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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 218, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/218\hfil Nonlinear fractional  Schr\"{o}dinger equations]
{Existence of solutions to nonlinear fractional
 Schr\"{o}dinger equations with singular potentials}

\author[Q. Wang, D. Zhao, K. Wang \hfil EJDE-2016/218\hfilneg]
{Qingxuan Wang, Dun Zhao, Kai Wang}

\address{Qingxuan Wang (corresponding author)\newline
School of Mathematics and Statistics,
Lanzhou University, Lanzhou 730000, China}
\email{wangqx12@lzu.edu.cn, Fax +86 931 8912481}

\address{Dun Zhao \newline
School of Mathematics and Statistics,
Lanzhou University, Lanzhou 730000, China}
\email{zhaod@lzu.edu.cn}

\address{Kai Wang \newline
School of Mathematics and Statistics,
Lanzhou University, Lanzhou 730000, China}
\email{wangk2010@lzu.edu.cn}

\thanks{Submitted November 22, 2015. Published August 16, 2016.}
\subjclass[2010]{35R11}
\keywords{Nonlinear fractional Schr\"{o}dinger equation; ground state;
\hfill\break\indent  positive solution; weakly continuous}

\begin{abstract}
 We study the eigenvalue problem
 \[
 (-\Delta)^s u(x)+ V(x)u(x)-K(x)|u|^{p-2}u(x)
 =\lambda u(x) \quad \text{in } \mathbb{R}^N,
 \]
 where $s\in(0,1)$, $N>2s$, $2<p<2^{*}=\frac{2N}{N-2s}$, $V(x)$ is
 indefinite and allowed to be unbounded from below, and $K(x)$ is
 nonnegative and allowed to be unbounded from above.
 When $\lambda <\lambda_0=\inf \sigma((-\Delta)^s +V(x))$
 (the lowest spectrum of the operator $(-\Delta)^s +V(x))$,
 we  obtain a positive ground state solution by using the
 constrained minimization method. Also we discuss the regularity  of
 solutions.
\end{abstract}

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

\section{Introduction and statement of main results}

In this article, we consider standing waves of the
nonlinear fractional Schr\"{o}dinger equation
\begin{equation} \label{0}
i\psi_{t}=(-\Delta)^s \psi + V(x)\psi -K(x) |\psi|^{p-2}\psi \quad
\text{in } \mathbb{R}^{N} ,
\end{equation}
where $(x,t)\in \mathbb{R}^N \times (0, \infty)$, $0< s<1$, $V(x)$ and
$K(x)$ are some real functions. The operator $(-\Delta)^s$ is the
fractional Laplacian of order $s$.

This equation was introduced by Laskin
\cite{Laskin2000,Laskin2002}, and comes from fractional quantum
mechanics for the study of particles on stochastic fields modelled
by L\'{e}vy process. The L\'{e}vy process is widely used to model
a variety of processes, such as turbulence,  financial dynamics,
biology and physiology, see \cite{KBS1987,MS1995,WD1994}.  When
$s=1$, the L\'{e}vy process becomes the Brownian motion, and the
equation \eqref{0} reduces to the classical Schr\"{o}dinger equation
 \begin{equation}
 i\psi_{t}=-\Delta \psi + V(x)\psi -K(x) |\psi|^{p-2}\psi \quad \text{in }
\mathbb{R}^{N}.
 \end{equation}
Standing wave solutions to this equation are solutions of the form
$\psi(x,t)=e^{-i\lambda t}u(x)$ where $u(x)$ satisfies the  equation
\begin{equation} \label{Lapla}
 -\Delta u + (V(x)-\lambda)u -K(x) |\psi|^{p-2}u=0 \quad \text{in } \mathbb{R}^{N},
\end{equation}
which has been extensively studied in the past 20 years.
We mention some earlier work here. Oh \cite{Oh1990} studied positive
multi-lump bound states, and it was assumed that $K(x)\equiv \gamma$ for
some $\gamma>0$, and $V(x)$ belongs to a class of potentials $(V)_a$ for
some $a$ and $\lambda<a$ ($V\in (V)_a$ if either $V(x)\equiv a$ or $V(x)>a$
for all $x\in \mathbb{R}^N$ and $(V(x)-a)^{-1/2}\in Lip(\mathbb{R}^N)$).
 Rabinowitz \cite{PR1992} investigated the ground state solutions of
the problem \eqref{Lapla} under the condition
$\inf_{\mathbb{R}^N}V(x)>\lambda$ and after this
Byeon and  Wang \cite{ByWa2002} considered the case
$\inf_{\mathbb{R}^N}V(x)= \lambda$ which they call it critical frequency case.

  Our goal is to look for  standing wave solutions  of the form
$\psi(x,t)=e^{-i\lambda t}u(x)$  to equation \eqref{0} for fractional
order $s\in (0,1)$. Precisely, we will investigate the  problem.
\begin{equation}\label{equa}
\begin{gathered}
 (-\Delta)^s u(x)+ (V(x)-\lambda) u(x)-K(x)|u|^{p-2}u(x)=0 \quad
 \text{in } \mathbb{R}^N,\\
 u(x)\in H^{s}( \mathbb{R}^{N}).
\end{gathered}
\end{equation}
Where $s\in (0,1)$, $2<p< 2^{*}= \frac{2N}{N-2s}$, $N>2s$,
$\lambda\in \mathbb{R}$,  $V(x)$ and $K(x)$ are real
functions satisfying the following conditions:
\begin{itemize}
\item[(A1)] $V(x): \mathbb{R}^{N}\to \mathbb{R}$,
$V(x)\in L^{\frac{N}{2s}}( \mathbb{R}^{N})+L^{\infty}( \mathbb{R}^{N})$;

\item[(A2)] for any $\epsilon>0$,  the Lebesgue measure
$|\{x: |V(x)|> \epsilon \}|<\infty$.

\item[(A3)]  $K(x)\geq 0$, $K(x)\not\equiv0$, $K(x)\in
L^{\frac{2^{*}}{2^{*}-p}}( \mathbb{R}^{N})+L^{\infty}( \mathbb{R}^{N})$;

\item[(A4)]  for any $\epsilon>0$, the Lebesgue measure
$|\{x: |K(x)|> \epsilon \}|<\infty$.

\item[(A5)]   $V(x)\in L^{\tilde{q}}( \mathbb{R}^{N})
+L^{\infty}(\mathbb{R}^{N})$, $K(x)\in
L^{\tilde{r}}( \mathbb{R}^{N})+L^{\infty}( \mathbb{R}^{N})$
for $\tilde{q}>\frac{N}{2s}$ and $\tilde{r}>\frac{2^{*}}{2^{*}-p}$.
\end{itemize}

We remark that in \cite{Laskin2002}, Laskin investigated the
fractional Hydrogen-like atom where $V(x)=-\frac{Ze^2}{|x|}$ (for
$N=3$ and $1/2 <s<1$), and evaluated the corresponding
energy spectrum. It is easy to check that such potential satisfies
condition (A1).

 In recent years, there have been a few
results for nonlinear fractional Schr\"{o}dinger equations like
\eqref{equa}.  Teng \cite{Teng2015} investigated
multiple solutions of the  equation
\begin{equation} \label{generalequ}
(-\Delta)^s u + V(x)u= f(x, u),
\end{equation}
for $V(x)\in C(\mathbb{R}^N)$, $\operatorname{ess\,inf} V(x)>0$, and
$f\in C(\mathbb{R}^N\times \mathbb{R})$. Secchi \cite{Secc2013} studied
the ground state solutions of \eqref{generalequ} for the case that
$V\in C^1 (\mathbb{R}^N)$, $\inf_{x\in \mathbb{R}^N} V(x)= V_0>0$,
and $f\in C^{1}(\mathbb{R}^N\times \mathbb{R})$ satisfying
Ambrosetti-Rabinowitz condition.
 In \cite{Cheng2012}, ground states and bound states of \eqref{generalequ}
are obtained by assuming that $V(x)>1$ and $\lim_{|x|\to +\infty}V(x)= +\infty$,
and the nonlinearity is $f(t)=|t|^{p-1}t$.
 Chang \cite{XjC2013} investigated the ground state solutions for asymptotically
linear fractional Schr\"{o}dinger equations.
 In particular, Felmer \cite{Felmer2012} studied the existence of  positive
solutions of \eqref{generalequ} for $V(x)\equiv 1$ and $f(x, u)$ is superlinear
and has subcritical growth with respect to $u$ such that there exist
$1<p<(N+2s)/(N-2s)$, so that
\begin{equation} \label{nonlinea}
f(x,\xi)\leq C(1+|\xi|)^{p} \quad \text{for all $\xi\in \mathbb{R}$ and a.e.
$x\in \mathbb{R}^N$.}
\end{equation}
Furthermore, they  discuss the regularity, decay and symmetry
properties of solutions.

The nonlinearity $K(x) |u|^{p-2}u(x)$ in this paper is quite
different from \eqref{nonlinea}, since $K(x)$ may not be bounded
by a constant $C$. For example, $K(x)=\frac{1}{|x-x_0|^{\alpha}}$
for $0<\alpha<\frac{(2^{*}-p)N}{2^{*}}$, satisfies
(A3), (A4), and has singular point
$x_0\in \mathbb{R}^N$.
On the other hand, since $V(x)$  is
indefinite, it is hard to use usual mountain pass arguments to
obtain ground state solutions(\cite{Felmer2012,Secc2013,XjC2013}),
here we will use the constrained minimization method to obtain the
ground state solutions.


We say that $u\in H^{s}(\mathbb{R}^{N})$ is a weak solution of \eqref{equa},
if for any  $\phi\in H^{s}(\mathbb{R}^N)$,
\[
\int_{\mathbb{R}^N}(-\Delta)^{s/2}\overline{u}\cdot(-\Delta)^{s/2}\phi\,dx
+\int_{\mathbb{R}^N}(V(x)-\lambda)\overline{u}\cdot\phi\,dx
=\int_{\mathbb{R}^N}K(x)|u|^{p-2}\overline{u}\cdot\phi\,dx,
\]
where $\overline{u}(x)$ is conjugation of $u(x)$ in the complex space
$H^{s}(\mathbb{R}^N)$.

Solutions of  \eqref{equa} correspond to the critical points of the
energy functional
\begin{equation}  \label{energy}
\mathcal{I}(u)= \frac{1}{2}[\,\int_{ \mathbb{R}^{N}}|(-\Delta)^{s/2} u|^{2}\,dx
+\int_{ \mathbb{R}^{N}}(V(x)-\lambda)|u|^{2}\,dx]-\frac{1}{p}\int_{ \mathbb{R}
^{N}}K(x)|u|^{p}\,dx.
\end{equation}
And \emph{a ground state} of \eqref{equa} is a solution that minimizes
the energy functional on the Nehari manifold
\begin{equation}\label{Neh}
\mathcal{N}=\big\{u\in{H^{s}(\mathbb{R}^{N})\setminus\{0\}:
\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2}u|^{2}+(V(x)
-\lambda)|u|^{2}\,dx=\int_{\mathbb{R}^N}K(x)|u|^{p}\,dx}\big\}.
\end{equation}
Now we state our main result.

\begin{theorem}\label{mainresult}
Let $s\in (0,1)$, $2< p< 2^{*}$ ($2^{*}=\frac{2N}{N-2s}$),
$N>2s$. Assume that {\rm (A1)--(A4)} are satisfied.
Let
\begin{align*}
\lambda_0&=\inf \sigma((-\Delta)^{s}+V(x)) \\
&=\inf\{ \int_{ \mathbb{R}^{N}}|(-\Delta)^{s/2} \psi|^{2}+V(x)|\psi|^{2}\, dx:
 \psi\in H^{s}(\mathbb{R}^N), \|\psi\|_{L^2}=1\},
\end{align*}
and assume that $\lambda\leq 0$ and $\lambda<\lambda_0$.
Then \eqref{equa} admits at least one nonnegative weak solution such
that this solution is a ground state.
\end{theorem}

To prove the positive property of nonnegative weak solutions, we need
to take advantage of the representation formula
$$
u=\mathcal{K}^{\mu}\ast f= \int_{\mathbb{R}^N}K(x-\xi)f(\xi)\, d\xi,
$$
for some $\mu >0 $, and that $u$ satisfies the equation
$$
(-\Delta)^{s}u + \mu u= f\ \ \ \text{in $\mathbb{R}^N$},
$$
where $\mathcal{K}^{\mu}$ is the Bessel kernel
$$
\mathcal{K}^{\mu}=\mathcal{F}^{-1}\big(\frac{1}{\mu+|\xi|^{2s}}\big).
$$
We have the following positive property.

\begin{theorem}\label{positive-result}
Under the assumptions of Theorem \ref{mainresult}, let $w(x)$ be a nonnegative
ground state solution obtain in Theorem \ref{mainresult}.
If we further assume that $V(x)$ is bound from above, then $w(x)$ can be
 chosen positive in $\mathbb{R}^N$.
\end{theorem}


The next step is to prove regularity of the weak solutions.
Inspired by ideas in \cite{Felmer2012}, We also use the representation
formula above to discuss the regularity. We have the following result.

\begin{theorem}\label{mainregular}
 Let $u(x)\in H^{s}(\mathbb{R}^N)$ be a solution of \eqref{equa},
assume that $\lambda<0$ and {\rm (A5)} holds, i.e.,
$V(x)\in L^{\tilde{q}}( \mathbb{R}^{N})+L^{\infty}( \mathbb{R}^{N})$,
$K(x)\in L^{\tilde{r}}( \mathbb{R}^{N})+L^{\infty}( \mathbb{R}^{N})$ for
 $\tilde{q}>\frac{N}{2s}$ and $\tilde{r}>\frac{2^{*}}{2^{*}-p}$.
Then $u\in C^{0,\alpha}(\mathbb{R}^N)$ for some $\alpha\in (0,1)$.
 Moreover, $u(x)\to 0$ as $|x|\to \infty$.
\end{theorem}

We remark that having the regularities above, by the same arguments as in
\cite[Theorem 1.5]{Felmer2012}, it is easy to show that the
positive ground state solutions $u(x)$ behave at infinity like
$\frac{1}{|x|^{N+2s}}$.

The rest of the article is originated as follows.
In section \ref{Preliminary} we give some preliminary and show some properties of
the operator $(-\Delta)^{s}+V(x)$. In section \ref{Weak-converge}
we will show that weak convergence in $H^s (\mathbb{R}^N)$ implies strong
convergence on finite measure sets, which is important to prove our result.
 In section \ref{weak-continuity} we prove that weak continuity of the potential
energies.  In section \ref{theorem-1} we prove the Theorem \ref{mainresult}.
In section \ref{theorem-2-3} we give the proof of Theorem \ref{positive-result}
and \ref{mainregular}.

\subsection*{Notation}
 To coincide with the book \cite{Lieb}, the Banach spaces
$L^{p}( \mathbb{R}^{N})$, $ H^{s}(\mathbb{R}^{N})$
used here are  complex Banach spaces. And
 the inner product is defined by
\begin{equation}\label{inner-p}
(f(x),g(x))= \int_{\mathbb{R}^N}\overline{f(x)} g(x)\,dx, \quad
\text{for any $f(x), g(x) \in L^2(\mathbb{R}^N)$},
\end{equation}
where $\overline{f}$ denotes conjugation of $f(x)$.

 $\widehat{u}$ denotes the Fourier transform of $u\in L^2(\mathbb{R}^N)$.
\[
L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N):=\{u=u_0 +u_1:
 u_0\in L^{q}(\mathbb{R}^N), u_1\in L^{\infty}(\mathbb{R}^N)\}.
\]
$\rightharpoonup$ denotes weakly converge.
$C^{0,\alpha}(\mathbb{R}^N)$ denotes H\"{o}lder continuous with
exponent $\alpha\in (0, 1)$.


\section{Preliminaries}\label{Preliminary}

The fractional Laplacian $(-\Delta)^{s}$ of a rapidly decaying test function $u$
is defined as
\begin{equation} \label{fracLap}
(-\Delta)^s u(x) = C_{N,s} \operatorname{P.V.}
 \int\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dx\,dy,
\end{equation}
where P.V. denotes the principal value of the singular integral,
and $C_{N,s}$ is a constant.

We recall that the fractional Sobolev space $W^{s,p}(\mathbb{R}^N)$
(e.g., see \cite{Secc2013}) is defined for any $p\in[1,\infty)$ and
$s\in (0,1)$ as
$$
 W^{s,p}(\mathbb{R}^N)=\big\{u\in L^{p}(\mathbb{R}^N) :
 \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{|u(x)-u(y)|^p}{|x-y|^{s\,p+N}}\,dxdy< \infty \big\},
$$
endowed with the norm
$$
\|u\|_W^{s,p}=\Big(\int_{\mathbb{R}^N} |u|^{p}dx
+ \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{|u(x)-u(y)|^p}{|x-y|^{s\,p+N}}\,dxdy\Big)^{1/p}.
$$
When $p=2$, these spaces are also denoted by $H^s(\mathbb{R}^{N})$.

When $p=2$, there is an equivalent definition of fractional Sobolev
spaces based on Fourier analysis that
\begin{gather*}
H^s(\mathbb{R}^{N})=\big\{u\in L^{2}(\mathbb{R}^N):
\int (1+|\xi|^{2s})|\widehat{u}(\xi)|^2d\xi < \infty \big\}, \\
\widehat{(-\Delta)^{s}u}=|\xi|^{2s}\widehat{u},\quad   \text{for }
 u\in H^{s}(\mathbb{R}^N),
\end{gather*}
and the norm can be equivalently written
$$
\|u\|_{H^s}=\Big(\|u\|^{2}_{L^2}+\int |\xi|^{2s}|\widehat{u}(\xi)|^2d\xi
\Big)^{1/2}
=\Big(\|u\|^{2}_{L^2}+ \|(-\Delta)^{s/2}u\|^{2}_{L^2}\Big)^{1/2}
$$
Therefore, we see that $H^s(\mathbb{R}^N)$ is just $L^2(\mathbb{R}^N,\,d\mu)$,
 where $\mu$ is a measure defined by
$$
\mu(dx)=(1+|x|^{2s})dx.
$$
  A sequence $f^{j}(x)$ \textit{converges weakly} to $f(x)$
(we write $f^j\rightharpoonup f$) in $H^{s} (\mathbb{R}^N)$ in the following
sense (see \cite[\S 7.18 ]{Lieb} or \cite{CotTav2004}):
for any $g(x)\in H^{s} (\mathbb{R}^N)$, when $j\to\infty$, one has
 \begin{equation}\label{2.1}
 \frac{1}{(2\pi)^{N/2}}\int_{\mathbb{R}^N}[\widehat{f^j}(\xi)
-\widehat{f}(\xi)]\widehat{g}(\xi)(1+|\xi|^{2s})\,d\xi\to 0.
 \end{equation}

The following lemma is obvious. There are more details about
 $H^{1/2}(\mathbb{R}^N)$ in \cite{Lieb}, so the case for general $s\in(0,1)$
is just the same arguments to $H^{1/2}(\mathbb{R}^N)$.

\begin{lemma}\label{bound}
If a sequence $f^j$ converges weakly to $f$ in $H^{s} (\mathbb{R}^N)$.
Then, there exists a constant $M$ independent of number $j$, such that
\begin{equation} \label{2.2}
\|f^j\|_{H^s}\leq M,\quad \|f\|_{H^s}\leq M\,.
\end{equation}
\end{lemma}

\begin{proof}
Since $H^s(\mathbb{R}^N)$ is just $L^2(\mathbb{R}^N,\,d\mu)$, thus by
 uniform boundedness principle and Lower semicontinuity of $L^p$-norm
 respectively, we  obtain \eqref{2.2}.
\end{proof}

Now we review the Sobolev inequality and Sobolev-Gagliardo-Nirenberg
inequality for fractional Sobolev spaces, we only show the case for
$H^{s}(\mathbb{R}^N)$.

\begin{lemma}[Sobolev inequality \cite{Stein1970}] \label{Sob}
Let $s\in (0,1)$ be such that $N> 2s$. Then
$$
\|u\|_{L^{2^{*}}}\leq S_{N,s}\|(-\Delta)^{s/2}u\|_{L^2}
$$
for every $u\in H^{s}(\mathbb{R}^N)$, where $S_{N,s}$ is sharp constants
depending only on $N$,$s$, and
$$
2^{*}=\frac{2N}{N-2s}
$$
is the factional critical exponent.
\end{lemma}


\begin{lemma}[Sobolev-Gagliardo-Nirenberg inequality \cite{Secc2013}] \label{GN}
Let $q\in(2, 2^{*})$. Then there exists a constant $C>0$ such that
$$
\|u\|^{q}_{L^q}\leq C\|u\|^{\frac{(q-2)N}{2s}}_{H^s}
\|u\|^{q-\frac{(q-2)N}{2s}}_{L^2}
$$
for every $u\in H^s(\mathbb{R}^N)$.
\end{lemma}

Next we show some properties of fractional Schr\"{o}dinger operator
$(-\Delta)^s + V(x)$.  For any $\psi(x) \in H^{s}(\mathbb{R}^N)$,
 let $\lambda_0$ be defined in Theorem \ref{mainresult}, define
\begin{equation}
\mathcal{E}(\psi):=\|(-\Delta)^{s/2}\psi\|^{2}_{L^2}
+ \int_{\mathbb{R}^N}V(x)|\psi|^2\,dx.
\end{equation}
then  $\lambda_0=\inf\{\mathcal{E}(\psi): \psi\in H^{s}(\mathbb{R}^N),
\|\psi\|_{L^2}=1\}$. We have the following theorem.

\begin{theorem}\label{finitebound}
For $s\in (0,1)$, $N>2s$, if
$V(x)\in L^{\frac{N}{2s}}(\mathbb{R}^{N})+L^{\infty}(\mathbb{R}^{N})$, then
\begin{itemize}
\item[(i)] $\lambda_0$ is finite.
\item[(ii)] $\|(-\Delta)^{s/2}\psi\|^{2}_{L^2}\leq C \mathcal{E}(\psi)
+ D\|\psi\|^{2}_{L^2}$  for $\psi\in H^{s}(\mathbb{R}^N)$ and suitable
constants $C$ and $D$.
\end{itemize}
\end{theorem}

\begin{proof}
Since $V(x)\in L^{\frac{N}{2s}}(\mathbb{R}^{N})+L^{\infty}(\mathbb{R}^{N})$,
we can write $V(x)=v(x)+w(x)$ with
$v(x)\in L^{\frac{N}{2s}}(\mathbb{R}^{N})$ and $w(x)\in L^{\infty}(\mathbb{R}^N)$.

First we claim that we can choose $v(x)$ satisfying
$\|v(x)\|_{L^{\frac{N}{2s}}}\leq \frac{1}{2}(S_{N,s})^{-2}$.
In fact, for $M>0$, define $S_{v}(M)$ by
$$
S_{v}(M)=\{x\in \mathbb{R}^{N}:|v(x)|> M\},
$$
then  by Chebyshev inequality (see \cite{J2001})
\begin{equation}\label{cheby-ineq}
|S_{v}(M)|\leq \Big(\frac{C\|v\|_{L^\frac{N}{2s}}}{M}\Big)^{N/(2s)}.
\end{equation}
Let $\chi_{A}$ be the characteristic function on subset $A\subset \mathbb{R}^N$.
Decompose $v(x)$ into
$$
v(x)= \chi_{S_{v}(M)}v(x)+(1-\chi_{S_{v}(M)})v(x).
$$
Let $v_1 = \chi_{S_{v}(M)}v(x)$, $v_{2}= (1-\chi_{S_{v}(M)})v(x)$,
then $v_1\in L^{\frac{N}{2s}}(\mathbb{R}^N)$,
$v_{2}\in L^{\infty}(\mathbb{R}^N)$, and  by \eqref{cheby-ineq} we have
$\|v_1\|_{L^\frac{N}{2s}}< \frac{1}{2}(S_{N,s})^{-2}$ for large enough $M$.
Replace $v(x)$ by $v_1$, then the claim holds.

For any function $\psi\in H^{s}(\mathbb{R}^N)$, combing with
$\|v(x)\|_{L^{\frac{N}{2s}}}\leq \frac{1}{2}(S_{N,s})^{-2}$ and using
 H\"{o}lder inequality and Sobolev inequality (Lemma \ref{Sob}), we have
\begin{align*}
\Big|\int_{\mathbb{R}^N}v(x)|\psi|^2\,dx \Big|
&\leq \|v(x)\|_{L^{\frac{N}{2s}}}\|\psi\|^{2}_{L^{2^{*}}}\\
&\leq S^{2}_{N,s}\|v(x)\|_{L^{\frac{N}{2s}}}\|(-\Delta)^{s/2}\psi\|^{2}_{L^2}\\
&\leq \frac{1}{2}\|(-\Delta)^{s/2}\psi\|^{2}_{L^2},
\end{align*}
it follows that
\begin{equation} \label{12}
\begin{aligned}
\mathcal{E}(\psi)
&=\|(-\Delta)^{s/2}\psi\|^{2}_{L^2}+\int_{\mathbb{R}^N}v(x)|\psi|^2\,dx
 +\int_{\mathbb{R}^N}w(x)|\psi|^2\,dx \\
&\geq \frac{1}{2}\|(-\Delta)^{s/2}\psi\|^{2}_{L^2}
 - \|w(x)\|_{L^{\infty}}\|\psi\|^{2}_{L^2}
\geq - \|w(x)\|_{L^{\infty}}\|\psi\|^{2}_{L^2}\,,
\end{aligned}
\end{equation}
and we see that $- \|w(x)\|_{L^{\infty}}$ is a lower bound to $\lambda_0$,
i.e., (i) holds.  Furthermore, the first inequality of \eqref{12} implies
$$
\|(-\Delta)^{s/2}\psi\|^{2}_{L^2}\leq 2 (\mathcal{E}(\psi)
+ \|w(x)\|_{L^{\infty}}\|\psi\|^{2}_{L^2}),
$$
i.e., (ii) holds.
\end{proof}

\section{Weak convergence implies strong convergence on small sets}
\label{Weak-converge}

Consider the semigroup $\{e^{-(-\Delta)^s t}\}_{t>0}$.
We know that, for any function $f(x)\in H^{s}(\mathbb{R}^{N})$,
$$
\widehat{e^{-(-\Delta)^s t}f(\xi)}= e^{-|\xi|^{2s}t} \hat{f}(\xi).
$$
Now we define the heat kernel for $s\in (0,1)$, $t>0$, and
$x\in \mathbb{R}^N$ as
\begin{equation}\label{hxt}
\mathcal{H}(x,t)=\frac{1}{(2\pi)^{N/2}}
\int_{\mathbb{R}^N}e^{i\,x\cdot\xi -t|\xi|^{2s}}\,d\xi,
\end{equation}
and we know that
\begin{equation}\label{3.2}
e^{-(-\Delta)^s t}f(x)=\int_{\mathbb{R}^N}
\mathcal{H}(x-y,t)f(y)\,dy.
\end{equation}
It is well known that $\mathcal{H}(x,t)$ has the following
properties, see \cite[Appendix A]{Felmer2012} and references
therein.

\begin{lemma}\label{kernel}
$\mathcal{H}(x,t)$ is radially symmetric in $x$, and  there exists
two constants $c_1$ and $c_2$ such that
\begin{equation}
c_1 \min\big\{t^{-\frac{N}{2s}}, t|x|^{-N-2s}\big\}
\leq \mathcal{H}(x,t)
\leq c_2 \min\big\{t^{-\frac{N}{2s}}, t|x|^{-N-2s}\big\}.
\end{equation}
\end{lemma}

Now we  use the properties of semigroup with respect to
$(-\Delta)^s$ to prove that weak convergence in
$H^{s}(\mathbb{R}^N)$ implies strong convergence on any finite
measure set (not just on a bounded  domain $\Omega\in
\mathbb{R}^N$, see compact embeddings in
\cite{PalaPis2014,PalaSaVald2013} ).
 This result can also be found in \cite{CotTav2004},
 here we give a different proof  along  the ideas in \cite[Theorem8.6]{Lieb}.

\begin{theorem}\label{smaset}
Let $\{f^{j}\}\subset H^{s}(\mathbb{R}^N)$ such that
$f^{j}$ converges weakly to $f$ in $H^{s}(\mathbb{R}^{N})$.
 Let $A\subset \mathbb{R}^N$ be any set of finite Lebesgue measure,
i.e., $|A|<\infty$, and let $\chi_{A}$ be its characteristic
function. Then
$$
\chi_{A}f^j\to \chi_{A}f\quad  \text{strongly in } L^{q}(\mathbb{R}^N)
$$
for $1\leq q< 2^{*}=\frac{2N}{N-2s}$, when $N> 2s$.
\end{theorem}

\begin{proof}
We  take three steps to prove the theorem.
\smallskip

\noindent\textbf{Step 1.}
We claim that, for any $f\in H^{s}(R^N)$,
\begin{equation}\label{4.1}
\|f-e^{-(-\Delta)^s t}f\|_{L^2}\leq
\|(-\Delta)^{s/2}f\|_{L^2}\sqrt{t}.
\end{equation}
In fact, we know that
$$
1-\exp[-(|\xi|)^{2s}t]\leq \min\big\{1, (|\xi|)^{2s}t \big\}
\leq |\xi|^s \sqrt{t},
 $$
and it follows that
\begin{align*}
\|f-e^{-(-\Delta)^s t}f\|^{2}_{L^2}
&= \int_{\mathbb{R}^N}|\hat{f}(\xi)|^2 (1-\exp[-(|\xi|)^{2s}t])^2\,d\xi\\
&\leq \int_{\mathbb{R}^N}|\hat{f}(\xi)|^2(|\xi|^s
\sqrt{t})^2\,d\xi = \|(-\Delta)^{s/2}f\|^{2}_{L^2}\,t,
\end{align*}
this proves \eqref{4.1}.
\smallskip

\noindent\textbf{Step 2.}
We first prove that $\chi_{A}f^j\to \chi_{A}f$ strongly in $L^{2}(\mathbb{R}^N)$.
Let $g^j := e^{-(-\Delta)^s t}f^j$, by Lemma \ref{bound},  we
note that
\begin{gather}\label{3.5}
 \|(-\Delta)^{s/2}f^j\|_{L^2} \leq \|f^j\|_{H^s}\leq M,\\
 \|(-\Delta)^{s/2}f\|_{L^2}  \leq \|f\|_{H^s}\leq M,
 \end{gather}
where $M$ is a constant independent of $j$. Then by \eqref{4.1} we
have
\begin{gather*}
\|f^j-g^j\|_{L^2}=\|f^j -e^{-(-\Delta)^s t}f^j\|_{L^2}\leq M\sqrt{t},\\
\|f-g\|_{L^2}=\|f-e^{-(-\Delta)^s t}f\|_{L^2}\leq M\sqrt{t}.
\end{gather*}
Simply note that
\begin{align*}
\|\chi_{A}(f^j-f)\|_{L^2}
&\leq \|\chi_{A}(f^j-g^j)\|_{L^2}+\|\chi_{A}(g^j-g)\|_{L^2}+\|\chi_{A}(g-f)\|_{L^2}\\
&\leq 2M\sqrt{t}+ \|\chi_{A}(g^j-g)\|_{L^2}\,.
\end{align*}
For $\epsilon >0 $ given, first choose $t>0$ (depending on
$\epsilon$) such that $2M\sqrt{t}<\epsilon/2$ and if for
$j$ (depending on $\epsilon$) we have
$\|\chi_{A}(g^j-g)\|_{L^2}<\epsilon/2$, then we have
$\|\chi_{A}(f^j-f)\|_{L^2}< \epsilon$. Therefore, it remains to
prove that $\chi_{A}g^j\to \chi_{A}g$ strongly in
$L^{2}(\mathbb{R}^N)$.

To prove $\chi_{A}g^j\to \chi_{A}g$ strongly in $L^{2}(\mathbb{R}^N)$,
first  we note that,   if $|y-x|\geq t^{\frac{1}{2s}}$, we have
$t|x-y|^{-N-2s}\leq t^{-\frac{N}{2s}}$, and then by Lemma \ref{kernel}, we have
 \begin{equation}
\begin{aligned}
  \mathcal{H}(x-y,t)
&\leq c_2\,t|x-y|^{-N-2s}= 2c_{2}\frac{t}{2|x-y|^{N+2s}}   \\
  &\leq 2c_{2}\frac{t}{t^{\frac{N+2s}{2s}}+|x-y|^{N+2s}}\,.
\end{aligned}
  \end{equation}
Then,  for every fix $x$, we have
$\mathcal{H}(x-y,t)\in L^{(2^{*})'}(\mathbb{R}^N)$,
where $(2^{*})'= 2N/(N+2s)$, is dual
index to $2^{*}$. In fact, let $B(x, t^{\frac{1}{2s}})$ denote a
ball center at $x$ and has radius $t^{\frac{1}{2s}}$,
\begin{align*}
&\int_{\mathbb{R}^{N}}(\mathcal{H}(x-y,t))^{(2^{*})'}\,dy\\
&=\int_{B(x,t^{\frac{1}{2s}})}(\mathcal{H}(x-y,t))^{(2^{*})'}\,dy
 + \int_{\mathbb{R}^{N}\setminus{B(x, t^{\frac{1}{2s}})}}
 (\mathcal{H}(x-y,t))^{(2^{*})'}\,dy\\
&\leq \int_{B(x,t^{\frac{1}{2s}})}(c_2\,t^{-\frac{N}{2s}})^{(2^{*})'}dy
 +\int_{\mathbb{R}^{N}\setminus{B(x, t^{\frac{1}{2s}})}}
 (c_2t|x-y|^{-N-2s})^{(2^{*})'}\,dy\\
&\leq M_1+ \int_{\mathbb{R}^{N}}(2c_{2}\frac{t}{t^{\frac{N+2s}{2s}}
 +|x-y|^{N+2s}})^{(2^{*})'}\,dy\\
&\leq M_1+ \int_{\mathbb{R}^{N}}(2c_{2}\frac{t}{t^{\frac{N+2s}{2s}}
 +|y|^{N+2s}})^{(2^{*})'}\,dy\leq M_2,
\end{align*}
where $M_2$ is a constant independent of $x$.

 Since for every $x$, we have $\mathcal{H}(x-y,t)\in L^{(2^{*})'}(\mathbb{R}^N)$,
by H\"{o}lder inequality
 $$
\chi_{A}|g^{j}(x)|\leq \|\mathcal{H}(x-y,t)\|_{(2^{*})'}\|f^j\|_{2^{*}} \chi_{A}(x).
$$
Using Lemma \ref{Sob} and \eqref{3.5},
$\|f^j\|_{2^{*}}\leq S_{N,s}\|(-\Delta)^{s/2}f^j\|_{L^2}\leq S_{N,s} M$.
Hence $\chi_{A}g^j$ is dominated by a constant multiple of the square
integrable function $\chi_{A}(x)$. On the other hand, if
$g^{j}(x)$ converges pointwise to $g(x)$ for every
$x\in \mathbb{R}^N$, Then by general dominated convergence theorem, we
have $\chi_{A}g^j\to \chi_{A}g$ strongly in
$L^{2}(\mathbb{R}^{N})$. Next we shall prove $g^{j}(x)$ converges
pointwise for every $x\in \mathbb{R}^N$. We note that, for fixed
$x$,
$$
\widehat{H(x-y,t)}(\xi)=(e^{-ix\cdot \xi})e^{-t|\xi|^{2s}},
$$
and
\begin{align*}
g^{j}(x)
&= e^{-(-\Delta)^s t}f^j(x) =\int_{\mathbb{R}^N} \mathcal{H}(x-y,t)f^{j}(y)\,dy\\
&=\int_{\mathbb{R}^N}\widehat{H(x-y,t)}(\xi)\hat{f^j}(\xi)\,d\xi
= \int_{\mathbb{R}^N} (e^{-ix\cdot \xi})e^{-t|\xi|^{2s}}\hat{f^j}(\xi)\,d\xi\\
&=\int_{\mathbb{R}^N} \frac{(e^{-ix\cdot
\xi})e^{-t|\xi|^{2s}}}{1+|\xi|^{2s}}\hat{f^j}(\xi)(1+|\xi|^{2s})\,d\xi.
\end{align*}
Let $h(y)$ be a function satisfying
$\hat{h}(\xi)=\frac{(e^{-ix\cdot
\xi})e^{-t|\xi|^{2s}}}{1+|\xi|^{2s}}$, it is easy to see that
$h(y)\in H^{s}(\mathbb{R}^{N})$. Since $f^{j}$ converges weakly to
$f$ in $H^{s}(\mathbb{R}^{N})$, by \eqref{2.1}, then we have
$g^{j}(x)$ converges pointwise to $g(x)$ for every $x\in
\mathbb{R}^N$. Hence  we complete the proof of Step 2.
\smallskip

\noindent\textbf{Step 3.}
The inequality
$$
\|\chi_{A}(f^j-f)\|_{L^q}\leq \|\chi_{A}\|_{L^r}\|\chi_{A}(f-f^j)\|_{L^2}
$$
for $1/q=1/r + 1/2$ proves the theorem for $1\leq q \leq 2$. Again
by H\"{o}lder inequality, Lemma \ref{Sob},
\begin{align*}
\|\chi_{A}(f^j-f)\|_{L^q}
&\leq \|\chi_{A}(f^j-f)\|^{\alpha}_{L^2}\|\chi_{A}(f-f^j)\|^{1-\alpha}_{L^{2^{*}}}\\
&\leq \|\chi_{A}(f^j-f)\|^{\alpha}_{L^2}\|f-f^j\|^{1-\alpha}_{L^{2^{*}}}\\
&\leq \|\chi_{A}(f^j-f)\|^{\alpha}_{L^2}(S_{N,s})^{1-\alpha}\|
 (-\Delta)^{s/2}(f-f^j)\|^{1-\alpha}_{L^2}\\
&\leq \|\chi_{A}(f^j-f)\|^{\alpha}_{L^2}(2M\,S_{N,s})^{1-\alpha},
\end{align*}
where $\alpha=(1/q-1/2^{*})(1/2-1/2^{*})$, and this proves the
theorem for $2\leq q < 2^{*}$. The proof is complete.
\end{proof}

\section{Weak continuity of the potential energies}
\label{weak-continuity}

\begin{lemma}\label{weakcontinuous}
Let $2\leq q < 2^{*}$,  $F_{\psi}:=\int_{ \mathbb{R}^{N}}F(x)|\psi|^{q}\,dx$,
$F(x)$ be a real function on $\mathbb{R}^{N}$ such that
$F(x)\in L^{\frac{2^{*}}{2^{*}-q}}( \mathbb{R}^{N})+L^{\infty}( \mathbb{R
}^{N})$ and $|\{x: |F(x)|> \epsilon \}|<\infty$ for any $\epsilon> 0$.
Then $F_{\psi}$ is weakly continuous in $H^{s}(\mathbb{R}^{N})$, i.e.,
if $\psi^{j}\rightharpoonup \psi$ as $j\to \infty$ in
$H^{s}(\mathbb{R}^{N})$, then $F_{\psi^{j}}\to F_{\psi}$ as $j\to \infty$.
\end{lemma}

\begin{proof}
Note that by assumption, $\|\psi^{j}\|_{H^{s}}$ is uniformly bounded,
i.e., there is a constant $M>0$ independent of $j$ such that
$\|\psi^{j}\|_{H^{s}}\leq M$ for all $j$. For any $\delta>0$, define
\[
F^{\delta}(x)= \begin{cases}
F(x)  & \text{if } |F(x)|\leq \frac{1}{\delta}, \\
0  & \text{if } |F(x)| > \frac{1}{\delta}.
\end{cases}
\]

First we claim that $F-F^{\delta}\in L^{\frac{2^{*}}{2^{*}-q}}(\mathbb{R}^N)$.
Indeed, let $\Omega=\{x: |F(x)| > \frac{1}{\delta}\}$, by assumption above we
know $|\Omega|<\infty$. Writing $F(x)= f_1(x)+ f_2(x)$ with
$f_1\in L^{\frac{2^{*}}{2^{*}-q}}(\mathbb{R}^N)$ and
$f_2(x)\in L^{\infty}(\mathbb{R}^N)$, then we have
 $$
F-F^{\delta}= \chi_{\Omega}F(x)=\chi_{\Omega}f_1(x)+ \chi_{\Omega}f_2(x)\, ,
$$
where $\chi_{\Omega}$ be the characteristic function on $\Omega$.
Since $|\Omega|<\infty$, by H\"{o}lder inequality, we have
$\chi_{\Omega}f_2(x)\in L^{\frac{2^{*}}{2^{*}-q}}(\Omega)$.
It follows that $\chi_{\Omega}f_2(x)\in L^{\frac{2^{*}}{2^{*}-q}}(\mathbb{R}^N)$,
thus the claim holds.

 Moreover, $F-F^{\delta}\to 0$ in $L^{\frac{2^{*}}{2^{*}-q}}(\mathbb{R}^N)$
as $\delta\to 0$(by dominated convergence). Since $\|\psi^{j}\|_{H^{s}}\leq M$,
by Sobolev inequality (Lemma \ref{Sob})
\begin{equation}  \label{22}
\|\psi^{j}\|_{L^{2^{*}}}\leq S_{N,s}\|(-\Delta)^{s/2} \psi^{j}\|_{L^2}
\leq S_{N,s}\|\psi^{j}\|_{H^{s}}\leq C_1.
\end{equation}
By H\"{o}lder inequality, we have
\[
\int(F-F^{\delta})|\psi^{j}|^{q}\leq \|F-F^{\delta}\|_{\frac{2^{*}}{2^{*}-q}
}\|\psi^{j}\|^{q}_{2^{*}}\leq C_1\|F-F^{\delta}\|_{\frac{2^{*}}{2^{*}-q
}}=C_{\delta},
\]
with $C_{\delta}$ independent of $j$, moreover, $C_{\delta}\to 0$ as
$\delta\to 0$. Thus, our goal of showing that
$F_{\psi^{j}}\to F_{\psi}$ as $j\to\infty$ would be achieved if we can
prove that $F^{\delta}_{\psi^{j}}\to F^{\delta}_{\psi}$ as
$j\to\infty$ for each $\delta > 0$.

To prove that $F^{\delta}_{\psi^{j}}\to F^{\delta}_{\psi}$ as $j\to\infty$,
now fix $\delta$ and define the set
\[
A_{\epsilon}=\{x:|F^{\delta}(x)|>\epsilon\}
\]
for $\epsilon>0$. By assumption, $|A_{\epsilon}|<\infty$. Then
\begin{equation}  \label{23}
F^{\delta}_{\psi^{j}}=\int_{A_{\epsilon}}F^{\delta}|\psi^{j}|^{q}+
\int_{A^{c}_{\epsilon}}F^{\delta}|\psi^{j}|^{q}.
\end{equation}
Since $2\leq q<2^{*}$, $\|\psi^{j}\|_{2}\leq \|\psi^{j}\|_{H^{s}}\leq M$,
by Lemma \ref{GN}, we have
\begin{equation}  \label{24}
\|\psi^{j}\|_{L^q}\leq C\|\psi^{j}\|^{\frac{(k-2)N}{2sq}}_{H^s}
\|\psi^{j}\|^{1-\frac{(q-2)N}{2sq}}_{L^2}\leq CM\, ,
\end{equation}
by weak lower semicontinuity of the norm, we also have
\begin{equation}  \label{25}
\|\psi\|_{L^q}\leq \liminf_{j\to\infty}\|\psi^{j}\|_{L^q}\leq CM.
\end{equation}
Then
\[
\int_{A^{c}_{\epsilon}}F^{\delta}|\psi^{j}|^{q}\leq \epsilon \int_{ \mathbb{R%
}^{N}}|\psi^{j}|^{q}\leq\, \epsilon(C\,M)^{q}\,,
\]
i.e., the last term of \eqref{23} tend to zero as $\epsilon\to 0$,
and hence it suffices to show that the first term of \eqref{23} converges to
$\int_{A_{\epsilon}}F^{\delta}|\psi|^{q}$.

This is accomplished as follows. By Theorem \ref{smaset}
(in the Appendix below), on any finite measure set (that we take to be
$A_{\epsilon}$) $\psi^{j}\to \psi$ strongly in
$L^{r}(A_{\epsilon})$, for $r\in[1,2^{*})$. Here we can choose
$r=q$. Since $q\geq 2$, using the inequality
\[
\big| |\psi^{j}|^{q}-|\psi|^{q}\big|\leq C_{q} (|\psi^{j}|^{q-1} +
|\psi|^{q-1})|\psi^{j}-\psi|,
\]
where $C_{q}$ is a constant only dependent of $q$, and by \eqref{24},
\eqref{25} and H\"{o}lder inequality, we have
\begin{align*}
\int_{A_{\epsilon}}\big||\psi^{j}|^{q}-|\psi|^{q}\big|
&\leq \int_{A_{\epsilon}}C_{q} (|\psi^{j}|^{q-1} + |\psi|^{q-1})|\psi^{j}-\psi|
\\
&\leq  C_{q}\||\psi^{j}|^{q-1} + |\psi|^{q-1}\|_{L^{\frac{q}{q-1}}(A_{\epsilon})
}\|\psi^{j}-\psi\|_{L^{q}(A_{\epsilon})} \\
&\leq C_{3}\|\psi^{j}-\psi\|_{L^{q}(A_{\epsilon})},
\end{align*}
so $|\psi^{j}|^{q}\to |\psi|^{q}$ strongly in $L^{1}( A_{\epsilon})$.
Since $F^{\delta}\in L^{\infty}(\mathbb{R}^N)$ (see the definition above),
we conclude that
\[
\int_{A_{\epsilon}}F^{\delta}|\psi^{j}|^{q}\to
\int_{A_{\epsilon}}F^{\delta}|\psi|^{q},\quad\text{as } j\to \infty.
\]
This completes the proof.
\end{proof}

\section{Proof of Theorem \ref{mainresult}} \label{theorem-1}%\label{03}

We will give the proof by a series of lemmas. Firstly, for any
 $\beta> 0$, we set
\[
\Sigma_{\beta}:=\big\{u\in H^{s}( \mathbb{R}^{N}):\int_{ \mathbb{R}
^{N}}K(x)|u|^{p}\,dx  =\beta\big\}.
\]

\begin{lemma} \label{302}
Assume that $K(x)$ satisfies {\rm (A3)}, then $\Sigma_{\beta}$ is not empty.
\end{lemma}

\begin{proof}
Since $K(x)\geq 0$ and $K(x)\not \equiv0$, for any fixed
$u\in H^{s}(\mathbb{R}^{N})\setminus \{0\}$, we have
\[
\int_{\mathbb{R}^{N}}K(x)|u|^{p}\,dx>0.
\]
Write $K(x)= K_1+K_{2}$ with
$K_1\in L^{\frac{2^{*}}{2^{*}-p}}(\mathbb{R}^{N})$ and
$K_{2}\in L^{\infty} (\mathbb{R}^{N})$. For any fixed
$u\in H^{s}(\mathbb{R}^{N})\backslash \{0\}$, Since $2<p<2^{*}$,
by  H\"{o}lder inequality, Lemma \ref{Sob}, Lemma \ref{GN} we have
\begin{align*}
\int_{\mathbb{R}^{N}}K(x)|u|^{p}\,dx
&\leq\|K_1\|_{L^{\frac{2^{*}}{2^{*}-p}}
}\|u\|^{p}_{L^{2^{*}}}+ \|K_{2}\|_{L^{\infty}}\|u\|^{p}_{L^p} \\
&\leq C_1\|(-\Delta)^{s/2} u\|^{p}_{L^2} +
C_{2}\|u\|^{\frac{(p-2)N}{2s}}_{H^s}\,\|u\|^{p-\frac{(p-2)N}{2s}}_{L^2}
< \infty,
\end{align*}
where $C_1$ and $C_{2}$ are some constants. Then we can choose $t>0$
such that $tu(x)\in \Sigma_{\beta}$, where
\[
t=\Big(\frac{\beta}{\int_{\mathbb{R}^{N}}K(x)|u|^{p}\,dx}\Big)^{1/p}.
\]
\end{proof}

Let $\mathcal{I}(u)$ be the energy functional defined by \eqref{energy},
 we want to consider the  minimizing problem
\[
\inf_{\Sigma_\beta}\mathcal{I}(u)
= \frac{1}{2}\,\inf_{\Sigma_\beta}
\Big\{\int_{ \mathbb{R}^{N}}(|(-\Delta)^{s/2} u|^{2}
+(V(x)-\lambda)|u|^{2})\,dx\,
\Big\} -\frac{1}{p}\beta.
\]
Let
\begin{equation}  \label{ju}
\mathcal{J}(u)=\int_{ \mathbb{R}^{N}}(|(-\Delta)^{s/2} u|^{2}
+(V(x)-\lambda)|u|^{2})\,dx,
\end{equation}
with $m_{\beta}=\inf_{u\in \Sigma_\beta}\mathcal{J}(u)$,
so we have
\[
\inf_{u\in\Sigma_\beta} \mathcal{I}(u)=\frac{1}{2}m_{\beta}-\frac{1}{p}\beta.
\]
Thus minimizing $\mathcal{I}(u)$ on $\Sigma_\beta$ is equivalent to considering
 just $m_{\beta}$.

\begin{lemma} \label{minibound}
With the assumptions of Theorem \ref{mainresult}, let
$\{u_{k}\}_{k}\subset \Sigma_\beta$ be a minimizing sequence
for $m_{\beta}$.  Then $\{u_{k}\}$ is bounded in $H^{s}(\mathbb{R}^{N})$.
\end{lemma}

\begin{proof}
Since $\{u_{k}\}_{k}$ is a minimizer sequence for $m_{\beta}$, it follows that
\[
\lim_{k \to\infty} \mathcal{J}(u_{k})= m_{\beta}.
\]
Then, $\mathcal{J}(u_{k})$ is bounded by a constant independent of
$k$, i.e., $\mathcal{J}(u_{k})\leq M$.
Since $V(x)\in L^{\frac{N}{2s}}( \mathbb{R}^{N})+L^{\infty}( \mathbb{R}^{N})$
by Theorem \ref{finitebound}, we know
that $\lambda_0$ is finite. By the assumption $\lambda <\lambda_0$ 
in Theorem \ref{mainresult},  we have
\[
\mathcal{J}(u_{k})\geq \lambda_0\int_{ \mathbb{R}^{N}}|u_{k}|^{2}\,dx
-\lambda\int_{\mathbb{R}^{N}}|u_{k}|^{2}\,dx
=(\lambda_0-\lambda)\|u_{k}\|^{2}_{2}\,,
\]
it follows that
$\|u_{k}\|_{2}\leq M/(\lambda_0-\lambda)$.
i.e., $\{u_{k}\}_{k}$ is bounded in $L^{2}(\mathbb{R}^{N})$.

Since $\lambda\leq 0$, by (ii) of Theorem \ref{finitebound}, we have
\begin{align*}
\|(-\Delta)^{s/2} u_{k}\|^{2}_{L^2}
&\leq C\mathcal{E}(u_{k})+ D\|u_{k}\|^{2}_{L^2} \\
&\leq C(\mathcal{E}(u_{k})-\lambda\|u_{k}\|^{2}_{L^2})+D
\|u_{k}\|^{2}_{L^2} \\
&= C \mathcal{J}(u_{k})+ D\|u_{k}\|^{2}_{L^2}\\
&\leq C\,M+D\,M/(\lambda_0-\lambda).
\end{align*}
Therefore, $\{u_{k}\}_{k}$ is bounded in $H^{s}(\mathbb{R}^{N})$.
\end{proof}

\begin{lemma} \label{existnonega}
 With the assumptions of Theorem \ref{mainresult}, for every $\beta > 0$,
 $m_{\beta}$ is attained by a nonnegative function, namely there exists
$u_0\in \Sigma_\beta$, $u_0(x)\geq 0$ a.e. in $\mathbb{R}^{N}$, such that
\[
m_{\beta}=\mathcal{J}(u_0).
\]
Moreover, $m_{\beta}> 0$.
\end{lemma}

\begin{proof}
Let $\{u_{k}\}_{k}\subset \Sigma_\beta$ be a minimizing sequence for
$m_{\beta}$. In Section \ref{Preliminary}, we know that
$\|(-\Delta)^{s/2}u_{k}\|^{2}_{L^{2}}$ is equivalent to
$$
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}
\frac{|u_{k}(x)-u_{k}(y)|^2}{|x-y|^{2s+N}}\,dx\,dy,
$$
it follows that
$\|(-\Delta)^{s/2}|u_{k}|\|^{2}_{L^2}\leq\|(-\Delta)^{s/2}u_{k}\|^{2}_{L^2} $,
hence the sequence $\{|u_{k}|\}_{k}$ is still a minimizing
sequence  and  we can assume from the beginning
that $u_{k}\geq 0$ a.e. in $\mathbb{R}^{N}$ for all $k$.
By Lemma \ref{minibound},
this minimizing sequence is bounded in $H^{s}( \mathbb{R}^{N})$, so up to
subsequences,
\[
u_{k}\rightharpoonup u_0 \quad\text{in } H^{s}( \mathbb{R}^{N}),
\]
by Lemma \ref{weakcontinuous}, we have
\[
\int_{ \mathbb{R}^{N}}K(x)|u_{k}|^{p}\,dx\to \int_{ \mathbb{R}%
^{N}}K(x)|u_0|^{p}\,dx,
\]
and then
\[
\int_{ \mathbb{R}^{N}}K(x)|u_0|^{p}\,dx=\int_{ \mathbb{R}^{N}}K(x)|u_{k}|^{p}%
\,dx=\beta,
\]
thus $u_0\in \Sigma_{\beta}$.

Applying Lemma \ref{weakcontinuous} for $p=2$, we have
\[
\int_{ \mathbb{R}^{N}}V(x)|u_{k}|^{2}\,dx \to \int_{ \mathbb{R}%
^{N}}V(x)|u_0|^{2}\,dx.
\]
Since $\lambda\leq 0$, by weak lower semicontinuity of the norm, it follows that
\begin{align*}
&\int_{ \mathbb{R}^{N}}|(-\Delta)^{s/2} u_0|^{2}\,dx +\int_{ \mathbb{R}%
^{N}}V(x)|u_0|^{2}\,dx-\lambda\int_{ \mathbb{R}^{N}}|u_0|^{2}\,dx\\
&=\|(-\Delta)^{s/2}
u_0\|^{2}_{L^2}+(-\lambda)\|u_0\|^{2}_{L^2}
+\int_{ \mathbb{R}^{N}}V(x)|u_0|^{2}\,dx \\
&\leq \liminf_{k\to\infty}\,[\,\|
(-\Delta)^{s/2}u_{k}\|^{2}_{L^2}+(-\lambda)\|u_{k}\|^{2}_{L^2}
+\int_{ \mathbb{R}^{N}}V(x)|u_{k}|^{2}\,dx]
= m_{\beta},
\end{align*}
together with $u_0\in \Sigma_{\beta}$, this shows that
\[
m_{\beta}=\int_{ \mathbb{R}^{N}}|(-\Delta)^{s/2} u_0 |^{2}\,dx
+\int_{ \mathbb{R}^{N}}V(x)|u_0|^{2}\,dx
-\lambda\int_{ \mathbb{R}^{N}}|u_0|^{2}\,dx = \mathcal{J}(u_0).
\]
Note that $u_0\in \Sigma_{\beta}$ implies that $u_0\not\equiv 0$,
then from the definition of $\lambda_0$ given in Theorem \ref{mainresult}
it follows that
\[
m_{\beta}=\mathcal{J}(u_0)\geq(\lambda_0-\lambda)\|u_0\|^{2}_{L^2}>0.
\]
This completes the proof.
\end{proof}

\begin{lemma} \label{lagrangelem}
With the assumptions of Theorem \ref{mainresult}, let $u_0$ be a
minimizer for $m_{\beta}$. Then $u_0$ satisfies
\begin{equation}  \label{lagrange}
\begin{aligned}
&\int_{ \mathbb{R}^{N}}(-\Delta)^{s/2} \overline{u_0}\cdot(-\Delta)^{s/2} v\,dx
 +\int_{\mathbb{R}^N} (V(x)-\lambda)\overline{u_0}\cdot v\,dx \\
&=\frac{m_{\beta}}{\beta} \int_{ \mathbb{R}^{N}}K(x)|u_0|^{p-2}\overline{u_0}
\cdot v\,dx
\end{aligned}
\end{equation}
for all $v\in H^{s}( \mathbb{R}^{N})$.
\end{lemma}

\begin{proof}
Let $\mathcal{J}(u_0)$ be energy functional defined by \eqref{ju}.
Fix $v(x)\in H^{s}( \mathbb{R}^{N})$, for $\varepsilon\in \mathbb{R}$
small enough, when $r\in(-\varepsilon,\varepsilon)$, the function $u_0+rv$
is not identically zero. Therefore there exists a function
$t(r):(-\varepsilon,\varepsilon)\to(0,\infty)$ such that
\[
\int_{ \mathbb{R}^{N}}|K(x)t(r)(u_0+rv)|^{p}\,dx=\beta.
\]
Precisely,
\[
t(r)=\Big(\frac{\beta}{\int_{ \mathbb{R}^{N}}|K(x)(u_0+rv)|^{p}\,dx}
\Big)^{1/p}.
\]
Note that the map $r\mapsto t(r)(u_0+rv)$ defines a curve on
$\Sigma_{\beta}$ that passes through $u_0$ when $r=0$. The function $t(r)$
is differentiable on $(-\varepsilon,\varepsilon)$,
\[
t'(r)=-\beta^{1/p}\Big(\int_{ \mathbb{R}^{N}}|K(x)(u_0+rv)|^{p}\,dx
\Big)^{-\frac{1}{p}-1}\operatorname{Re}(K(x)|u_0+rv|^{p-2}(u_0+rv), v),
\]
where $\operatorname{Re}$ denotes real part of inner product
 $(\cdot,\cdot)$ (defined in \eqref{inner-p}), and
\[
\operatorname{Re}(|u_0+rv|^{p-2}(u_0+rv), v)
=\operatorname{Re} \int_{ \mathbb{R}^{N}}K(x)|(u_0+rv)|^{p-2}
\overline{(u_0+rv)}\cdot v\,dx.
\]
Then we have
\begin{equation}\label{t0}
t(0)=1\quad\text{and}\quad
t'(0)=-\beta^{-1}\operatorname{Re}(K(x)|u_0|^{p-2}u_0, v).
\end{equation}
We define $\gamma:(-\varepsilon,\varepsilon)\to R$ as
\begin{align*}
\gamma(r)
&=\mathcal{J}(t(r)(u_0+rv))
=t^{2}(r)\mathcal{J}(u_0+rv) \\
&=t^{2}(r)((-\Delta)^{s/2} (u_0+rv),(-\Delta)^{s/2}
(u_0+rv))\\
&\quad +t^{2}(r)((V(x)-\lambda)(u_0+rv),(u_0+rv)).
\end{align*}
Since $t(r)(u_0+rv)\in\Sigma_{\beta}$ for every $r\in(-\varepsilon,\varepsilon)
$, the point $r=0$ is a local minimum for $\gamma$, such that
\begin{equation}\label{r0}
\gamma(0)=\mathcal{J}(u_0)=m_{\beta}.
\end{equation}
The function $\gamma$ is differentiable and
\begin{align*}
\gamma'(r)=&2t(r)t'(r)\mathcal{J}(u_0+rv)\\
&+2t^{2}(r)\operatorname{Re}[((-\Delta)^{s/2}
(u_0+rv), (-\Delta)^{s/2} v)+( (V(x)-\lambda)(u_0+rv), v)].
\end{align*}
by \eqref{t0}, \eqref{r0}, then
\begin{equation}  \label{38}
\begin{aligned}
0=\gamma'(0)
 &=2t(0)t'(0)\mathcal{J}(u_0)+2t^{2}(0)\operatorname{Re}\big[((-\Delta)^{s/2}
u_0, (-\Delta)^{s/2} v)\\
&\quad +( (V(x)-\lambda)u_0, v)\big]   \\
&=-2\beta^{-1}\operatorname{Re}(K(x)|u_0|^{p-2}u_0,v)m_{\beta} \\
&\quad +2\operatorname{Re}[((-\Delta)^{s/2}
u_0, (-\Delta)^{s/2}v)+( (V(x)-\lambda)u_0,v)].
\end{aligned}
\end{equation}
Since $v$ is an arbitrary complex function in $H^{s}( \mathbb{R}^{N})$, it
follows that
\[ % \label{39}
-\beta^{-1}(K(x)|u_0|^{p-2}u_0, v)m_{\beta}+[( (-\Delta)^{s/2}
u_0, (-\Delta)^{s/2}v)+( (V(x)-\lambda)u_0, v)]=0,
\]
i.e. \eqref{lagrange} holds.
\end{proof}


Let $u_0$ be a minimizer for $m_{\beta}$. Set $u_0(x)=c\,w(x)$, where
$c\in \mathbb{R}$ will be determined later. By Lemma \ref{lagrangelem},
$w(x)$ satisfies
\[
c[( (-\Delta)^{s/2}w, (-\Delta)^{s/2}v) + ( (V(x) - \lambda)w, v)]
=\frac{m_{\beta}}{\beta}\,c^{p-1}(K(x)
|w|^{p-2}w, v)
\]
for all $v\in H^{s}(\mathbb{R}^{N})$.
Choosing $c=(\frac{\beta}{m_{\beta}})^{\frac{1}{p-2}}$, we see that $w(x)$
is nonnegative by Lemma \ref{existnonega} and satisfies
\[
\big( (-\Delta)^{s/2}w, (-\Delta)^{s/2}v\big)
+ \big( (V(x) - \lambda)w, v\big)
=\big(K(x) |w|^{p-2}w, v\big)\quad \forall v\in H^{s}( \mathbb{R}^{N}),
\]
namely $w(x)$ is a weak (nonzero) solution of \eqref{equa}, such that
 \begin{equation}\label{u_0-w}
 u_0(x)=(\frac{\beta}{m_{\beta}})^{\frac{1}{p-2}} w(x).
 \end{equation}
Thus we obtain the existence of the solution.

Let $\mathcal{N}$ be the Nehari manifold defined by \eqref{Neh},
note that $w(x)\in \mathcal{N}$. We mention in Section 1 that a ground
state of \eqref{equa} is a solution that minimizes the energy functional
$\mathcal{I}(u)$ on the Nehari manifold $\mathcal{N}$, next we will prove
that $w(x)$ is a ground state, that is, we need to prove that
\begin{equation}
\mathcal{I}(w)\leq \mathcal{I}(\phi), \quad \text{for any } \phi\in \mathcal{N}.
\end{equation}
For any function $\phi \in \mathcal{N}$, then by the definition of $\mathcal{N}$
we have
\begin{equation}\label{i-phi}
\mathcal{I}(\phi)=\big(\frac{1}{2}-\frac{1}{p}\big)\mathcal{J}(\phi),
\end{equation}
where $\mathcal{J}(\phi)$ is energy functional defined in \eqref{ju}.

 Fix any $\phi\in \mathcal{N}$ and let
$\theta :=\int_{ \mathbb{R}^{N}}K(x)|\phi|^{p}\,dx$, then
$\phi\in \Sigma_{\theta}$. Let $v_0 = \tilde{c}w(x)$ with
$\tilde{c}=(\frac{\theta}{m_{\theta}})^{\frac{1}{p-2}}$,
we claim that $v_0$ is a minimizer for $m_\theta$.
Indeed, for any $u\in \Sigma_\beta$, the scaling
 $v=(\frac{\theta}{\beta})^{1/p}u\in \Sigma_\theta$, then
$\mathcal{J}(v)=(\frac{\theta}{\beta})^{2/p}\mathcal{J}(u)$, it follows that
\begin{equation}\label{mb-mr}
\frac{m_{\beta}}{\beta^{2/p}}=\frac{m_{\theta}}{\theta^{2/p}}, \quad
\text{for any $\theta>0$ such that $\theta\neq\beta$}.
\end{equation}
 Note that, by \eqref{u_0-w} we know
$w(x)=(\frac{m_{\beta}}{\beta})^{\frac{1}{p-2}}u_0$,
then by \eqref{mb-mr} we have
 \begin{equation}
 v_0=\tilde{c}w(x)=(\frac{\theta}{m_{\theta}})^{\frac{1}{p-2}}
(\frac{m_{\beta}}{\beta})^{\frac{1}{p-2}}u_0=(\frac{\theta}{\beta})^{1/p}u_0.
 \end{equation}
Since $u_0$ is the minimizer for $m_\beta$, it follows that $u_0\in \Sigma_\beta$,
and that that $v_0\in \Sigma_\theta$. Moreover, using \eqref{mb-mr} again,
\[
\mathcal{J}(v_0)=(\frac{\theta}{\beta})^{2/p}\mathcal{J}(u_0)
= (\frac{\theta}{\beta})^{2/p}m_\beta= m_\theta,
\]
thus $v_0$ is the minimizer for $m_\theta$.

Since $w\in \mathcal{N}$, $v_0$,
$\phi\in \Sigma_{\theta}$ and $v_0$ is the minimizer for $m_\theta$,
by \eqref{i-phi} we have
\begin{align*}
\mathcal{I}(w)
&=(\frac{1}{2}-\frac{1}{p})\mathcal{J}(w) =(\frac{1}{2}-\frac{1%
}{p})\mathcal{J}(\tilde{c}^{-1}v_0) \\
&=(\frac{1}{2}-\frac{1}{p})\tilde{c}^{-2}\mathcal{J}(v_0)=(\frac{1}{2}-\frac{1}{p}%
)(\frac{m_{\theta}}{\theta})^{\frac{2}{p-2}}\mathcal{J}(v_0)\\
&\leq (\frac{1}{2}-\frac{1}{p}%
)(\frac{m_{\theta}}{\theta})^{\frac{2}{p-2}}\mathcal{J}(\phi)= (\frac{m_{\theta}}{\theta})^{\frac{2}{p-2}}\mathcal{I}(\phi),
\end{align*}
  hence to prove $\mathcal{I}(w)\leq \mathcal{I}(\phi)$, it is sufficient
to show that $\frac{m_{\theta}}{\theta}\leq 1$.
Since $\phi\in \mathcal{N}\cap \Sigma_{\theta}$, we obtain
\[
\mathcal{J}(\phi)=\int_{\mathbb{R}^{N}}|(-\Delta)^{s/2} \phi|^{2}\,dx
+\int_{ \mathbb{R}^{N}}(V(x)-\lambda)|\phi|^{2}\,dx
=\int_{ \mathbb{R}^{N}}K(x)|\phi|^{p}\,dx=\theta.
\]
Thus
\[
m_{\theta}=\inf_{u\in \Sigma_{\theta}}\mathcal{J}(u)
\leq \mathcal{J} (\phi)=\theta,
\]
i.e., $\frac{m_{\theta}}{\theta}\leq 1$. Thus $w(x)$ is a ground
state of \eqref{equa}.  This completes the proof of Theorem \ref{mainresult}.


\section{Proof of Theorems \ref{positive-result} and  \ref{mainregular}}
 \label{theorem-2-3}

In this section we prove that weak solutions of \eqref{equa} are of class
$C^{0,\alpha}(\mathbb{R}^N)$ for some $\alpha\in(0,1)$.
First we give some properties of
$L^{q}(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$ which will be used below.

\begin{proposition}\label{sumspace}
The space $L^{q}(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$ has  following properties.
\begin{itemize}
\item[(i)]  $L^{r}(\mathbb{R}^{N})\subset L^{q}(\mathbb{R}^{N})
+L^{\infty }(\mathbb{R}^{N})$
for any $1\leq q\leq r\leq \infty$.

\item[(ii)]   $L^{r}(\mathbb{R}^{N})+ L^{\infty}(\mathbb{R}^{N})
\subset L^{q}(\mathbb{R}^{N})+L^{\infty }(\mathbb{R}^{N})$ for any
$1\leq q\leq r\leq \infty$.
\end{itemize}
\end{proposition}

\begin{proof}
(i) Let $f(x) \in L^{r}(\mathbb{R}^{N})$,  for a given constant $M>0$ we have
$f(x)=f_0+f_1$, where
\begin{align*}
f_0=\chi_{\{x: |f(x)|>M\}}f(x),\quad
f_1=\chi_{\{x: |f(x)|\leq M\}}f(x).
\end{align*}
by the Chebyshev inequality \cite{J2001}
\[
|\{x:|f(x)|> M\}|\leq \bigg(\frac{C\|f\|_{L^r}}{M}\bigg)^{r}<\infty.
\]
Since $q\leq r$, then  $L^{r}(\{x:|f(x)|> M\})\subset L^{q}(\{x:|f(x)|> M\}) $,
then $f_0\in L^{q}(\mathbb{R}^N) $. It is obvious to see that
$f_1\in L^{\infty}(\mathbb{R}^N)$.
Then $f(x)\in L^{q}(\mathbb{R}^{N})+L^{\infty }(\mathbb{R}^{N})$.
Therefore, the case (i) holds.

The case (ii) is easy to obtain from  case (i).
\end{proof}

Recall that the definition of fractional Sobolev spaces
(e.g. see \cite{Felmer2012}) for $p\geq 1$ and $\beta>0$:
$$
\mathcal{L}^{\beta,p}=\{u\in L^{p}(\mathbb{R}^N)|
 \mathcal{F}^{-1}[(1+|\xi|^{2})^{\beta/2}\widehat{u}]\in L^{p}(\mathbb{R}^N) \},
$$
and associated to the fractional Laplacian, the space
$$
\mathcal{W}^{\beta, p}=\{\in L^{p}(\mathbb{R}^N)|
 \mathcal{F}^{-1}[(1+|\xi|^{\beta})\widehat{u}]\in L^{p}(\mathbb{R}^N) \}.
$$
 The following two theorems are basic results for these spaces which can
be found in \cite{Felmer2012}.

\begin{theorem}[\cite{Felmer2012}]
Assume that $p\geq 1$ and $\beta > 0$.  The following hold:
\begin{itemize}
\item[(i)]  $\mathcal{L}^{\beta, p} = \mathcal{W}^{\beta,p}$, and
$\mathcal{L}^{n,p}=W^{n,p}(\mathbb{R}^{N} )$ for all $n\in \mathbb{N}$,
where $W^{n,p}$ is the usual Sobolev space.

\item[(ii)] For $\alpha\in (0,1)$ and $2\alpha<\beta$, we have
$(-\Delta)^{\alpha}: W^{\beta,p}\to W^{\beta-2\alpha, p}$.

\item[(iii)] For $\alpha, \gamma\in(0,1)$ and $0<\mu\leq \gamma-2\alpha$, we have
$$
(-\Delta)^{\alpha}: C^{0,\gamma}(\mathbb{R}^N )\to C^{0,\mu}(\mathbb{R}^N )
\quad \text{if } 2\alpha<\gamma,
$$
and, for $0\leq\mu\leq 1+\gamma-2\alpha$,
$$
(-\Delta)^{\alpha}: C^{1,\gamma}(\mathbb{R}^N )\to C^{0,\mu}(\mathbb{R}^N )\quad
\text{if } 2\alpha>\gamma.
$$
\end{itemize}
\end{theorem}

\begin{theorem}[\cite{Felmer2012}] \label{embed}
\begin{itemize}
\item[(i)]  If $0\leq \alpha$, and either $1<p\leq q\leq Np/(N-\alpha p)<\infty$
or $p=1$ and $1\leq q< N/(N-\alpha)$, then
$\mathcal{L}^{\alpha,p}$ is continuously embedded in $L^{q}(\mathbb{R}^N )$.

\item[(ii)]
 Assume that $0\leq \alpha\leq2$ and $\alpha>N/p$. If $\alpha-N/p>1$ and
$0<\mu\leq \alpha-N/p -1$, then $\mathcal{L}^{\alpha,p}$ is continuously
embedded in $C^{1,\mu}(\mathbb{R}^N )$. If $\alpha-N/p<1$ and
$0<\mu\leq \alpha-N/p$, then $\mathcal{L}^{\alpha,p}$ is continuously embedded in
$C^{0,\mu}(\mathbb{R}^N )$.
\end{itemize}
\end{theorem}

Let $\mathcal{H}(x,t)$ be defined in \eqref{hxt} (in the Appendix
below), then we define the kernel $\mathcal{K}$,
$\mathcal{K}^{\mu}$ with $\mu>0$ as
\begin{equation}
\mathcal{K}(x)=\int^{\infty}_0e^{-t}\mathcal{H}(x,t)\,dt,
\quad
\mathcal{K}^{\mu}(x)=\int^{\infty}_0e^{-\mu t}\mathcal{H}(x,t)\,dt.
\end{equation}
By the rescaling property of $\mathcal{H}(x,t)$,
$$
\mathcal{H}(x,\frac{t}{\mu})=\mu^{\frac{N}{2s}}\mathcal{H}(\mu^{\frac{1}{2s}}\,x,t),$$
we have
\begin{equation}\label{rescale}
\mathcal{K}^{\mu}(x)=\mu^{\frac{N}{2s}-1}\mathcal{K}(\mu^{\frac{1}{2s}}x).
\end{equation}
On the other hand, In the Appendix of \cite{Felmer2012}, we know that
$\mathcal{K}(x)=\mathcal{F}^{-1}\big(\frac{1}{1+|\xi|^{2s}}\big)$,
then in the same way,  we have
 \begin{equation}
\mathcal{K}^{\mu}(x)=\mathcal{F}^{-1}\big(\frac{1}{\mu+|\xi|^{2s}}\big).
\end{equation}
The following theorem can be found in \cite{Felmer2012}.

\begin{theorem}[\cite{Felmer2012}] \label{kx}
Let $N\geq 2$ and $s\in(0,1)$. Then we have the following:
\begin{itemize}
\item[(i)] $\mathcal{K}$ is positive, radically symmetric and smooth in
$\mathbb{R}^N\setminus\{0\}$. Moreover, it is nonincreasing as a function
of $r=|x|$.

\item[(ii)] For appropriate constants $C_1$ and $C_{2}$,
\begin{equation} \label{k7.2}
\begin{gathered}
\mathcal{K}(x)\leq \frac{C_1}{|x|^{N+2s}}\quad \text{if } |x|\geq 1, \\
 \mathcal{K}(x)\leq \frac{C_{2}}{|x|^{N-2s}} \quad \text{if } |x|\leq 1.
\end{gathered}
\end{equation}
\end{itemize}
\end{theorem}

\begin{corollary}\label{kmu}
For $N\geq 2$ and  $s\in (0,1)$, we have $\mu>0$ and
 $\mathcal{K}^{\mu}$  satisfies Theorem \ref{kx} (i)-(ii).
\end{corollary}

Since \eqref{rescale} holds, then it is easy to verify the above corollary.

\begin{proof}[Proof of Theorem \ref{positive-result}]
 Since $V(x)$ is bound from above, then there exists a constant $M>0$ 
such that $V(x)\leq M$.  Note that $u(x)\in H^{s}(\mathbb{R}^N)$ 
is a nonnegative solution of \eqref{equa} satisfying
$$
(-\Delta)^s u(x) + V(x)u(x) -K(x) |u|^{p-2}u(x) =\lambda u(x),
$$
then
$$
(-\Delta)^s u(x) + (M-\lambda)u(x)=(M-V(x))u(x) +K(x) |u|^{p-2}u(x).
$$
Let $\mu_0 =M-\lambda$, since $\lambda\leq 0$, we have $\mu_0 >0$. 
Let $h(x)=(M-V(x))u(x) + K(x) |u|^{p-2}u(x)$, then we have
$$
u(x) = \mathcal{K}^{\mu_0} \ast h(x).
$$
Note that  $u(x)$ is  nonnegative and nontrivial, $V(x)\leq M$, $K(x)\neq 0$,  
we have $h(x)\geq 0$ such that $h(x)\neq 0$. By the corollary \ref{kmu}, 
we know that $K^{\mu_0}$ is positive, it follows that $u(x)$ is positive 
in $\mathbb{R}^N$. The proof is complete.
\end{proof}

To discuss the regularity of the weak solution \eqref{equa}, first we discuss 
the following result about liner equations.

\begin{theorem}\label{regularity}
Let $s\in (0,1)$, assume that $u\in H^{s}(\mathbb{R}^N)$, $N>2s$ such that
\begin{equation}
(-\Delta)^{s}u(x)+ \mu u(x)= V(x)u(x) \quad \text{in } \mathbb{R}^N,
\end{equation}
for $\mu >0$, $V(x)\in L^{q}(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$ with 
$q>\frac{N}{2s}$. Then $u\in C^{0, \alpha}(\mathbb{R}^N)$ for some 
$\alpha\in (0,1)$. Moreover, $u(x)\to 0$ as $|x|\to \infty$.
\end{theorem}

\begin{proof}
First we know that $u\in H^{s}(\mathbb{R}^N)=\mathcal{W}^{s,2}$. 
Let $1=r_0>r_1>r_2>\cdots$, and consider $B_i =B(0,r_i )$, the ball of 
radius $r_i$ and centered at the origin. We define $h(x)=V(x)u(x)$, 
since $V(x)\in L^{q}(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$, we have 
$V(x)=V_1 +V_2 $ such that $V_1\in L^{q}(\mathbb{R}^N)$ and 
$V_2 \in L^{\infty}(\mathbb{R}^N)$, then $h(x)=h_1 +h_2$ with $h_1 =V_1 u(x)$ 
and $h_2 =V_2 u(x)$.  Since $u\in H^{s}(\mathbb{R}^N)$, by Sobolev inequality
 we have $u\in L^{2^{*}}(\mathbb{R}^N)$ with $2^{*}=2N/(N-2s)$. 
Since $V_1\in L^{q}(\mathbb{R}^N)$, by H\"{o}lder inequality, then we have 
$h_1\in L^{k_0  }(\mathbb{R}^N)$ with $k_0 =(1/q +1/2^{*})^{-1}$. 
Therefore, $h(x)=h_1 +h_2$ with $h_1 \in L^{k_0}(\mathbb{R}^N)$ and 
$h_2 \in L^{2^{*}}(\mathbb{R}^N)$.

Now let $\eta_1 \in C^{\infty}$ with $0\leq\eta_1\leq 1$, with support in 
$B_0$ and such that $\eta_1 \equiv 1$ in $B_{1/2}$, where 
$B_{1/2}=B(0,r_{1/2})$ with $r_1 <r_{1/2}<r_0$. Let $u_1$ be the 
solution of the equation
\begin{equation}\label{u1}
(-\Delta)^s u_1 +\mu u_1 =\eta_1 h(x) \quad \text{in } \mathbb{R}^N,
\end{equation}
then
\begin{equation}
(-\Delta)^s(u- u_1) +\mu(u- u_1) =(1-\eta_1 )h(x) \quad \text{in } \mathbb{R}^N,
\end{equation}
so that
\begin{equation}
u-u_1 = \mathcal{K}^\mu \ast \{(1-\eta_1 )h\}.
\end{equation}
Using the H\"{o}lder inequality and \eqref{k7.2} we have 
\begin{equation}\label{cutequ}
\begin{aligned}
&|u(x)-u_1 (x)| \\
&\leq C\big\{\|\mathcal{K}^{\mu}\|_{L^{l_0}
(B^{c}_{1/2})}\|(1-\eta_1)h_1 \|_{L^{k_0}}
+\|\mathcal{K}^{\mu}\|_{L^{l_1}(B^{c}_{1/2})}\|
(1-\eta_1)h_2 \|_{L^{2^{*}}}\big\},
\end{aligned}
\end{equation}
for all $x\in B_1 $, where $l_0 =k_0 /(k_0 -1)$, $k_0$ is given above, 
and $l_1 = 2^{*}/(2^{*}-1)$. In view of this inequality we have to concentrate 
our attention in $u_1 (x)$.

Since $B_0$ is bound and $V(x)\in L^q (\mathbb{R}^N) + L^{\infty}(\mathbb{R}^N)$, 
we obtain that $\eta_1 V(x)\in L^{q}(B_0 )$. With the assumption $q>\frac{N}{2s}$, 
we have $\eta_1 V(x)\in L^{q_0}(B_0 )$ for  
$\frac{N}{2s}<q_0\leq\min\{q, \frac{N}{s}\}$. 
Since $u\in L^{2^{*}}(\mathbb{R}^N)$, by H\"{o}lder inequality, we have 
$\eta_1 V(x)u\in L^{k_1}(\mathbb{R}^N)$, for $k_1=(1/q_0 +1/2^{*})^{-1}$ 
such that $k_1>1$. Since $\eta_1 $ has support in $B_0$, we have  
$\eta_1 V(x)u\in L^{p_1}(\mathbb{R}^N)$, for any $1<p_1< \min \{k_1, N/(2s)\}$. 
Note that $u_1 $ satisfies \eqref{u1}, thus by the definition of the space 
$\mathcal{W}^{2s, p_1}$, we have $u_1 \in \mathcal{W}^{2s, p_1}$. 
Then, using Sobolev embedding of the Theorem \ref{embed} (i) and \eqref{cutequ}, 
we have $u\in L^{q_1}(B_1)$ for  $ q_1= p_1 N/(N-2s\,p_1)$.

Now we repeat the procedure, but consider a smooth function $\eta_2$ 
such that $0\leq \eta_2\leq 1$, with support in $B_1$ and $\eta_2\equiv 1$ 
in $B_{3/2}$, where $B_{3/2}=B(0, r_{3/2})$ with $r_2< r_{3/2}<r_1$. We also have
$\eta_2 V(x)\in L^{q_0}(B_1 )$ for any $\frac{N}{2s}<q_0\leq\min\{q, \frac{N}{s}\}$,
we can set $\frac{1}{q_0 } =\frac{2s}{N}-\epsilon$ with $0<\epsilon\leq \frac{s}{N}$. 
By H\"{o}lder inequality  again, we have $\eta_2 V(x)u\in L^{p_2} (B_2 )$ for any
 $$
1\leq p_2< p_1/(1-\epsilon)\quad  \text{where }  p_{2}=(1/q_0 +1/q_1 )^{-1}.
$$
 Proceeding as above, with the obvious changes we obtain that
$$
u_2=\mathcal{K}^{\mu}\ast (\eta_{2}h(x)),
$$
satisfying $u_2\in \mathcal{W}^{2s, p_2}$.  
Then we have $u\in L^{q_2}(B_2)$ for  $ q_2= p_2 N/(N-2s\,p_2)$.

 Repeating the argument, for  sequences $\eta_j$, $p_j$ and 
$ q_j= p_j N/(N-2s\,p_j)$, we have $\eta_j V(x)u\in L^{p_j}(B_j)$ for any
 $$
1\leq p_j< p_{j-1}/(1-\epsilon)\quad  \text{where } 
  p_{j}=(1/q_0 +1/q_{j} )^{-1}.
$$
 It follows that for some finite $j$, $\eta_j V(x)u\in L^{p_j} (B_j )$ 
such that $p_j>N/(2s)$. Then by Theorem \ref{embed}(ii), we have 
$u_j\in C^{0, \alpha}(\mathbb{R}^N)$ for some $\alpha\in (0,1)$. 
Since $u_j$ satisfies the inequality that similar to \eqref{cutequ}, 
we have $u\in C^{0, \alpha}(B_{j+1})$.

 The ball $B_j$ is centered at the origin, but we may arbitrarily move it
 around $\mathbb{R}^N$. Covering $\mathbb{R}^N$ with these balls, we obtain 
that $u\in C^{0, \alpha}(\mathbb{R}^N)$.
 Finally, the fact that $u\in L^{2^{*}}(\mathbb{R}^N)\cap C^{0,\alpha}(\mathbb{R}^N)$ 
implies that $u(x)\to 0$ as $|x|\to \infty$, completing the proof.
\end{proof}


\begin{proof}[Proof of Theorem \ref{mainregular}]
Note that $u(x)$ satisfies
$$
(-\Delta)^s u(x) + V(x)u(x) -K(x) |u|^{p-2}u(x) =\lambda u(x),
$$
for $2<p< 2^*$.
Let $\widetilde{V}(x)=-V(x)+ K(x)|u|^{p-2}$, then the equation becomes
$$
(-\Delta)^s u(x)-\lambda u(x) = \widetilde{V}(x)u(x).
$$
We claim that $\widetilde{V}(x)\in L^{l}(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$ 
for some $l>\frac{N}{2s}$.

Since the condition (A5) holds, 
$K(x)\in L^{\tilde{r}}(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$ 
for $\tilde{r}>\frac{2^{*}}{2^{*}-p}$, then  $K(x)=K_1 + K_2$ with 
$K_1\in L^{\tilde{r}}(\mathbb{R}^N)$ and $K_2 \in L^{\infty}(\mathbb{R}^N)$. Then
Since $u\in L^{2^{*}}(\mathbb{R}^N)$, we have 
$K_2 |u|^{p-2}\in L^{r_0}(\mathbb{R}^N)$ for 
$r_0=\frac{2^{*}}{p-2}>\frac{2^{*}}{2^{*}-2}=\frac{N}{2s}$.
 By H\"{o}lder inequality, we have 
$K_1 |u|^{p-2}\in L^{r_1}(\mathbb{R}^N)$ with 
$r_1=(\frac{1}{\tilde{r}}+\frac{p-2}{2^{*}})^{-1}$ such that 
$r_1>\frac{N}{2s}$. Then by Proposition \ref{sumspace} (i),
 we have $K(x)|u|^{p-2}\in L^{l_1}(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$ 
with $l_1=\min\{r_0, r_1\}$ such that $l_1>\frac{N}{2s}$. 
Then by Proposition \ref{sumspace} (ii), we have 
$\widetilde{V}(x)\in L^{l}(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$ with 
$l=\min\{\tilde{q},l_1\}$(where $\tilde{q}$ given in (A5)), such that
 $l>\frac{N}{2s}$. Then by Theorem \ref{regularity}, we obtain the regular 
result of Theorem \ref{mainregular}.
\end{proof}

\subsection*{Acknowledgments}
This work was supported by the NSFC under grants
No. 11475073 and No. 11325417.

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