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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 127, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/127\hfil Ground states]
{Ground states for the fractional Schr\"{o}dinger equation}

\author[B. Feng \hfil EJDE-2013/127\hfilneg]
{Binhua Feng}  % in alphabetical order

\address{Binhua Feng \newline
School of Mathematics and Statistics, Lanzhou University\\
Lanzhou 730000,  China\newline
Tel: +86-0931-8912483; Fax: +86-0931-8912481}
\email{binhuaf@163.com}

\thanks{Submitted November 2, 2012. Published May 27, 2013.}
\subjclass[2000]{35J60, 35Q55}
\keywords{Fractional Laplacians; nonlinear Schr\"{o}dinger
equation; ground states; \hfill\break\indent  concentration compactness}

\begin{abstract}
 In this article, we show the existence of ground state solutions
 for the nonlinear Schr\"{o}dinger equation with fractional Laplacian
 \[
 (-\Delta )^\alpha u+ V(x)u =\lambda
 |u|^{p}u\quad\text{in $\mathbb{R}^N$ for $\alpha \in (0,1)$}.
 \]
 We use the concentration compactness principle in fractional
 Sobolev spaces $H^\alpha$ for $\alpha \in (0,1)$.
 Our results  generalize the corresponding results in the case $\alpha =1$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

This article is devoted to the study of existence of ground
state solutions for the  fractional nonlinear
Schr\"{o}dinger equation
\begin{equation}\label{1.1'}
\begin{gathered}
(-\Delta )^\alpha u+ V(x)u  =\lambda |u|^{p}u \quad
\text{in }\mathbb{R}^N,\\
u\in H^\alpha(\mathbb{R}^N),\quad u\neq 0,
\end{gathered}
\end{equation}
where $0<\alpha <1$, $0<p<\frac{2N}{N-2\alpha}$, $N\geq 2$,
$\lambda >0$ and $V$ is a positive continuous function. The fractional
Laplacian can be characterized as
$\mathcal{F}((-\Delta)^\alpha u )(\xi)=|\xi|^{2\alpha}\mathcal{F}(u)(\xi)$,
 where $\mathcal{F}$ is the Fourier transform.

Equation \eqref{1.1'} arises in the study of the fractional
Schr\"{o}dinger equation
\begin{equation}\label{1.2}
i\psi_{t}+(-\Delta )^\alpha \psi+ (V(x)+\omega)\psi=\lambda |\psi|^{p}\psi ,
\end{equation}
when looking for standing waves solutions that have the form
$\psi(x,t)=e^{i\omega t}u(x)$, where $\omega \in \mathbb{R}$ and
$u \in H^\alpha$, $u\neq 0$. This equation is a fundamental equation of
fractional quantum mechanics; see, e.g., \cite{La1,La3,La2}.
 The fractional quantum mechanics has been
discovered as a result of expanding the Feynman path integral, from
the Brownian-like to the L\'{e}vy-like quantum mechanical paths.
In recent years, more and more attention has been focusing on the study of the
fractional Schr\"{o}dinger equation \eqref{1.2} from a pure
mathematical point of view, see \cite{DX,GH,GX,OCJ,OJ,WX} and the
references therein.

 As is known, for $\alpha =1$ the L\'{e}vy motion becomes Brownian
motion and the fractional Schr\"{o}dinger equation becomes the
standard Schr\"{o}dinger equation
\begin{equation}\label{1.1}
i\psi_{t}+\Delta \psi+(V(x)+\omega)\psi=\lambda |\psi|^{p}\psi.
\end{equation}
The corresponding elliptic equation reads
\begin{equation}\label{1.1a}
-\Delta u+V(x)u=\lambda |u|^{p}u.
\end{equation}
 There exist a considerable
amount of results in the physics and mathematics literature for
those classical Schr\"{o}dinger equations. In particular, the
standing wave solutions have been investigated by many authors. We
only refer to \cite{BL,LL,Li,We,Ca2003}.

Recently, fractional nonlinear
Schr\"{o}dinger equation \eqref{1.1'} has begun to receive increasing attention; for example, see \cite{Ch,DPV,FQT,guo}. In \cite{FQT}, by using
mountain pass lemma, the authors derived the existence of positive
solutions to nonlinear fractional Schr\"{o}dinger equation
\begin{equation} \label{1.3}
(-\Delta )^\alpha u+u=f(x,u).
\end{equation}
In particular, they used a comparison argument to overcome the
difficulty that the Palais-Smale sequences might lose compactness in
the whole space $\mathbb{R}^N$. They also analyzed regularity,
decay, and symmetry properties of these solutions. Those results
heavily rely on the representation formula
\[
u=\mathcal{K}*f=\int_{\mathbb{R}^N}\mathcal{K}(x-\xi)f(\xi)d\xi,
\]
for solutions of the equation
\[
(-\Delta )^\alpha u+u  =f\quad\text{in }\mathbb{R}^N,
\]
where $\mathcal{K}$ is the Bessel kernel
\[
\mathcal{K}=\mathcal{F}^{-1}\Big(\frac{1}{1+|\xi|^{2\alpha}}\Big).
\]
However, there is not a similar representation formula for equation
\eqref{1.1'} with general potential $V$.
In \cite{DPV}, following the argument of \cite{BL}, the existence
and symmetry results for bound state solutions to \eqref{1.1'} with
$V\equiv 1$ have been obtained by applying symmetric decreasing
rearrangement. However, this method fails to work for equation
\eqref{1.1'} except that potential $V$ is a spherically symmetric
function. The existence of bound state solutions to \eqref{1.1'}
with unbounded potential have been derived by Lagrange multiplier
method and Nehari's manifold approach in \cite{Ch}. It is worth
noticing that under the assumption that potential $V(x)\to
\infty$ as $|x|\to\infty$, the embedding
$H\hookrightarrow L^q(\mathbb{R}^N)$ is compact, where
$H=\{u \in L^2;\int_{\mathbb{R}^N}|\xi|^{2\alpha}|\hat{u}(\xi)|^2d\xi+\int
_{\mathbb{R}^N} V(x)|u(x)|^2 dx< \infty\}$ and
$2\leq q<\frac{2N}{N-2\alpha}$. However, when potential $V$ is a bounded
function, it is well known that the embedding $H\hookrightarrow
L^q(\mathbb{R}^N)$ is not compact. Therefore, how to overcome lack of
compactness in variational problem, which is of
particular interest, is one of main technique challenges in this
paper.

Motivated by the above discussion, the goal of the current paper is
to consider the existence of ground states to \eqref{1.1'} with a
bounded potential $V$. Equation \eqref{1.1'} involves the fractional
Laplacian $(-\Delta)^\alpha$, $0<\alpha <1$, which is a nonlocal
operator. A general approach to deal with this problem is to
transform (1.1) into a local problem via the Dirichlet-Neumann map,
see \cite{CS,Si}. That is, for $u\in H^\alpha$, one considers the
problem
\begin{equation}
\begin{gathered}
-\operatorname{div}(y^{1-2\alpha }\nabla v)=0 \quad\text{in }\mathbb{R}^{N+1}_{+},\\
v(x,0) =u \quad \text{on }\mathbb{R}^N,
\end{gathered}
\end{equation}
from where the fractional Laplacian is obtained by
\[
(-\Delta )^\alpha u(x)=-b_{\alpha}\lim_{y\to
0^+}y^{1-2\alpha}v_y,
\]
where $b_{\alpha}$ is an appropriate constant.

In this article, we prefer to investigate equation \eqref{1.1'}
directly in $H^\alpha(\mathbb{R}^N)$. This enables us to prove the
existence of solutions to \eqref{1.1'} by an analogue argument as the
case $\alpha=1$. It is well known that concentration compactness
principle due to Lions is a powerful tool which is designed to pass
to limit in variational problem with lack of compactness.
Recently, a version of concentration compactness of Lions was used in \cite{FQT} to treat fractional nonlinear
Schr\"{o}dinger equation \eqref{1.3}. Thus, there is no doubt that the method can be adapted and applied to deal with our problem.

 Firstly, we consider the case where $V$ is
a constant. Without loss of generality, we may assume $\lambda =1$ and
consider the following minimization problem:
\begin{equation}\label{402}
-c =\inf\{E(u);~u\in M\},
\end{equation}
where the constricted set is
\[
M:=\{u\in H^\alpha;\int_{\mathbb{R}^N}|u(x)|^{2}dx=\mu \},
\]
and energy is
\[
E(u)=\frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}(\xi)|^2d\xi-\frac{1}{p+2}\int_{\mathbb{R}^N}|u(x)|^{p+2}dx.
\]
Any minimizer $u \in H^\alpha$ of \eqref{402} has to satisfy the
corresponding Euler-Lagrange equation
\begin{equation}\label{402'}
(-\Delta )^\alpha u+ \omega u  = |u|^{p}u,
\end{equation}
with some Lagrange multiplier $\omega >0$. Notice that the existence
of bound state solutions of \eqref{402'} have been obtained in \cite{DPV},
 by using concentration compactness principle in fractional Sobolev spaces, we
obtain the existence of ground state solutions to \eqref{402'}. More
precisely, our result is as follows:

\begin{theorem} \label{thm1.1}
Let $0<p<4\alpha/N$, $N\geq 2$, and $\mu >0$.
 Then, the minimizing problem \eqref{402} has a positive ground
state solution $u \in H^\alpha$,   and it satisfies \eqref{402'} for some
$\omega >0$. Moreover,   every minimizing sequence $(u_n)_{n>0}$ of
\eqref{402} is relatively compact in
  $H^\alpha$ up to translations; i.e., there exist a subsequence
  $(u_{n_k})_{k>0}$
and $(y_k)_{k>0}\subset \mathbb{R}^N$ such that
$u_{n_k}(\cdot-y_k)\to u$ in $H^\alpha$ as $k\to
\infty$. In particular, $u$ is a solution of \eqref{402}.
\end{theorem}

Next, let us consider general situation.
Let $V:\mathbb{R}^N \mapsto \mathbb{R}$ be a continuous function $V>0$
and assume that
\begin{equation}\label{302}
 \lim_{|x|\to\infty }V(x)=V_\infty >0.
\end{equation}
Note that solutions of \eqref{1.1'} correspond to critical points of
the functional
\begin{equation}\label{3022}
E_1(u)=\frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}(\xi)|^2d\xi+\frac{1}{2}\int_{\mathbb{R}^N}V(x)|u(x)|^2dx
\end{equation}
on $H^\alpha$, restricted to the unit sphere
\[
M_1=\{u\in H^\alpha;\int_{\mathbb{R}^N}|u(x)|^{p+2}dx=1\}
\]
in $L^{p+2}$. In addition, if $V(x)=V_\infty $, then $E_1 $ is
invariant under translations
\[
u\mapsto u_{x_0}=u(x-x_0).
\]
In general, for any $u\in H^\alpha$, after a substitution of
variables
\begin{align*}
E_1(u_{x_0})
&=\frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}|^2d\xi+\frac{1}{2}\int_{\mathbb{R}^N}V(x+x_0)|u|^2dx\\
&\to \frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}|^2d\xi+\frac{1}{2}\int_{\mathbb{R}^N}V_\infty |u|^2dx
\end{align*}
as $|x_0|\to\infty$. Hence, we call
\[
E_1^\infty (u):=\frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}|^2d\xi+\frac{1}{2}\int_{\mathbb{R}^N}V_\infty |u|^2dx.
\]
the functional at infinity associated with $E_1$.

\begin{theorem} \label{thm1.2}
Let $0<p<4\alpha /(N-2\alpha)$, $N\geq 2$ and $V$ satisfy \eqref{302}.
Assume
\begin{equation}\label{303}
c=\inf_{u\in M_1}E_1 (u)< \inf_{u\in M_1}E_1^\infty (u):=c^\infty.
\end{equation}
Then, there exists a positive solution $u\in H^\alpha$ of
\eqref{1.1'} for some $\lambda >0$. Moreover, condition \eqref{303}
 is necessary and
sufficient for the relative compactness of all minimizing
sequences for $E_1$ in $M_1$.
\end{theorem}

\noindent\textbf{Notation.} Throughout this paper,
$C> 0$ will stand for a constant that may be different from
line to line when it does not cause any confusion. Since we deal
with $\mathbb{R}^N$, we often use the abbreviations
$L^{r}=L^{r}(\mathbb{R}^N)$, $H^\alpha =H^\alpha(\mathbb{R}^N)$.

\section{Preliminaries}

In this section, we state and prove some preliminary results that
will be used later. First, we recall some definitions
about the fractional Laplacian operator. For any $\alpha
\in(0,1)$, the fractional Sobolev space $H^\alpha$ is defined by
\[
H^\alpha=\big\{u\in
L^2;\int_{\mathbb{R}^N}(1+|\xi|^{2\alpha})|\hat{u}(\xi)|^2d\xi<\infty
\big\},
\]
endowed with the norm
\[
\|u\|_{H^\alpha}=\big\{\int_{\mathbb{R}^N}
|u|^2dx+\iint_{\mathbb{R}^N\times \mathbb{R}^N
}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy\big\}^{1/2},
\]
where the symbol $\hat{}$ stands for Fourier transform and the term
\[
[u]_{H^\alpha}=\|(-\Delta)^{\frac{\alpha}{2}}u\|_{L^2}
:=\Big\{\iint_{\mathbb{R}^N\times \mathbb{R}^N
}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy\Big\}^{1/2}
\]
is the so-called Gagliardo semi-norm of $u$. Therefore,
we often use the explicit formula
\[
\langle (-\Delta)^\alpha u,u\rangle :=\iint_{\mathbb{R}^N\times
\mathbb{R}^N
}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy.
\]
Next, we recall the definition of weak solutions $u\in H^\alpha$ to
\eqref{1.1'}.

\begin{definition} \label{def2.1} \rm
We say that $u\in H^\alpha$ is a weak solution of \eqref{1.1'} if
\[
\int_{\mathbb{R}^N}|\xi|^{2\alpha
}\hat{u}\hat{\varphi}d\xi+\int_{\mathbb{R}^N}V(x)u\varphi dx =
\int_{\mathbb{R}^N} |u|^p u\varphi dx,
\]
for any $\varphi \in C_c^\infty(\mathbb{R}^N)$.
\end{definition}

 Following the argument of \cite[Theorem 7.13]{LL},
we can obtain the following lemma
which can be used to derive the existence of positive solution to
\eqref{1.1'}.

\begin{lemma} \label{lem2.2}
Let $f,g$ are two real-valued functions in $H^\alpha$ with $f\neq
0$. Then
\[
\big\langle (-\Delta)^\alpha \sqrt{f^2+g^2},\sqrt{f^2+g^2}\big\rangle
\leq
\langle (-\Delta)^\alpha f,f\rangle+\langle (-\Delta)^\alpha g,g\rangle.
\]
Equality holds if and only if $f$ has a definite sign and
$g(x)=Cf(x)$ for a.e $x\in \mathbb{R}^N$ for some constant $C$.
\end{lemma}

By the similar proof as \cite[Theorem 7.16]{LL}, we have the following lemma:

\begin{lemma} \label{lem2.3}
Let $\psi$ be a bounded function in $C^\infty(\mathbb{R}^N)$ with
bounded derivatives and $f \in H^\alpha$. Then the pointwise product
of $\psi$ and $f$,
\[
(\psi \cdot f)(x)=\psi(x) f(x),
\]
is also a function in $H^\alpha$ and
\[
\|\psi f\|_{H^\alpha} \leq C(\|\psi\|_{L^\infty}+\|\nabla
\psi\|_{L^\infty})\|f\|_{H^\alpha}.
\]
\end{lemma}

Finally, we extend the concentration compactness principle in
$H^1$ to fractional Sobolev spaces $H^\alpha$. The proof follows the
argument of \cite[Proposition 1.7.6]{Ca2003}, but we need to
operate some modifications due to the non-locality of the fractional
operators $(-\Delta)^\alpha$.

\begin{lemma} \label{lem2.4}
Let $N\geq 2$. Suppose $(u_n)_{n> 0}\subset H^\alpha$ and satisfy
\begin{gather}\label{201}
\int _{\mathbb{R}^N} |u_n(x)|^2 dx =\mu >0,\\
\label{202}
\sup_{n>0}\|u_n\|_{H^\alpha}<\infty .
\end{gather}
Then there exists a subsequence $(u_{n_k})_{k> 0}$, for which one of
the following properties holds.
\begin{itemize}
\item[(i)] Compactness: There exists a sequence $(y_k)_{k> 0}$ in
$\mathbb{R}^N$ such that, for any $\varepsilon >0$, there exists $0<r<\infty $
with
\begin{equation}\label{203}
\int _{|x-y_k|\leq r} |u_{n_k}(x)|^2 dx \geq \mu -\varepsilon.
\end{equation}

\item[(ii)] Vanishing: For all $r<\infty$, it follows that
\begin{equation*}
\lim_{k\to\infty }\sup_{y\in \mathbb{R}^N}\int _{|x-y|\leq
r} |u_{n_k}(x)|^2 dx =0.
\end{equation*}

\item[(iii)] Dichotomy: There exist a constant $\beta \in (0,\mu )$ and two
bounded sequences $(v_k)_{k> 0}$, $(w_k)_{k> 0}\subset H^\alpha$ such that
\begin{gather}\label{204}
\operatorname{supp}v_k\cap \operatorname{supp} w_k=\emptyset, \\
\label{204'}
|v_k|+|w_k|\leq |u_{n_k}|, \\
\label{205}
\|v_k\|_{L^2}^2\to \beta,\quad \|w_k\|_{L^2}^2\to(\mu -\beta )
\quad\text{as } k\to\infty, \\
\label{206}
\|u_{n_k}-v_k-w_k\|_{L^p}\to 0 \quad \text{for }2\leq p < \frac{2N}{N-2\alpha},\\
\label{207}
\liminf_{k\to\infty}\big\{\langle (-\Delta)^\alpha
u_{n_k}, u_{n_k}\rangle-\langle (-\Delta)^\alpha v_k,
v_k\rangle-\langle (-\Delta)^\alpha w_k, w_k\rangle\big\}\geq 0.
\end{gather}
\end{itemize}
\end{lemma}

\begin{proof}
We proceed along the lines of \cite[Proposition 1.7.6]{Ca2003}.
Let $(u_n)_{n> 0}\subset H^\alpha$ satisfy \eqref{201}
and \eqref{202}. We define the sequence, $(Q_n)_{n> 0}$, of L\'{e}vy
concentration functions by
\begin{equation}\label{208}
Q_n(r):=\sup_{y\in \mathbb{R}^N}\int
_{|x-y|<r}|u_n(x)|^2dx,\quad \text{for all }r \geq 0.
\end{equation}
By a similar argument as that of
\cite[Proposition 1.7.6]{Ca2003}, there exists a subsequence, 
$(Q_{n_{k}})_{k> 0}$, such that
\[
Q_{n_{k}}(r)\to Q(r)\quad\text{as $k\to\infty$ for all $r\geq 0$},
\]
where $Q(r)$ is a nonnegative, nondecreasing function. Clearly, we
see that
\begin{equation}\label{209}
\beta:=\lim_{r\to\infty}Q(r)\in [0,\mu ].
\end{equation}
If $\beta=0$, then the vanishing property is a direct consequence of the
definition of $Q_n(r)$. If $\beta =\mu$, then compactness holds, see
\cite{Ca2003} for details.

Assume that $\beta \in (0,\mu)$ holds. We fix $\phi, \varphi \in
C^\infty (\mathbb{R}^N)$ such that $0\leq \phi, \varphi \leq 1$ and
\begin{gather*}
\phi (x)\equiv 1\quad\text{for }0\leq |x|\leq 1, \quad
\phi (x)\equiv 0\quad\text{for }|x|\geq 2, \\
\varphi (x)\equiv 0 \quad\text{for }0\leq |x|\leq 2, \quad
\varphi (x)\equiv 1\quad\text{for }|x|\geq 3.
\end{gather*}
We set $v_k=\phi _k u_{n_k}$ and $w_k=\varphi _k u_{n_k}$, where
\[
\phi _k(x)=\phi (\frac{x- y_k}{r_k}),\quad
\varphi_k(x)=\varphi(\frac{x- y_k}{r_k}).
\]
Therefore, properties \eqref{204} and \eqref{204'} are immediate.
By using Lemma \ref{lem2.3}, we find that $(v_k)_{k>0}$ and
$(w_k)_{k>0}$ are two bounded sequences $H^\alpha$. Following the
argument of \cite[Proposition 1.7.6]{Ca2003}, we can obtain
\eqref{205}. \eqref{206} follows by interpolation inequality.
Finally, we show \eqref{207}. One easily verifies that
\begin{equation} \label{210}
\begin{aligned}
|v_k(x)-v_k(y)|^2&=|\phi _k(x) u_{n_k}(x)-\phi _k(y) u_{n_k}(y)|^2
 \\
&\leq  \frac{1}{2}|\phi _k(x)-\phi
_k(y)|^2(|u_{n_k}(x)|^2+|u_{n_k}(y)|^2)  \\
&\quad +\frac{1}{2}(|\phi _k(x)|^2+|\phi
_k(y)|^2)|u_{n_k}(x)-u_{n_k}(y)|^2.
\end{aligned}
\end{equation}
Similarly,
\begin{equation} \label{211}
\begin{aligned}
|w_k(x)-w_k(y)|^2 &\leq  \frac{1}{2}|\varphi_k(x)-\varphi
_k(y)|^2(|u_{n_k}(x)|^2+|u_{n_k}(y)|^2)  \\
 &\quad +\frac{1}{2}(|\varphi _k(x)|^2+|\varphi
_k(y)|^2)|u_{n_k}(x)-u_{n_k}(y)|^2.
\end{aligned}
\end{equation}
Combining \eqref{210} and \eqref{211}, we derive
\begin{equation} \label{212}
\begin{aligned}
&\langle (-\Delta)^\alpha u_{n_k}, u_{n_k}\rangle-\langle
(-\Delta)^\alpha v_k, v_k\rangle-\langle (-\Delta)^\alpha w_k,
w_k\rangle \\
&= \frac{1}{2}\iint_{\mathbb{R}^N\times \mathbb{R}^N
}\frac{|u_{n_k}(x)-u_{n_k}(y)|^2-|v_k(x)-v_k(y)|^2
-|w_k(x)-w_k(y)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy  \\
&\geq -\iint_{\mathbb{R}^N\times
\mathbb{R}^N }\frac{|\varphi_k(x)-\varphi
_k(y)|^2|u_{n_k}(x)|^2+|\phi _k(x)-\phi
_k(y)|^2|u_{n_k}(x)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy.
\end{aligned}
\end{equation}
Therefore,  to prove \eqref{207}, it suffices to show that the
last term in \eqref{212} converges to zero as $k$ approaches infinity.
Indeed, note the fact that
$\operatorname{supp} \phi _k \cap \operatorname{supp} \varphi_k
=\emptyset$, by mean value theorem, we can estimate as follows:
\begin{align*}
&\iint_{\mathbb{R}^N\times \mathbb{R}^N}\frac{|\phi_k(x)-\phi
 _k(y)|^2|u_{n_k}(x)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy \\
&\leq \int_{|x-y|\leq r_k }\frac{|\phi_k(x)-\phi
 _k(y)|^2|u_{n_k}(x)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy\\
&\quad +\int_{|x-y|>r_k}\frac{|\phi_k(x)-\phi
 _k(y)|^2|u_{n_k}(x)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy \\
&\leq \frac{1}{r_k^2}\int_{|x-y|\leq r_k
 }\frac{|u_{n_k}(x)|^2}{|x-y|^{N+2\alpha-2}}\,dx\,dy\\
&\quad +\frac{1}{r_k^\alpha}
 \int_{|x-y|> r_k}\frac{|\phi_k(x)-\phi
 _k(y)|^2|u_{n_k}(x)|^2}{|x-y|^{N+\alpha}}\,dx\,dy
 \\
&\leq \frac{1}{r_k^2}\int_{\mathbb{R}^N}|u_{n_k}(x)|^2dx
 \int_{|z|\leq r_k }|z|^{-(N+2\alpha-2)}dz\\
&\quad +\frac{C}{r_k^\alpha}
 \int_{\mathbb{R}^N}|u_{n_k}(x)|^2dx \int_{|x-y|>
 r_k}\frac{1}{|x-y|^{N+\alpha}}dy
 \\
&\leq C(r_k^{-2\alpha}+r_k^{-\alpha})\int_{\mathbb{R}^N}|u_{n_k}(x)|^2dx
\to 0\quad \text{as }k\to\infty.
\end{align*}
Similarly,
\[
\iint_{\mathbb{R}^N\times \mathbb{R}^N
}\frac{|\varphi_k(x)-\varphi
_k(y)|^2|u_{n_k}(x)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy \leq
C(r_k^{-2\alpha}+r_k^{-\alpha})\int_{\mathbb{R}^N}|u_{n_k}(x)|^2dx
\]
which converges to zero as $k\to\infty$.
This completes the proof.
\end{proof}

\section{Proof of main results}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
We proceed in three steps.

\noindent\textbf{Step 1.} $0<c<\infty$. It is clear that $M\neq \emptyset$. Let
$u\in M$ and $\gamma >0$, set
$u_\gamma=\gamma^{N/2}u(\gamma x)$. It is straightforward to check that
$u_\gamma \in M$ and
\[
E(u_\gamma)=\frac{\gamma^{2\alpha}}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}|^2d\xi
-\frac{\gamma^{\frac{Np}{2}}}{p+2}\int_{\mathbb{R}^N}|u|^{p+2}dx.
\]
Since $Np<4\alpha$, we derive $E(u_\gamma)<0$ for $\gamma$ small,
and so $c>0$.

On the other hand, we deduce from an interpolation inequality, the
embedding theorem $\|u\|_{L^{\frac{2N}{N-2\alpha}}}\leq
C\|(-\Delta)^{\alpha/2}u\|_{L^2}$ and the Young inequality with
$\varepsilon$ that
\begin{equation} \label{403}
\begin{aligned}
E(u)&\geq \frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}|^2d\xi-C\Big(\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}|^2d\xi\Big)^{\frac{Np}{4\alpha}}\|u\|_{L^2}^{p+2-\frac{Np}{2\alpha}}
 \\
&\geq \frac{1-\varepsilon}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}|^2d\xi-K
\end{aligned}
\end{equation}
for some $0<K<\infty$. This implies $c <\infty$.


\noindent\textbf{Step 2.}
 Estimates on the minimizing sequence $(u_n)_{n>0}$ of
\eqref{402}. Due to $u_n \in M$, $(u_n)_{n>0}$ is bounded in $L^2$.
It follows from \eqref{403} that $(u_n)_{n>0}$ is bounded in
$H^\alpha$. Furthermore, thanks to $c>0$, we obtain 
$E(u_n)\leq -c/2$ for n sufficiently large. We consequently derive that
\begin{equation}\label{404}
\int_{\mathbb{R}^N}|u_n|^{p+2}dx\geq \frac{p+2}{2}c.
\end{equation}

\noindent\textbf{Step 3.} Conclusion.
Let $(u_n)_{n>0}$ be a minimizing sequence of \eqref{402}. Note
that by scaling we may assume $\mu=1$. 
In view of Lemma \ref{lem2.2}, $(|u_n|)_{n>0}$ is also a minimizing 
sequence of \eqref{402}. So, without loss of generality,
 we may suppose that $u_n$ is nonnegative. Let us now apply 
Lemma \ref{lem2.4}  to the minimizing sequence $(u_n)_{n>0}$.

Firstly, we claim vanishing cannot occur. Indeed, if not, applying
\cite[Lemma 2.2]{FQT}, we have $u_k\to0$ in $L^{p+2}$, which
is a contradiction with \eqref{404}.

Next, we show dichotomy cannot occur. If not, there exist a constant 
$\beta \in (0,1)$, two sequences $(v_k)_{k>0}$ and $(w_k)_{k>0}$ 
introduced in Lemma \ref{lem2.4}. It follows from \eqref{206} and 
\eqref{207} that
\[
\liminf_{k\to\infty}(E(u_k)-E(v_k)-E(w_k))\geq 0.
\]
Hence,
\begin{equation}\label{405}
\limsup_{k\to\infty}(E(v_k)+E(w_k))\leq -c.
\end{equation}

On the other hand, given $u \in H^\alpha$ and $a>0$, we have
\[
E(u)=\frac{1}{a^2}E(au)+\frac{a^p-1}{p+2}\int_{\mathbb{R}^N}|u|^{p+2}dx.
\]
Applying the above inequality with $v_k$ and $a_k=1/\|v_k\|_{L^2}$,
due to $a_kv_k \in M$, we obtain
\begin{equation}\label{406}
E(v_k)\geq
\frac{-c}{a_k^2}+\frac{a_k^p-1}{p+2}\int_{\mathbb{R}^N}|v_k|^{p+2}dx.
\end{equation}
Similarly,
\begin{equation}\label{407}
E(w_k)\geq
\frac{-c}{b_k^2}+\frac{b_k^p-1}{p+2}\int_{\mathbb{R}^N}|w_k|^{p+2}dx,
\end{equation}
where $b_k=1/\|w_k\|_{L^2}$. Therefore, collecting \eqref{406} and
\eqref{407}, we see that
\[
E(v_k)+E(w_k)\geq -c({a_k^{-2}}
+{b_k^{-2}})+\frac{a_k^p-1}{p+2}\int_{\mathbb{R}^N}|v_k|^{p+2}dx+
\frac{b_k^p-1}{p+2}\int_{\mathbb{R}^N}|w_k|^{p+2}dx.
\]
Note that $a_k^{-2}\to\beta$ and $b_k^{-2}\to1-\beta $
by \eqref{205}. It follows from $0<\beta<1$ that
\[
\theta:=\min\{\beta^{-p/2},(1-\beta)^{-p/2}\}>1.
\]
Therefore, we deduce from \eqref{206} and \eqref{404} that
\[
\liminf_{k\to\infty}(E(v_k)+E(w_k))\geq
-c+\frac{\theta-1}{p+2}\liminf_{k\to
\infty}\int_{\mathbb{R}^N}|u_k|^{p+2}dx\geq
-c+\frac{\theta-1}{2}>-c,
\]
which  contradicts \eqref{405}.

Finally, since we have ruled out both vanishing and dichotomy, then
we conclude that indeed compactness occur. 
Applying Lemma \ref{lem2.4}, we deduce
that for some sequence $(y_k)\subset \mathbb{R}^N$ and some $u \in
H^\alpha$, such that $u_{n_k}(\cdot-y_k)\to u$ in $L^2$ and
in $L^{p+2}$. Together with the weak lower semicontinuity of the
$H^\alpha$ norm, this implies
\[
E(u)\leq \lim_{k\to\infty}E(u_{n_k})=-c.
\]
In view of definition of $c$, we have $E(u)=-c$. In particular,
$E(u_{n_k})\to E(u)$ and it holds that
\[
\frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
}|\hat{u}_{n_k}|^2d\xi\to
\frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha }|\hat{u}|^2d\xi
\quad\text{as } k\to\infty,
\]
which implies $u_{n_k}(\cdot-y_k)\to u$ in $H^\alpha$.
This completes the proof.
\end{proof}



\begin{proof}[Proof of Theorem \ref{thm1.2}]
Firstly, we show that \eqref{303} is necessary for the relative
compactness of all minimizing sequences for $E_1$ in $M_1$. Indeed, if
$c\geq c^\infty$, let $(u_n)_{n>0}$ be a minimizing sequence for
$E_1^\infty$. Then $(\tilde{u}_n)_{n>0}$, defined by
$\tilde{u}_n=u_n(\cdot+x_n)$, is also a minimizing sequence for
$E_1^\infty$.  One easily verifies that
\[
|E_1(\tilde{u}_n)-E_1^\infty(\tilde{u}_n)|=\frac{1}{2}
\int_{\mathbb{R}^N}(V(x)-V_\infty )|\tilde{u}_n|^2dx\to
0\quad\text{as }|x_n|\to\infty.
\]
This implies $(\tilde{u}_n)$ is a minimizing sequence for $E_1$.

On the other hand, it is straightforward to check that
\[
\tilde{u}_n\to0\quad\text{in $L^p_{\rm loc}$ as $|x_n|\to \infty$}.
\]
Thus, $(\tilde{u}_n)_{n>0}$ cannot be relatively compact, which is a
contradiction with the fact $\|\tilde{u}_n\|_{L^p}=1$.

We now show that condition \eqref{303} is sufficient. Let
$(u_n)_{n>0}$ be a minimizing sequence for $E_1$ in $M_1$ such that
\[
E_1(u_n)\to c\quad\text{as }n\to\infty.
\]
By Lemma \ref{lem2.2}, we may assume $u_n$ is nonnegative and 
$u_n\rightharpoonup u$ weakly in $L^{p+2}$. In
view of assumption of $V$, $V$ is positive on $ \mathbb{R}^N$. Hence
we have
\begin{equation}\label{305'}
\|u_n\|_{H^\alpha}^2\leq C_1E_1(u_n)\leq C< \infty,
\end{equation}
and we may assume that $u_n\rightharpoonup u$ weakly in $H^\alpha$
and pointwise almost everywhere. Denote $u_n=v_n+u$. Applying
Bresiz-Lieb lemma, we have
\[
\Big|\int_{\mathbb{R}^N}|u_n|^{p+2}dx-\int_{\mathbb{R}^N}|v_n|^{p+2}dx-
\int_{\mathbb{R}^N}|u|^{p+2}dx\Big|\to 0\quad\text{as }n\to
\infty.
\]
Therefore,
\[
\int_{\mathbb{R}^N}|v_n|^{p+2}dx+
\int_{\mathbb{R}^N}|u|^{p+2}dx\to 1 \quad\text{as }n\to\infty.
\]
It follows that
\begin{equation} \label{304}
\begin{aligned}
E_1(u_n)&=\frac{1}{2}\iint_{\mathbb{R}^N\times \mathbb{R}^N
}\frac{|u_n(x)-u_n(y)|^2}{|x-y|^{N+2\alpha}}\,dx\,dy+\frac{1}{2}
\int_{\mathbb{R}^N}V(x)|u_n|^2dx
 \\
&= \frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha }|\hat{v}_n|^2d\xi +
 \frac{1}{2}\int_{\mathbb{R}^N}|\xi|^{2\alpha
 }|\hat{u}|^2d\xi\\
&\quad +\iint_{\mathbb{R}^N\times \mathbb{R}^N
 }\frac{(v_n(x)-v_n(y))(u(x)-u(y))}{|x-y|^{N+2\alpha}}\,dx\,dy \\
&\quad +\frac{1}{2}\int_{\mathbb{R}^N}V(x)(|u|^2+2u v_n+|v_n|^2)dx \\
&=E_1(v_n)+ E_1(u)+\langle (-\Delta)^\alpha v_n,
u\rangle +\int_{\mathbb{R}^N}V(x)u v_ndx.
\end{aligned}
\end{equation}
We deduce from $v_n\rightharpoonup 0$ in $H^\alpha$ that
\begin{equation}\label{305}
\langle (-\Delta)^\alpha v_n,u\rangle
+\int_{\mathbb{R}^N}V(x)u v_n\,dx\to0.
\end{equation}
Moreover, for any $\varepsilon >0$, let
\[
\Omega_\varepsilon=\{x\in \mathbb{R}^N;|V(x)-V_\infty|\geq
\varepsilon\}.
\]
In view of the definition of $V$, $\Omega_\varepsilon$ is a bounded
compact subset in $\mathbb{R}^N$, we obtain
\begin{equation} \label{306}
\begin{aligned}
 \int_{\mathbb{R}^N}(V(x)-V_\infty)| v_n|^2dx
&\leq  V_\infty \int_{\Omega_\varepsilon}| v_n|^2dx
 +\varepsilon \int_{\mathbb{R}^N}| v_n|^2dx  \\
&\leq  C\varepsilon +o(1),
\end{aligned}
\end{equation}
where $o(1)$ denotes error terms such that $o(1)\to0$ as
$n\to\infty$. Collecting \eqref{304}-\eqref{306}, we have
\begin{equation} \label{307}
E _1(u_n)=E_1(u)+E_1(v_n)+o(1)
=E_1(u)+E_1^\infty(v_n)+o(1).
\end{equation}
By homogeneity, and setting $\beta=\int_{\mathbb{R}^N}|u|^{p+2}dx$, if
$\beta >0$
\begin{equation}\label{308}
E_1(u)=\beta^{2/(p+2)}E_1(\beta^{-1/(p+2)}u)\geq \beta^{2/(p+2)}c
\end{equation}
if $0\leq \beta<1$
\begin{equation}\label{309}
E_1^\infty(v_n)=(1-\beta)^{2/(p+2)}E_1^\infty((1-\beta)^{-1/(p+2)}u)\geq
(1-\beta)^{2/(p+2)}c^\infty+o(1).
\end{equation}
Therefore, we deduce \eqref{307}-\eqref{309} that for all $\beta
\in [0,1]$
\begin{align*}
c&=E_1(u_n)+\circ(1)=E_1(u)+E_1^\infty(v_n)+o(1) \\
 &\geq \beta^{2/(p+2)}c+(1-\beta)^{2/(p+2)}c^\infty+o(1),
\end{align*}
which, together with \eqref{303}, implies $\beta \in \{0,1\}$.
If $\beta=0$, then
\[
c \geq c^\infty+o(1).
\]
Thus $\beta =1$; that is, $u_n\to u$ in $L^{p+2}$ and
$u\in M_1$. By convexity of $E_1$,
\[
E_1(u)\leq \liminf_{n\to\infty}E_1(u_n)=c,
\]
and $u$ minimizes $E_1$ in $M_1$. Hence, $E_1(u_n)\to E_1(u)$. Combining
this, \eqref{305'} and \eqref{307}, we have
\[
\|u_n-u\|_{H^\alpha}^2\leq
C_1E_1(u_n-u)=C_1(E_1(u_n)-E_1(u))+o(1).
\]
This implies $u_n\to u$ in $H^\alpha $ as $n\to\infty$.
The existence of solution to \eqref{1.1'} follows by Lagrange
multiplier methods for some $\lambda  >0$.
\end{proof}

\subsection*{Acknowledgments}
This work is supported by the Program for the
Fundamental Research Funds for the Central Universities,
by grants 11031003, 11171028 from the NSFC, and by the Program for NCET.


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\end{document}