4\alpha-3}{2}}u+6F(t^{\frac{4\alpha-3}{2}}u)]\,\mathrm{d}x 
 =0 \\
& \Leftrightarrow  J(u_t)=0 \\
&\Leftrightarrow  u_t\in \mathcal{M}.
\end{aligned}
 \end{equation}
 It is easy to verify, using (A1), (A2), (A3), (A4) and (A11), that 
$g(0)=0$, $g(t)>0$ for $t>0$ small and $g(t)<0$
 for $t$ large. Therefore $\max_{t\in [0, \infty)}g(t)$ is achieved at
 $t_0=t(u)>0$ so that
 $g'(t_0)=0$ and $u_{t_0}\in \mathcal{M}$.

 Next we claim that $t(u)$ is unique for any 
$u\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$.
 In fact, for any given $u\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$, 
let $t_1, t_2>0$ such that $u_{t_1}, u_{t_2}
 \in \mathcal{M}$. Then $J(u_{t_1})=J(u_{t_2})=0$. Jointly with \eqref{B31}, we have
 \begin{equation}\label{B411}
\begin{aligned}
 I(u_{t_1})
 & \ge  I(u_{t_2})+\frac{t_1^{4\alpha}-t_2^{4\alpha}}{4\alpha t_1^{4\alpha}}J(u_{t_1})
 +\frac{(1-\theta)a(t_1^{2\alpha}-t_2^{2\alpha})^2}{4t_1^{2\alpha}}
 \|(-\Delta)^{\alpha/2} u_{t_1}\|_2^2 \\
 & =  I(u_{t_2})+\frac{(1-\theta)a(t_1^{2\alpha}-t_2^{2\alpha})^2}
 {4t_1^{2\alpha}}\|(-\Delta)^{\alpha/2} u_{t_1}\|_2^2
\end{aligned}
 \end{equation}
 and
 \begin{equation}\label{B42}
\begin{aligned}
 I(u_{t_2})
 & \ge  I(u_{t_1})+\frac{t_2^{4\alpha}-t_1^{4\alpha}}
 {4\alpha t_2^{4\alpha}}J(u_{t_2})
 +\frac{a(t_2^{2\alpha}-t_1^{2\alpha})^2}{4t_2^{2\alpha}}
 \|(-\Delta)^{\alpha/2} u_{t_2}\|_2^2 \\
 & =  I(u_{t_1})+\frac{(1-\theta)a(t_2^{2\alpha}-t_1^{2\alpha})^2}
 {4t_2^{2\alpha}}\|(-\Delta)^{\alpha/2} u_{t_2}\|_2^2.
\end{aligned}
 \end{equation}
Inequalitites \eqref{B411} and \eqref{B42} imply $t_1=t_2$. 
Therefore, $t(u)> 0$ is unique for any $u\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$.
\end{proof}


\begin{lemma} \label{lem3.7}
 Assume that {\rm (A1)-(A4), (A10)--(A12)} hold. Then
 $$
 \inf_{u\in \mathcal{M}}I(u)
 :=m_0=\inf_{u\in H^{\alpha}(\mathbb R^3)\setminus \{0\}}\max_{t > 0}I(u_t).
 $$
\end{lemma}

Note that Corollary \ref{coro3.4} and Lemma \ref{lem3.6} imply the above lemma.

\begin{lemma} \label{lem3.8}
 Assume that {\rm (A1)-(A4), (A10)--(A12)} hold. Then 
\begin{itemize} 
\item[(i)]  there exists $\rho_0>0$ such that $\|u\|\ge \rho_0$ for all 
$u\in \mathcal{M}$;

 \item[(ii)] $m_0=\inf_{u\in \mathcal{M}}I(u)>0$.
\end{itemize}
\end{lemma}

\begin{proof} (i) Since $J(u)=0$ for all $u\in \mathcal{M}$, by (A3), 
\eqref{Cv}, \eqref{Ju}, \eqref{Jinf}, \eqref{SC1} and Lemma \ref{lem2.1}, one has
%\label{B62}
\begin{align*}
 \omega_1\|u\|^2
 & \le  \alpha a\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{1}{2}\int_{\mathbb R^3}[4\alpha V(x)+(\nabla V(x),x)]u^2\,\mathrm{d}x
 +\alpha b\|(-\Delta)^{\alpha/2} u\|_2^4 \\
 & =  \frac{1}{2}\int_{\mathbb R^3}\left[(4\alpha-3)f(u)u+6F(u)\right]\,\mathrm{d}x  \\
 & \le  \frac{\omega_1}{2}\|u\|^2+C_1\|u\|^p.
\end{align*}
This implies
 \begin{equation}\label{B63}
 \|u\|\ge \rho_0:=\big(\frac{\omega_1}{2C_1}\big)^{1/(p-2)}, \quad 
 \forall \ u\in \mathcal{M}.
 \end{equation}

 
(ii). By (A2), there exists $R>0$ such that $V(x)\ge \frac{V_\infty}{2}$ for 
$|x|\ge R$. From \eqref{Cv}, there exists $C_2>0$ such that
 \begin{equation}\label{B64}
 |F(t)|\le \frac{1}{4}\min\Big\{aS_\alpha\Big(\frac{3}{4\pi R^3}\Big)^{2\alpha/3},
 V_\infty\Big\}|t|^2+C_2|t|^{2_\alpha^*}, \quad \forall  t\in \mathbb{R}.
 \end{equation}
For $u\in \mathcal{M}$, let 
$t_u =(\frac{aS_\alpha^{\frac{2^*_\alpha}{2}}}{12C_2})^{\frac{3-2\alpha}
{4\alpha^2}}\|(-\Delta)^{\alpha/2} u\|_2^{-\frac{1}{\alpha}}$.
Using  H\"older's inequality and the fractional Sobolev inequality, we obtain
 \begin{equation}\label{T32}
\begin{aligned}
 \int_{|t_u x|<R}u^2\,\mathrm{d}x
&\le \Big(\frac{4\pi R^3}{3t_u^{3}}\Big)^{2\alpha/3}
 \Big(\int_{|t_u x|<R}u^{2_\alpha^*}\,\mathrm{d}x\Big)^{2/2_\alpha^*} \\
&\le \Big(\frac{4\pi R^3}{3t_u^{3}}\Big)^{2\alpha/3}
S_\alpha^{-1}\|(-\Delta)^{\alpha/2} u\|_2^2.
\end{aligned}
 \end{equation}
Then from \eqref{B31}, \eqref{B33}, \eqref{B64}, \eqref{T32} and the fractional 
Sobolev inequality, we have
 \begin{align*}
 I(u)&\ge I(u_{t_u}) \\
 &= \frac{at_u^{2\alpha}}{2}\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{t_u^{4\alpha}}{2}\int_{\mathbb R^3}V(t_ux)u^2\,\mathrm{d}x
 +\frac{bt_u^{4\alpha}}{4}\|(-\Delta)^{\alpha/2} u\|_2^4 \\
&\quad -t_u^3\int_{\mathbb R^3}F(t_u^{\frac{4\alpha-3}{2}}u)\,\mathrm{d}x \\
&\ge \frac{at_u^{2\alpha}}{4}\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{aS_\alpha}{4}\Big(\frac{3}{4\pi R^3}\Big)^{2\alpha/3}t_u^{4\alpha}
 \int_{|t_u x|<R}u^{2}\,\mathrm{d}x \\
&\quad +\frac{V_\infty t_u^{4\alpha}}{4}\int_{|t_u x|\ge R}u^2\,\mathrm{d}x
 -t_u^3\int_{\mathbb R^3}F(t_u^{\frac{4\alpha-3}{2}}u)\,\mathrm{d}x \\
&\ge  \frac{at_u^{2\alpha}}{4}\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{1}{4}\min\Big\{aS_\alpha\Big(\frac{3}{4\pi R^3}\Big)^{2\alpha/3},
 V_\infty\Big\}t_u^{4\alpha}\|u\|_2^2 \\
&\quad -t_u^3\int_{\mathbb R^3} F(t_u^{\frac{4\alpha-3}{2}}u)\,\mathrm{d}x \\
&\ge \frac{at_u^{2\alpha}}{4}\|(-\Delta)^{\alpha/2} u\|_2^2
 -C_2t_u^{\frac{6\alpha}{3-2\alpha}}\|u\|_{2^*_\alpha}^{2^*_\alpha} \\
&\ge \frac{at_u^{2\alpha}}{4}\|(-\Delta)^{\alpha/2} u\|_2^2
 -C_2S_\alpha^{-{\frac{2^*_\alpha}{2}}}t_u^{\frac{6\alpha}{3-2\alpha}}\|
 (-\Delta)^{\alpha/2} u\|_2^{2^*_\alpha} \\
&= \frac{a}{6}\Big(\frac{aS_\alpha^{\frac{2^*_\alpha}{2}}}{12C_2}
 \Big)^{\frac{3-2\alpha}{2\alpha}}, \quad \forall  u\in \mathcal{M}.
 \end{align*}
This shows that $m_0=\inf_{u\in \mathcal{M}}I(u)>0$.
\end{proof}


 \begin{lemma} \label{lem3.9}
 Assume that {\rm (A3)}  and {\rm (A4)} hold. 
If $u_n\rightharpoonup \bar{u}$ in $H^{\alpha}(\mathbb R^3)$, then along
 a subsequence of $\{u_n\}$, 
 \begin{equation}\label{N60}
 \lim_{n \to\infty}\sup_{\varphi\in H^{\alpha}(\mathbb R^3), 
\|\varphi\|\le 1}\big|\int_{\mathbb R^3}[f(u_n)-f(u_n-\bar{u})
 -f(\bar{u})]\varphi\,\mathrm{d}x\big|=0.
 \end{equation}
\end{lemma}

The above lemma was proved in \cite[Page 77-80]{Di} and a simpler 
and more direct proof has been given in \cite[Lemma 2.7]{TA}.

 \begin{lemma} \label{lem3.10}
 Assume that {\rm (A1),  (A3),  (A4), (A10)} hold. 
If $u_n\rightharpoonup \bar{u}$ in $H^{\alpha}(\mathbb R^3)$, then 
 \begin{gather}\label{F60}
 I(u_n)=I(\bar{u})+I(u_n-\bar{u})+\frac{\alpha b}{2}
\|(-\Delta)^{\alpha/2}\bar{u}\|_2^2\|(-\Delta)^{\alpha/2} (u_n-\bar{u})\|_2^2+o(1),\\
 \label{F62}
\begin{aligned}
 \langle I'(u_n), u_n\rangle
&=\langle I'(\bar{u}), \bar{u}\rangle+\langle I'(u_n-\bar{u}), u_n-\bar{u}\rangle \\
&\quad +2\alpha b\|(-\Delta)^{\alpha/2}\bar{u}\|_2^2
 \|(-\Delta)^{\alpha/2}(u_n-\bar{u})\|_2^2+o(1),
\end{aligned} \\
\label{F63}
 J(u_n)=J(\bar{u})+J(u_n-\bar{u})+2\alpha b\|(-\Delta)^{\alpha/2}\bar{u}\|_2^2
\|(-\Delta)^{\alpha/2} (u_n-\bar{u})\|_2^2+o(1).
 \end{gather}
\end{lemma}

 The proof of the above lemma is similar to the one in \cite[Lemma 2.10]{TC}, 
we omit it here.

\begin{lemma} \label{lem3.11}
Assume that {\rm (A3), (A4), (A11), (A12)} hold. Then $m^\infty$ is achieved.
\end{lemma}

 
\begin{proof}  Let $t\to 0$ in \eqref{B20}, then we have
 \begin{equation}\label{B70}
 f(w)w-2F(w)\ge 0, \quad  \forall  w\in \mathbb{R}.
 \end{equation}
We introduce a new functional $\Phi_\infty: H^{\alpha}(\mathbb R^3)\to \mathbb{R}$ as follows
 \begin{equation}\label{Ps}
\Phi_\infty(u) = \frac{a}{4}\|(-\Delta)^{\alpha/2} u\|_2^2
+\frac{4\alpha-3}{8\alpha}\int_{\mathbb R^3}[f(u)u-2F(u)]\,\mathrm{d}x.
 \end{equation}
For any $u\in \mathcal{M}^\infty$, we have $\Phi_\infty(u)=I^\infty(u)\ge m^\infty$.
 Let $\{u_n\}\subset \mathcal{M}^\infty$ be such that $I^\infty(u_n)\to m^\infty$. 
Since $J^\infty(u_n)=0$, then it
 follows from \eqref{IIn} with $t\to 0$ that
 \begin{equation}\label{B74}
 m^\infty+o(1)= I^\infty(u_n)\ge \frac{a}{4}\|(-\Delta)^{\alpha/2} u_n\|_2^2.
 \end{equation}
This shows that $\{\|(-\Delta)^{\alpha/2} u_n\|_2\}$ is bounded. 
Next, we prove that $\{\|u_n\|\}$ is also bounded.
 By \eqref{Cv}, \eqref{Jinf} and Sobolev embedding theorem, it holds
 \begin{equation}\label{B75}
\begin{aligned}
 &\min\{\alpha a, 2V_\infty\}\|u_n\|^2 \\
 & \le \int_{\mathbb R^3}(\alpha a|(-\Delta)^{\alpha/2} u_n|^2
 +2V_\infty u_n^2)\,\mathrm{d}x
 +\alpha b\Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u_n|^2\,\mathrm{d}x\Big)^2 \\
 & =  \frac{1}{2}\int_{\mathbb R^3}\left[(4\alpha-3)f(u_n)u_n+6F(u_n)\right]
 \,\mathrm{d}x  \\
 & \le \frac{1}{2}\min\{\alpha a, 2V_\infty\}\|u_n\|^2
 +C_3\|u_n\|_{2^*_\alpha}^{2^*_\alpha} \\
 & \le \frac{1}{2}\min\{\alpha a, 2V_\infty\}\|u_n\|^2
 +C_3S_\alpha^{-\frac{2^*_\alpha}{2}}\|(-\Delta)^{\alpha/2} u_n\|_2^{2^*_\alpha}.
\end{aligned}
 \end{equation}
This shows that $\{u_n\}$ is bounded in $H^{\alpha}(\mathbb R^3)$.
 By Lemma \ref{lem2.2}, one easily prove that
 $$
 \delta:=\limsup_{n\to\infty}\sup_{y\in \mathbb R^3}\int_{B_1(y)}|u_n|^2
\,\mathrm{d}x>0.
 $$
Going to a subsequence, if necessary, we may assume the existence of 
$\{y_n\}\subset \mathbb R^3$ such that
 $$
 \int_{B_1(y_n)}|u_n|^2\,\mathrm{d}x> \frac{\delta}{2}.
 $$
Let $\widetilde{u}_n(x)=u_n(x+y_n)$. Then $\|\widetilde{u}_n\|=\|u_n\|$,
 \begin{gather}\label{B79}
 \int_{B_1(0)}|\widetilde{u}_n|^2\,\mathrm{d}x> \frac{\delta}{2}, \\
\label{D71}
 I^\infty(\widetilde{u}_n)\to m^\infty, \ \ \ \ \ J^\infty(\widetilde{u}_n)= 0.
 \end{gather}
Then, there exists $\bar{u}\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$ such that
 \begin{equation}\label{D72}
 \begin{gathered}
 \widetilde{u}_n\rightharpoonup \bar{u}, \quad \text{in }  H^{\alpha}(\mathbb R^3); \\
 \widetilde{u}_n\to \bar{u}, \quad \text{in }  L_{\mathrm{loc}}^s(\mathbb R^3), \;
 \forall \ s\in [1, 2^*_\alpha);\\
 \widetilde{u}_n\to \bar{u}, \quad \text{a.e. on }  \mathbb R^3.
 \end{gathered}
 \end{equation}
Let $\hat{u}_n=\widetilde{u}_n-\bar{u}$. Then \eqref{D72} and Lemma \ref{lem3.10} yield
 \begin{gather}\label{D73}
 \Phi_\infty(\widetilde{u}_n)=\Phi_\infty(\bar{u})+\Phi_\infty(\hat{u}_n)+o(1),\\
\label{D74}
 J^\infty(\widetilde{u}_n) = J^\infty(\bar{u})+J^\infty(\hat{u}_n)
+2\alpha b\|(-\Delta)^{\alpha/2}\bar{u}\|_2^2\|(-\Delta)^{\alpha/2} 
\hat u_n\|_2^2+o(1).
 \end{gather}
From \eqref{Iinf}, \eqref{Jinf}, \eqref{D71}, \eqref{D73} and \eqref{D74}, one has
 \begin{equation}\label{D75}
 \Phi_\infty(\hat{u}_n)=m^\infty-\Phi_\infty(\bar{u})+o(1), \quad
J^\infty(\hat{u}_n) \le -J^\infty(\bar{u})+o(1).
 \end{equation}
If there exists a subsequence $\{\hat{u}_{n_i}\}$ of $\{\hat{u}_n\}$ such that 
$\hat{u}_{n_i}=0$, then going to this subsequence, we have
 \begin{equation}\label{D76}
 I^\infty(\bar{u})=m^\infty, \quad  J^\infty(\bar{u})=0,
 \end{equation}
which implies that Lemma \ref{lem3.11} holds. 
Next, we assume that $\hat{u}_n\ne 0$. In view of Lemma \ref{lem3.6},
 there exists $t_n>0$ such that $(\hat{u}_n)_{t_n}\in \mathcal{M}^\infty$. 
We claim that $J^\infty(\bar{u})\le 0$.
Otherwise, if $J^\infty(\bar{u})>0$, then \eqref{D75} implies 
$J^\infty(\hat{u}_n) < 0$ for large $n$. From \eqref{Iinf}, \eqref{Jinf}, \eqref{B31},
 \eqref{Ps} and \eqref{D75}, we obtain
 \begin{align*}
 m^\infty-\Phi_\infty(\bar{u})+o(1)
 & =  \Phi_\infty(\hat{u}_n) \\
 & =  \frac{a}{4}\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{4\alpha-3}{8\alpha}\int_{\mathbb R^3}[f(u)u-2F(u)]\,\mathrm{d}x \\
 & =  I^\infty(\hat{u}_n)-\frac{1}{4\alpha}J^\infty(\hat{u}_n) \\
 & \ge  I^\infty((\hat{u}_n)_{t_n})-\frac{t_n^{4\alpha}}{4\alpha}J^\infty(\hat{u}_n) \\
 & \ge  m^\infty-\frac{t_n^{4\alpha}}{4\alpha}J^\infty(\hat{u}_n)\ge m^\infty,
 \end{align*}
which implies $J^\infty(\bar{u})\le 0$ due to $\Phi_\infty(\bar{u})>0$. 
Since $\bar{u}\ne 0$, in view of Lemma \ref{lem3.6}, there exists
 $\bar{t}>0$ such that $\bar{u}_{\bar{t}}\in \mathcal{M}^\infty$. 
From \eqref{Iinf}, \eqref{Jinf}, \eqref{B31}, \eqref{B70},
 the weak semicontinuity of norm and Fatou's lemma, one has
 \begin{align*}
m^\infty
 & =  \lim_{n\to\infty} 
\big[I^\infty(\widetilde{u}_n)-\frac{1}{4\alpha}J^\infty(\widetilde{u}_n)\big] \\
 & =  \lim_{n\to\infty} \Big\{ \frac{a}{4}\|(-\Delta)^{\alpha/2} \widetilde{u}_n\|_2^2 +\frac{4\alpha-3}{8\alpha}\int_{\mathbb R^3}[f(\widetilde{u}_n)\widetilde{u}_n-2F(\widetilde{u}_n)]\,\mathrm{d}x\Big\} \\
 & \ge  \frac{a}{4}\|(-\Delta)^{\alpha/2} \bar{u}\|_2^2
 +\frac{1}{8\alpha}\int_{\mathbb R^3}[(4\alpha-3)f(\bar{u})\bar{u}-2F(\bar{u})]\,\mathrm{d}x \\
 & =  I^\infty(\bar{u})-\frac{1}{4\alpha}J^\infty(\bar{u}) \\
 & \ge  I(\bar{u}_{\bar{t}})
 -\frac{\bar{t}^{4\alpha}}{4\alpha}J^\infty(\bar{u}) \\
 & \ge  m^\infty-\frac{\bar{t}^{4\alpha}}{4\alpha}J^\infty(\bar{u})\ge m^\infty,
 \end{align*}
 which implies
 \begin{equation*}\label{B792}
 J^\infty(\bar{u})=0, \quad  I^\infty(\bar{u})=m^\infty.
 \end{equation*}
\end{proof}

 \begin{lemma} \label{lem3.12}
 Assume that {\rm (A1)--(A4), (A10)--(A12)} hold. 
If $\bar{u}\in \mathcal{M}$  and $I(\bar{u})=m_0$, then $\bar{u}$ is a critical 
point of $I$.
\end{lemma}

\begin{proof} 
 Assume that $I'(\bar{u})\ne 0$. Then there exist $\delta>0$ and $\varrho>0$ 
such that
 \begin{equation}\label{B81}
 \|u-\bar{u}\|\le 3\delta\Rightarrow \|I'(u)\|\ge \varrho.
 \end{equation}
Firstly, in the same way as \cite[Lemma 2.13]{TC}, we can prove that
 \begin{equation}\label{B821}
 \lim_{t\to 1}\|\bar{u}_t-\bar{u}\|=0.
 \end{equation}
Thus, there exists $\delta_1>0$ such that
 \begin{equation}\label{B822}
 |t^{\alpha}-1|<\delta_1\Rightarrow \|\bar{u}_t-\bar{u}\|< \delta.
 \end{equation}
In view of Lemma \ref{lem3.2}, one has
 \begin{equation}\label{B832}
\begin{aligned}
 I(\bar{u}_t)
 & \le  I(\bar{u})-\frac{a(1-\theta)(1-t^{2\alpha})^2}{4}
 \|(-\Delta)^{\alpha/2} \bar{u}\|_2^2  \\
 & =  m_0-\frac{a(1-\theta)(1-t^{2\alpha})^2}{4}\|(-\Delta)^{\alpha/2} \bar{u}\|_2^2,
 \quad  \forall  t> 0.
\end{aligned}
 \end{equation}
It follows from \eqref{B41} that there exist $T_1\in (0,1)$ and $T_2\in(1,\infty)$ 
such that
 $$
J(\bar{u}_{T_1})>0,\quad  J(\bar{u}_{T_2})<0.
 $$
Let
\begin{align*}
 \varepsilon:=\min\Big\{&\frac{a(1-\theta)(1-T_1^{2\alpha})^2\|(-\Delta)^{\alpha/2}
 \bar{u}\|_2^2}{24}, \\
&\frac{a(1-\theta)(1-T_2^{2\alpha})^2\|(-\Delta)^{\alpha/2} \bar{u}\|_2^2}{24},
 1, \varrho\delta/8\Big\}.
\end{align*}
 The rest of the proof is similar to the proof of \cite[Lemma 2.13]{TA2}.
\end{proof}

Now we can draw a conclusion on the existence of a ground state solution 
of Nehari-Pohozaev type to the ``limit problem" of Problem \eqref{KE}
\begin{equation}\label{lim}
 \begin{gathered}
 \Big(a+b\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)
 (-\Delta)^\alpha u+V_\infty u=f(u), \quad x\in \mathbb R^3; \\
 u\in H^{\alpha}(\mathbb R^3).
 \end{gathered}
\end{equation}

\begin{theorem} \label{thm3.13}
 Assume that $f$ satisfies {\rm  (A3), (A4), (A11), (A12)} hold. 
Then Problem \eqref{lim} has a solution 
$\bar{u}\in H^\alpha(\mathbb R^3)\backslash\{0\}$ such that 
$I^\infty(\bar{u})=\inf_{\mathcal{M}^\infty}I^\infty>0$.
\end{theorem}

 \begin{corollary} \label{coro3.14}
 Let $\widetilde{f}(t)=0$ for $t<0$ and $\widetilde{f}(t)=f(t)$
 for $t\ge 0$, use $\widetilde{f}$ to take place of $f$ in \eqref{lim} and assume
{\rm (A3), (A4), (A11), (A12)} hold, then Problem \eqref{lim} has a positive 
solution $\bar{u}\in  H^\alpha(\mathbb R^3)\backslash\{0\}$ such that 
$I^\infty(\bar{u})=\inf_{\mathcal{M}^\infty}I^\infty>0$.
\end{corollary}

\begin{lemma} \label{lem3.15}
 Assume that {\rm (A1)--(A4), (A10)--(A12)} hold. Then $m_0< m^\infty$.
\end{lemma}

\begin{proof} 
 In view of Theorem \ref{thm3.13} and Corollary \ref{coro3.14}, $I^\infty$ has a minimizer 
$u^\infty>0$ on $\mathcal{M}^\infty$, i.e.
 $$
u^\infty\in \mathcal{M}^\infty \quad \text{and}\quad  m^\infty=I^\infty(u^\infty).
$$
In view of Lemma \ref{lem3.6}, there exists $t_0>0$ such that 
$(u^\infty)_{t_0}\in\mathcal{M}$. Thus, it follows from (A2), 
\eqref{IU}, \eqref{Iinf}, \eqref{IIn} that
$$
m^\infty=I^\infty(u^\infty)\ge I^\infty((u^\infty)_{t_0})>I((u^\infty)_{t_0})\ge m_0.
$$
\end{proof}

\begin{lemma} \label{lem3.16}
Assume that {\rm (A1)--(A4), (A10)--(A12)} hold. Then $m_0$ is achieved.
\end{lemma}

The proof of this lemma is analogous to the one of \cite[Lemma 3.2]{TC}, 
so we omit it here.
 
\begin{proof}[Proof of Theorem \ref{thm1.2}]
It is a direct corollary of Lemmas \ref{lem3.8}, \ref{lem3.12} and \ref{lem3.16}.
\end{proof}


\section{Proof of Theorem \ref{thm1.3}}

To obtain the boundedness of the Palais-Smale sequence, we adopt a monotonicity 
technique due to Jeanjean.

 \begin{proposition}[\cite{JL}] \label{prop4.1}
Let $X$ be a Banach space and let $\Lambda\subset \mathbb R^+$ be an interval.
 We consider a family
 $\{\Phi_{\lambda}\}$ of $C^1$-functionals on $X$ of the form
 $$
 \Phi_{\lambda}(u)=A(u)-\lambda B(u), \quad \forall  \lambda\in J,
 $$
where $B(u) \ge 0$  for all $u \in X$, and such that either $A(u)\to +\infty$ 
or $B(u)\to +\infty$, as  $\|u\| \to \infty$. We assume that there are two 
points $v_1, v_2$ in $X$ such that
 \begin{equation}\label{cm}
 c_{\lambda}:=\inf_{\gamma\in \Gamma}\max_{t\in [0, 1]}
\Phi_{\lambda}(\gamma(t))>\max\{\Phi_{\lambda}(v_1), \Phi_{\lambda}(v_2)\},
 \end{equation}
where
 $$
 \Gamma=\{\gamma\in C([0, 1], X): \gamma(0)=v_1, \gamma(1)=v_2\}.
 $$
Then, for almost every $\lambda\in J$, there is a bounded (PS)$_{c_{\lambda}}$ 
sequence for $\Phi_{\lambda}$; that is,  there exists a sequence such that
\begin{itemize}
\item[(i)]  $\{u_n(\lambda)\}$ is bounded in $X$;

\item[(ii)]  $\Phi_{\lambda}(u_n(\lambda))\to c_{\lambda}$;

\item[(iii)]  $\Phi_{\lambda}'(u_n(\lambda))\to 0$ in $X^*$, where $X^*$ is 
the dual of $X$.

\end{itemize} 
\end{proposition}

 To apply Proposition \ref{prop4.1}, we introduce two families of perturbed functionals 
 \begin{equation}\label{Ilu}
\begin{aligned}
 I_{\lambda}(u)
&=\frac{1}{2}\int_{\mathbb R^3}(a|(-\Delta)^{\alpha/2} u|^2+V(x)u^2)
\,\mathrm{d}x \\
&\quad +\frac{b}{4}\Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)^2
 -\lambda\int_{\mathbb R^3}F(u)\,\mathrm{d}x
\end{aligned}
 \end{equation}
 and
 \begin{equation}\label{Il}
\begin{aligned}
 I_{\lambda}^{\infty}(u)
&=\frac{1}{2}\int_{\mathbb R^3}(a|(-\Delta)^{\alpha/2} u|^2
 +V_{\infty}u^2)\,\mathrm{d}x
 +\frac{b}{4}\Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)^2\\
&\quad -\lambda\int_{\mathbb R^3}F(u)\,\mathrm{d}x,
\end{aligned}
 \end{equation}
for $\lambda\in [1/2, 1]$.

 
 \begin{lemma}[\cite{Gu}]\label{lem4.2}
Assume that {\rm (A1)--(A4)} amd {\rm (A9)} hold. Let $u$ be a critical point of
 $I_{\lambda}$ in $H^{\alpha}(\mathbb R^3)$, then we have the following 
Pohozaev type identity
 \begin{equation}\label{Pl}
\begin{aligned}
 P_{\lambda}(u)
 &:=  \frac{(3-2\alpha)a}{2}\|(-\Delta)^{\alpha/2} u\|_2^2+\frac{1}{2}\int_{\mathbb R^3}\left[3V(x)+(\nabla V(x), x)\right]u^2\,\mathrm{d}x \\
 & \quad +\frac{(3-2\alpha)b}{2}\Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} 
u|^2\,\mathrm{d}x\Big)^2-3\lambda\int_{\mathbb R^3}F(u)\,\mathrm{d}x=0.
\end{aligned}
 \end{equation}
Note that $P_\lambda(u)=\frac{\,\mathrm{d}}{\,\mathrm{d}t}I_{\lambda}(u(x/t))|_{t=1}$.
\end{lemma}

We set  $J_{\lambda}(u):=\frac{4\alpha-3}{2}\langle I_{\lambda}'(u), 
u\rangle+P_{\lambda}(u)$, then
 \begin{equation}\label{JL}
\begin{aligned}
 J_{\lambda}(u)
 & =  \alpha a\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{1}{2}\int_{\mathbb R^3}[4\alpha V(x)+(\nabla V(x), x)]u^2\,\mathrm{d}x \\
 & \quad + \alpha b\|(-\Delta)^{\alpha/2} u\|_2^4
 -\frac{\lambda}{2}\int_{\mathbb R^3}\left[(4\alpha-3)f(u)u+6F(u)\right]\,
 \mathrm{d}x,
\end{aligned}
 \end{equation}
for $\lambda\in [1/2, 1]$. Correspondingly, we  let
 \begin{equation}\label{JlL}
\begin{aligned}
 J_{\lambda}^{\infty}(u)
 & =  \alpha a\|(-\Delta)^{\alpha/2} u\|_2^2+2\alpha V_{\infty}\|u\|_2^2
 +\alpha b\|(-\Delta)^{\alpha/2} u\|_2^4 \\
 & \quad -\frac{\lambda}{2}\int_{\mathbb R^3}\left[(4\alpha-3)f(u)u
 +6F(u)\right]\,\mathrm{d}x
\end{aligned}
 \end{equation}
 for $\lambda\in [1/2, 1]$. Set
 $$
 \mathcal{M}_{\lambda}^{\infty}:=\{u\in H^{\alpha}(\mathbb R^3)\setminus \{0\}: 
J_{\lambda}^{\infty}(u)=0\}, \quad
 m_{\lambda}^{\infty}:=\inf_{u\in \mathcal{M}_{\lambda}^{\infty}}
I_{\lambda}^{\infty}(u).
 $$

By Corollary \ref{coro3.3}, we have the following lemma.

 \begin{lemma} \label{lem4.3}
 Assume that {\rm (A3), (A4), (A12)} hold. Then
 \begin{equation}\label{G45}
 I_{\lambda}^{\infty}(u)
  \ge  I_{\lambda}^{\infty}(u_t)+\frac{1-t^{4\alpha}}{4\alpha}
J_{\lambda}^{\infty}(u)+\frac{a(1-t^{2\alpha})^2}{4}\|
(-\Delta)^{\alpha/2} u\|_2^2, 
\end{equation}
for all $u\in H^{\alpha}(\mathbb R^3)$, $t>0$, $\lambda\ge 0$.
\end{lemma} 

 In view of Theorem \ref{thm3.13} and Corollary \ref{coro3.14}, $I_1^{\infty}$
 has a minimizer $u_1^{\infty}>0$  on $\mathcal{M}_1^{\infty}$, i.e.,
 \begin{equation}\label{G50}
 u_1^{\infty}\in \mathcal{M}_1^{\infty}, \quad 
(I_1^{\infty})'(u_1^{\infty})=0, \quad 
 m_1^{\infty}=I_1^{\infty}(u_1^{\infty}).
 \end{equation}

Under the assumptions of Theorem \ref{thm1.2}, we first show that 
$I_\lambda$ with $\lambda\in [\frac{1}{2}, 1]$ has the mountain pass geometry.

 \begin{lemma} \label{lem4.4}
 Assume that {\rm (A1)--(A4), (A9), (A11), (A12)} hold. Then
\begin{itemize}
\item[(i)]  there exists $T_0 >0$ independent of $\lambda$ such that 
$I_{\lambda}((u_1^{\infty})_{T_0 })<0$
 for all $\lambda\in [1/2, 1]$;

 \item[(ii)]  there exists $\kappa_0>0$ independent of $\lambda$ such that for 
all $\lambda\in [\frac{1}{2}, 1]$,
 $$
c_{\lambda}:=\inf_{\gamma\in \Gamma}\max_{\varsigma\in [0, 1]}
I_{\lambda}(\gamma(\varsigma))>\max\{I_{\lambda}(0), I_{\lambda}((u_1^{\infty}
)_{T_0 })\}
$$
 where
 $$
 \Gamma=\{\gamma\in C([0, 1], H^{\alpha}(\mathbb R^3)):
 \gamma(0)=0, \gamma(1)=(u_1^{\infty})_{T_0}\};
 $$

 \item[(iii)]  $c_{\lambda}$ and $m_{\lambda}^{\infty}$
 are non-increasing on $\lambda\in [\frac{1}{2}, 1]$.
\end{itemize}
\end{lemma}

\begin{proof}  
(i) For fixed $u\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$ and any 
$\lambda\in [\frac{1}{2},1]$, one has
\begin{align*}
I_\lambda(u)\leq I_{\frac{1}{2}}(u)
&=\frac{1}{2}\int_{\mathbb R^3}(a|(-\Delta)^{\alpha/2} u|^2+V(x)u^2)\,\mathrm{d}x
 +\frac{b}{4}\Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)^2 \\
 &\quad -\frac{1}{2}\int_{\mathbb R^3}F(u)\,\mathrm{d}x.
\end{align*}
It is easy to verify that
\begin{align*}
I_{\frac{1}{2}}(u_t)
& =  \frac{at^{2\alpha}}{2}\|(-\Delta)^{\alpha/2} u\|_2^2
 +\frac{t^{4\alpha}}{2}\int_{\mathbb R^3}V(tx)u^2\,\mathrm{d}x
 +\frac{bt^{4\alpha}}{4}\|(-\Delta)^{\alpha/2} u\|_2^4 \\
& \quad -\frac{1}{2}t^3\int_{\mathbb R^3}F(t^{\frac{4\alpha-3}{2}}u)\,\mathrm{d}x.
\end{align*}
Note that $F(t^{\frac{4\alpha-3}{2}}u)/|t^{\frac{4\alpha-3}{2}}u|^2\to +\infty$.
 Then by Fatou's lemma, $I_{\frac{1}{2}}(u_t)\to -\infty$ as 
$t\to +\infty$. Then, combining Lemma \ref{lem3.6}, there exists $T_0>0$ sufficiently large, 
independent of $\lambda\in[\frac{1}{2},1]$, such that 
$I_{\lambda}((u_1^{\infty})_{T_0 })<0$.

(ii) In view of \eqref{Ilu}, \eqref{Cv} and the Sobolev embedding, 
there exist $\sigma_1, \sigma_2>0$ such that
 \begin{align*}
 I_{\lambda}(u) 
&\geq \frac{1}{2}\int_{\mathbb R^3}(a|(-\Delta)^{\alpha/2} u|^2+V(x)u^2)\,\mathrm{d}x
 +\frac{b}{4}\Big(\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x\Big)^2 \\
&\quad -\lambda\int_{\mathbb R^3}(\epsilon |u|^2+C_\epsilon|u|^p)\,\mathrm{d}x\\
&\geq  \sigma_1\|u\|^2-\sigma_2\|u\|^{p},
\end{align*}
Recalling that $p>2$, then there exist $\rho_1, \kappa_0>0$ independent of 
$\lambda$, such that for $\|u\|=\rho_1, I_\lambda(u)\geq \kappa_0$. 
For any $\lambda\in[\frac{1}{2},1]$ and $\gamma\in \Gamma$, it is easily 
seen that $\|\gamma(1)\|>\rho_1$. By continuity, there exists 
$\bar{\varsigma}\in(0,1)$ such that $\|\gamma(\bar{\varsigma})\|=\rho_1$, 
which implies that 
$$
c_{\lambda}\geq\inf_{\gamma\in \Gamma}I_{\lambda}
(\gamma(\bar{\varsigma}))\geq \kappa_0>\max\left\{I_{\lambda}(0), 
I_{\lambda}((u_1^{\infty})_{T_0 })\right\},  \quad 
\forall \lambda\in [\frac{1}{2},1].
$$

(iii) For any $u\in H^\alpha(\mathbb R^3)\backslash\{0\}$, since 
$c_\lambda\leq \max_{t>0}I_\lambda(u_t)\leq \max_{t>0}I_\frac{1}{2}(u_t)$ 
for all $\lambda\in [\frac{1}{2},1]$,
we obtain the conclusion.
\end{proof}

\begin{lemma} \label{lem4.5}
 Assume that {\rm  (A1)--(A4), (A9), (A11), (A12)}  hold. Then
 there exists $\lambda_1\in [\frac{1}{2}, 1)$ such that 
$c_{\lambda}<m_{\lambda}^{\infty}$ for $\lambda\in [\lambda_1, 1]$.
\end{lemma}


This lemma can be proved similarly as in \cite[Lemma 4.5]{TC}, 
we omit the proof here.
In what follows we use profile decomposition to obtain the compactness 
for any bounded (PS) sequence of the perturbed functional, which is 
crucial in our proof.

\begin{lemma} \label{lem4.6}
 Assume that {\rm (A1)--(A4), (A9), (A11)} hold. 
Let $\{u_n\}$ be a bounded (PS)$_c$
sequence for $I_{\lambda}$ with $\lambda\in [\frac{1}{2}, 1]$. 
Then there exist a subsequence of $\{u_n\}$, still denoted by
 $\{u_n\}$, and $u_0\in H^{\alpha}(\mathbb R^3)$ such that 
$B^2:=\lim_{n\to\infty}\|(-\Delta)^{\alpha/2} u\|_2^2$ exist, 
$u_n\rightharpoonup u_0$ in
$H^{\alpha}(\mathbb R^3)$ and $\mathcal{I}_{\lambda}'(u_0)=0$, where
 \begin{equation}\label{Up}
 \mathcal{I}_{\lambda}(u) 
 = \frac{a+bB^2}{2}\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x
 +\frac{1}{2}\int_{\mathbb R^3}V(x)u^2\,\mathrm{d}x
 -\lambda\int_{\mathbb R^3}F(u)\,\mathrm{d}x,
 \end{equation}
 and either
\begin{itemize}
\item[(i)]  $u_n\to u_0$ in $H^{\alpha}(\mathbb R^3)$; or
\item[(ii)]  there exist an integer $l\in \mathbb{N}$ and 
$w^1, \ldots, w^l\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$ such that
 $(\mathcal{I}_{\lambda}^{\infty})'(w^k)=0$ for $1\le k\le l$, and
\begin{gather*}
 c+\frac{bB^4}{4}= \mathcal{I}_{\lambda}(u_0)
+\sum_{k=1}^{l}\mathcal{I}_{\lambda}^{\infty}(w^k);
 B^2= \|(-\Delta)^{\alpha/2} u_0\|_2^2+\sum_{k=1}^{l}\|(-\Delta)^{\alpha/2}
 w^k\|_2^2,
\end{gather*}
 where
 \begin{equation}\label{Upi}
 \mathcal{I}_{\lambda}^{\infty}(u)
 = \frac{a+bB^2}{2}\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x
 +\frac{V_{\infty}}{2}\int_{\mathbb R^3}u^2\,\mathrm{d}x
 -\lambda\int_{\mathbb R^3}F(u)\,\mathrm{d}x.
 \end{equation}
\end{itemize}
\end{lemma}
 
 Since \eqref{N60} and \eqref{F60} hold, we can prove the above lemma as in
 \cite[Lemma 3.4]{LG}. We omit it here.

 \begin{lemma} \label{lem4.7}
 Assume that {\rm (A1)--(A4), (A9), (A11)} hold. Then for almost every
 $\lambda\in [\lambda_1,1]$, there exists 
$u_{\lambda}\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$ such that 
 \begin{equation}\label{D31}
 I_{\lambda}'(u_{\lambda})=0, \quad  I_{\lambda}(u_{\lambda}) = c_{\lambda}.
 \end{equation}
\end{lemma}

\begin{proof} 
Under Assumptions (A3), (A4), (A11), Lemma \ref{lem4.4} implies that 
$I_{\lambda}(u)$ satisfies the assumptions of Proposition \ref{prop4.1}
with $X=H^{\alpha}(\mathbb R^3)$ and $\Phi_{\lambda}=I_{\lambda}$. 
So for almost every $\lambda\in [1/2,1]$, there exists a bounded
 sequence $\{u_n(\lambda)\} \subset H^{\alpha}(\mathbb R^3)$ 
(for simplicity, we denote the sequence by $\{u_n\}$ instead of
 $\{u_n(\lambda)\}$ such that
 \begin{equation}\label{PS}
 I_{\lambda}(u_n)\to c_{\lambda}>0, \quad \|I_{\lambda}'(u_n)\| \to 0.
 \end{equation}
By Lemma \ref{lem4.6}, there exist a subsequence of $\{u_n\}$, still denoted by 
$\{u_n\}$, and $u_{\lambda}\in H^{\alpha}(\mathbb R^3)$ such
 that $B_{\lambda}^2:=\lim_{n\to\infty}\|(-\Delta)^{\alpha/2} u_n\|_2^2$ 
exists, $u_n\rightharpoonup u_{\lambda}$ in $H^{\alpha}(\mathbb R^3)$ and
 $(\widehat{\mathcal{I}}_{\lambda}) '(u_{\lambda})=0$, and either (i) or (ii) 
occurs, where
 \begin{equation}\label{Up1}
 \widehat{\mathcal{I}}_{\lambda}(u) 
= \frac{a+bB_{\lambda}^2}{2}\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x
 +\frac{1}{2}\int_{\mathbb R^3}V(x)u^2\,\mathrm{d}x
-\lambda\int_{\mathbb R^3}F(u)\,\mathrm{d}x.
\end{equation}

 If (ii) occurs, i.e. there exists $l\in \mathbb{N}$ and 
$w^1, \ldots, w^l\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$ such that
 $(\widehat{\mathcal{I}}_{\lambda}^{\infty})'(w^k)=0$ for $1\le k\le l$,
 \begin{gather}\label{Ab1}
 c_{\lambda}+\frac{bB_{\lambda}^4}{4}
= \widehat{\mathcal{I}}_{\lambda}(u_{\lambda})
+\sum_{k=1}^{l}\widehat{\mathcal{I}}_{\lambda}^{\infty}(w^k), \\
\label{Ab2}
 B_{\lambda}^2= \|(-\Delta)^{\alpha/2} u_{\lambda}\|_2^2
+\sum_{k=1}^{l}\|(-\Delta)^{\alpha/2} w^k\|_2^2,
 \end{gather}
 where
 \begin{equation}\label{Up2}
 \widehat{\mathcal{I}}_{\lambda}^{\infty}(u) 
= \frac{a+bB_{\lambda}^2}{2}\int_{\mathbb R^3}|(-\Delta)^{\alpha/2} u|^2\,\mathrm{d}x
 +\frac{1}{2}\int_{\mathbb R^3}V_{\infty}u^2\,\mathrm{d}x
 -\lambda\int_{\mathbb R^3}F(u)\,\mathrm{d}x.
 \end{equation}
 Since $(\widehat{\mathcal{I}}_{\lambda})'(u_{\lambda})=0$, then we have 
the Pohozaev identity of the functional $\widehat{\mathcal{I}}_{\lambda}$
 \begin{equation}\label{Plt}
\begin{aligned}
 \widetilde{P}_{\lambda}(u_{\lambda})
 & := \frac{(3-2\alpha)(a+bB_{\lambda}^2)}{2}\|(-\Delta)^{\alpha/2} 
 u_{\lambda}\|_2^2 \\
&\quad +\frac{1}{2}\int_{\mathbb R^3}\left[3V(x)
 +(\nabla V(x), x)\right]u_{\lambda}^2\,\mathrm{d}x 
 -3\lambda\int_{\mathbb R^3}F(u_{\lambda})\,\mathrm{d}x 
  = 0.
\end{aligned}
 \end{equation}
From (A9) and fractional Hardy inequality,
 \begin{equation}\label{D32}
 a\|(-\Delta)^{\alpha/2}u_{\lambda}\|_2^2 
\ge \frac{a\Gamma^2(\frac{1+2\alpha}{2})}{\pi}\int_{\mathbb R^3}
\frac{u_{\lambda}^2}{|x|^{2\alpha}}\,\mathrm{d}x
 \ge \frac{1}{2\alpha}\int_{\mathbb R^3}(\nabla V(x), x)u_{\lambda}^2\,\mathrm{d}x.
 \end{equation}
From \eqref{B70}, \eqref{Up1}, \eqref{Plt} and \eqref{D32} it follows that
 \begin{equation}\label{D33}
\begin{aligned}
 \widehat{\mathcal{I}}_{\lambda}(u_{\lambda})
 & =  \widehat{\mathcal{I}}_{\lambda}(u_{\lambda})-\frac{1}{4\alpha}
\big[\frac{4\alpha-3}{2}\langle\widehat{\mathcal{I}}_{\lambda}'(u_{\lambda}), 
u_{\lambda}\rangle  +\widetilde{P}_{\lambda}(u_{\lambda})\big] \\
& =  \frac{a+bB_{\lambda}^2}{4}\|(-\Delta)^{\alpha/2} u_{\lambda}\|_2^2
 -\frac{1}{8\alpha}\int_{\mathbb R^3}(\nabla V(x), x)u_{\lambda}^2\,\mathrm{d}x \\
 &\quad +\frac{\lambda(4\alpha-3)}{8\alpha}\int_{\mathbb R^3}
 [f(u_{\lambda})u_{\lambda}-2F(u_{\lambda})]\,\mathrm{d}x \\
& \ge  \frac{bB_{\lambda}^2}{4}\|(-\Delta)^{\alpha/2} u_{\lambda}\|_2^2.
\end{aligned}
 \end{equation}
Since $(\widehat{\mathcal{I}}_{\lambda}^{\infty})'(w^k)=0$, 
 we have the Pohozaev identity of the functional 
$\widehat{\mathcal{I}}_{\lambda}^{\infty}$,
 \begin{equation}\label{D34}
\begin{aligned}
 \widetilde{P}_{\lambda}^{\infty}(u_{\lambda})
&:= \frac{(3-2\alpha)}{2}(a+bB_{\lambda}^2)\|(-\Delta)^{\alpha/2}u_{\lambda}\|_2^2
 +\frac{3}{2}\int_{\mathbb R^3}V_{\infty}u_{\lambda}^2\,\mathrm{d}x \\
&\quad -3\lambda\int_{\mathbb R^3}F(u_{\lambda})\,\mathrm{d}x=0.
\end{aligned}
 \end{equation}
Thus, from \eqref{JlL}, \eqref{Ab2}, \eqref{Up2} and \eqref{D34}, we have
 \begin{equation}\label{D35}
\begin{aligned}
 0 & =  \frac{4\alpha-3}{2}\langle(\widehat{\mathcal{I}}_{\lambda}^{\infty})'(w^k),
  w^k\rangle+\widetilde{P}_{\lambda}^{\infty}(w^k) \\
& =  \alpha(a+bB_{\lambda}^2)\|(-\Delta)^{\alpha/2} w^k\|_2^2
 +2\alpha V_{\infty}\|w^k\|_2^2 \\
&\quad -\frac{\lambda}{2}\int_{\mathbb R^3}[(4\alpha-3)f(w^k)w^k+6F(w^k)
 ]\,\mathrm{d}x \\
& \ge  J_{\lambda}^{\infty}(w^k).
\end{aligned}
 \end{equation}
Since $w^k\in H^{\alpha}(\mathbb R^3)\setminus \{0\}$, in view of 
Lemma \ref{lem3.6}, there exists $t_k>0$ such that 
$(w^k)_{t_k}\in \mathcal{M}_{\lambda}^{\infty}$.
 From \eqref{Il}, \eqref{JlL}, \eqref{G45}, \eqref{Up2} and \eqref{D35}, 
one has
 \begin{equation}\label{D36}
\begin{aligned}
&\widehat{\mathcal{I}}_{\lambda}^{\infty}(w^k) \\
& =  \widehat{\mathcal{I}}_{\lambda}^{\infty}(w^k)
 -\frac{1}{4\alpha}\big[\frac{4\alpha-3}{2}
 \langle(\widehat{\mathcal{I}}_{\lambda}^{\infty})'
 (w^k), w^k\rangle+\widetilde{P}_{\lambda}^{\infty}(w^k)\big] \\
& = \frac{a+bB_{\lambda}^2}{4}\|(-\Delta)^{\alpha/2} w^k\|_2^2
 +\frac{\lambda(4\alpha-3)}{8\alpha}\int_{\mathbb R^3}[f(w^k)w^k
 -2F(w^k)]\,\mathrm{d}x \\
& =  \frac{bB_{\lambda}^2}{4}\|(-\Delta)^{\alpha/2} w^k\|_2^2
 +I_{\lambda}^{\infty}(w^k)
 -\frac{1}{4\alpha}J_{\lambda}^{\infty}(w^k) \\
& \ge  \frac{bB_{\lambda}^2}{4}\|(-\Delta)^{\alpha/2} w^k\|_2^2
 +I_{\lambda}^{\infty}((w^k)_{t_k})
 -\frac{t_k^{4\alpha}}{4\alpha}J_{\lambda}^{\infty}(w^k) \\
& \ge  \frac{bB_{\lambda}^2}{4}\|(-\Delta)^{\alpha/2} w^k\|_2^2
 +m_{\lambda}^{\infty}.
\end{aligned}
\end{equation}
 It follows from \eqref{Ab1}, \eqref{Ab2}, \eqref{D33} and \eqref{D36} that
 \begin{align*}
 c_{\lambda}+\frac{bB_{\lambda}^4}{4}
 & =  \widehat{\mathcal{I}}_{\lambda}(u_{\lambda})+\sum_{k=1}^{l}
 \widehat{\mathcal{I}}_{\lambda}^{\infty}(w^k)\\
 & \ge  lm_{\lambda}^{\infty}+\frac{bB_{\lambda}^2}{4}
 \big[\|(-\Delta)^{\alpha/2} u_{\lambda}\|_2^2+\sum_{k=1}^{l}\|
 (-\Delta)^{\alpha/2} w^k\|_2^2\big]\\
 & \ge  m_{\lambda}^{\infty}+\frac{bB_{\lambda}^4}{4}, \quad 
\forall  \lambda\in [\lambda_1, 1],
 \end{align*}
 which contradicts  Lemma \ref{lem4.5}. Thus $u_n\to u_{\lambda}$ 
in $H^{\alpha}(\mathbb R^3)$ and $I_{\lambda}(u_{\lambda}) = c_{\lambda}$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.3}]
 In view of Lemma \ref{lem4.7}, there exist two sequences of
 $\{\lambda_n\}\subset [\lambda_1, 1]$
 and $\{u_{\lambda_n}\}\subset H^{\alpha}(\mathbb R^3)$, denoted by 
$\{u_n\}$, such that
 \begin{equation}\label{Q00}
 \lambda_n\to 1, \quad  I_{\lambda_n}'(u_n)=0, \quad 
 I_{\lambda_n}(u_n) = c_{\lambda_n}.
 \end{equation}
From (A9), \eqref{FH}, \eqref{B70}, \eqref{Ilu}, \eqref{JL} and \eqref{Q00}, 
one has
 \begin{equation}\label{Q01}
\begin{aligned}
 c_{1/2} 
&\ge c_{\lambda_n}
 =I_{\lambda_n}(u_n)-\frac{1}{4\alpha}J_{\lambda_n}(u_n) \\
 &= \frac{a}{4}\|(-\Delta)^{\alpha/2} u_n\|_2^2-\frac{1}{8\alpha}
 \int_{\mathbb R^3}(\nabla V(x), x)u_n^2\,\mathrm{d}x  \\ 
&\quad +\frac{\lambda_n(4\alpha-3)}{8\alpha}\int_{\mathbb R^3}
 [f(u_n)u_n-2F(u_n)]\,\mathrm{d}x \\
 &\ge  \frac{(1-\theta)a}{4}\|(-\Delta)^{\alpha/2} u_n\|_2^2.
\end{aligned}
 \end{equation}
This shows that $\{\|(-\Delta)^{\alpha/2} u_n\|_2\}$ is bounded. 
Next, we show that $\{u_n\}$ is bounded in $H^{\alpha}(\mathbb R^3)$.
Combining Lemma \ref{lem2.1}, \eqref{Cv}, \eqref{Ilu}, \eqref{Q00}, \eqref{Q01} 
and the fractional Sobolev embedding theorem, we have
 \begin{align*}%\label{Q03}
 \omega_1'\|u_n\|^2
& \le  \int_{\mathbb R^3}\left[a|(-\Delta)^{\alpha/2} u_n|^2+V(x)u_n^2\right]
 \,\mathrm{d}x\\
& \le  2c_{\lambda_n}+2\lambda_n\int_{\mathbb R^3}F(u_n)\,\mathrm{d}x\\
& \le  2c_{1/2}+\frac{\omega_1'}{4}\|u_n\|^2+C_5\|u_n\|_{2^*_\alpha}^{2^*_\alpha} \\
& \le  2c_{1/2}+\frac{\omega_1'}{4}\|u_n\|^2
 +C_5S_\alpha^{-{\frac{2^*_\alpha}{2}}}\|(-\Delta)^{\alpha/2} u_n\|_2^{2^*_\alpha}.
 \end{align*}
where $\omega_1'>0$ is a constant. This shows that $\{u_n\}$ is bounded in 
$H^{\alpha}(\mathbb R^3)$. The rest of the proof is the same as the one in \cite{Gu},
 so we omit it.
\end{proof}

\subsection*{Acknowledgments}
This work is partially supported by the NNFC of China (No. 11501190, 11571370).
The authors are grateful to the referee for valuable comments and suggestions, 
which allows us to improve this article.

 \begin{thebibliography}{00}

 \bibitem{AC} C. O. Alves, F. Corr\'ea, T. F. Ma;
\emph{Positive solutions for a quasilinear elliptic equation of Kirchhoff type},
 Comput. Math. Appl., 49 (2005), 85-93.

 \bibitem{AI} V. Ambrosio, T. Isernia;
\emph{A multiplicity result for a fractional Kirchhoff equation in $\mathbb R^N$ 
with a general nonlinearity}, Comm. Contemporary Math., 2017, 1750054, 1-17.

 \bibitem{AA} A. Arosio, S. Panizzi;
\emph{On the well-posedness of the Kirchhoff string}, 
Trans. Amer. Math. Soc., 348 (1996), 305-330.

 \bibitem{BL} H. Berestycki, P. L. Lions;
\emph{Nonlinear scalar field equations I. Existence of a ground state}, 
Arch. Ration. Mech. Anal., 82 (1983), 313-346.

 \bibitem{BD} K. Bogdan, B.Dyda;
\emph{The best constant in a fractional Hardy inequality}, 
Math. Nachr., 284(2011), 629-638.

 \bibitem{CS} L. Caffarelli, L. Silvestre;
\emph{An extension problem related to the fractional Laplacian}, 
Comm. Partial Differential Equations, 32 (2007), 1245-1260.

 \bibitem{CX} X. J. Chang;
\emph{Ground States of some Fractional Schr\"odinger Equations on $\mathbb R^N$}, 
Proc. Edinburgh Math. Soc., 58 (2015), 305-321.

 \bibitem{CT} S. T. Chen, X. H. Tang;
\emph{Improved results for Klein-Gordon-Maxwell systems with general nonlinearity},
 Discrete Contin. Dyn. Syst., Ser. A, 38 (2018), 2333-2348.

 \bibitem{CBT} B. T. Cheng, X. Wu;
\emph{Existence results of positive solutions of Kirchhoff type problems}, 
Nonlinear Anal. 71 (2009), 4883-4892.

 \bibitem{DY} Y. B. Deng, S.J. Peng, W. Shuai;
\emph{Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type 
problems in $\mathbb R^3$}, J. Functional Analysis, 269 (2015), 3500-3527.

\bibitem{Di} Y. H. Ding;
\emph{Variational Methods for Strongly Indefinite Problems},
 World Scientific, Singapore, 2007.

 \bibitem{TJ} P. Felmer, A. Quaas, J. Tan;
\emph{Positive solutions of the nonlinear Schr\"odinger equation with the fractional 
Laplacian}, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.

\bibitem{FV} A. Fiscella, E. Valdinoci;
\emph{A critical Kirchhoff type problem involving a nonlocal operator}, 
Nonlinear Anal., 94 (2014), 156�C170.

\bibitem{Gu} Z. J. Guo;
\emph{Ground states for Kirchhoff equations without compact condition}, 
J. Differential Equations, 259 (2015), 2884-2902.

\bibitem{HZ2} X. M. He, W. M. Zou;
\emph{Existence and concentration behavior of positive solutions for a 
Kirchhoff equation in $\mathbb R^3$}, J. Differential Equations, 2 (2012), 
1813-1834.

\bibitem{JL} L. Jeanjean;
\emph{On the existence of bounded Palais-Smale sequence and application to
 a Landesman-Lazer  type problem set on $\mathbb R^N$}, 
Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999): 787-809.

 \bibitem{JH} H. Jin, W. Liu;
\emph{Fractional Kirchhoff equation with a general critical nonlinearity},
 Appl. Math. Lett., 74(2017), 140-146.

\bibitem{Kirchhoff} G. Kirchhoff;
\emph{Mechanik}, Teubner, leipzig, 1883.

\bibitem{LG} G. B. Li, H. Y. Ye;
\emph{Existence of positive ground state solutions for the nonlinear Kirchhoff 
type equations in $\mathbb R^3$}, J. Differential Equations, 257 (2014), 566-600.

 \bibitem{LJL} J. L. Lions;
\emph{On some questions in boundary value problems of mathematical physics}, 
in: Contemporary Developments
 in Continuum Mechanics and Partial Differential Equations, 
Proc. Internat. Sympos. Inst. Mat, Univ. Fed. Rio de Janeiro, 1997,
 in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, 284-346.

 \bibitem{LZ} Z. S. Liu, M. Squassina, J. J. Zhang;
\emph{Ground states for fractional Kirchhoff equations with critical nonlinearity 
in low dimension}, J. Nonlinear Differ. Equ. Appl., 24 (2017), 50.

 \bibitem{Hi} E. D. Nezza, G. Palatucci, E. Valdinoci;
\emph{Hitchhiker's guide to the fractional Sobolev spaces},
 Bull. Sci. Math., 136 (2012), 521-573.

 \bibitem{PXZ} P. Pucci, M. Q. Xiang, B. L. Zhang;
\emph{Multiple solutions for nonhomogeneous Schr\"odinger-Kirchhoff type equations 
involving the fractional p-Laplacian in $\mathbb R^ N$}, 
Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.

\bibitem{RD} D. Ruiz;
\emph{The Schr\"odinger-Poisson equation under the effect 
of a nonlinear local term}, J. Funct. Anal., 237 (2006), 655-674.

\bibitem{SS} S. Secchi;
\emph{Ground state solutions for nonlinear fractional Schr\"odinger equations 
in $\mathbb R^N$}, J. Math. Phys., 54 (2013), 031501.

\bibitem{Ta} X. H. Tang, B. T. Cheng;
\emph{Ground state sign-changing solutions for Kirchhoff type problems in 
bounded domains}, J. Differential Equations, 261 (2016), 2384-2402.

\bibitem{TC} X. H. Tang, S. T. Chen;
\emph{Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems
 with general potentials}, Calc. Var. Partial Differential Equations, 56 (2017), 110.

\bibitem{TA} X. H. Tang, S. T. Chen;
\emph{Ground state solutions of Nehari-Pohozaev type for Schr\"odinger-Poisson
 problems with general potentials}, Discrete Contin. Dyn. Syst., 37 (2017), 
4973-5002.

\bibitem{TA2} X. H. Tang, S. T. Chen;
\emph{Berestycki-Lions conditions on ground state solutions for a Nonlinear 
Schr\"odinger equation with variable potentials}, 
arXiv preprint, arXiv:1803.01130, 2018.

\bibitem{TLY} X. H. Tang, X. Y. Lin, J. S. Yu;
\emph{Nontrivial solutions for Schr\"odinger equation with local super-quadratic 
conditions}, J. Dyn. Differ. Equ., (2018), 1-15.

 \bibitem{WM} M. Willem;
\emph{Minimax Theorems, Birkh\"auser}, Boston, 1996.

 \bibitem{Y} H. Y. Ye;
\emph{Positive high energy solution for Kirchhoff equation in $\mathbb R^3$ 
with superlinear nonlinearity via Nehari-Pohozaev manifold}, 
Discrete Contin. Dyn. Syst., 35 (2015), 3857-3877.

 \bibitem{Zh1} B. L. Zhang, M. Fereara;
\emph{Multiplicity of solutions for a class of superlinear non-local 
fractional equations}, Calc. Var. Partial Differential Equations, 60 (2015), 583-595.

 \bibitem{ZTZ} W. Zhang, X. H. Tang, J. Zhang;
\emph{Infinitely many radial and non-radial solutions for a fractional 
Schr\"odinger equation}, Compu. Math. App., 71 (2016), 737-747.

\end{thebibliography}

\end{document}