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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 22, pp. 1--12.\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-2018/22\hfil Critical Sobolev-Hardy exponents]
{Fractional elliptic problems with two critical Sobolev-Hardy exponents}

\author[W. Chen \hfil EJDE-2018/22\hfilneg]
{Wenjing Chen}

\address{Wenjing Chen \newline
School of Mathematics and Statistics,
Southwest University,
Chongqing 400715, China}
\email{wjchen@swu.edu.cn}

\dedicatory{Communicated by Giovanni Molica Bisci}

\thanks{Submitted July 1, 2017. Published January 15, 2018.}
\subjclass[2010]{35J20, 35J60, 47G20}
\keywords{Fractional elliptic problems; mountain pass lemma;
\hfill\break\indent critical fractional Hardy-Sobolev exponent;
 concentration compactness principle}

\begin{abstract}
 By using the mountain pass lemma and a concentration compactness principle,
 we obtain the existence of positive solutions to the fractional elliptic
 problem with two critical Hardy-Sobolev exponents at the origin.
\end{abstract}

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

\section{Introduction}\label{intro}

In this article, we study the following doubly critical problem involving 
the fractional Laplacian
\begin{equation}\label{frac1}
(-\Delta)^s u-\gamma\frac{u}{|x|^{2s}}
=\frac{|u|^{2_s^\ast(\alpha)-2}u}{|x|^\alpha}
+ \frac{|u|^{2_s^\ast(\beta)-2}u}{|x|^\beta},\quad
u>0,\quad  \text{in }\mathbb{R}^n,
\end{equation}
where $s\in(0,1)$, $0< \alpha,\beta< 2s<n$  with $\alpha\neq \beta$,
$ \gamma < \gamma_H$ with 
\[
\gamma_H =4^s \frac{\Gamma^2(\frac{n+2s}{4})}{\Gamma^2(\frac{n-2s}{4})} 
\]
being the fractional best Hardy constant on $\mathbb{R}^n$,
and $ 2_s^\ast(\alpha)=2(n-\alpha)/(n-2s)$ is the fractional critical 
Hardy-Sobolev exponent. The operator $(-\Delta)^{s}$ is  the fractional 
Laplacian defined as
\begin{align*}
(-\Delta )^su(x)=c_{n,s}\operatorname{pv}\int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy ,
\quad s\in(0,1),
\end{align*}
where pv stands for the Cauchy principle value and 
\[
c_{n,s}=2^{2s-1}\pi^{-\frac{n}{2}}
\frac{\Gamma\left(\frac{n+2s}{2}\right)}{|\Gamma(-s)|}
\]
is the normalization constant so that the identity
$$
 (-\Delta)^{s}u= \mathcal{F}^{-1}(|\xi|^{2s}(\mathcal{F}u)) \quad 
\forall\xi\in\mathbb{R}^n,\; s\in(0,1),\; u\in\mathcal{S}(\mathbb{R}^n),
$$
holds, here $ \mathcal{F}u$ denotes the Fourier transform of $u$, 
$\mathcal{F}u(\xi)=\int_{\mathbb{R}^n} e^{-2\pi i x\cdot\xi} u(x)\,dx$,
and $ \mathcal{S}(\mathbb{R}^n)$ the Schwartz class, see \cite{Hitchhikers} 
and references therein for the basics on the fractional Laplacian.

In previous twenty years, the nonlocal elliptic problems have been 
investigated by many researchers, for example, 
\cite{fiscella,MB1,sv,svmountain,svlinking}
for the subcritical case, 
\cite{ao,CD1,hmv,F2,MB2,servadeivaldinociBN,servadeivaldinociBNLOW} 
for the critical case, \cite{CD2,CD3,CM} for the existence of solutions 
to fractional Laplacian system.
Moreover, a great attention has been devoted to study the existence of solutions 
for the nonlocal problems with Hardy potential or nonlinearity term,
we refer to see 
\cite{APP,fethi,Barrios-Medina-Peral,Cotsiolis-Tavoularis,Fall,Fall-Minlend-Thiam,
wangyang,Yang,Yang2}
and the references therein. In particular, the existence of solutions to the 
 problem
\begin{equation} \label{one}
({-}{ \Delta})^{s}u- \gamma \frac{u}{|x|^{2s}}
= {\frac{u^{2_{s}^*(\alpha)-1}}{|x|^\alpha}} ,\quad
 u>0 \quad     \text{in }  \mathbb{R}^n,
\end{equation}
corresponds to the  minimization problem
\begin{equation}\label{hsc}
\mu_{s,\gamma,\alpha}(\mathbb{R}^n)
=\inf_{u\in H^s(\mathbb{R}^n)\backslash\{0\}}
\frac{\int_{\mathbb{R}^n} |({-}{ \Delta})^{s/2}u|^2\,dx
- \gamma \int_{\mathbb{R}^n} \frac{|u|^{2}}{|x|^{{2s}}}\,dx}
{\big(\int_{\mathbb{R}^n} \frac{|u|^{2_s^\ast(\alpha)}}{|x|^{\alpha}}\,dx
\big)^\frac{2}{{2_s^\ast(\alpha)}}}.
\end{equation}
 Fall et al.\ \cite{Fall-Minlend-Thiam}  proved the existence of extremals
for $\mu_{s,0,\alpha}(\mathbb{R}^n)$ in the case $s=\frac{1}{2}$.
Yang \cite{Yang} proved that there exists a positive, radially symmetric and
non-increasing extremal for $\mu_{s,0,\alpha}(\mathbb{R}^n)$ when $s \in (0,1)$.
Asymptotic properties of the positive solutions was given by Lei  \cite{Lei}
and  Yang-Yu \cite{Yang-Yu}.
The existence of extremals for $\mu_{s,\gamma,\alpha}(\mathbb{R}^n)$ in \eqref{hsc},
when $\alpha \in [0,2s)$ and $ \gamma\in (-\infty, \gamma_H)$,
 was recently studied by  Ghoussoub and Shakerian in \cite{Ghoussoub-Shakerian}.
Moreover, the authors in \cite{Ghoussoub-Shakerian} used the mountain pass
lemma to establish  the existence of a nontrivial  weak solution  to the  problem
\[
(-\Delta)^s u-\gamma\frac{u}{|x|^{2s}}
= |u|^{2_s^\ast -2}u + \frac{|u|^{2_s^\ast(\alpha)-2}u}{|x|^\alpha},\quad
u>0,\quad \text{in }\mathbb{R}^n.
\]
Furthermore, the authors in \cite{Yang2} showed the existence of nontrivial
solutions for fractional elliptic
problem in $\mathbb{R}^n$ with the critical nonlocal Hartree term  and
critical fractional Hardy-Sobolev term.

It is worth pointing out that in the local case, i.e. $s=1$, 
the existence and multiplicity of solutions for the Laplacian problems 
with Hardy terms have been extensively studied,
we refer the reader to  
\cite{bome,Catrina-Wang,Chern-Lin,Filippucci-Pucci-Robert,Ghoussoub-Yuan} 
and references therein.

The aim of this paper is to consider the existence of nontrivial weak solutions 
of \eqref{frac1}, which has a single pole with different powers of singularity 
and fractional critical Hardy-Sobolev exponents.
We get the existence of nontrivial weak solutions of our problem by the 
Mountain Pass Lemma  with concentration-compactness principle.
Our result can be stated as follows.

\begin{theorem}\label{fracmain}
Let $0<s<1$, $ 0 < \alpha,\beta < 2s <n$ with $\alpha\neq \beta$, and 
$\gamma<\gamma_H$.
Then problem \eqref{frac1} admits a nontrivial solution.
\end{theorem}

This article is organized as follows: in Section \ref{pre}, 
we give some preliminaries about fractional Laplacian harmonic extension 
and function space, and also the fractional Hardy-Sobolev inequality. 
We prove the compactness of the energy in Section \ref{com}. 
Section \ref{proof} is concerned with the proof of our main result.


\section{Preliminary results}\label{pre}

In this section, we first introduce suitable function spaces for the variational
 principles that will be needed in the sequel.
Caffarelli and Silvestre in \cite{Caffarelli-Silvestre} showed that the 
fractional Laplacian operator can be realized
in a local way by using one more variable and the so-called $s-$harmonic extension, 
that is, for a function $u \in H^{s}(\mathbb{R}^n)$, we say that  $U=E_{s}(u)$ is its 
$s$-harmonic extension to the upper half-space, $\mathbb{R}_+^{n+1}$, i.e.\ 
it is a solution to the  problem
\[
\operatorname{div}(y^{1-2s} \nabla U)=0 \quad \text{in } \mathbb{R}_+^{n+1}, \\
U= u \quad \text{on }   \mathbb{R}^n \times \{y=0\}.
\]
Define the space  $X^{s} (\mathbb{R}_+^{n+1})$ as the closure of  
$C_0^{\infty}(\overline{\mathbb{R}_+^{n+1})}$ with the norm
$$
\|U\|_{X^{s}({\mathbb{R}_+^{n+1}})}
:=\Big( k_{s}  \int_{\mathbb{R}_+^{n+1}} y^{1-2s} | \nabla U(x,y) |^2 \,dx\,dy
 \Big)^{1/2},
$$
 where $k_s=\frac{\Gamma(s)}{2^{1-2s}\Gamma(1-{s})}$ is a normalization constant 
chosen in such a way that the extension operator
 $U: {H^{s}(\mathbb{R}^n) \to X^{s} (\mathbb{R}_+^{n+1})}$  is an isometry, that is, 
for any $ u \in H^{s}(\mathbb{R}^n)$, we have
\begin{equation} \label{extension norm}
\|U\|_{X^{s} (\mathbb{R}^{n+1}_+)} = \|u \|_{H^{s}(\mathbb{R}^n)}
=\| (-\Delta)^{s/2} u \|_{L^2(\mathbb{R}^n)}.
\end{equation}
Conversely,  for a function $U \in X^{s} (\mathbb{R}_+^{n+1})$, 
we denote its trace on $\mathbb{R}^n \times  \{y = 0\}$ as $u=\text{Tr}(U):=U(\cdot,0)$.
This trace operator is also well defined and satisfies
\begin{equation}\label{trace inequality between extension norm and fractional sobolev}
\|u\|_{H^{s}(\mathbb{R}^n)}=\|U(\cdot,0) \|_{H^{s}(\mathbb{R}^n)} \le \|U\|_{X^{s} (\mathbb{R}_+^{n+1})}.
\end{equation}
Caffarelli and Silvestre \cite{Caffarelli-Silvestre} showed that the extension 
function $U:=E_{s}(u)$ is related to the fractional Laplacian of the original 
function $u$ in the following way:
\begin{equation*}
(-\Delta)^{s}u(x)=\frac{\partial U}{\partial \nu^{s}}
:= - k_{s} \lim_{y \to 0^+} y^{1-2s} \frac{\partial U}{\partial y}(x,y).
\end{equation*}
Thus,  problem \eqref{frac1} can  be written as the  local problem
\begin{equation} \label{frac1a}
\begin{gathered}
 - \operatorname{div}(y^{1-2s}\nabla  U)=0 \quad \text{in }  \mathbb{R}^{n+1}_+ \\
\frac{\partial U}{\partial \nu^{s}}= \gamma \frac{u}{|x|^{2s}}  
+ \frac{|u|^{{2_{s}^*(\alpha)}-2}u}{|x|^\alpha}
+ \frac{|u|^{{2_{s}^*(\beta)}-2}u}{|x|^\beta}  \quad\text{on }  \mathbb{R}^n,
\end{gathered}
\end{equation}
where and in the follows $u=U(\cdot,0)$.
A function $U \in X^{s}(\mathbb{R}_+^{n+1}) $ is said to be a weak solution to 
\eqref{frac1a}, if for all $\Psi \in X^{s}(\mathbb{R}_+^{n+1})$,
\begin{align*}
  k_{s} \int_{\mathbb{R}_+^{n+1}} y^{1-2s} 
\langle \nabla U, \nabla \Psi \rangle  \,dx\,dy &
= \int_{\mathbb{R}^n} \gamma \frac{u}{|x|^{2s}} \psi\,dx
 +  \int_{\mathbb{R}^n} \frac{|u|^{{2_{s}^*(\alpha)}-2}u}{|x|^\alpha} \psi\,dx \\
&\quad + \int_{\mathbb{R}^n} \frac{|u|^{{2_{s}^*(\beta)}-2}u}{|x|^\beta} \psi\,dx,
\end{align*}
where $\psi=\Psi(\cdot,0)$.
The energy functional corresponding to \eqref{frac1a} is
\begin{align*}
J(U)&= \frac{1}{2} \| U\|^2_{X^{s} (\mathbb{R}_+^{n+1})}
  - \frac{\gamma}{2}\int_{\mathbb{R}^n}\frac{|u|^{2}}{|x|^{2s}}\,dx
 -\frac{1}{2_{s}^*(\alpha)}\int_{\mathbb{R}^n} \frac{|u|^{2_{s}^*(\alpha)}}{|x|^{s}}\,dx \\
&\quad -\frac{1}{2_{s}^*(\beta)}\int_{\mathbb{R}^n} \frac{|u|^{2_{s}^*(\beta)}}{|x|^{s}}\,dx.
\end{align*}
We note that for any weak solution $U\in X^{s}(\mathbb{R}_+^{n+1})$ to  \eqref{frac1a}, 
the function $u=U(\cdot,0) $  is in $H^{s}(\mathbb{R}^n)$ and is a weak solution to 
problem \eqref{frac1}. Hence the associated trace of any critical point $U$ of $J$ 
 in $X^{s}(\mathbb{R}_+^{n+1})$  is a weak solution  for  \eqref{frac1}.
Let us recall the following results.

\begin{lemma}\label{INEQUALITY} 
Assume that $0<s<1$.

(i) (The fractional  Hardy inequality  \cite{Frank-Lieb-Seiringer}) 
For all $u \in H^{s} (\mathbb{R}^n)$, we have
\begin{equation}\label{fhi}
\gamma_H \int_{\mathbb{R}^n}\frac{|u|^2}{|x|^{2s}}\,dx 
\leq \int_{\mathbb{R}^n} |({-}{ \Delta})^{s/2}u|^2\,dx,
\end{equation}
where $\gamma_H= 4^{ s} \frac{\Gamma^2(\frac{n+2s}{4})}{\Gamma^2(\frac{n-2s}{4})}$
is the best constant in the above inequality on $\mathbb{R}^n$.

(ii) (The fractional Hardy-Sobolev inequality \cite{Ghoussoub-Shakerian}) 
Assume $ 0 \le \alpha \le 2s <n$. Then, there exist positive constants 
$ c$ and $C$, such that for all $u \in H^{s} (\mathbb{R}^n)$,
\begin{equation} \label{Fractional H-S inequality}
\Big(\int_{\mathbb{R}^n} \frac{|u|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
\Big)^\frac{2}{{2_{s}^*(\alpha)}} 
\leq c \int_{\mathbb{R}^n} |({-}{ \Delta})^{s/2}u|^2\,dx.
\end{equation}
Moreover, if $\gamma < \gamma_H$, then
\begin{equation} \label{fractional H-S-M inequality}
C\Big(\int_{\mathbb{R}^n} \frac{|u|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
\Big)^\frac{2}{{2_{s}^*(\alpha)}} 
\leq \int_{\mathbb{R}^n} |({-}{ \Delta})^{s/2}u|^2\,dx  
- \gamma \int_{\mathbb{R}^n} \frac{|u|^{2}}{|x|^{2s}}\,dx,
\end{equation}
for all $u \in H^{s} (\mathbb{R}^n)$.
 \end{lemma}


\begin{remark} \rm
One can use \eqref{extension norm} to rewrite inequalities \eqref{fhi},
 \eqref{Fractional H-S inequality}  and  \eqref{fractional H-S-M inequality} 
as the following trace class inequalities:
\begin{gather} \label{fractional Trace Hardy inequality}
\gamma_H \int_{\mathbb{R}^n} \frac{|u|^{2}}{|x|^{2s}} \,dx
 \leq  \| U\|^2_{X^{s} (\mathbb{R}_+^{n+1})}, \\
 \label{fractional Trace H-S inequality}
\Big(\int_{\mathbb{R}^n} \frac{|u|^{2_{s}^*(\alpha)}}{|x|^{\alpha}} 
\,dx\Big)^\frac{2}{{2_{s}^*(\alpha)}} 
\leq c  \| U\|^2_{X^{s} (\mathbb{R}_+^{n+1})}, \\
\label{fractional Trace H-S-M inequality }
C\Big(\int_{\mathbb{R}^n} \frac{|u|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}
 \,dx\Big)^\frac{2}{{2_{s}^*(\alpha)}} 
\leq  \| U\|^2_{X^{s} (\mathbb{R}_+^{n+1})} - \gamma\int_{\mathbb{R}^n}
\frac{|u|^{2}}{|x|^{2s}}\,dx.
\end{gather}
\end{remark}

In what follows, we will denote by $X^{s} (\mathbb{R}_+^{n+1})$ the closure of 
 $C_0^{\infty}(\overline{\mathbb{R}_+^{n+1})}$ for the following norm
\begin{equation} \label{norm}
\|U\|:=\Big( k_{s}  \int_{\mathbb{R}_+^{n+1}} y^{1-2s} | \nabla U |^2 \,dx\,dy 
- \gamma\int_{\mathbb{R}^n}\frac{|u|^{2}}{|x|^{2s}}\,dx\Big)^{1/2} \quad 
\text{for  all } \gamma<\gamma_H.
\end{equation}
Note that inequality \eqref{fractional Trace Hardy inequality} asserts that 
$X^{s} (\mathbb{R}_+^{n+1})$ is embedded in the weighted space $L^2(\mathbb{R}^n, |x|^{-2s})$ 
and this embedding is continuous.
Set $\gamma_+ = \text{max} \{\gamma,0\}$ and 
$\gamma_- = - \text{max} \{\gamma,0\}$. The following inequalities hold for 
any $u \in X^{s} (\mathbb{R}_+^{n+1})$,
\begin{equation}\label{comparable norms}
(1-\frac{\gamma_+}{\gamma_H}) \|U\|^2_{X^{s} (\mathbb{R}_+^{n+1})} 
\le \|U\|^2 \le (1+\frac{\gamma_-}{\gamma_H}) \|U\|^2_{X^{s} (\mathbb{R}_+^{n+1})} .
\end{equation}
Thus, $\| \cdot \|$ is equivalent to the norm $\| \cdot\ \|_{X^{s} (\mathbb{R}_+^{n+1})}$.


The best constant $\mu_{s,\gamma,\alpha}(\mathbb{R}^n)$ in inequality
 \eqref{fractional H-S-M inequality}  can be written as
\begin{gather*}
S(n,s,\gamma,\alpha)= \inf_{U\in X^{s} (\mathbb{R}_+^{n+1})\setminus 
\{0\}}I_{\gamma,\alpha}(U),\quad \text{with}\\
 I_{\gamma,\alpha}(U)= \frac{k_{s}\int_{\mathbb{R}_+^{n+1}} y^{1-2s} |\nabla U|^2  \,dx\,dy 
- \gamma \int_{\mathbb{R}^n} \frac{|u|^2}{|x|^{2s}}\,dx}{(\int_{\mathbb{R}^n} 
\frac{|u|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx)^\frac{2}{2_{s}^*(\alpha)}}.
\end{gather*}
If $S(n,s,\gamma,\alpha)$ is attained at some function $U\in X^{s} (\mathbb{R}_+^{n+1})$, 
then $u =  U(.,0)$ will be a function in $ H^{s}(\mathbb{R}^n)$, where
 $\mu_{s,\gamma,\alpha}(\mathbb{R}^n)$ is attained.
Recently, Ghoussoub and Shakerian \cite{Ghoussoub-Shakerian} proved the extremal 
function  of $S(n,s,\gamma,\alpha)$ is attained as following.

\begin{lemma}[\cite{Ghoussoub-Shakerian}] \label{extremal}
 Suppose  $0<s<1$, $ 0 \le \alpha < 2s <n$, and $\gamma < \gamma_H $. Then
\begin{enumerate}
\item If $ \{ \alpha > 0 \}$  or $\alpha=0 $ and 
$\gamma \ge 0$, then $S(n,s,\gamma,\alpha)$ is attained in 
$X^{s} (\mathbb{R}_+^{n+1})$ by $W_{\gamma,\alpha}$.

\item If $\alpha=0$ and $\gamma < 0$, then there are no extremals for
 $S(n,s,\gamma, \alpha)$ in $X^{s} (\mathbb{R}_+^{n+1})$.
 \end{enumerate}
\end{lemma}


\section{Compactness lemmas}\label{com}

In this section, we study the compactness properties of the functional
\begin{equation}\label{energya}
J(U)= \frac{1}{2} \| U\|^2 -\frac{1}{2_{s}^*(\alpha)}
\int_{\mathbb{R}^n} \frac{|u|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
-\frac{1}{2_{s}^*(\beta)}\int_{\mathbb{R}^n} \frac{|u|^{2_{s}^*(\beta)}}{|x|^{\beta}}\,dx
\end{equation}
for $U \in X^{s} (\mathbb{R}_+^{n+1})$,
where again $u:= U(\cdot,0)$.  From Lemma \ref{INEQUALITY}, we have that  
$J \in C^1(X^{s} (\mathbb{R}_+^{n+1}))$.

\begin{definition}\label{defps}\rm
Let $c\in\mathbb{R}$ , $E$ be a Banach space and $J\in  C^1(E,\mathbb{R})$.

(i) $\{u_k\}$ is a $(PS)_c$ sequence in $E$ for $J$ if $J(u_k)=c+o(1)$ and 
$J'(u_k)=o(1)$ strongly in $E^\ast$ as $k\to\infty$.

(ii) We say that $J$ satisfies the $(PS)_c$ condition if any $(PS)_c$ sequence
 $\{u_k\}$ for $J$ in $E$ has a convergent subsequence.
\end{definition}

\begin{proposition} \label{psbound}
Suppose  $  0 < \alpha,\beta<2s$ and $\gamma < \gamma_H $, then the functional 
$J$ defined in \eqref{energya} satisfies the Palais-Smale condition $(PS)_c$ 
for $c<c_\ast$, where
\begin{equation} \label{Definition of C^star}
c_\ast: = \min  \Big\{ \frac{2s-\alpha}{2(n-\alpha)} 
 S(n,s,\gamma,\alpha)^{\frac{n-\alpha}{2s-\alpha}} ,
\frac{2s-\beta}{2(n-\beta)} S(n,s,\gamma,\beta)^{\frac{n-\beta}{2s-\beta}} \Big\}.
\end{equation}
\end{proposition}

\begin{proof}
Let $\{U_k\}_{k\in\mathbb{N}}$ be the Palais-Smale sequence of the functional $J$, 
i.e.
$$
J(U_k)\to c,\quad J'(U_k)\to0\quad \text{in } 
 (X^s(\mathbb{R}^{n+1}_+))' \text{ as } k\to\infty.
$$
Then
\begin{equation}\label{psc1a}
\begin{aligned}
J(U_k)&= \frac{1}{2} \| U_k\|^2 -\frac{1}{2_{s}^*(\alpha)}
\int_{\mathbb{R}^n} \frac{|u_k|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
-\frac{1}{2_{s}^*(\beta)}\int_{\mathbb{R}^n} 
\frac{|u_k|^{2_{s}^*(\beta)}}{|x|^{\beta}}\,dx \\
&=c+o_k(1),
\end{aligned}
\end{equation}
and
\begin{equation}\label{psc2}
\langle J'(U_k),U_k\rangle
= \|U_k\|^2 - \int_{\mathbb{R}^n} \frac{|u_k|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
- \int_{\mathbb{R}^n} \frac{|u_k|^{2_{s}^*(\beta)}}{|x|^{\beta}}\,dx
= o_k(1) \|w_k\|,
\end{equation}
where again $u_k =U_k(\cdot,0)$ and $o_k(1)\to0$ as $k\to\infty$. 
From \eqref{psc1a} and \eqref{psc2}, we have
\begin{align*}% \label{psc3}
c+o_k(1)\|U_k\|&= J(U_k)-  \frac{1}{2}\langle J'(U_k),U_k\rangle \\
&= \Big(\frac{1}{2} -\frac{1}{2_{s}^*(\alpha)}\Big)
\int_{\mathbb{R}^n} \frac{|u_k|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
+\Big(\frac{1}{2} -\frac{1}{2_{s}^*(\beta)}\Big)
\int_{\mathbb{R}^n} \frac{|u_k|^{2_{s}^*(\beta)}}{|x|^{\beta}}\,dx.
\end{align*}
Since $2_{s}^*(\alpha)>2$, $2_{s}^*(\beta)>2$, we have
\begin{equation}\label{psc3}
\int_{\mathbb{R}^n} \frac{|u_k|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx\leq C+o_k(1)\|U_k\|,\quad \ \ \
\int_{\mathbb{R}^n} \frac{|u_k|^{2_{s}^*(\beta)}}{|x|^{\beta}}\,dx\leq C+o_k(1)\|U_k\|.
\end{equation}
By \eqref{psc2} and \eqref{psc3}, we obtain
\begin{equation}\label{psc4}
   \|U_k\|^2+o_k(1)\|U_k\|
=  \int_{\mathbb{R}^n} \frac{|u_k|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
+ \int_{\mathbb{R}^n} \frac{|u_k|^{2_{s}^*(\beta)}}{|x|^{\beta}}\,dx
\leq C+o_k(1)\|U_k\|,
\end{equation}
which implies that $\{U_k\}_{k\in\mathbb{N}}$ is bounded in $X^s(\mathbb{R}^{n+1}_+)$.
It follows that there exists a subsequence, still denote by $U_k$,  
such that $U_k\rightharpoonup U$
in $X^s(\mathbb{R}^{n+1}_+)$.
For any $\Psi \in C^\infty_0(\mathbb{R}^{n+1}_+)$, we have
\begin{equation}\label{Psida}
\begin{aligned}
&o_k(1)\\
 & = \langle J'(U_k) , \Psi\rangle \\
 &= k_{s} \int_{\mathbb{R}_+^{n+1}} y^{1-2s}  \langle \nabla U_k , \nabla \Psi \rangle \,dx\,dy  -\gamma \int_ {\mathbb{R}^n}  \frac{ u_k(x)  }{|x|^{2s}}\psi (x)\,dx \\
&\quad - \int_ {\mathbb{R}^n}  \frac{|u_k(x)|^{2_{s}^*(\alpha)-2} u_k(x) }{|x|^\alpha}\psi(x)\,dx- \int_ {\mathbb{R}^n}  \frac{|u_k(x)|^{2_{s}^*(\beta)-2} u_k(x) }{|x|^\beta}\psi(x)\,dx .
\end{aligned}
\end{equation}
Since $ U_k \rightharpoonup U \text{ in } X^s(\mathbb{R}^{n+1}_+)$ as ${k \to \infty}$, 
we have that
\begin{align*}
&\int_{\mathbb{R}_+^{n+1}} y^{1-2s}  \langle \nabla U_k , \nabla \Psi \rangle \,dx\,dy
 - \gamma\int_ {\mathbb{R}^n}  \frac{ u_k(x)  }{|x|^{2s}} \psi(x)\,dx\\
& \to \int_{\mathbb{R}_+^{n+1}} y^{1-2s}  \langle \nabla U , \nabla \Psi \rangle \,dx\,dy
-  \gamma\int_ {\mathbb{R}^n}  \frac{ u(x)  }{|x|^{2s}}\psi(x)\,dx,
\end{align*}
for all $\Psi \in C^\infty_0(\mathbb{R}^{n+1}_+)$, where $u= U(\cdot,0)$.

Moreover, the boundedness of $U_k$ in $X^s(\mathbb{R}^{n+1}_+) $ implies that   
$|u_k|^{2_{s}^*(\alpha)-2} u_k $ and  
$ |u_k|^{2_{s}^*(\beta)-2} u_k$ are bounded in  
 $ L^{\frac{2_{s}^*(\alpha)}{2_{s}^*(\alpha)-1}}(\mathbb{R}^n, |x|^{-\alpha})$ and
$ L^{\frac{2_{s}^*(\beta)}{2_{s}^*(\beta)-1}}(\mathbb{R}^n, |x|^{-\beta})$ respectively.
 Therefore, 
\begin{gather*}
|u_k|^{2_{s}^*(\alpha)-2} u_k  \rightharpoonup |u|^{2_{s}^*(\alpha)-2} u \quad 
\text{in }    L^{\frac{2_{s}^*(\alpha)}{2_{s}^*(\alpha)-1}}(\mathbb{R}^n, |x|^{-\alpha}), \\
|u_k|^{2_{s}^*(\beta)-2} u_k  \rightharpoonup |u|^{2_{s}^*(\beta)-2} u \quad 
\text{in }    L^{\frac{2_{s}^*(\beta)}{2_{s}^*(\beta)-1}}(\mathbb{R}^n, |x|^{-\beta}).
\end{gather*}
Thus, taking limits as ${k \to \infty}$ in \eqref{Psida}, we obtain 
\begin{equation}\label{cadr}
\begin{aligned}
0 &= \langle J'(U) , \Psi \rangle \\
 &= k_{s} \int_{\mathbb{R}_+^{n+1}} y^{1-2s}  \langle \nabla U  , \nabla \Psi \rangle 
\,dx\,dy  - \gamma\int_ {\mathbb{R}^n}  \frac{ u (x)  }{|x|^{2s}}\psi(x)\,dx \\
&\quad - \int_ {\mathbb{R}^n}  \frac{|u (x)|^{2_{s}^*(\alpha)-2} u (x) }{|x|^\alpha}
\psi(x)\,dx- \int_ {\mathbb{R}^n}  \frac{|u (x)|^{2_{s}^*(\beta)-2} u (x) }{|x|^\beta}
\psi(x)\,dx .
\end{aligned}
\end{equation}
Hence $U$ is a weak solution of \eqref{frac1a}.

The set $\mathbb{R}^n\cup \{\infty\}$ is compact for the standard topology which means 
that the measures can be identified as the dual space $C(\mathbb{R}^n\cup \{\infty\})$. 
For example, $\delta_\infty$ is well defined and $\delta_\infty=\varphi(\infty)$. 
By the concentration compactness principle \cite{PL1,PL2}, there exist a subsequence,
still denoted by $U_k$ and real numbers $\mu_0,\mu_\infty$, 
$\nu_0,\nu_\infty$, $\eta_0,\eta_\infty$ and $\zeta_0,\zeta_\infty$ such that
\begin{gather}\label{cons1}
\|U_k\|_{X^{s} (\mathbb{R}_+^{n+1})}^2\rightharpoonup d\mu
\geq \|U\|_{X^{s} (\mathbb{R}_+^{n+1})}^2+\mu_0\delta_0+\mu_\infty\delta_\infty, \\
\label{cons20}
|u_k|^{2}|x|^{-{2s}}\rightharpoonup d\nu
= |u|^{2}|x|^{-{2s}}+\nu_0\delta_0+\nu_\infty\delta_\infty, \\
\label{cons2}
|u_k|^{2_s^\ast(\alpha)}|x|^{-\alpha}\rightharpoonup d\eta
= |u|^{2_s^\ast(\alpha)}|x|^{-\alpha}+\eta_0\delta_0+\eta_\infty\delta_\infty, \\
\label{cons3}
|u_k|^{2_s^\ast(\beta)}|x|^{-\beta}\rightharpoonup d\zeta
= |u|^{2_s^\ast(\beta)}|x|^{-\beta}+\zeta_0\delta_0+\zeta_\infty\delta_\infty,
\end{gather}
where $\delta_0$ and $\delta_\infty$ are the Dirac mass at the origin and 
infinity respectively.

For $\varrho>0$, define $B_\varrho^+:= \{(x,y) \in \mathbb{R}_+^{n+1}: |(x,y)| < \varrho \}$, 
$B_\varrho:= \{x \in \mathbb{R}^n: |x| < \varrho\}$ and let 
$\Phi \in C_0^{\infty}(\mathbb{R}_+^{n+1})$ be a cut-off function such that $\Phi\equiv 1$  
in $B^+_{\frac{1}{2}}$ and $0 \le \Phi\le 1$ in $\mathbb{R}_+^{n+1}$.
We use $\Phi  U_k$ as test function, we have
\begin{equation}\label{testphif}
\begin{aligned}
&\langle J'(U_k) ,\Phi U_k  \rangle   \\
&= k_{s} \int_{\mathbb{R}_+^{n+1}} y^{1-2s}  \langle \nabla U_k  , \nabla (\Phi  U_k) \rangle 
 \,dx\,dy  -\gamma \int_ {\mathbb{R}^n}  \frac{ u_k(x)^2  \phi(x)  }{|x|^{2s}}\,dx \\
&\quad - \int_ {\mathbb{R}^n}  \frac{|u_k(x)|^{2_{s}^*(\alpha) }  \phi(x)  }{|x|^\alpha }\,dx
 - \int_ {\mathbb{R}^n}  \frac{|u_k(x)|^{2_{s}^*(\beta) }  \phi(x)  }{|x|^\beta}\,dx\\
&= k_{s} \int_{\mathbb{R}_+^{n+1}} y^{1-2s}  |\nabla U_k|^2\Phi (x)  \,dx\,dy
  - \gamma\int_ {\mathbb{R}^n}  \frac{ u_k(x)^2  \phi(x)  }{|x|^{2s}}\,dx\\
&\quad +k_{s} \int_{\mathbb{R}_+^{n+1}} y^{1-2s} U_k \langle\nabla U_k,\nabla\Phi  
 \rangle \,dx\,dy\\
&\quad - \int_ {\mathbb{R}^n}  \frac{|u_k(x)|^{2_{s}^*(\alpha) }  \phi(x)  }{|x|^\alpha}\,dx
- \int_ {\mathbb{R}^n}  \frac{|u_k(x)|^{2_{s}^*(\beta) }  \phi(x)  }{|x|^\beta}\,dx ,
\end{aligned}
\end{equation}
where $\phi=\Phi(\cdot,0)$.
First, we have 
\[
\lim_{\varrho\to0}\lim_{k\to\infty}\Big(k_{s} \int_{\mathbb{R}_+^{n+1}} y^{1-2s} 
U_k \langle\nabla U_k,\nabla\Phi  \rangle \,dx\,dy\Big)=0.
\]
Moreover, from \eqref{cons1}-\eqref{cons3}, we obtain
\begin{gather*}
\lim_{\varrho\to0}\lim_{k\to\infty}\Big(k_{s} \int_{\mathbb{R}_+^{n+1}} y^{1-2s}  
|\nabla U_k|^2\Phi   \,dx\,dy \Big)\geq \mu_0, \\
\lim_{\varrho\to0}\lim_{k\to\infty}  \int_ {\mathbb{R}^n}  
\frac{ u_k(x)^2  \phi(x)  }{|x|^{2s}}\,dx= \nu_0, \\
\lim_{\varrho\to0}\lim_{k\to\infty} \int_ {\mathbb{R}^n}  \frac{|u_k |^{2_{s}^*(\alpha) }
  \phi(x)  }{|x|^\alpha}\,dx=\eta_0,\quad
\lim_{\varrho\to0}\lim_{k\to\infty} \int_ {\mathbb{R}^n}  \frac{|u_k |^{2_{s}^*(\beta) }
  \phi(x)  }{|x|^\beta}\,dx=\zeta_0.
\end{gather*}
Thus we obtain
\begin{equation}\label{testphifga}
\lim_{\varrho\to0}\lim_{k\to\infty} \langle J'(U_k) , \Phi U_k  \rangle 
\geq \mu_0-\gamma \nu_0-\eta_0-\zeta_0.
\end{equation}
By the fractional Hardy-Sobolev inequalities, we have
\begin{equation}\label{testphifgg}
\eta_0^{\frac{2}{2_s^\ast(\alpha)}}S(n,s,\gamma,\alpha)\leq  \mu_0-\gamma \nu_0,\quad
 \zeta_0^{\frac{2}{2_s^\ast(\beta)}}S(n,s,\gamma,\beta)\leq  \mu_0-\gamma \nu_0.
\end{equation}
By \eqref{testphifga} and \eqref{testphifgg}, we find
\begin{equation}\label{testphifggh}
\eta_0^{\frac{2}{2_s^\ast(\alpha)}}S(n,s,\gamma,\alpha)
\leq  \eta_0+\zeta_0,\quad \zeta_0^{\frac{2}{2_s^\ast(\beta)}}S(n,s,\gamma,\beta)
\leq  \eta_0+\zeta_0.
\end{equation}
So
\begin{gather*}
\eta_0^{\frac{2}{2_s^\ast(\alpha)}}\Big(1-S(n,s,\gamma,\alpha)^{-1} 
\eta_0^{\frac{2_s^\ast(\alpha)-2}{2_s^\ast(\alpha)}}\Big)
\leq  S(n,s,\gamma,\alpha)^{-1} \zeta_0, \\
\zeta_0^{\frac{2}{2_s^\ast(\beta)}}\Big(1-S(n,s,\gamma,\beta)^{-1} 
\zeta_0^{\frac{2_s^\ast(\beta)-2}{2_s^\ast(\beta)}}\Big)
\leq  S(n,s,\gamma,\beta)^{-1} \eta_0.
\end{gather*}
Since $\{U_k\}_{k\in\mathbb{N}}$ is bounded in $X^s(\mathbb{R}^{n+1}_+)$, 
we have $\eta_0\leq c_1$ and $\zeta_0\leq c_2$ for positive constants 
$c_1,c_2$,  thus
\begin{gather*}
\eta_0^{\frac{2}{2_s^\ast(\alpha)}}\Big(1-S(n,s,\gamma,\alpha)^{-1} 
c_1^{\frac{2_s^\ast(\alpha)-2}{2_s^\ast(\alpha)}}\Big)
\leq  S(n,s,\gamma,\alpha)^{-1} \zeta_0, \\
\zeta_0^{\frac{2}{2_s^\ast(\beta)}}\Big(1-S(n,s,\gamma,\beta)^{-1} 
c_2^{\frac{2_s^\ast(\beta)-2}{2_s^\ast(\beta)}}\Big)
\leq  S(n,s,\gamma,\beta)^{-1} \eta_0.
\end{gather*}
Therefore, there exist constants $A=A(\alpha,2_s^\ast(\alpha),c_1)$ and 
$B=B(\beta,2_s^\ast(\beta),c_2)$ such that
\begin{equation*}
\eta_0^{\frac{2}{2_s^\ast(\alpha)}}\leq A\zeta_0,\quad  \text{and}\quad
 \zeta_0^{\frac{2}{2_s^\ast(\beta)}}\leq B\eta_0.
\end{equation*}
In particular, we have that
either $\eta_0=0$ and $ \zeta_0=0$, or
\[
\eta_0\geq S(n,s,\gamma,\alpha)^{\frac{n-\alpha}{2s-\alpha}}, \quad
\zeta_0\geq S(n,s,\gamma,\beta)^{\frac{n-\beta}{2s-\beta}}.
\]
On the other hand, we know that
\begin{equation}\label{Psida2}
\begin{aligned}
c&=J(U_k)-\frac{1}{2} \langle J'(U_k) , U_k \rangle + o_k(1) \\
&\geq \frac{2s-\alpha}{2(n-\alpha)}
\Big( \int_ {\mathbb{R}^n}  \frac{|u_k(x)|^{2_{s}^*(\alpha) } }{|x|^\alpha}\,dx+\eta_0\Big)\\
&\quad +\frac{2s-\beta}{2(n-\beta)}
\Big( \int_ {\mathbb{R}^n}  \frac{|u_k(x)|^{2_{s}^*(\beta) } }{|x|^\beta}\,dx+\zeta_0\Big)\\
&\geq  \frac{2s-\alpha}{2(n-\alpha)}\eta_0+\frac{2s-\beta}{2(n-\beta)}\zeta_0.
\end{aligned}
\end{equation}
By the assumption that $c<c_\ast$, we obtain that $\eta_0=0$,  $\zeta_0=0$.

For the concentration at infinity, we define 
$B_R^+:= \{(x,y) \in \mathbb{R}_+^{n+1}: |(x,y)| < R \}$, 
$B_R:= \{x \in \mathbb{R}^n: |x| < R\}$ and let $\Psi \in C_0^{\infty}(\mathbb{R}_+^{n+1})$
 be a cut-off function such that $\Psi=0$  in $ B^+_{R}$ and $\Psi\equiv 1$  
in $\mathbb{R}_{+}^{n+1}\backslash B^+_{2R}$ and $0 \le \Psi\le 1$ in $\mathbb{R}_+^{n+1}$.
Consider
\begin{gather*}
\mu_\infty=\lim_{R\to\infty}\lim_{k\to\infty}\sup\Big(k_{s} 
 \int_{\mathbb{R}_+^{n+1}\backslash B_{2R}^+} y^{1-2s}  |\nabla U_k|^2\Psi   \,dx\,dy \Big) ,\\
\nu_\infty=\lim_{R\to\infty}\lim_{k\to\infty}\sup  
\int_{\mathbb{R}^{n}\backslash B_{2R}}  \frac{ u_k(x)^2  \psi(x)  }{|x|^{2s}}\,dx ,\\
\eta_\infty=\lim_{R\to\infty}\lim_{k\to\infty}\sup  
\int_{\mathbb{R}^{n}\backslash B_{2R}}   \frac{|u_k(x)|^{2_{s}^*(\alpha) } 
 \psi (x) }{|x|^\alpha}\,dx,\\
\zeta_\infty=\lim_{R\to\infty}\lim_{k\to\infty}\sup  
\int_{\mathbb{R}^{n}\backslash B_{2R}}    \frac{|u_k(x)|^{2_{s}^*(\beta) }  
\psi(x)  }{|x|^\beta}\,dx.
\end{gather*}
By the same arguments as the concentration at the origin, we can get the following 
facts: either $\eta_\infty=0$ and $\zeta_\infty=0$, or
\[
\eta_\infty\geq S(n,s,\gamma,\alpha)^{\frac{n-\alpha}{2s-\alpha}},\quad
\zeta_\infty\geq S(n,s,\gamma,\beta)^{\frac{n-\beta}{2s-\beta}}.
\]
As for \eqref{Psida2}, we obtain
\begin{equation}\label{Psidaa}
c \geq  \frac{2s-\alpha}{2(n-\alpha)}\eta_\infty
+\frac{2s-\beta}{2(n-\beta)}\zeta_\infty.
\end{equation}
By the assumption that $c<c_\ast$, we   obtain that $\eta_\infty=0$,  
$\zeta_\infty=0$. Therefore, up to a subsequence
$\{U_k\}_k$ converges strongly to $U$ in $X^s(\mathbb{R}^{n+1}_+)$.
\end{proof}

Let  $W_{\gamma,\alpha}$ be the extremal function of $S(n,s,\gamma,\alpha)$ 
in $X^{s} (\mathbb{R}_+^{n+1})$, whose existence was obtained by  Ghoussoub and Shakerian 
in \cite{Ghoussoub-Shakerian} for $\alpha>0$ or $\alpha=0$ and 
$0\leq \gamma<\gamma_H$.


\begin{lemma}\label{comcle}
Let $0<s<1$, $ 0 < \alpha,\beta < 2s <n$,
and $\gamma<\gamma_H$.
Then
$$
\sup_{t\geq0}J(tW_{\gamma,\vartheta})<c_\ast \quad \text{for }
 \vartheta=\alpha,\beta,
$$
where $c_\ast$ is defined in Proposition \ref{psbound}.
\end{lemma}

\begin{proof}
For $\vartheta=\alpha$, we have
\[
J(tW_{\gamma,\alpha})= \frac{t^2}{2} \|W_{\gamma,\alpha}\|^2 
-\frac{t^{2_{s}^*(\alpha)}}{2_{s}^*(\alpha)}
 \int_{\mathbb{R}^n} \frac{|w_{\gamma,\alpha}|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
-\frac{t^{2_{s}^*(\beta)}}{2_{s}^*(\beta)}
 \int_{\mathbb{R}^n} \frac{|w_{\gamma,\alpha}|^{2_{s}^*(\beta)}}{|x|^{\beta}}\,dx.
\]
where $w_{\gamma,\alpha}:= Tr (W_{\gamma,\alpha})=W_{\gamma,\alpha}(\cdot,0)$. 
By construction, we have that
\begin{align*}\label{psc1hj}
J(tW_{\gamma,\alpha})\leq f_\alpha(t):= \frac{t^2}{2} \|W_{\gamma,\alpha}\|^2 
-\frac{t^{2_{s}^*(\alpha)}}{2_{s}^*(\alpha)}\int_{\mathbb{R}^n} 
\frac{|w_{\gamma,\alpha}|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
\end{align*}
Straightforward computations yield that $f_\alpha(t)$ attains  its maximum 
at the point
$$
\tilde{t} =  \Big(  \frac{\|W_{\gamma,\alpha}\|^2}{\int_{\mathbb{R}^n} 
\frac{|w_{\gamma,\alpha}|^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx} 
 \Big)^\frac{1}{2^*_s(\alpha) -2}.
$$
It follows that
\[
\sup_{t\ge0}f_\alpha(t)
 = \frac{2s-\alpha}{2(n-\alpha)} 
\Big( \frac{\|W_{\gamma,\alpha}\|^2}{(\int_{\mathbb{R}^n}
 |w_{\gamma,\alpha}|^{2_s^\ast(\alpha)}/|x|^\alpha\,dx)^\frac{2}{2^*_s(\alpha)}}
\Big)^{\frac{n-\alpha}{2s-\alpha}}.
\]
Since $W_{\gamma,\alpha}$ is an extremal for $S(n,s,\gamma,\alpha)$ on
 $X^{s} (\mathbb{R}_+^{n+1})$, we obtain that
\begin{equation}\label{fg5}
\sup_{t\geq0}J(tW_{\gamma,\alpha})\leq \sup_{t\geq0}f_\alpha(t) 
= \frac{2s-\alpha}{2(n-\alpha)}S(n,s,\gamma,\alpha)^{\frac{n-\alpha}{2s-\alpha}}.
\end{equation}
We now need to show that equality does not hold in \eqref{fg5}. Indeed, otherwise
we would have that $\sup_{t\geq0}J(tW_{\gamma,\alpha})= \sup_{t\geq0}f_\alpha(t)$. 
Consider  $t_1$ (resp. $t_2 > 0$) where
$\sup_{t\ge0} J(tW_{\gamma,\alpha})$  (resp. 
$\sup_{t\ge0}f_\alpha(t)$) is attained. We obtain
$$
f_\alpha(t_1) - \frac{t_1^{2_{s}^*(\beta)}}{2_{s}^*(\beta)}
\int_{\mathbb{R}^n} \frac{|w_{\gamma,\alpha}|^{2_{s}^*(\beta)}}{|x|^{\beta}}\,dx 
= f_\alpha(t_2),
$$
which means that $f_\alpha(t_1) > f_\alpha(t_2)$ since  $t_1 > 0$. 
This contradicts  the fact that $t_2$ is a maximum point of $f_\alpha(t)$, 
hence the strict inequality holds in \eqref{fg5}.

Similarly, for $\vartheta=\beta$, we obtain
\[
\sup_{t\geq0}J(tW_{\gamma,\beta})< \sup_{t\geq0}f_\beta(t) 
= \frac{2s-\beta}{2(n-\beta)}S(n,s,\gamma,\beta)^{\frac{n-\beta}{2s-\beta}}.
\]
This completes the proof.
\end{proof}


\section{Proof of main result}\label{proof}

\begin{proof}[Proof of Theorem \ref{fracmain}]
For any $U\in X^{s} (\mathbb{R}_+^{n+1})$, the energy functional to problem \eqref{frac1a} 
is
\[ % \label{psc1}
J(U) = \frac{1}{2} \|U \|^2 -\frac{1}{2_{s}^*(\alpha)}\int_{\mathbb{R}^n} 
\frac{|u |^{2_{s}^*(\alpha)}}{|x|^{\alpha}}\,dx
-\frac{1}{2_{s}^*(\beta)}\int_{\mathbb{R}^n} \frac{|u |^{2_{s}^*(\beta)}}{|x|^{\beta}}\,dx,
\]
where again $u:= Tr (U)=U(\cdot,0)$. By fractional Hardy-Sobolev inequality, 
we have 
\begin{align*}
J(U) &\ge  \frac{1}{2} \| U\|^2  -\frac{1}{2_s^\ast(\alpha)} 
 S(n,s,\gamma,\alpha)^{-\frac{2_s^\ast(\alpha)}{2}} \| U\|^{2_s^\ast(\alpha)} \\
& -\frac{1}{2_s^\ast(\beta)} S(n,s,\gamma,\beta)^{-\frac{2_s^\ast(\beta)}{2}} 
 \| U\|^{2_s^\ast(\beta)} \\
& = \Big( \frac{1}{2}-\frac{1}{2_s^\ast(\alpha)} S(n,s,\gamma,
 \alpha)^{-\frac{2_s^\ast(\alpha)}{2}} \|U\|^{2_s^\ast(\alpha)-2} \\
&\quad -\frac{1}{2_s^\ast(\beta)} S(n,s,\gamma,\beta
 )^{-\frac{2_s^\ast(\beta)}{2}} \|U\|^{2_s^\ast(\beta)-2}\Big) \|U\|^2.
\end{align*}
Since $\alpha,\beta \in (0,2s)$, we have that  $2_{s}^*(\alpha)>2,2_{s}^*(\beta)>2$. 
By \eqref{comparable norms}, we then get that there exists $R>0$ such that  
$J(U) \ge \rho$ for all $U \in X^{s} (\mathbb{R}_+^{n+1})$ with 
$\|U\|_{X^{\alpha} (\mathbb{R}_+^{n+1})} = R$.
Moreover, for $\vartheta=\alpha$ or $\vartheta=\beta$,
$$
J(tW_{\gamma,\vartheta}) = \frac{t^2}{2} \| W_{\gamma,\vartheta}\|^2  
 -\frac{ t^{ 2_s^\ast(\alpha)}  }{2_s^\ast(\alpha)}\int_{\mathbb{R}^n}
 \frac{|w_{\gamma,\vartheta}|^{2_s^\ast(\alpha)}}{|x|^{\alpha}}\,dx
-\frac{ t^{ 2_s^\ast(\beta)}  }{2_s^\ast(\beta)}
 \int_{\mathbb{R}^n} \frac{|w_{\gamma,\vartheta}|^{2_s^\ast(\beta)}}{|x|^{\alpha}}\,dx,
$$
hence $ \lim_{t \to +\infty} J(tW_{\gamma,\vartheta})=  -\infty$ , then there 
exists $t_0>0$ such that $\|t_0 W_{\gamma,\vartheta}\|>R$  and 
$J(t_0 W_{\gamma,\vartheta})<0$. Set
$$
c_\vartheta:=\inf_{g\in \Gamma_\vartheta}\max_{t\in[0,1]}J(g(t)),
$$
where
$$
\Gamma_\vartheta:=\big\{g\in C^0([0,1],X^{s} (\mathbb{R}_+^{n+1})):
 g(0)=0,\ g(1)=  t_0 W_{\gamma,\vartheta}\big\}.
$$
Thus by Mountain Pass Lemma, there exists a sequence 
$\{U_k\}$ in $X^{s} (\mathbb{R}_+^{n+1})$ such that
$$
J(U_k)\to c,\quad J'(U_k)\to0\quad \text{in }  (X^s(\mathbb{R}^{n+1}_+))' \text{ as }
 k\to\infty.
$$
By Lemma \ref{comcle}, we have
$$
0<c\leq\sup_{t\in[0,1]}J(tt_0W_{\gamma,\vartheta})
\leq \sup_{t>0}J(t W_{\gamma,\vartheta})<c_\ast.
$$
By Proposition \ref{psbound}, we deduce that $\{U_k\}$ has a subsequence, 
still denote by $\{U_k\}$ , such that $ U_k \to U$ strongly in $X^s(\mathbb{R}^{n+1}_+)$. 
Thus $U$ is a nontrivial solution of problem \eqref{frac1a}, and
$u:= Tr (U)=U(\cdot,0)$ is a nontrivial solution of problem \eqref{frac1}.
\end{proof}

\subsection*{Acknowledgments}
This research was supported by the NSFC No.\ 11501468, by the Chongqing Research Program 
of Basic Research and Frontier Technology cstc2016jcyjA0323, 
by the Fundamental Research Funds for the Central Universities XDJK2017C049,
 and by the innovation support program of Chongqing cx2017070.


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