1. Introduction
In this paper, we study the following problem
\begin{equation}
\begin{cases}
( -\Delta ) ^su=|u|^{2_{s}^{*}-2-\varepsilon}u,& \text{in } \varOmega ,\\
u=0,& \text{on } \Sigma,\\
\end{cases}
\end{equation} where 
$2_{s}^{*}=\frac{2N}{N-2s}$,
$s\in(\frac{1}{2},1)$, 
$N \gt 2s$, Ω is a bounded domain in 
$\mathbb{R}^N$, ɛ is a small parameter and 
$(-\Delta)^{s}$ is a fractional Laplacian operator with the suitable boundary Σ, which is given according to the definition of the fractional Laplacian 
$(-\Delta)^{s}$ (see below).
For the classical case s = 1, the problem
\begin{equation}
\begin{cases}
-\Delta u=|u|^{2^{*}-2-\varepsilon}u,& \text{in } \varOmega ,\\
u=0,& \text{on } \partial \Omega\\,
\end{cases}
\end{equation} has been studied extensively when 
$\varepsilon \geq0$. It is well-known that there is only a trivial solution for (1.2) when ɛ = 0 and Ω is star-shaped, whereas there is always a non-trivial solution uɛ for (1.2) with ɛ > 0. Hence, it is natural to ask about the asymptotic behaviour of uɛ as ɛ → 0. Atkinson and Peletier [Reference Atkinson and Peletier1] made a first study when Ω is a ball in 
$\mathbb{R}^{N}$ by using the ODE method. Brezis and Peletier [Reference Brezis and Peletier3] revisit this problem when Ω is a ball in 
$\mathbb{R}^{3}$ and obtain that
\begin{equation*}\lim_{\varepsilon \to 0} \varepsilon u_{\varepsilon}^{2}(0) = \frac{32\sqrt{3}}{\pi},\end{equation*}and
\begin{equation*}\lim_{\varepsilon \to 0} \varepsilon^{-\frac12} u_{\varepsilon}(x) =\frac{3^{\frac{1}{4}}}{4} \sqrt{\frac{\pi}{2}} \bigl(\frac{1}{|x|}-1\bigr),\end{equation*} by PDE methods, which can be used to handle the problem (1.2) in non-spherical domains. In [Reference Brezis and Peletier3], Brezis and Peletier proposed a conjecture, called the Brezis–Peletier conjecture, that the asymptotic behaviour of uɛ should also hold in non-spherical domains. Interestingly, Han [Reference Han11] and Rey [Reference Rey13] independently gave a firm answer about such conjecture. Both of them proved that uɛ blows up and concentrates near the critical point of the Robin function 
$\phi(x)=h(x, x)$, where
\begin{equation*}h(x, y)= \frac{1}{(N-2)\omega_{N-1}|x-y|^{N-2}}- g(x, y),\end{equation*} 
$\omega_{N-1}$ is the area of the unit sphere in 
$\mathbb{R}^{N}$ and 
$g(x, y)$ is the Green function of 
$-\Delta$ with homogeneous Dirichlet boundary condition.
For the case of 
$0 \lt s \lt 1$, the operator 
$(-\Delta)^{s}$ is defined in two different types: the spectral fractional Laplacian and the restricted fractional Laplacian, see [Reference Barrios, Colorado, de Pablo and Sánchez2, Reference Choi, Kim and Lee6, Reference Dávila, López Ríos and Sire7, Reference Guo, Li, Pistoia and Yan10, Reference Long, Yan and Yang12, Reference Tan14, Reference Tan and Xiong16, Reference Yan, Yang and Yu17] and reference therein.
The spectral fractional Laplacian. To define the operator 
$(-\Delta)^{s}$, let 
$\{(\lambda_{i}, \phi_{i})\}_{i=1}^{+\infty}$ be the sequence of eigenvalues and corresponding normalized eigenfunctions of the classic Laplacian problem with homogeneous Dirichlet boundary condition, namely,
\begin{equation*}\left\{\begin{array}{lll}
-\Delta \phi_{i}=\lambda_{i}\phi_{i}, & \text{in } \Omega,\\
\phi_{i}=0, & \text{on } \partial \Omega
\end{array}
\right.
\end{equation*} with 
$\|\phi_{i}\|_{L^{2}(\Omega)} = 1$. Then, the fractional Laplacian operator 
$(-\Delta)^{s}$ is defined by
\begin{equation*}u=\sum_{i=1}^{\infty}u_{i}\phi_{i} \to (-\Delta)^{s}u = \sum_{i=1}^{\infty} u_{i} \lambda_{i}^{s} \phi_{i}. \end{equation*}In the case, the homogeneous boundary condition is given by
\begin{equation*}u=0, \text{on } \partial \Omega,\end{equation*} that is, 
$\Sigma := \partial \Omega$.
The restricted fractional Laplacian. The fractional Laplacian operator can be defined by the integral representation, that is,
\begin{equation*}(-\Delta)^{s} u(x) =C_{N, s} P.V. \int_{\mathbb{R}^{N}} \frac{u(x) -u(y)}{|x-y|^{N+2s}}dy, \end{equation*}where 
$C_{N,s}$ is the normalized constant and 
$P.V.$ is the Cauchy principal value. In this way, u(x) should be defined in 
$\mathbb{R}^{N}$, so it is restricted to be zero outside Ω, that is, u = 0, in 
$\mathbb{R}^{N}\setminus \Omega$. In this case, 
$\Sigma :=\mathbb{R}^{N}\setminus \Omega$.
The existence and non-existence results about solutions to equations are crucial. At first, by Pohozaev identity, problem (1.1) still has no bounded solution when Ω is star-shaped and ɛ = 0 (cf. [Reference Tan15]). Then, the existence of solutions and the asymptotic behaviour of solutions to (1.1) become natural questions. Especially, the fractional case for the Brezis–Peletier conjecture, usually called the fractional Brezis–Peletier conjecture, are interesting and challenging. For the fractional Brezis–Peletier conjecture, Choi, Kim, and Lee [Reference Choi, Kim and Lee6] gave an affirmative answer by investigating the asymptotic behaviour of solutions for the following equation
\begin{equation*}
\begin{cases}
( -\Delta ) ^su=|u|^{2_{s}^{*}-2-\varepsilon}u,& \text{in } \varOmega ,\\
u=0,& \text{in } \partial\varOmega,
\end{cases}
\end{equation*}as ɛ → 0 and getting some analogous results with the works of [Reference Han11, Reference Rey13]. Afterwards, Dávila et al. [Reference Dávila, López Ríos and Sire7] revisited the fractional Brezis–Peletier conjecture by a new framework and constructed solitary bubbling solutions concentrated near the critical point of the Robin function for (1.1) by the Lyapunov-Schmidt reduction method. Moreover, it is noted that to the best of our knowledge, it seems that there is no result concerning the non-degeneracy and local uniqueness of bubble solutions for (1.1) until now.
The main motivation of this paper is to revisit the fractional Brezis–Peletier conjecture by showing that problem (1.1) has a single bubbling solution centred around the non-degenerate critical point of the Robin function. Specifically, we will still employ the Lyapunov-Schmidt reduction method to construct such bubbling solutions for (1.1). However, as one will see, we do not follow the method with [Reference Dávila, López Ríos and Sire7] to enlarge the domain, such that the domain is 
$\varepsilon^{-\frac{1}{N-2s}} \varOmega$. Here, we will use the projection of the solution for the limiting problem to construct the bubbling solutions in the domain Ω. Moreover, based on the existence and asymptotic behaviour of bubbling solutions, we make the further study by Pohozaev identities and obtain the non-degeneracy and local uniqueness of the above bubbling solutions for (1.1).
Before stating our main results, let us introduce some notations. Since our results can be proved by using the unified way for two different operators, we will only consider the problem with the restricted fractional Laplacian
\begin{equation}
\begin{cases}
( -\Delta ) ^su=|u|^{2_{s}^{*}-2-\varepsilon}u,& \text{in } \varOmega ,\\
u=0,& \text{in } \mathbb{R}^{N} \setminus \varOmega.
\end{cases}
\end{equation} In this case, the boundary condition 
$\Sigma =\mathbb{R}^{N} \setminus \varOmega $.
For 
$\varOmega \subseteq \mathbb{R}^N$, we set
\begin{align*}
H_{0}^{s}(\varOmega) =\left\{u\in H^s( \mathbb{R}^N ) \,:\,u=0\,\,\text{in }\,\,\mathbb{R}^N\setminus \varOmega \right\},
\end{align*}with the norm
\begin{align*}
\left\| u \right\| _{H_{0}^{s}( \varOmega )}=\left( \int_{\mathbb{R}^N\times \mathbb{R}^N}{\frac{\left(u(x)-u(y)\right)^2}{| x-y |^{N+2s}}} \right) ^{\frac{1}{2}}.
\end{align*}By [Reference Di Nezza, Palatucci and Valdinoci9], we have the following identity
\begin{align*}
\left\| u \right\| _{H_{0}^{s}\left( \varOmega \right)}^{2}=\left\| (-\Delta) ^{\frac{s}{2}}u \right\| _{L^2(\mathbb{R}^N)}^{2}.
\end{align*}It is well-known that
\begin{align*}
U_{\zeta ,\mu}( x ) =\frac{C_{N,s}\,\mu ^{\frac{N-2s}{2}}}{( 1+\mu ^2\left| x-\zeta \right|^2 ) ^{\frac{N-2s}{2}}} \, \, \, \, \, \, \, \, \Bigl( \text{for any } \zeta \in \mathbb{R}^{N} \text{and } \mu \in \mathbb{R}_{+}\Bigr),
\end{align*}solves the equation
\begin{align*}
( -\Delta ) ^su=u^{2_{s}^{*}-1}, \, u \gt 0 \,\, \text{in }\,\,\, \mathbb{R}^N,
\end{align*}where
\begin{align*}
C_{N,s}=2^{\frac{N-2s}{2}}\frac{\Gamma(\frac{N+2s}{2})}{\Gamma(\frac{N-2s}{2})}.
\end{align*} As 
$U_{\zeta ,\mu}$ does not belong to 
$H_{0}^{s}\left( \varOmega \right)$, in order to obtain an approximate solution for problem (1.3), we need to make a projection as follows. Let 
$PU_{\zeta ,\mu}$ be the solution of the following problem
\begin{equation}
\begin{cases}
( -\Delta ) ^sw=U_{\zeta ,\mu}^{2_{s}^{*}-1},& \text{in } \varOmega ,\\
w=0,& \text{in } \mathbb{R}^N\setminus \varOmega .\\
\end{cases}
\end{equation} Denote 
$G(x,y)$ be the Green function for 
$(-\Delta) ^s$ in Ω with homogeneous Dirichlet boundary condition of the form
\begin{align*}
G(x,y)=S(x,y)-H(x,y),
\end{align*}where
\begin{align*}
S(x,y) =\frac{a_{N,s}}{\left| x-y \right|^{N-2s}},\, a_{N,s}=\frac{2^{1-2s}\Gamma(\frac{N-2s}{2})}{2\pi^{\frac{N}{2}}\Gamma(s)},
\end{align*} and 
$H(x,y)$ is the regular part of 
$G(x,y)$. We call 
$H(x,x)$ the Robin function h(x).
One of our main results is the following existence result of single bubbling solutions:
Theorem 1.1. Suppose that 
$s\in \left( \frac{1}{2},1 \right)$, 
$2s+1 \lt N\leq6s$, and 
$x_0\in \varOmega$ is a non-degenerate critical point of the Robin function h(x). Then, there exists 
$\varepsilon _0 \gt 0$, such that for any 
$\varepsilon \in (0,\varepsilon _0)$, (1.3) has a solution of the form
\begin{equation}
u_{\varepsilon}=PU_{\zeta _{\varepsilon},\mu _{\varepsilon}}+\omega _{\varepsilon},
\end{equation} satisfying that as 
$\varepsilon \rightarrow 0 $,
\begin{equation}
\zeta _{\varepsilon}\rightarrow x_0,\, \varepsilon^{\frac{1}{N-2s}}\mu _{\varepsilon}\rightarrow C \,and\, \left\| \omega _{\varepsilon} \right\| _{H_{0}^{s}(\varOmega )}\rightarrow 0,
\end{equation}where C is a positive constant.
The other of our main results are the non-degeneracy and local uniqueness of the above bubbling solutions in Theorem 1.1.
Theorem 1.2. Let 
$s\in \left( \frac{1}{2},1 \right)$ and 
$2s+1 \lt N \lt 6s$. Suppose that x 0 and uɛ are given in Theorem 1.1, then there exists 
$\varepsilon_{0} \gt 0$, such that for 
$\varepsilon \in (0, \varepsilon_{0})$, uɛ is non-degenerate, i.e.
\begin{equation}
(-\Delta )^s\xi -(2_{s}^{*}-1-\varepsilon )|u_{\varepsilon}|^{2_{s}^{*}-2-\varepsilon}\xi =0, \,\,in\,\varOmega,\,\, \xi \in H_{0}^{s}( \varOmega ) \,\, \Rightarrow \,\, \xi =0.
\end{equation} Moreover, assume that there exist two solutions 
$u_{\varepsilon}^{(1)}$ and 
$u_{\varepsilon}^{(2)}$ for problem (1.3) as in Theorem 1.1. Then, 
$u_{\varepsilon}^{(1)} \equiv u_{\varepsilon}^{(2)}$ for 
$\varepsilon \in (0, \varepsilon_{0})$.
Remark 1.3. In the above theorems, we have imposed 
$s \in (\frac12, 1)$ due to considerations of two aspects. On one hand, we need 
$1-2s \lt 0$ in (2.27), which makes sure that we can prove the existence of the solution and its concentrating point is the critical point of the Robin function. On the other hand, as 
$\nabla \bar u_{\varepsilon}$ could be estimated and used in the Pohozaev identities, its representation in (3.14) should be summable. As it is known, the singularity of the gradient of the Green function can be treated as 
$|\nabla_{X} \bar G(X, y)| \sim |X-(y,0)|^{-N+2s -1}$. To satisfy the integrability, s must be greater than 
$\frac12$. Moreover, for the requirement of the dimension 
$N \in (2s+1, 6s]$, this is a technical issue. If 
$N \gt 6s$, we are unable to guarantee that the main term in (2.19), whose order is 
$\mu^{-N+2s}$, can control the “higher order” term in (2.23) with the order 
$\mu^{-\frac{(N+2s)^{2}}{2(N-2s)}+1}$. Simultaneously, the reason why 
$N \gt 2s+1$ is to ensure that the main term in (2.22) is dominated, that is, the higher-order term 
$\varepsilon^{2} \mu (\ln\mu)^{2} \sim \mu^{-2(N-2s)+1}(\ln\mu)^{2} \lt \lt \mu^{-(N-2s)}$ when µ is large sufficiently, which implies 
$N \gt 2s+1$. Therefore, these assumptions on s and N are put forward.
Remark 1.4. If Ω is the unit ball 
$B_{1}(0)$, the Robin function h(x) has the unique critical point 
$x_{0}=0$, which is non-degenerate. The fact implies that there exists a domain Ω, such that the Robin function h(x) has a non-degenerate point.
The idea of the proof is inspired by [Reference Caffarelli and Silvestre4, Reference Deng, Lin and Yan8, Reference Guo, Li, Pistoia and Yan10]. Since this is the nonlocal problem, we use the method, introduced by Caffarelli and Silvestre [Reference Caffarelli and Silvestre4], to transfer into a local Neumann problem. However, the Neumann problem is in the upper half space, which brings some difficulties, such as, the properties of the Green’s function. On the other hand, we will use various kinds of Pohozaev identities to show the local uniqueness result. In addition, since the parameter ɛ appears in the exponent, the estimates are quite different to handle with the work of [Reference Deng, Lin and Yan8, Reference Guo, Li, Pistoia and Yan10]. In fact, the same strategies can be used to deal with the non-degeneracy and uniqueness results for the classical problem (1.2), that is, the local case 
$(s=1)$.
This paper is organized as follows. In 
$\S$ 2, by using the well-known Lyapunov-Schmidt reduction method, we revisit the existence of a positive single bubbling solution. In addition, we give the proof of the non-degeneracy and the local uniqueness of the solution in 
$\S$ 3 and 
$\S$ 4, respectively. In Appendix A and Appendix B, we introduce several identities involving the Green’s functions and several Pohozaev identities for proving the non-degeneracy and the local uniqueness of the solution, respectively.
2. Revisit on the existence of bubbling solutions
Define the Sobolev space 
$D^s(\mathbb{R}^N)$ as the completion of the 
$C_{0}^{\infty}(\mathbb{R}^N)$ with respect to the norm 
$\left\| u \right\| _{D^s(\mathbb{R}^N)}=\left\| (-\Delta )^{\frac{s}{2}}u \right\| _{L^2(\mathbb{R}^N)}$. By [Reference Capozzi, Fortunato and Palmieri5], for any 
$u\in D^s(\mathbb{R}^N)$, we can define 
$\bar{u}\in D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})$ as the s-harmonic extension of u, which solve
\begin{equation}
\begin{cases}
Div( x_{N+1}^{1-2s}\nabla \bar{u} ) =0,& \text{in}\,\,\mathbb{R}_{+}^{N+1},\\
\bar{u}=u, & \text{on}\,\,\mathbb{R}^N,\\
\end{cases}
\end{equation} and where 
$D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})$ is the completion of 
$C_{0}^{\infty}(\overline{\mathbb{R}_{+}^{N+1}})$ with respect to the norm
\begin{align*}
\left\| U \right\| _{D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})}=\left( \int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\left| \nabla U \right|^2}dX \right) ^{\frac{1}{2}}.
\end{align*} Then, 
$ \left( -\Delta \right) ^su=-\underset{{x_{N+1}\rightarrow 0^+}}{\lim}x_{N+1}^{1-2s}\frac{\partial\bar{u}}{\partial x_{N+1}}$. Thus, the nonlocal problem (1.3) can also be reformulated to be the following local problem:
\begin{equation}
\begin{cases}
Div( x_{N+1}^{1-2s}\nabla \bar{u} ) =0,& \text{in}\,\,\mathbb{R}_{+}^{N+1},\\
\bar{u}=0,& \text{on}\,\,\mathbb{R}^N\setminus \varOmega ,\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim}x_{N+1}^{1-2s}\partial _{\nu}\bar{u}=|u|^{2_{s}^{*}-2-\varepsilon}u,& \text{on}\,\,\varOmega .\\
\end{cases}
\end{equation}Its energy functional is given by
\begin{equation}
I\left( \bar{u} \right) =\frac{1}{2}\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2dX}-\frac{1}{2_{s}^{*}-\varepsilon}\int_{\varOmega}{|u|^{2_{s}^{*}-\varepsilon}dx}, \,\,\,\, \,\,\,\bar{u}\in D_{0}^{s}( \varOmega ),
\end{equation} where 
$D_{0}^{s}(\varOmega) $ is the completion of 
$\left\{U \in C_{0}^{\infty}(\overline{\mathbb{R}_{+}^{N+1}}):U =0\,\,in\,\,\mathbb{R}^N\setminus \varOmega \right\} $ with respect to the norm
\begin{equation}
\left\| U \right\| _{D_{0}^{s}(\varOmega )}=\left( \int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\left| \nabla U \right|^2}dX \right) ^{\frac{1}{2}},
\end{equation}and the corresponding inner product is given by
\begin{align*}
\left \lt U,V \right \gt _{D_{0}^{s}(\varOmega )}=\int_{\mathbb{R}_{+}^{N+1}} x_{N+1}^{1-2s} \nabla U\nabla VdX,\,\,\,for\,\,all\,\,u,v\in D_{0}^{s}(\varOmega ).
\end{align*} For simplicity, for any 
$\zeta \in \varOmega$ and µ > 0 large, we use 
$\bar{P}_{\zeta ,\mu}$ to denote the s-harmonic extension 
$\overline{PU_{\zeta ,\mu}}$. In this section, we prove the existence part of Theorem 1.1 by applying the Lyapunov-Schmidt reduction method to the local problem (2.2), that is, we will prove the following result.
Theorem 2.1. Assume that 
$s\in \left( \frac{1}{2},1 \right)$ and 
$2s+1 \lt N\leq6s$. Then, for any non-degenerate critical point 
$x_0\in \varOmega $ of the Robin function h(x), there exists 
$\varepsilon _0 \gt 0$ such that for any 
$\varepsilon \in (0,\varepsilon _0)$, (2.2) has a solution of the form
\begin{align*}
\bar{u}_{\varepsilon}=\bar{P}_{\zeta _{\varepsilon},\mu _{\varepsilon}}+\bar{\omega} _{\varepsilon},
\end{align*} satisfying that as 
$\varepsilon \rightarrow 0 $,
\begin{align*}
\zeta _{\varepsilon}\rightarrow x_0,\, \mu _{\varepsilon}\rightarrow +\infty \text{ and } \left\|\bar{\omega} _{\varepsilon} \right\|_{D_{0}^{s}(\varOmega )}\rightarrow 0.
\end{align*} We denote 
$\psi _{\zeta ,\mu}(x)=U_{\zeta ,\mu}(x)-PU_{\zeta ,\mu}(x)$ and give the following estimate for 
$\psi _{\zeta ,\mu}(x)$, which will be very useful throughout the paper and whose proof is given in [Reference Guo, Li, Pistoia and Yan10].
Lemma 2.2. We have
\begin{align*}
\psi _{\zeta ,\mu}(x)=\frac{AH(x,\zeta )}{\mu ^{\frac{N-2s}{2}}}+O(\mu ^{{-}\frac{N+2s}{2}}),
\end{align*}
\begin{align*}
\frac{\partial \psi _{\zeta ,\mu}(x)}{\partial \mu}=-\frac{A(N-2s)H(x,\zeta )}{2\mu ^{\frac{N-2s}{2}+1}}+O(\mu ^{-\frac{N+2s}{2}-1}),
\end{align*}and
\begin{align*}
\nabla _{\zeta}\psi _{\zeta ,\mu}(x)=\frac{A\nabla _{\zeta}H(x,\zeta )}{\mu ^{\frac{N-2s}{2}}}+O(\mu ^{-\frac{N+2s}{2}}),
\end{align*} if 
$d(\zeta ,\partial \varOmega )\ge c_0 \gt 0$, where 
$A=\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-1}dy}$.
Let
\begin{align*}
E_{\zeta ,\mu}=\left\{\bar{\omega}\,: \bar{\omega}\in D_{0}^{s}(\varOmega ),\, \left \lt \bar{\omega},\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu} \right \gt _{D_{0}^{s}( \varOmega )} =\left \lt \bar{\omega},\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _j} \right \gt _{D_{0}^{s}( \varOmega )} =0,\,\, j=1,2,\cdots ,N \right\}
\end{align*}and
\begin{align*}
J( \zeta ,\mu ,\bar{\omega} ) =I( \bar{P}_{\zeta ,\mu}+\bar{\omega} ) , \,\, \bar{\omega}\in E_{\zeta ,\mu ,\varepsilon}.
\end{align*} We expand 
$J( \zeta ,\mu ,\bar{\omega} )$ as follows:
\begin{equation}
J(\zeta ,\mu ,\bar{\omega})=J(\zeta ,\mu ,0)+\left \lt l_{\varepsilon},\bar{\omega} \right \gt _{D_{0}^{s}( \varOmega )}+\frac{1}{2}\left \lt L_{\varepsilon}\bar{\omega},\bar{\omega} \right \gt _{D_{0}^{s}( \varOmega )} +R_{\varepsilon}(\bar{\omega}), \,\, \bar{\omega}\in E_{\zeta ,\mu ,\varepsilon},
\end{equation}where
\begin{align*}
\left \lt l_{\varepsilon},\bar{\omega} \right \gt _{D_{0}^{s}( \varOmega )} &=\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{P}_{\zeta ,\mu}\nabla \bar{\omega}dX}-\int_{\varOmega}{(PU_{\zeta ,\mu})^{2_{s}^{*}-1-\varepsilon}\omega dx}\\
&=\int_{\varOmega}{U_{\zeta ,\mu}^{2_{s}^{*}-1}\omega dx}-\int_{\varOmega}{(PU_{\zeta ,\mu})^{2_{s}^{*}-1-\varepsilon}\omega dx},
\end{align*}
\begin{align*}
\left \lt L_{\varepsilon}\bar{\omega},\bar{\phi} \right \gt _{D_{0}^{s}( \varOmega )} =\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{\omega}\nabla \bar{\phi}dX}-( 2_{s}^{*}-1-\varepsilon ) \int_{\varOmega}{(PU_{\zeta ,\mu})^{2_{s}^{*}-2-\varepsilon}\omega \phi dx},
\end{align*}and
\begin{align*}
R_{\varepsilon}(\bar{\omega})=&-\frac{1}{2_{s}^{*}-\varepsilon}\int_{\varOmega}\Big( |PU_{\zeta ,\mu}+\omega |^{2_{s}^{*}-\varepsilon}-(PU_{\zeta, \mu})^{2_{s}^{*}-\varepsilon}-( 2_{s}^{*}-\varepsilon )(PU_{\zeta, \mu})^{2_{s}^{*}-1-\varepsilon}\omega\\
&-\frac{( 2_{s}^{*}-\varepsilon )( 2_{s}^{*}-1-\varepsilon)}{2}(PU_{\zeta ,\mu})^{2_{s}^{*}-2-\varepsilon}\omega ^2 \Big) dx.
\end{align*} Denote a projection 
$Q_{\zeta, \mu}$ from 
$D_{0}^{s}(\varOmega) $ to 
$E_{\zeta, \mu}$ as follows:
\begin{align*}
Q_{\zeta ,\mu}\bar u=\bar u-\sum_{i=1}^N{C_{i}\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i}}-C_{N+1}\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu},
\end{align*}where Ci is chosen in such a way that
\begin{align}
\left \lt Q_{\zeta, \mu}\bar u,\frac{\partial \bar{P}_{\zeta, \mu}}{\partial \zeta _j} \right \gt _{D_{0}^{s}( \varOmega )} =\left \lt Q_{\zeta, \mu}\bar u,\frac{\partial \bar{P}_{\zeta, \mu}}{\partial \mu} \right \gt _{D_{0}^{s}( \varOmega )} =0, j=1,\cdots ,N.
\end{align} By a standard argument, we need to prove the invertibility of the operator 
$Q_{\zeta, \mu }L_{\varepsilon}$. Before that, we check that the projection 
$Q_{\zeta, \mu}$ is bounded.
Lemma 2.3. There is a constant C > 0, such that
\begin{align*}
\left\| Q_{\zeta, \mu}\bar u \right\|_{D_{0}^{s}( \varOmega )} \le C\left\| \bar u \right\|_{D_{0}^{s}( \varOmega )} , \,\,\, \bar u\in D_{0}^{s}( \varOmega ).
\end{align*}Proof. By (2.6), we have
\begin{equation}
\begin{aligned}
\begin{cases}
\sum\limits_{i=1}^N{C_{i}\left \lt \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i},\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _j} \right \gt _{D_{0}^{s}( \varOmega )}}+C_{N+1}\left \lt \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu},\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _j} \right \gt _{D_{0}^{s}( \varOmega )} =\left \lt \bar u,\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _j} \right \gt _{D_{0}^{s}( \varOmega )},\,\,\, j=1,\cdots ,N,\\
\sum\limits_{i=1}^N{C_{i}\left \lt \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i},\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu} \right \gt _{D_{0}^{s}( \varOmega )}}+C_{N+1}\left \lt \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu},\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu} \right \gt _{D_{0}^{s}( \varOmega )} =\left \lt \bar u,\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu} \right \gt _{D_{0}^{s}( \varOmega )}.\\
\end{cases}
\end{aligned}
\end{equation}We claim that (2.7) is solvable. Indeed, since
\begin{equation*}
\begin{cases}
Div(x_{N+1}^{1-2s}\nabla \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i})=0,& \text{in}\,\,\mathbb{R}_{+}^{N+1},\\
\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i}=0,& \text{on}\,\,\mathbb{R}^N\setminus \varOmega ,\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim} x_{N+1}^{1-2s}\partial _{\nu}\left( \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i} \right) =\left( 2_{s}^{*}-1 \right) U_{\zeta ,\mu}^{2_{s}^{*}-2}\frac{\partial U_{\zeta ,\mu}}{\partial \zeta _i},& \text{on}\,\,\varOmega \\
\end{cases}
\end{equation*}and
\begin{equation*}
\begin{cases}
Div(x_{N+1}^{1-2s}\nabla \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu})=0,& \text{in}\,\,\mathbb{R}_{+}^{N+1},\\
\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu}=0,& \text{on}\,\,\mathbb{R}^N\setminus \varOmega ,\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim} x_{N+1}^{1-2s}\partial _{\nu}\left( \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu} \right) =\left( 2_{s}^{*}-1 \right) U_{\zeta ,\mu}^{2_{s}^{*}-2}\frac{\partial U_{\zeta ,\mu}}{\partial \mu},& \text{on}\,\,\varOmega ,\\
\end{cases}
\end{equation*}using Lemma 2.2, we have the following estimates:
\begin{align*}
\left \lt \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i},\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _j} \right \gt _{D_{0}^{s}( \varOmega )} &=\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i}\nabla \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _j}dX}\\
&=\int_{\varOmega}{(2_{s}^{*}-1)U_{\zeta ,\mu}^{2_{s}^{*}-2}\frac{\partial U_{\zeta ,\mu}}{\partial \zeta _i}\frac{\partial PU_{\zeta ,\mu}}{\partial \zeta _j}dx}\\
&=\begin{cases}
\frac{N^2-4s^2}{N}C_{N,s}^{2_{s}^{*}}\mu ^2\int_{\mathbb{R}^N}{\frac{\left| z \right|^2}{\left( 1+\left| z \right|^2 \right) ^{N+2}}dz}+O\left( \mu ^{-( N-2s )} \right) ,& \, i=j,\\
O\left( \mu ^{-(N-2s)} \right) ,& \, i\ne j,\\
\end{cases}
\end{align*}
\begin{align*}
\left \lt \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i},\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu} \right \gt _{D_{0}^{s}( \varOmega )} &=\int_{\varOmega}{(2_{s}^{*}-1)U_{\zeta ,\mu}^{2_{s}^{*}-2}\frac{\partial U_{\zeta ,\mu}}{\partial \mu}\frac{\partial PU_{\zeta ,\mu}}{\partial \zeta _i}dx}\\
&=O\left(\mu ^{-(N-2s+1)}\right)
\end{align*}and
\begin{align*}
\left \lt \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu},\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu} \right \gt _{D_{0}^{s}( \varOmega )}=&\int_{\varOmega}{(2_{s}^{*}-1)U_{\zeta ,\mu}^{2_{s}^{*}-2}\frac{\partial U_{\zeta ,\mu}}{\partial \mu}\frac{\partial PU_{\zeta ,\mu}}{\partial \mu}dx}\\
=&\frac{N^2-4s^2}{4(N+1)}C_{N,s}^{2_{s}^{*}}\mu ^{-2}\int_{\mathbb{R}^N}{\frac{\left| z \right|^4}{\left( 1+\left| z \right|^2 \right) ^{N+2}}dz}+O\left(\mu ^{-(N-2s+2)}\right).
\end{align*}Hence, (2.7) is solvable and we have
\begin{align}
C_{i}=O( \mu ^{-1} ) \left\| \bar{u} \right\|_{D_{0}^{s}( \varOmega )} ,C_{N+1}=O( \mu ) \left\| \bar{u} \right\|_{D_{0}^{s}( \varOmega )} ,\,\,\, i=1,\cdots ,N.
\end{align}From (2.8), we have
\begin{align*}
\left\| Q_{\zeta, \mu}\bar{u} \right\|_{D_{0}^{s}( \varOmega )} \le \left\| \bar{u} \right\|_{D_{0}^{s}( \varOmega )} +\sum_{i=1}^N{\left| C_{i} \right|\left\| \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i} \right\|_{D_{0}^{s}( \varOmega )}}+\left| C_{N+1} \right|\left\| \frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu} \right\|_{D_{0}^{s}( \varOmega )} \le C\left\|\bar{u} \right\|_{D_{0}^{s}( \varOmega )}.
\end{align*} Now, we prove the operator 
$Q_{\zeta, \mu}L_{\varepsilon}$ is invertible.
Lemma 2.4. For any given 
$a_1 \gt a_0 \gt 0$, there exists 
$\varepsilon _0,\theta _0 \gt 0$ and ρ > 0, such that for any 
$\varepsilon \in ( 0,\varepsilon _0 ]$, 
$\mu \in \left[ a_0\varepsilon ^{-\frac{1}{N-2s}},a_1\varepsilon ^{-\frac{1}{N-2s}} \right]$ and 
$\zeta \in B_{\theta _0}( x_0 )$, the operator 
$Q_{\zeta, \mu}L_{\varepsilon}$ is invertible in 
$E_{\zeta, \mu}$. Moreover, one has
\begin{align*}
\left\| Q_{\zeta, \mu} L_{\varepsilon}\bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} \ge \rho \left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} , \,\, \forall \bar{\omega}\in E_{\zeta, \mu}.
\end{align*}Proof. We argue it by contradiction. Suppose there are 
$\varepsilon_n\rightarrow 0$, 
$\zeta_n\rightarrow x_0$, 
$\mu _n\in \left[ a_0\varepsilon _{n}^{-\frac{1}{N-2s}},a_1\varepsilon _{n}^{-\frac{1}{N-2s}} \right] $ and 
$\bar{\omega}_n\in E_{\zeta _n, \mu _n}$ such that
\begin{align*}
\left\| Q_{\zeta _n, \mu _n}L_{\varepsilon _n}\bar{\omega}_n \right\|_{D_{0}^{s}( \varOmega )} \lt \frac{1}{n}\left\| \bar{\omega}_n \right\|_{D_{0}^{s}( \varOmega )}.
\end{align*} We may assume 
$\left\| \bar{\omega}_n \right\|_{D_{0}^{s}( \varOmega )}=1$, then 
$\left\| Q_{\zeta _n, \mu _n}L_{\varepsilon _n}\bar{\omega}_n \right\|_{D_{0}^{s}( \varOmega )} \lt \frac{1}{n}$, that is, for any 
$\bar{\phi}\in E_{\zeta _n, \mu _n}$,
\begin{align}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{\omega}_n\nabla \bar{\phi}dX-}( 2_{s}^{*}-1-\varepsilon _n ) \int_{\varOmega}{( PU_{\zeta _n,\mu _n} ) ^{2_{s}^{*}-2-\varepsilon _n}\omega _n\phi dx}=o_n( 1 ) \left\| \bar{\phi} \right\|_{D_{0}^{s}( \varOmega )} .
\end{align}We note that
\begin{equation}
\begin{aligned}
U_{\zeta _n ,\mu _n}^{-\varepsilon _n}&=\exp \left\{-\varepsilon _n \ln U_{\zeta _n ,\mu _n} \right\}\\
&=\exp \left\{-\frac{N-2s}{2}\varepsilon _n \ln \mu _n \right\} \cdot \exp \left\{\frac{N-2s}{2}\varepsilon _n \ln \left( 1+\mu _n ^2\left| x-\zeta _n \right|^2 \right) \right\}\\
&=1+O\bigl( \varepsilon _n \ln \mu _n \bigr),
\end{aligned}
\end{equation}uniformly on Ω.
Take 
$\bar{\phi}=\bar{\omega}_n$ in (2.9), we get
\begin{align}
1-o_n(1)&=(2_{s}^{*}-1-\varepsilon _n)\int_{\varOmega}{(PU_{\zeta _n,\mu _n})^{2_{s}^{*}-2-\varepsilon _n}\omega _{n}^{2}dx}\nonumber\\
&\le C(1+\varepsilon _n\ln \mu _n)\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _{n}^{2}dx}\nonumber\\
&\le C\int_{\mu _n(\varOmega -\zeta _n)}{U_{0,1}^{2_{s}^{*}-2}\Big(\mu _{n}^{-\frac{N-2s}{2}}\omega _n(\mu _{n}^{-1}x+\zeta _n)\Big)^2dx}.
\end{align} Denote 
$\tilde{\omega}_n(x)=\mu _{n}^{-\frac{N-2s}{2}}\omega _n(\mu _{n}^{-1}x+\zeta _n)$, 
$\tilde{\bar{\omega}}_n(x)=\mu _{n}^{-\frac{N-2s}{2}}\bar{\omega}_n(\mu _{n}^{-1}x+\zeta _n, \mu _{n}^{-1}x_{N+1})$, it is easy to prove that 
$\tilde{\bar{\omega}}_n(x)$ is s-harmonic extension of 
$\tilde{\omega}_n(x)$, namely 
$\bar{\tilde{\omega}}_n(x)=\tilde{\bar{\omega}}_n(x)$. By 
$\left\| \tilde{\bar{\omega}}_n \right\| _{D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})}\le C$, there exists 
$\bar{\omega}\in D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})$, such that
\begin{align*}
\tilde{\bar{\omega}}_n&\rightharpoonup \bar{\omega}\,\, in\,\, D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1}),\\
\tilde{\omega}_n&\rightharpoonup \omega\,\, in\,\, L^{2_{s}^{*}}(\mathbb{R}^N),\\
\tilde{\omega}_n& \rightarrow \omega\,\, in\,\, L_{loc}^{p}(\mathbb{R}^N),\, p\in [ 1,2_{s}^{*} ).
\end{align*}Moreover, from (2.11), we get, for R > 0,
\begin{align*}
1-o_n(1)&\le C\Big(\int_{\mu _n(\varOmega -\zeta _n)\setminus B_R(0)}{U_{0,1}^{2_{s}^{*}}dx}\Big)^{\frac{2_{s}^{*}}{2_{s}^{*}-2}}\|\tilde{\omega}_n\|_{L^{2_s^*}(\mathbb{R}^N)}^2+C\int_{B_R(0)}{\tilde{\omega} _{n}^{2}dx}\\
&\le C\Big(\int_{\mu _n(\varOmega -\zeta _n)\setminus B_R(0)}{U_{0,1}^{2_{s}^{*}}dx}\Big)^{\frac{2_{s}^{*}}{2_{s}^{*}-2}}+C\int_{B_R(0)}{\omega ^2dx}+o_n(1)\\
&=o_R(1)+C\int_{B_R(0)}{\omega ^2dx}+o_n(1).
\end{align*}To obtain a contradiction from the above formula, we only need to prove ω = 0.
Next, we will prove
\begin{align}
\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial x _i}\omega dx}=\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg|_{\mu =1}\omega dx}=0, \, i=1,\cdots ,N.
\end{align} By 
$\bar{\omega}_n\in E_{\zeta _n, \mu _n}$,
\begin{align*}
0=\left \lt \bar{\omega}_n,\frac{\partial \bar{P}_{\zeta _n,\mu _n}}{\partial \zeta _{n,i}} \right \gt _{D_{0}^{s}( \varOmega )}=&(2_{s}^{*}-1)\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\frac{\partial U_{\zeta _n,\mu _n}}{\partial \zeta _{n,i}}\omega _ndx}\\
=&(2_{s}^{*}-1)\mu _{n}\int_{\mu _n(\varOmega -\zeta _n)}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial x _i}\tilde{\omega}_ndz}\\
=&(2_{s}^{*}-1)\mu _{n}\Big(\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial x _i}\omega dz}+\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial x _i}(\tilde{\omega}_n-\omega )dz}\\
&-\int_{\mathbb{R}^N\setminus \mu _n(\varOmega-\zeta _n)}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial x _i}\tilde{\omega}_ndz}\Big)\\
=&(2_{s}^{*}-1)\mu _{n}\Big(\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial x _i}\omega dz+o_{n}(1)}\Big),
\end{align*}we have
\begin{align*}
\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial x _i}\omega dx}=0, \, i=1,\cdots ,N.
\end{align*}Similarly, we also have
\begin{align*}
\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg|_{\mu =1}\omega dx}=0.
\end{align*} On the other hand, by (2.9), we know that for any 
$\bar{\varphi}\in E_{\zeta _n, \mu _n}$,
\begin{align*}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{\omega}_n\nabla \bar{\varphi}dX-}( 2_{s}^{*}-1)\int_{\varOmega}{( PU_{\zeta _n,\mu _n} ) ^{2_{s}^{*}-2-\varepsilon _n}\omega _n\varphi dx}=o_n( 1 ) \left\| \bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )}.
\end{align*}Here, by Lemma 2.2 and (2.10), we have
\begin{align*}
& \int_{\varOmega}{(PU_{\zeta _n,\mu _n})^{2_{s}^{*}-2-\varepsilon _n}\omega _n\varphi dx} \\&\quad =\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2-\varepsilon _n}\omega _n\varphi dx}+\begin{cases}
O(\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-3-\varepsilon _n}\left| \psi _{\zeta _n,\mu _n}\omega _n\varphi \right|dx}),& 2_{s}^{*}-2 \gt 1,\\[2mm]
O(\int_{\varOmega}{\psi _{\zeta _n,\mu _n}^{2_{s}^{*}-2-\varepsilon _n}\left| \omega _n\varphi \right|dx}),& 2_{s}^{*}-2\le 1\\
\end{cases}
\\& \quad
= \int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _n\varphi dx}+o_n(1)\left\| \bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )}\\
&\quad \quad +\begin{cases}
O(\mu _{n}^{-\frac{N-2s}{2}}\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-3}\left| \omega _n\varphi \right|dx}),& N \lt 6s,\\[1mm]
O(\mu _{n}^{-2s}\int_{\varOmega}{\left| \omega _n\varphi \right|dx}),& N\ge 6s
\end{cases}\\
&\quad = \int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _n\varphi dx}+o_n(1)\left\| \bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )} .
\end{align*}Thus, we obtain
\begin{align}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{\omega}_n\nabla \bar{\varphi}dX-}( 2_{s}^{*}-1) \int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _n\varphi dx}=o_n( 1 ) \left\| \bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )},
\end{align} for any 
$\bar{\varphi} \in E_{\zeta _n, \mu _n}$.
Moreover, we expect that
\begin{align}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{\omega}_n\nabla \bar{\varphi}dX-}( 2_{s}^{*}-1) \int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _n\varphi dx}=o_n( 1 ) \left\| \bar{\varphi} \right\| _{D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})},
\end{align} for any 
$\bar{\varphi}\in D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})$.
Thus, if (2.14) is correct, we have, for any 
$\bar{\phi}\in D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})$,
\begin{align}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \tilde{\bar{\omega}}_n\nabla \bar{\phi}dX-}(2_{s}^{*}-1)\int_{\varOmega}{U_{0,1}^{2_{s}^{*}-2}\tilde{\omega}_n\phi dx}=o_n(1)\left\| \bar{\phi} \right\| _{D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})}.
\end{align} Let 
$n\rightarrow \infty$ in (2.15), we can get
\begin{align*}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{\omega}\nabla \bar{\phi}dX-}(2_{s}^{*}-1)\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\omega \phi dx}=0.
\end{align*} This means that 
$\bar{\omega}\in D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})$ satisfies
\begin{align*}
\begin{cases}
Div(x_{N+1}^{1-2s}\nabla \bar{\omega})=0,& \text{in}\,\,\mathbb{R}_{+}^{N+1},\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim} x_{N+1}^{1-2s}\partial _{\nu}\bar{\omega}=(2_{s}^{*}-1)U_{0,1}^{2_{s}^{*}-2}\omega,& \text{on}\,\,\mathbb{R}^N.\\
\end{cases}
\end{align*} It follows from the non-degeneracy of 
$U_{0,1}$ that
\begin{align*}
\omega =\sum_{i=1}^N{\alpha _i\frac{\partial U_{0,1}}{\partial x _i}}+\alpha _{N+1}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg| _{\mu =1},
\end{align*} where αi are some constants for 
$i=1, 2, \cdots,N+1$. By (2.12), we conclude ω = 0.
To finish the proof of this Lemma, it remains to prove (2.14). We select a family of cut-off functions 
$\eta _k\in C_{0}^{\infty}(\varOmega )$ satisfying 
$0\le \eta _k\le 1$ and 
$\eta _k\equiv 1 \text{on }\,\, \varOmega _k$, where 
$\varOmega _k\rightarrow \varOmega$ as 
$k\rightarrow \infty $. Then for any 
$\bar{\varphi}\in D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})$, we have 
$\overline{\eta _k\varphi }\in D_{0}^{s}(\varOmega )$ and it is easy to check that
\begin{align*}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{\omega}_n\nabla \overline{\eta _k\varphi }dX}&\rightarrow \int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{\omega}_n\nabla \bar{\varphi}dX},\\
\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _n\eta _k\varphi dx}&\rightarrow \int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _n\varphi dx},
\end{align*}and
\begin{align*}
\left\| \overline{\eta _k\varphi } \right\|_{D_{0}^{s}( \varOmega )} &\rightarrow \left\| \bar{\varphi} \right\|_{D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})},
\end{align*} as 
$k\rightarrow \infty $. Therefore, in order for (2.14) to be true for any 
$\bar{\varphi}\in D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})$, we just need to prove that (2.14) is true for any 
$\bar{\varphi}\in D_{0}^{s}(\varOmega )$. For any 
$\bar{\varphi}\in D_{0}^{s}(\varOmega )$, we know 
$Q_{\zeta _n, \mu _n}\bar{\varphi}=\bar{\varphi}-\sum\limits _{i=1}^N{C_{i}\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i}}-C_{N+1}\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu}\in E_{\zeta _n, \mu _n}$. Then by 
$\bar{\omega}_n\in E_{\zeta _n,\mu _n}$ and (2.13), we can get
\begin{align*}
&\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla \bar{\omega}_n\nabla \bar{\varphi}dX}-(2_{s}^{*}-1)\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _n\varphi dx}\\
=&o_{n}(1)\left\| Q_{\zeta _n, \mu _n}\bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )} -(2_{s}^{*}-1)\sum_{i=1}^N{C_{i}\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\frac{\partial PU_{\zeta _n,\mu _n}}{\partial \zeta _{n,i}}\omega _ndx}}\\
&-(2_{s}^{*}-1)C_{N+1}\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\frac{\partial PU_{\zeta _n,\mu _n}}{\partial \mu _n}\omega _ndx}\\
=&o_{n}(1)\left\| Q_{\zeta _n,\mu _n}\bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )} -(2_{s}^{*}-1)\sum_{i=1}^N{C_{i}\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\frac{\partial \psi _{\zeta _n,\mu _n}}{\partial \zeta _{n,i}}\omega _ndx}}\\
&-(2_{s}^{*}-1)C_{N+1}\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\frac{\partial \psi _{\zeta _n,\mu _n}}{\partial \mu _n}\omega _ndx}.
\end{align*} Moreover, by (2.8) and Lemma 2.2, we obtain 
$i=1,\cdots,N $,
\begin{align*}
C_{i}\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\frac{\partial \psi _{\zeta _n,\mu _n}}{\partial \zeta _{n,i}}\omega _ndx}=O\left(\mu _{n}^{-\frac{N-2s}{2}-1}\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _ndx}\right)\left\| \bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )} =o_{n}(1)\left\| \bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )},
\end{align*}and
\begin{align*}
C_{N+1}\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\frac{\partial \psi _{\zeta _n,\mu _n}}{\partial \mu _n}\omega _ndx}=O\left(\mu _{n}^{-\frac{N-2s}{2}}\int_{\varOmega}{U_{\zeta _n,\mu _n}^{2_{s}^{*}-2}\omega _ndx}\right)\left\| \bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )} =o_{n}(1)\left\| \bar{\varphi} \right\|_{D_{0}^{s}( \varOmega )} .
\end{align*} Therefore, we conclude (2.14) is true for any 
$\bar{\varphi}\in D^1(x_{N+1}^{1-2s},\mathbb{R}_{+}^{N+1})$.
In order to using contraction mapping theorem, we need to estimate 
$\left\| l_{\varepsilon} \right\|_{D_{0}^{s}( \varOmega )}$, 
$\left\| R_{\varepsilon}^{\prime}(\bar{\omega}) \right\|_{D_{0}^{s}( \varOmega )}$ and 
$\left\| R_{\varepsilon}^{\prime\prime}(\bar{\omega}) \right\|_{D_{0}^{s}( \varOmega )}$.
Lemma 2.5. There is a constant C > 0, such that
\begin{align*}
\left\| l_{\varepsilon} \right\|_{D_{0}^{s}( \varOmega )} \le \begin{cases}
C\left( \mu ^{-(N-2s)}+\varepsilon \ln \mu \right) ,& \,N \lt 6s,\\
C\left( \mu ^{-4s}(\ln \mu )^{\frac{2}{3}}+\varepsilon \ln \mu \right) ,& \,N=6s,\\
C\left( \mu ^{-\frac{N+2s}{2}}+\varepsilon \ln \mu \right),& \,N \gt 6s.\\
\end{cases}
\end{align*}Proof. According to Lemma 2.2 and (2.10), we have
\begin{align*}
\left| ( PU_{\zeta ,\mu} ) ^{2_{s}^{*}-1-\varepsilon}-U_{\zeta ,\mu}^{2_{s}^{*}-1} \right|&\le \left| ( PU_{\zeta ,\mu} ) ^{2_{s}^{*}-1-\varepsilon}-U_{\zeta ,\mu}^{2_{s}^{*}-1-\varepsilon} \right|+\left| U_{\zeta ,\mu}^{2_{s}^{*}-1-\varepsilon}-U_{\zeta ,\mu}^{2_{s}^{*}-1} \right|
\\
&=O\left( U_{\zeta ,\mu}^{2_{s}^{*}-2-\varepsilon}\left| \psi _{\zeta ,\mu} \right| \right) +O\left( \varepsilon \ln \mu U_{\zeta ,\mu}^{2_{s}^{*}-1} \right)
\\
&=O\left( \mu ^{-\frac{N-2s}{2}}U_{\zeta ,\mu}^{2_{s}^{*}-2} \right) +O\left( \varepsilon \mu ^{-\frac{N-2s}{2}}\ln \mu U_{\zeta ,\mu}^{2_{s}^{*}-2} \right) \\
& \quad +O\left( \varepsilon \ln \mu U_{\zeta ,\mu}^{2_{s}^{*}-1} \right)
\\
&=O\left( \mu ^{-\frac{N-2s}{2}}U_{\zeta ,\mu}^{2_{s}^{*}-2} \right) +O\left( \varepsilon \ln \mu U_{\zeta ,\mu}^{2_{s}^{*}-1} \right).
\end{align*}Then, it can be deduced that
\begin{align}
\left| \left \lt l_{\varepsilon},\bar{\omega} \right \gt \right|&=\left| \int_{\varOmega}{U_{\zeta ,\mu}^{2_{s}^{*}-1}\omega dx}-\int_{\varOmega}{(PU_{\zeta ,\mu})^{2_{s}^{*}-1-\varepsilon}\omega dx} \right| \nonumber\\[3mm]
&\le C\mu ^{-\frac{N-2s }{2}}\int_{\varOmega}{U_{\zeta ,\mu}^{2_{s}^{*}-2}\left| \omega \right|dx}+C\varepsilon \ln \mu \int_{\varOmega}{U_{\zeta ,\mu}^{2_{s}^{*}-1}\left| \omega \right|dx} \nonumber\\
&\le C\mu ^{-\frac{N-2s }{2}}\left( \int_{\varOmega}{U_{\zeta ,\mu}^{\frac{2_{s}^{*}( 2_{s}^{*}-2 )}{2_{s}^{*}-1}}dx} \right) ^{\frac{2_{s}^{*}-1}{2_{s}^{*}}}\left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} \nonumber\\
& \quad +C\varepsilon \ln \mu \left( \int_{\varOmega}{U_{\zeta ,\mu}^{2_{s}^{*}}dx} \right) ^{\frac{2_{s}^{*}-1}{2_{s}^{*}}}\left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )}.
\end{align}We can check that
\begin{align*}
\left( \int_{\varOmega}{U_{\zeta ,\mu}^{\frac{2_{s}^{*}(2_{s}^{*}-2)}{2_{s}^{*}-1}}dx} \right) ^{\frac{2_{s}^{*}-1}{2_{s}^{*}}}&=O\Bigg( \mu ^{2s-\frac{N+2s}{2}}\Big( \int_{\mu \left( \varOmega -\zeta \right)}{\frac{1}{\left( 1+\left| z \right|^2 \right) ^{\frac{4Ns}{N+2s}}}dz} \Big) ^{\frac{N+2s}{2N}} \Bigg)\\
&=O\Bigg( \mu ^{-\frac{N-2s}{2}}\Big( \int_0^{C\mu}{\frac{r^{N-1}}{\left( 1+r^2 \right) ^{\frac{4Ns}{N+2s}}}dr} \Big) ^{\frac{N+2s}{2N}} \Bigg)\\
&=\begin{cases}
O\left( \mu ^{-\frac{N-2s}{2}} \right) ,&\,\, N \lt 6s,\\
O\left( \mu ^{-2s}( \ln \mu ) ^{\frac{2}{3}} \right) ,& \,\, N=6s,\\
O\left( \mu ^{-2s} \right) ,& \,\, N \gt 6s\\
\end{cases}
\end{align*}and
\begin{align*}
\left(\int_{\varOmega}{U_{\zeta ,\mu}^{2_{s}^{*}}dx}\right) ^{\frac{1}{2_{s}^{*}}}\le \left(\int_{\mathbb{R}^N}{U_{\zeta ,\mu}^{2_{s}^{*}}dx} \right) ^{\frac{1}{2_{s}^{*}}}\le C.
\end{align*}Substituting them back to (2.16), we obtain the result.
Lemma 2.6. There is a constant C > 0, such that
\begin{align*}
\left\| R_{\varepsilon}^{\prime}(\bar{\omega}) \right\|_{D_{0}^{s}( \varOmega )} \le C(1+\varepsilon \ln \mu )\left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} ^{\min \{2_{s}^{*}-1,2 \}},
\end{align*}and
\begin{align*}
\left\| R_{\varepsilon}^{\prime\prime}(\bar{\omega}) \right\|_{D_{0}^{s}( \varOmega )} \le C(1+\varepsilon \ln \mu )\left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} ^{\min \{2_{s}^{*}-2,1 \}}.
\end{align*}Proof. If 
$2_{s}^{*}-1 \gt 2$, for sufficiently small ɛ, such that 
$2_{s}^{*}-1-\varepsilon \gt 2$, we have
\begin{align*}
\left| \left \lt R_{\varepsilon}^{\prime}(\bar{\omega}),\bar{\phi} \right \gt _{D_{0}^{s}( \varOmega )} \right|=&\bigg| \int_{\varOmega}\Big( \left| PU_{\zeta ,\mu}+\omega \right|^{2_{s}^{*}-2-\varepsilon}(PU_{\zeta ,\mu}+\omega)-(PU_{\zeta ,\mu})^{2_{s}^{*}-1-\varepsilon}\\
&-(2_{s}^{*}-1-\varepsilon )(PU_{\zeta ,\mu})^{2_{s}^{*}-2-\varepsilon}\omega \Big) \phi dx \bigg|\\
\le& C\int_{\varOmega}{\left| PU_{\zeta ,\mu} \right|^{2_{s}^{*}-3-\varepsilon}\omega ^2\left| \phi \right|dx}\\
\le& C( 1+\varepsilon \ln \mu )\left( \int_{\varOmega}{\left| U_{\zeta ,\mu} \right|^{2_{s}^{*}}dx} \right) ^{\frac{2_{s}^{*}-3}{2_{s}^{*}}}\left( \int_{\varOmega}{\left| \omega \right|^{2_{s}^{*}}dx} \right) ^{\frac{2}{2_{s}^{*}}}\left( \int_{\varOmega}{\left| \phi \right|^{2_{s}^{*}}dx} \right) ^{\frac{1}{2_{s}^{*}}}\\
\le& C( 1+\varepsilon \ln \mu )\left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} ^2\left\| \bar{\phi} \right\|_{D_{0}^{s}( \varOmega )},
\end{align*}and
\begin{align*}
\left| \left \lt R_{\varepsilon}^{\prime\prime}(\bar{\omega})\bar{\psi},\bar{\phi} \right \gt _{D_{0}^{s}( \varOmega )} \right|&=(2_{s}^{*}-2-\varepsilon)\left| \int_{\varOmega}{\left( \left| PU_{\zeta ,\mu}+\omega \right|^{2_{s}^{*}-2-\varepsilon}-(PU_{\zeta ,\mu})^{2_{s}^{*}-2-\varepsilon} \right) \phi \psi dx} \right|\\
&\le C\int_{\varOmega}{\left| PU_{\zeta ,\mu} \right|^{2_{s}^{*}-3-\varepsilon}\left| \omega \phi \psi \right|dx}\\
&\le C( 1+\varepsilon \ln \mu )\left( \int_{\varOmega}{\left| U_{\zeta ,\mu} \right|^{2_{s}^{*}}dx} \right) ^{\frac{2_{s}^{*}-3}{2_{s}^{*}}}\left( \int_{\varOmega}{\left| \omega \right|^{2_{s}^{*}}dx} \right) ^{\frac{1}{2_{s}^{*}}}\\
& \quad times \left( \int_{\varOmega}{\left| \phi \right|^{2_{s}^{*}}dx} \right) ^{\frac{1}{2_{s}^{*}}}\left( \int_{\varOmega}{\left| \psi \right|^{2_{s}^{*}}dx} \right) ^{\frac{1}{2_{s}^{*}}}\\
&\le C( 1+\varepsilon \ln \mu )\left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} \left\| \bar{\phi} \right\|_{D_{0}^{s}( \varOmega )} \left\| \bar{\psi} \right\|_{D_{0}^{s}( \varOmega )} .\\
\end{align*} If 
$2_{s}^{*}-1\leq2$, we have
\begin{gather*}
\left| \left \lt R_{\varepsilon}^{\prime}(\bar{\omega}),\bar{\phi} \right \gt _{D_{0}^{s}( \varOmega )} \right|\le C(1+\varepsilon \ln \mu )\left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} ^{2_{s}^{*}-1}\left\| \bar{\phi} \right\|_{D_{0}^{s}( \varOmega )},\\[1mm]
\left| \left \lt R_{\varepsilon}^{\prime\prime}(\bar{\omega})\bar{\psi},\bar{\phi} \right \gt _{D_{0}^{s}( \varOmega )} \right|\le C(1+\varepsilon \ln \mu )\left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} ^{2_{s}^{*}-2}\left\| \bar{\phi} \right\|_{D_{0}^{s}( \varOmega )} \left\| \bar{\psi} \right\|_{D_{0}^{s}( \varOmega )}.
\end{gather*}As a result, we complete the proof.
Proposition 2.7. For any given 
$a_1 \gt a_0 \gt 0$, there exists 
$\varepsilon _0,\theta _0 \gt 0$ and ρ > 0, such that for any 
$\varepsilon \in ( 0,\varepsilon _0 ]$, there is a C 1 map 
$\bar{\omega}_{\zeta ,\mu,\varepsilon}:B_{\theta _0}(x_0)\times \left[ a_0\varepsilon ^{-\frac{1}{N-2s}},a_1\varepsilon ^{-\frac{1}{N-2s}} \right] \rightarrow D_{0}^{s}(\varOmega )$, such that 
$\bar{\omega}_{\zeta ,\mu ,\varepsilon}\in E_{\zeta, \mu}$ and
\begin{align*}
\frac{\partial J}{\partial \bar{\omega}}=\sum_{i=1}^N{\beta _i\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \zeta _i}}+\beta _{N+1}\frac{\partial \bar{P}_{\zeta ,\mu}}{\partial \mu},
\end{align*} where βi are some constants for 
$i=1, 2, \cdots, N+1 $. Moreover,
\begin{align*}
\left\| \bar{\omega}_{\zeta ,\mu ,\varepsilon} \right\|_{D_{0}^{s}( \varOmega )} \le C\left\| l_{\varepsilon} \right\|_{D_{0}^{s}( \varOmega )}.
\end{align*}Proof. We know that finding a critical point for 
$J( \zeta ,\mu ,\bar{\omega} )\in E_{\zeta, \mu}$ is equivalent to solving
\begin{align*}
Q_{\zeta, \mu}\Big(L_{\varepsilon}\bar{\omega}+l_{\varepsilon}+R_{\varepsilon}^{\prime}(\bar{\omega})\Big)=0.
\end{align*} By Lemma 2.4, 
$Q_{\zeta, \mu}L_{\varepsilon}$ is invertible. Thus, we turn to solve
\begin{align*}
\bar{\omega}=A(\bar{\omega}):= -(Q_{\zeta, \mu}L_{\varepsilon})^{-1}Q_{\zeta, \mu}l_{\varepsilon}-(Q_{\zeta, \mu}L_{\varepsilon})^{-1}Q_{\zeta, \mu}R_{\varepsilon}^{\prime}(\bar{\omega}).
\end{align*}Set
\begin{align*}
S:= \left\{\bar{\omega}\in E_{\zeta ,\mu}:\left\| \bar{\omega} \right\|_{D_{0}^{s}( \varOmega )} \le \mu ^{-\frac{N+2s}{2}+\tau}+\varepsilon \mu ^{\tau}\ln \mu \right\},
\end{align*}where τ > 0 is some constant.
Now we verify that A is a contraction from S to S. By Lemmas 2.3-2.6, for any 
$\bar{\omega}\in S$, we have
\begin{align*}
\left\| A(\bar{\omega}) \right\|_{D_{0}^{s}( \varOmega )} &\le C\left\| l_{\varepsilon} \right\|_{D_{0}^{s}( \varOmega )} +C\left\| R_{\varepsilon}^{\prime}(\bar{\omega}) \right\|_{D_{0}^{s}( \varOmega )}\\
&\le C\left( \mu ^{-\frac{N+2s}{2}}+\varepsilon \ln \mu \right) +C(\mu ^{-\frac{N+2s}{2}+\tau}+\varepsilon \mu ^{\tau}\ln \mu )^{\min \{2_{s}^{*}-1,2\}}\\
& \lt \mu ^{-\frac{N+2s}{2}+\tau}+\varepsilon \mu ^{\tau}\ln \mu .
\end{align*} Thus, A maps S to S. On the other hand, for any 
$\bar{\omega}_1, \bar{\omega}_2\in S$, we have
\begin{align*}
\left\| A(\bar{\omega}_1)-A(\bar{\omega}_2) \right\|_{D_{0}^{s}( \varOmega )} &\le C\left\| R_{\varepsilon}^{\prime}(\bar{\omega}_1)-R_{\varepsilon}^{\prime}(\bar{\omega}_2) \right\|_{D_{0}^{s}( \varOmega )}\\
&\le C\left\| R_{\varepsilon}^{\prime}(\theta \bar{\omega}_1+(1-\theta )\bar{\omega}_2) \right\|_{D_{0}^{s}( \varOmega )} \left\| \bar{\omega}_1-\bar{\omega}_2 \right\|_{D_{0}^{s}( \varOmega )}\\
&\le C\Big(\left\| \bar{\omega}_1 \right\|_{D_{0}^{s}( \varOmega )} ^{\min \{2_{s}^{*}-2,1\}}+\left\| \bar{\omega}_2 \right\|_{D_{0}^{s}( \varOmega )} ^{\min \{2_{s}^{*}-2,1\}}\Big)\left\| \bar{\omega}_1-\bar{\omega}_2 \right\|_{D_{0}^{s}( \varOmega )}\\
&\le \frac{1}{2}\left\| \bar{\omega}_1-\bar{\omega}_2 \right\|_{D_{0}^{s}( \varOmega )} .
\end{align*}Hence, A is a contraction map in S and the result follows from the contraction mapping theorem.
In order to prove Theorem 2.1, we only need prove there exist 
$\zeta _{\varepsilon}\in B_{\theta _0}(x_0)$ and 
$\mu _{\varepsilon}\in \left[ a_0\varepsilon ^{-\frac{1}{N-2s}},a_1\varepsilon ^{-\frac{1}{N-2s}} \right]$ such that 
$\beta _i=0,\, i=1,\cdots,N+1$ in Proposition 2.7. Indeed, we need to solve the system
\begin{align}
\int_{\mathbb{R}_{+}^{N1}}{x_{N+1}^{1-2s}\nabla (\bar{P}_{\zeta, \mu}+\bar{\omega}_{\zeta, \mu,\varepsilon})\nabla \frac{\partial \bar{P}_{\zeta, \mu}}{\partial \zeta _{i}}dX}-\int_{\varOmega}{(PU_{\zeta, \mu}+\omega _{\zeta, \mu,\varepsilon})^{2_{s}^{*}-1-\varepsilon}\frac{\partial PU_{\zeta ,\mu}}{\partial \zeta _{i}}dx}=0,
\end{align}
\begin{align}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla (\bar{P}_{\zeta,\mu }+\bar{\omega}_{\zeta ,\mu,\varepsilon})\nabla \frac{\partial \bar{P}_{\zeta,\mu}}{\partial \mu}dX}-\int_{\varOmega}{(PU_{\zeta, \mu}+\omega _{\zeta, \mu, \varepsilon})^{2_{s}^{*}-1-\varepsilon}\frac{\partial PU_{\zeta ,\mu}}{\partial \mu}dx}=0,
\end{align} for 
$i=1,\cdots,N$.
First, we prove the following result.
Lemma 2.8. Let 
$\bar{\omega}_{\zeta, \mu, \varepsilon}$ be the function obtained in Proposition 2.7. Then for 
$\frac{1}{2} \lt s \lt 1$, it holds
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}_{+}^{N1}}{x_{N+1}^{1-2s}\nabla (\bar{P}_{\zeta, \mu}+\bar{\omega}_{\zeta , \mu, \varepsilon})\nabla \frac{\partial \bar{P}_{\zeta, \mu}}{\partial \zeta _{i}}dX}-\int_{\varOmega}{(PU_{\zeta, \mu}+\omega _{\zeta, \mu, \varepsilon})^{2_{s}^{*}-1-\varepsilon}\frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}dx}\\
=&\frac{2_{s}^{*}-1}{2}A^2\frac{\partial h(\zeta)}{\partial \zeta _{i}}\mu^{-(N-2s)}+\begin{cases}
O\Big(\varepsilon ^2\mu(\ln \mu )^2\Big),& 2s+1 \lt N\le 4s,\\
O\Big(\mu^{-N+1}\Big),& 4s \lt N\le 6s\\
\end{cases}
\end{aligned}
\end{equation}and
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\nabla (\bar{P}_{\zeta, \mu} +\bar{\omega}_{\zeta, \mu, \varepsilon})\nabla \frac{\partial \bar{P}_{\zeta, \mu}}{\partial \mu}dX}-\int_{\varOmega}{(PU_{\zeta, \mu} +\omega _{\zeta, \mu, \varepsilon})^{2_{s}^{*}-1-\varepsilon}\frac{\partial PU_{\zeta, \mu}}{\partial \mu}dx}\\
=&\frac{(N-2s)^2}{4}B\varepsilon \mu^{-1}-\frac{N-2s}{2}A^2h(\zeta)\mu^{-(N-2s+1)}+\begin{cases}
O\Big(\varepsilon ^2\mu^{-1}(\ln \mu )^2\Big),& 2s+1 \lt N\le 4s,\\
O\Big(\mu^{-(N+1)}\Big),& 4s \lt N\le 6s,\\
\end{cases}
\end{aligned}
\end{equation} where 
$B=\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}}dy}$.
Proof. We use 
$\partial \bar{P}_{\zeta, \mu}$ to denote 
$\frac{\partial \bar{P}_{\zeta, \mu}}{\partial \zeta _{i}}$ and 
$\frac{\partial \bar{P}_{\zeta, \mu}}{\partial \mu}$. By 
$\bar{\omega}_{\zeta ,\mu ,\varepsilon}\in E_{\zeta ,\mu ,\varepsilon}$, we have
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}_{+}^{N1}}{x_{N+1}^{1-2s}\nabla (\bar{P}_{\zeta, \mu}+\bar{\omega}_{\zeta, \mu, \varepsilon})\nabla \partial \bar{P}_{\zeta, \mu}dX}-\int_{\varOmega}{(PU_{\zeta, \mu}+\omega _{\zeta, \mu, \varepsilon})^{2_{s}^{*}-1-\varepsilon}\partial \bar{P}_{\zeta, \mu}dx}\\
=&\int_{\varOmega}{\left( U_{\zeta, \mu}^{2_{s}^{*}-1}-(PU_{\zeta, \mu})^{2_{s}^{*}-1-\varepsilon} \right) \partial PU_{\zeta, \mu}dx}\\
&-\int_{\varOmega}{\left( (PU_{\zeta, \mu} +\omega _{\zeta, \mu, \varepsilon})^{2_{s}^{*}-1-\varepsilon}-(PU_{\zeta, \mu })^{2_{s}^{*}-1-\varepsilon} \right)\partial PU_{\zeta, \mu } dx}.
\end{aligned}
\end{equation}Since
\begin{equation*}
U_{\zeta, \mu}^{-\varepsilon}=1-\frac{N-2s}{2}\varepsilon \ln \mu+\frac{N-2s}{2}\varepsilon \ln \left( 1+\mu^2\left| y-\zeta \right|^2 \right) +O\left( \varepsilon ^2(\ln \mu )^2 \right),
\end{equation*} by using Lemma 2.2, for 
$\frac{1}{2} \lt s \lt 1$, we have
\begin{align}
&\int_{\varOmega}{\left( U_{\zeta, \mu}^{2_{s}^{*}-1}-(PU_{\zeta, \mu})^{2_{s}^{*}-1-\varepsilon} \right) \frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}dx}\nonumber\\
=&\int_{\varOmega}{( U_{\zeta, \mu}^{2_{s}^{*}-1}-U_{\zeta , \mu}^{2_{s}^{*}-1-\varepsilon} ) \frac{\partial PU_{\zeta, \mu }}{\partial \zeta _{i}}dx}+\left( 2_{s}^{*}-1-\varepsilon \right) \int_{\varOmega}{(U_{\zeta, \mu })^{2_{s}^{*}-2-\varepsilon}\frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}\psi _{\zeta, \mu}dx}\nonumber\\
&+\begin{cases}
O\left( \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-3}\left| \frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}\psi _{\zeta, \mu}^{2} \right|dx} \right) ,& 2s+1 \lt N \lt 6s,\\
O\left( \int_{\varOmega}{\left| \frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}\psi _{\zeta, \mu}^{2_{s}^{*}-1-\varepsilon} \right|dx} \right) ,& N=6s
\end{cases}\nonumber\\
=&\frac{N-2s}{2}\varepsilon \ln \mu \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-1}\frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}dx}-\frac{N-2s}{2}\varepsilon \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-1}\frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}\ln \left( 1+\mu \left| x-\zeta \right| \right) dx}\nonumber\\
&+\left( 2_{s}^{*}-1-\varepsilon \right) \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-2-\varepsilon}\frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}\psi _{\zeta, \mu }dx}+\begin{cases}
O\left( \varepsilon ^2\mu(\ln \mu )^2 \right) ,& 2s+1 \lt N\le 4s,\\
O\left( \mu ^{-N+1} \right),& 4s \lt N\le 6s
\end{cases}\nonumber\\
=&\left( 2_{s}^{*}-1-\varepsilon \right) \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-2}\frac{\partial U_{\zeta, \mu}}{\partial \zeta _{i}}\psi _{\zeta, \mu}dx}+\begin{cases}
O\left( \varepsilon ^2\mu (\ln \mu )^2 \right) ,& 2s+1 \lt N\le 4s,\\
O\left( \mu^{-N+1} \right),& 4s \lt N\le 6s
\end{cases}\nonumber\\
=&A\left( 2_{s}^{*}-1-\varepsilon \right) \mu^{-\frac{(N-2s)}{2}}\int_{\varOmega}{H\left( x,\zeta \right) U_{\zeta, \mu}^{2_{s}^{*}-2}\frac{\partial U_{\zeta, \mu}}{\partial \zeta _{i}}dx}+\begin{cases}
O\left( \varepsilon ^2\mu(\ln \mu )^2 \right) ,& 2s+1 \lt N\le 4s,\\
O\left( \mu^{-N+1} \right),& 4s \lt N\le 6s
\end{cases}\nonumber\\
=&\frac{2_{s}^{*}-1}{2}A^2\frac{\partial h(\zeta )}{\partial \zeta _{i}}\mu^{-(N-2s)}+\begin{cases}
O\left( \varepsilon ^2\mu(\ln \mu )^2 \right) ,& 2s+1 \lt N\le 4s,\\
O\left( \mu^{-N+1} \right),& 4s \lt N\le 6s
\end{cases}
\end{align}and
\begin{align}
&\int_{\varOmega}{\left( (PU_{\zeta, \mu}+\omega _{\zeta,\mu,\varepsilon})^{2_{s}^{*}-1-\varepsilon}-(PU_{\zeta, \mu })^{2_{s}^{*}-1-\varepsilon} \right)\frac{\partial PU_{\zeta ,\mu}}{\partial \zeta _{i}} dx}\nonumber\\
=&\left( 2_{s}^{*}-1-\varepsilon \right) \int_{\varOmega}{(PU_{\zeta, \mu})^{2_{s}^{*}-2-\varepsilon}\frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}\omega _{\zeta, \mu,\varepsilon}dx}\nonumber\\
&+\begin{cases}
O\left( \int_{\varOmega}{(PU_{\zeta, \mu })^{2_{s}^{*}-3-\varepsilon}\left| \frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}\omega _{\zeta, \mu, \varepsilon}^{2} \right|dx} \right) ,& 2s+1 \lt N \lt 6s,\\
O\left( \int_{\varOmega}{\left| \frac{\partial PU_{\zeta,\mu}}{\partial \zeta _{i}}\omega _{\zeta, \mu,\varepsilon}^{2_{s}^{*}-1-\varepsilon} \right|dx} \right) ,& N=6s
\end{cases}\nonumber\\
=&\left( 2_{s}^{*}-1-\varepsilon \right) \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-2}\frac{\partial PU_{\zeta, \mu}}{\partial \zeta _{i}}\omega _{\zeta,\mu,\varepsilon}dx}+O\left( \varepsilon ^2\mu(\ln \mu)^2 \right)\nonumber\\
=&O\left( \varepsilon ^2\mu(\ln \mu)^2 \right) .
\end{align}Inserting (2.22) and (2.23) into (2.21), we complete the proof (2.19).
Similarly, when 
$\frac{1}{2} \lt s \lt 1$, we have
\begin{align}
&\int_{\varOmega}{\left( U_{\zeta, \mu}^{2_{s}^{*}-1}-(PU_{\zeta,\mu})^{2_{s}^{*}-1-\varepsilon} \right) \frac{\partial PU_{\zeta, \mu}}{\partial \mu}dx}\nonumber\\
=&\int_{\varOmega}{( U_{\zeta, \mu}^{2_{s}^{*}-1}-U_{\zeta, \mu}^{2_{s}^{*}-1-\varepsilon} ) \frac{\partial PU_{\zeta, \mu}}{\partial \mu}dx}+\left( 2_{s}^{*}-1-\varepsilon \right) \int_{\varOmega}{(U_{\zeta, \mu} )^{2_{s}^{*}-2-\varepsilon}\frac{\partial PU_{\zeta, \mu}}{\partial \mu}\psi _{\zeta, \mu}dx}\nonumber\\
&+\begin{cases}
O\left( \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-3}\left| \frac{\partial PU_{\zeta, \mu}}{\partial \mu}\psi _{\zeta,\mu}^{2} \right|dx} \right) ,& 2s+1 \lt N \lt 6s,\\
O\left( \int_{\varOmega}{\left| \frac{\partial PU_{\zeta,\mu}}{\partial \mu}\psi _{\zeta, \mu}^{2_{s}^{*}-1-\varepsilon} \right|dx} \right) ,& N=6s
\end{cases}\nonumber\\
=&\frac{N-2s}{2}\varepsilon \ln \mu\int_{\varOmega}{U_{\zeta,\mu}^{2_{s}^{*}-1}\frac{\partial PU_{\zeta, \mu}}{\partial \mu}dx}-\frac{N-2s}{2}\varepsilon \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-1}\frac{\partial PU_{\zeta,\mu}}{\partial \mu}\ln \left( 1+\mu\left| x-\zeta\right| \right) dx}\nonumber\\
&+\left( 2_{s}^{*}-1-\varepsilon \right) \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-2-\varepsilon}\frac{\partial PU_{\zeta, \mu}}{\partial \mu}\psi _{\zeta, \mu}dx}+\begin{cases}
O\left(\varepsilon ^2\mu^{-1}(\ln \mu )^2\right) ,& 2s+1 \lt N\le 4s,\\
O\left( \mu^{-(N+1)} \right),& 4s \lt N\le 6s
\end{cases}\nonumber\\
=&\frac{(N-2s)^2}{4N}B\varepsilon \mu^{-1}+\left( 2_{s}^{*}-1-\varepsilon \right) \int_{\varOmega}{U_{\zeta, \mu}^{2_{s}^{*}-2}\frac{\partial U_{\zeta, \mu}}{\partial \mu }\psi _{\zeta, \mu}dx}\nonumber\\
&+\begin{cases}
O\left(\varepsilon ^2\mu^{-1}(\ln \mu )^2\right) ,& 2s+1 \lt N\le 4s,\\
O\left( \mu^{-(N+1)} \right),& 4s \lt N\le 6s
\end{cases}\nonumber\\
=&\frac{(N-2s)^2}{4N}B\varepsilon \mu^{-1}+A\left( 2_{s}^{*}-1-\varepsilon \right) \mu^{-\frac{(N-2s)}{2}}\int_{\varOmega}{H\left( x,\zeta \right) U_{\zeta, \mu }^{2_{s}^{*}-2}\frac{\partial U_{\zeta, \mu}}{\partial \mu }dx}\nonumber\\
&+\begin{cases}
O\left( \varepsilon ^2\mu^{-1}(\ln \mu )^2 \right),& 2s+1 \lt N\le 4s,\\
O\left( \mu^{-(N+1)} \right),& 4s \lt N\le 6s
\end{cases}\nonumber\\
=&\frac{(N-2s)^2}{4N}B\varepsilon \mu^{-1}-\frac{N-2s}{2}A^2h(\zeta)\mu^{-(N-2s+1)}+\begin{cases}
O\left( \varepsilon ^2\mu^{-1}(\ln \mu )^2 \right) ,& 2s+1 \lt N\le 4s,\\
O\left( \mu^{-(N+1)} \right),& 4s \lt N\le 6s,
\end{cases}
\end{align}and
\begin{align}
\int_{\varOmega}{\left( (PU_{\zeta, \mu}+\omega _{\zeta, \mu, \varepsilon})^{2_{s}^{*}-1-\varepsilon}-(PU_{\zeta, \mu })^{2_{s}^{*}-1-\varepsilon} \right)\frac{\partial PU_{\zeta, \mu}}{\partial \mu} dx}=O\left( \varepsilon ^2\mu ^{-1}(\ln \mu )^2 \right) .
\end{align}Inserting (2.24) and (2.25) into (2.21), we complete the proof of (2.20).
Proof of Theorem 2.1
It follows from Lemma 2.8 that (2.17) and (2.18) are equivalent to
\begin{align}
\nabla h(\zeta)=\begin{cases}
O\left(\varepsilon ^2\mu^{N-2s+1}(\ln \mu )^2\right),& 2s+1 \lt N\le 4s,\\
O\left(\mu^{1-2s}\right),& 4s \lt N\le 6s\\
\end{cases}
\end{align}and
\begin{align}
\varepsilon =\frac{2NA^2h(\zeta )}{(N-2s)B}\mu^{-(N-2s)} +\begin{cases}
O\left(\varepsilon ^2(\ln \mu )^2\right),& 2s+1 \lt N\le 4s,\\
O\left(\mu^{-N}\right),& 4s \lt N\le 6s,\\
\end{cases}
\end{align} respectively. Since x 0 is a non-degenerate critical point of h(x), we can find 
$\zeta _{\varepsilon}\in B_{\theta _0}(x_0)$ and 
$\mu _{\varepsilon}\in \left[ a_0\varepsilon ^{-\frac{1}{N-2s}},a_1\varepsilon ^{-\frac{1}{N-2s}} \right]$, such that (2.17) and (2.18) hold.
3. Non-degeneracy of single bubbling solutions
In this section, the proof of the non-degeneracy result in Theorem 1.1 is equivalent to the following theorem.
Theorem 3.1. Let 
$\bar{u}_{\varepsilon}\in D_{0}^{s}( \varOmega ) $ be the solution of (2.2) constructed in Theorem 2.1. If 
$\bar{\xi}_{\varepsilon}\in D_{0}^{s}( \varOmega ) $ is a solution of the following problem
\begin{align}
\begin{cases}
Div(x_{N+1}^{1-2s}\nabla \bar{\xi})=0,& in\,\,\mathbb{R}_{+}^{N+1},\\
\bar{\xi}=0,& on\,\,\mathbb{R}^N\setminus \varOmega ,\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim} x_{N+1}^{1-2s}\partial _{\nu} \bar{\xi} =( 2_{s}^{*}-1-\varepsilon ) u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}\xi ,& on\,\,\varOmega ,\\
\end{cases}
\end{align} it has 
$\bar{\xi}_{\varepsilon}=0$.
We argue by contradiction. By the linearity, we suppose that 
$\bar{\xi}_{\varepsilon}\neq0$ is a solution of (3.1) satisfying 
$\left\| \bar{\xi}_{\varepsilon} \right\|_{L^{\infty}} =1$. Denote 
$\xi _{\varepsilon}=\underset{{x_{N+1}\rightarrow 0^+}}{\lim}\bar{\xi}_{\varepsilon}$. Then
\begin{align}
(-\Delta )^s\xi _{\varepsilon}=(2_{s}^{*}-1-\varepsilon )u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}\xi _{\varepsilon}.
\end{align} Note that 
$\bar{\xi}_{\varepsilon}$ is the s-harmonic extension of ξɛ. Therefore, we have
\begin{align}
\bar{\xi}_{\varepsilon}=\beta _{N,s}\int_{\mathbb{R}^N}{\frac{x_{N+1}^{1-2s}}{\left( \left| x-z \right|^2+x_{N+1}^{2} \right) ^{\frac{N+2s}{2}}}\xi _{\varepsilon}(z)dz,}
\end{align} where 
$\beta _{N,s} \gt 0$ is a constant. From (3.3), we know that if 
$\xi_{\varepsilon} \rightarrow 0$, 
$\bar{\xi}_{\varepsilon} \rightarrow 0$, which contradicts with 
$\left\| \bar{\xi}_{\varepsilon} \right\|_{L^{\infty}} =1$.
Therefore, we need to estimate the decay of ξɛ and the behaviour near the concentrating point. Before giving the proof, we will prove the following result.
Lemma 3.2. It holds
\begin{align*}
\bar{u}_{\varepsilon}\le C\bar{U}_{\zeta _{\varepsilon},\mu _{\varepsilon}}.
\end{align*}Proof. Denote 
$\tilde{\bar{u}}_{\varepsilon}=\mu _{\varepsilon}^{-\frac{N-2s}{2}}\bar{u}_{\varepsilon}(\mu _{\varepsilon}^{-1}X+(\zeta _{\varepsilon},0))$, we know it satisfies
\begin{align*}
\begin{cases}
Div(x_{N+1}^{1-2s}\nabla \tilde{\bar{u}}_{\varepsilon})=0,& \text{in}\,\,\mathbb{R}_{+}^{N+1},\\
\tilde{\bar{u}}_{\varepsilon}=0,& \text{on}\,\,\mathbb{R}^N\setminus \varOmega _{\varepsilon},\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim} x_{N+1}^{1-2s}\partial _{\nu}\tilde{\bar{u}}_{\varepsilon}=\mu _{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}\tilde{u}_{\varepsilon}^{2_{s}^{*}-1-\varepsilon},& \text{on}\,\,\varOmega _{\varepsilon},\\
\end{cases}
\end{align*} where 
$\tilde{u}_{\varepsilon}=\mu _{\varepsilon}^{-\frac{N-2s}{2}}u_{\varepsilon}(\mu _{\varepsilon}^{-1}x+\zeta _{\varepsilon})$ and 
$\varOmega _{\varepsilon}=\left\{x:\mu _{\varepsilon}^{-1}x+\zeta _{\varepsilon}\in \varOmega \right\}$. Then we only need to prove
\begin{align*}
\tilde{\bar{u}}_{\varepsilon}\le C\bar{U}_{0,1}.
\end{align*} Notice that 
$\tilde{\bar{u}}_{\varepsilon}$ is the extension of 
$\tilde{u}_{\varepsilon}$, so we only need to prove
\begin{align*}
\tilde{u}_{\varepsilon}\le CU_{0,1}.
\end{align*}On the one hand, for any small δ > 0, there is r > 0 small, such that
\begin{align*}
& \int_{B_r(z)}{\left| \mu _{\varepsilon}^{-\frac{N-2s}{2}}\tilde{u}_{\varepsilon}^{2_{s}^{*}-2-\varepsilon} \right|^{\frac{N}{2s}}}dx \\& \quad \le C\int_{B_r(z)}{\mu _{\varepsilon}^{-\frac{(N-2s)N}{4s}}U_{0,1}^{2_{s}^{*}-\frac{N}{2s}\varepsilon}}dx+C\int_{B_{\mu _{\varepsilon}^{-1}r}(\mu _{\varepsilon}^{-1}z+\zeta _{\varepsilon})}{|\omega _{\varepsilon}|^{2_{s}^{*}-\frac{N}{2s}\varepsilon}dx}
\\
& \quad \le C\left( \int_{B_r(z)}{U_{0,1}^{2_{s}^{*}}}dx \right) ^{\frac{2_{s}^{*}-\frac{N}{2s}\varepsilon}{2_{s}^{*}}}+C\left( \int_{\mathbb{R}^N}{|\omega _{\varepsilon}|^{2_{s}^{*}}dx} \right) ^{\frac{2_{s}^{*}-\frac{N}{2s}\varepsilon}{2_{s}^{*}}}
\\
& \quad \le \delta,
\end{align*} for any 
$B_{r}(z)\subset\varOmega _{\varepsilon}$. By Lemma C.2 in [Reference Guo, Li, Pistoia and Yan10], we obtain that 
$\tilde{u}_{\varepsilon}$ is bounded in 
$\left\{x\in \mathbb{R}^N:\left| x \right|\leq 2 \right\} $, which gives that 
$\tilde{u}_{\varepsilon}\le CU_{0,1}$ in 
$\left\{x\in \mathbb{R}^N:\left| x \right|\leq 2 \right\}$.
On the other hand, we use the Kelvin transformation 
$v(X)=\frac{1}{\left| X \right|^{N-2s}}\tilde{\bar{u}}_{\varepsilon}(\frac{X}{\left| X \right|^2})$. Then v satisfies
\begin{align*}
\begin{cases}
Div(x_{N+1}^{1-2s}\nabla v)=0,& \text{in} \,\,\mathbb{R}_{+}^{N+1},\\
v=0,& \text{on}\,\,\mathbb{R}^N\setminus D_{\varepsilon},\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim} x_{N+1}^{1-2s}\partial _{\nu}(v)=\mu _{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}\left| x \right|^{-(N-2s)\varepsilon}v^{2_{s}^{*}-1-\varepsilon},& \text{on}\,\,D_{\varepsilon},\\
\end{cases}
\end{align*} where 
$D_{\varepsilon}=\left\{x:\frac{x}{\left| x \right|^2}\in \varOmega _{\varepsilon} \right\}$.
We assert that if 
$\left| X \right|\le \frac{1}{2}$,
\begin{align}
v\le C.
\end{align} That is to say, when 
$\left| X \right|\le \frac{1}{2}$,
\begin{align*}
\tilde{\bar{u}}_{\varepsilon}\left( \frac{X}{\left| X \right|^2} \right) \le C\left| X \right|^{N-2s}.
\end{align*} As a result, when 
$\left| y \right| \geq 2$,
\begin{align*}
\tilde{u}_{\varepsilon}(y)\le C\frac{1}{\left| y \right|^{N-2s}}\le CU_{0,1}.
\end{align*}To complete the proof, we need to check (3.4). For any small δ > 0, there is r > 0 small, such that
\begin{align*}
\int_{B_r(0)}{\left| \mu _{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}\left| x \right|^{-(N-2s)\varepsilon}\left( v(x,0) \right) ^{2_{s}^{*}-2-\varepsilon} \right|^{\frac{N}{2s}}dx}\le& C\int_{B_r(0)}{\left| x \right|^{-2N}\left( \tilde{u}_{\varepsilon}(\frac{x}{\left| x \right|^2}) \right) ^{2_{s}^{*}-\frac{N}{2s}\varepsilon}dx}
\\
\le& C\int_{\mathbb{R}^N\setminus B_{\frac{1}{r}}(0)}{\left( \tilde{u}_{\varepsilon}(y) \right) ^{2_{s}^{*}-\frac{N}{2s}\varepsilon}dy}
\\
\le& \delta,
\end{align*}then applying Lemma C.2 in [Reference Guo, Li, Pistoia and Yan10], we obtain
\begin{align*}
\left\| v(x,0) \right\| _{L^{\infty}(B_{\frac{1}{2}}(0))}\le C\bigg(
\int_{\mathcal{B}_{1}^{+}(0)}{x_{N+1}^{1-2s}v^2\,dX} \bigg) ^{\frac{1}{2}}.
\end{align*}Moreover, by the Lemma B.1 in [Reference Yan, Yang and Yu17], we have
\begin{align*}
\int_{\mathcal{B}_{1}^{+}(0)}{x_{N+1}^{1-2s}v^2\, dX}\le C\int_{\mathcal{B}_{1}^{+}(0)}{x_{N+1}^{1-2s}\frac{v^2}{\left| X \right|^2}dX}{\le C\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}|\nabla v|^2dX}}\le C,
\end{align*}which implies (3.4).
Now, we estimate the decay of ξɛ.
Lemma 3.3. It holds
\begin{align*}
\left| \xi _{\varepsilon}(x) \right|\le \frac{C}{(1+\mu _{\varepsilon}\left| x-\zeta _{\varepsilon} \right|)^{N-2s}}.
\end{align*}Proof. By Lemma 3.2, we have
\begin{align}
\left| \xi _{\varepsilon}(x) \right|&= \left|(2_{s}^{*}-1-\varepsilon )\int_{\varOmega}{G(x,y)u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\xi _{\varepsilon}(y)dy}\right|
\nonumber\\
&\le C\int_{\varOmega}{\frac{1}{\left| x-y \right|^{N-2s}}\frac{\mu _{\varepsilon}^{2s}}{(1+\mu _{\varepsilon}^{2}\left| y-\zeta _{\varepsilon} \right|^2)^{2s}}dy}\nonumber\\
&\le C\int_{\mu _{\varepsilon}(\varOmega -\zeta _{\varepsilon})}{\frac{1}{\left| \mu _{\varepsilon}(x-\zeta _{\varepsilon})-y \right|^{N-2s}}\frac{1}{(1+\left| y \right|)^{4s}}dy}\nonumber\\
&\le \frac{C}{(1+\mu _{\varepsilon}\left| x-\zeta _{\varepsilon} \right|)^{2s}}.
\end{align}From (3.5), we obtain
\begin{align*}
\left| \xi _{\varepsilon}(x) \right|&\le C\int_{\varOmega}{\frac{1}{\left| x-y \right|^{N-2s}}\frac{\mu _{\varepsilon}^{2s}}{(1+\mu _{\varepsilon}\left| y-\zeta _{\varepsilon} \right|)^{6s}}dy}\\
&\le \frac{C}{(1+\mu _{\varepsilon}\left| x-\zeta _{\varepsilon} \right|)^{4s}}.
\end{align*}By the iteration process, we can prove
\begin{align*}
\left| \xi _{\varepsilon}(x) \right|\le \frac{C}{(1+\mu _{\varepsilon}\left| x-\zeta _{\varepsilon} \right|)^{N-2s}}.
\end{align*} From Lemma 3.3, we know that 
$\left| \xi _{\varepsilon}(x) \right|\ll 1$ when 
$x\in \varOmega \setminus B_{R\mu _{\varepsilon}^{-1}}(\zeta _{\varepsilon})$ and R > 0 is large. To obtain a contradiction, we only need to prove that 
$\left| \xi _{\varepsilon}(x) \right|\ll 1$ when 
$x\in B_{R\mu _{\varepsilon}^{-1}}(\zeta _{\varepsilon})$. For this purpose, we let 
$\dot{\tilde{\xi}}_{\varepsilon}(x)=\xi _{\varepsilon}(\mu _{\varepsilon}^{-1}x+\zeta _{\varepsilon})$.
Lemma 3.4. It holds
\begin{align*}
\dot{\tilde{\xi}}_{\varepsilon}(x)\rightarrow \sum_{j=1}^N{b_j\frac{\partial U_{0,1}}{\partial x _j}}+b_{N+1}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg|_{\mu =1}\,\,\, as\,\, \varepsilon \rightarrow 0,
\end{align*} uniformly in 
$C_{loc}^1(\mathbb{R}^N)$, where bj 
$(j=1,2,\cdots,N+1)$ is a constant.
Proof. We can see that
\begin{align*}
(-\Delta )^s\dot{\tilde{\xi}}_{\varepsilon}=(2_{s}^{*}-1-\varepsilon )\mu _{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}\tilde{u}_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}\dot{\tilde{\xi}}_{\varepsilon}, \,\, \text{in} \,\, \varOmega _{\varepsilon},
\end{align*} where 
$\tilde{u}_{\varepsilon}=\mu _{\varepsilon}^{-\frac{N-2s}{2}}u_{\varepsilon}(\mu _{\varepsilon}^{-1}x+\zeta _{\varepsilon})$ and 
$\varOmega _{\varepsilon}=\left\{x:\mu _{\varepsilon}^{-1}x+\zeta _{\varepsilon}\in \varOmega \right\}$.
Let 
$G_{\varepsilon}(y,x)$ be the Green’s function 
$(-\Delta )^s$ in 
$\varOmega _{\varepsilon}$. Then
\begin{align*}
\dot{\tilde{\xi}}_{\varepsilon}(x)=\int_{\varOmega _{\varepsilon}}{G_{\varepsilon}(y,x)\left( (2_{s}^{*}-1-\varepsilon )\mu _{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}\tilde{u}_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\dot{\tilde{\xi}}_{\varepsilon}(y) \right)}dy,
\end{align*}and
\begin{align}
\nabla \dot{\tilde{\xi}}_{\varepsilon}(x)=\int_{\varOmega _{\varepsilon}}{\nabla _xG_{\varepsilon}(y,x)\left( (2_{s}^{*}-1-\varepsilon )\mu _{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}\tilde{u}_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\dot{\tilde{\xi}}_{\varepsilon}(y) \right)}dy.
\end{align} Since 
$\left\| \dot{\tilde{\xi}}_{\varepsilon} \right\| _{L^{\infty}}\le 1$, 
$s \gt \frac{1}{2}$, using Lemma A.2 in [Reference Guo, Li, Pistoia and Yan10], we obtain from (3.6) that 
$\nabla \dot{\tilde{\xi}}$ is uniformly bounded in any bounded domain in 
$\mathbb{R}^N$. So we can assume that 
$\dot{\tilde{\xi}}_{\varepsilon}\rightarrow \xi _0$ in 
$C(B_{R}(0))$ for any R > 0. Using (3.6) again, we obtain that 
$\dot{\tilde{\xi}}_{\varepsilon}\rightarrow \xi _0$ in 
$C^{1}(B_{R}(0))$ for any R > 0.
It is easy to prove that ξ 0 solves
\begin{align*}
(-\Delta )^s\xi _0=(2_{s}^{*}-1)U_{0,1}^{2_{s}^{*}-2}\xi _0\,\, \text{in}\,\, \mathbb{R}^N.
\end{align*} By the non-degeneracy of 
$U_{0,1}$, we obtain that
\begin{align*}
\xi _0=\sum_{j=1}^N{b_j\frac{\partial U_{0,1}}{\partial x _j}}+b_{N+1}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg| _{\mu =1},
\end{align*} where bj 
$(j=1,2,\cdots,N+1)$ is a constant.
By Lemma 3.4, to prove 
$\left| \xi _{\varepsilon}(x) \right|\ll 1$ when 
$x\in B_{R\mu _{\varepsilon}^{-1}}(\zeta _{\varepsilon})$, we just have to prove that 
$b_j=0$, 
$j=1,2,\cdots,N+1$. In order to prove it using Pohozaev identities, we need to give some essential estimates.
In the following, denote 
$B_r(\zeta )=\{x\in \mathbb{R}^N,\left| x-\zeta \right| \lt r\}$, 
$\mathcal{B}_r(\zeta )=\{X\in \mathbb{R}^{N+1},\left| X-(\zeta ,0) \right| \lt r\}$, 
$\mathcal{B}_{r}^{+}(\zeta )=\mathcal{B}_r(\zeta )\cap \mathbb{R}_{+}^{N+1}$, and 
$\partial ^{\prime} \mathcal{B}_{r}^{+}(\zeta )=\partial \mathcal{B}_r(\zeta )\cap \mathbb{R}_{+}^{N+1}$.
Lemma 3.5. For any 
$X \in \partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta_{\varepsilon})$, we have
\begin{align}
\bar{u}_{\varepsilon}(X)=A_{\varepsilon} \bar{G}(X,\zeta _{\varepsilon})+O\left(\mu _{\varepsilon}^{-\frac{N+2s}{2}}\right),
\end{align}and
\begin{align}
\nabla \bar{u}_{\varepsilon}(X)=A_{\varepsilon}\nabla \bar{G}(X,\zeta _{\varepsilon})+O\left(\mu _{\varepsilon}^{-\frac{N+2s}{2}}\right),
\end{align} where 
$\bar{G}(X,\zeta)$ is the harmonic extension of 
$G(X,\zeta)$ and
\begin{align*}
A_{\varepsilon}=\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}.
\end{align*}Proof. By Green’s function, we see that
\begin{equation}
\begin{aligned}
\bar{u}_{\varepsilon}(x)=&\int_{\varOmega}{\bar{G}(X,y)u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}\\
=&\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\bar{G}(X,y)u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}+\int_{\varOmega \setminus B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\bar{G}(X,y)u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}\\
:=&F_1+F_2.
\end{aligned}
\end{equation}For F 1, we have
\begin{equation}
\begin{aligned}
F_1=&\bar{G}(X,\zeta _{\varepsilon})\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}+\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left \lt \partial _y\bar{G}(X,\zeta _{\varepsilon}),y-\zeta _{\varepsilon} \right \gt u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}\\
&+O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left| D_y^2\bar{G}(X,\zeta ) \right|\left| y-\zeta _{\varepsilon} \right|^2u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy} \bigg).
\end{aligned}
\end{equation}By Lemma A.1 in [Reference Guo, Li, Pistoia and Yan10], we can obtain that
\begin{align}
&\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left \lt \partial _y\bar{G}(X,\zeta _{\varepsilon}),y-\zeta _{\varepsilon} \right \gt u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}\nonumber\\
=&O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left| y-\zeta _{\varepsilon} \right|U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-2}(\left| \psi _{\zeta _{\varepsilon},\mu _{\varepsilon}} \right|+\left| \omega _{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right|)dy} \bigg)\nonumber\\
=&O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right) +\begin{cases}
O\left( \mu _{\varepsilon}^{-2s}\left\| \bar{\omega}_{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right\|_{D_{0}^{s}( \varOmega )} \right) ,\,& \frac{1}{2} \lt s\le \frac{3}{4},2s+1 \lt N\le 6s,\nonumber\\
O\left( \mu _{\varepsilon}^{-2s}\left\| \bar{\omega}_{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right\|_{D_{0}^{s}( \varOmega )} \right) ,\,& \frac{3}{4} \lt s \lt 1,6s-2 \lt N\le 6s,\\
O\left( \mu _{\varepsilon}^{-2s}(\ln \mu _{\varepsilon})^{\frac{N+2s}{2N}}\left\| \bar{\omega}_{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right\|_{D_{0}^{s}( \varOmega )} \right) ,\,& \frac{3}{4} \lt s \lt 1,N=6s-2,\\
O\left( \mu _{\varepsilon}^{-\frac{N-2s}{2}-1}\left\| \bar{\omega}_{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right\|_{D_{0}^{s}( \varOmega )} \right) ,\,& \frac{3}{4} \lt s \lt 1,2s+1 \lt N \lt 6s-2\\
\end{cases}\nonumber\\
=&O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right).
\end{align} Moreover, for 
$X \in \partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta_{\varepsilon})$, we have
\begin{equation}
\begin{aligned}
&\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left| D_{y}^{2}\bar{G}(X,\zeta ) \right|\left| y-\zeta _{\varepsilon} \right|^2u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}\\
=&O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left| y-\zeta _{\varepsilon} \right|^2U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-1}(y)dy} \bigg)\\
=&O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right).
\end{aligned}
\end{equation}Meanwhile, we can verify that
\begin{align}
|F_2|\le \int_{\varOmega \setminus B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\bar{S}(X,y)U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-1}(y)dy}=O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right).
\end{align}Therefore, by (3.9)-(3.13), we conclude (3.7).
Similarly, we have
\begin{align}
\nabla \bar{u}_{\varepsilon}(x)=&\int_{\varOmega}{\nabla_X \bar{G}(X,y)u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}\nonumber\\
=&\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\nabla_X\bar{G}(X,y)u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}+\int_{\varOmega \setminus B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\bar{G}(X,y)u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}\nonumber\\
=&\nabla_X\bar{G}(X,\zeta _{\varepsilon})\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}+\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left \lt \nabla_X \partial _y\bar{G}(X,\zeta _{\varepsilon}),y-\zeta _{\varepsilon} \right \gt u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}\nonumber\\
&+O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left| y-\zeta _{\varepsilon}\right|^2u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy} \bigg)+O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right)\nonumber\\
=&A_{\varepsilon}\nabla_X\bar{G}(X,\zeta _{\varepsilon})+O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right),
\end{align}where
\begin{align*}
A_{\varepsilon}=\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}.
\end{align*}We can estimate Aɛ as follow,
\begin{align}
A_{\varepsilon}=&\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-1-\varepsilon}dy}+O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-2-\varepsilon}(\left| \psi _{\zeta _{\varepsilon},\mu _{\varepsilon}} \right|+\left| \omega _{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right|)dy} \bigg)\nonumber\\
=&\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-1}dy}+O\left( \varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}}\ln \mu _{\varepsilon} \right) \nonumber\\
&+\left\{\begin{array}{ll}
O\left( \mu _{\varepsilon}^{-\frac{3(N-2s)}{2}} \right) +O\left( \varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}}\ln \mu _{\varepsilon} \right) ,& 2s+1 \lt N \lt 4s,\\
O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}}\ln \mu _{\varepsilon} \right) +O\left( \varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}}\ln \mu _{\varepsilon} \right) ,& N=4s,\\
O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right) +O\left( \varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}}\ln \mu _{\varepsilon} \right) ,& 4s \lt N \lt 6s,\\
O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right) +O\left( \varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}}(\ln \mu _{\varepsilon})^{\frac{5}{3}} \right) ,& N=6s
\end{array} \right.\nonumber\\
=&\mu _{\varepsilon}^{-\frac{N-2s}{2}}A+\begin{cases}
O\left( \varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}}\ln \mu _{\varepsilon} \right) ,& 2s+1 \lt N\le 4s,\\
O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right) ,& 4s \lt N\le 6s.
\end{cases}
\end{align}Lemma 3.6. For any 
$X \in \partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta_{\varepsilon})$, we have
\begin{align}
\bar{\xi}_{\varepsilon}(X)=B_{\varepsilon} \bar{G}(X,\zeta _{\varepsilon})+\sum_{i=1}^N{\partial _{\zeta _{\varepsilon ,i}}\bar{G}(X,\zeta _{\varepsilon})C_{i,\varepsilon}}+O\left( \mu _{\varepsilon}^{-N} \right),
\end{align}and
\begin{align}
\nabla \bar{\xi}_{\varepsilon}(X)=B_{\varepsilon}\nabla \bar{G}(X,\zeta _{\varepsilon})+\sum_{i=1}^N{\nabla \partial _{\zeta _{\varepsilon ,i}}\bar{G}(X,\zeta _{\varepsilon})C_{i,\varepsilon}}+O\left( \mu _{\varepsilon}^{-N} \right),
\end{align}where
\begin{align*}
B_{\varepsilon}=(2_{s}^{*}-1-\varepsilon )\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\xi _{\varepsilon}(y)dy},
\end{align*}and
\begin{align*}
C_{i,\varepsilon}=(2_{s}^{*}-1-\varepsilon )\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{(y_i-\zeta _{\varepsilon ,i})u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\xi _{\varepsilon}(y)dy}.
\end{align*}Proof. First, we see that
\begin{align}
\bar{\xi}_{\varepsilon}(X)=&(2_{s}^{*}-1-\varepsilon )\int_{\varOmega}{\bar{G}(X,y)u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\xi _{\varepsilon}(y)dy}\nonumber\\
=&(2_{s}^{*}-1-\varepsilon )\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\bar{G}(X,y)u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\xi _{\varepsilon}(y)dy}\nonumber\\ & +O\bigg( \int_{\varOmega \setminus B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\bar{G}(X,y)u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\xi _{\varepsilon}(y)dy} \bigg)\nonumber\\
:=&J_1+J_2.
\end{align}For J 2,
\begin{align}
J_2=O\bigg( \int_{\varOmega \setminus B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\frac{1}{\left| X-(y,0) \right|^{N-2s}}\frac{\mu _{\varepsilon}^{2s}}{(1+\mu _{\varepsilon}\left| y-\zeta _{\varepsilon} \right|)^{N+2s}}dy} \bigg) =O\left( \mu _{\varepsilon}^{-N} \right).
\end{align}For J 1,
\begin{align}
J_1=&B_{\varepsilon}\bar{G}(X,\zeta _{\varepsilon})+\sum_{i=1}^N{\partial _{\zeta _{\varepsilon ,i}}\bar{G}(X,\zeta _{\varepsilon})C_{i,\varepsilon}}+O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left| y-\zeta _{\varepsilon} \right|^2u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\xi _{\varepsilon}(y)dy} \bigg)\nonumber\\
=&B_{\varepsilon}\bar{G}(X,\zeta _{\varepsilon})+\sum_{i=1}^N{\partial _{\zeta _{\varepsilon ,i}}\bar{G}(X,\zeta _{\varepsilon})C_{i,\varepsilon}}+O\left( \mu _{\varepsilon}^{-N} \right),
\end{align}where
\begin{align*}
B_{\varepsilon}=(2_{s}^{*}-1-\varepsilon )\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\xi _{\varepsilon}(y)dy}
\end{align*}and
\begin{align*}
C_{i,\varepsilon}=(2_{s}^{*}-1-\varepsilon )\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{(y_i-\zeta _{\varepsilon ,i})u_{\varepsilon}^{2_{s}^{*}-2-\varepsilon}(y)\xi _{\varepsilon}(y)dy}.
\end{align*}Inserting (3.19) and (3.20) into (3.18), we conclude (3.16). Similarly, we can get (3.17).
Now, we can estimate Bɛ and 
$C_{i,\varepsilon}$ as follow,
\begin{align}
B_{\varepsilon}=&(2_{s}^{*}-1-\varepsilon )\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-2-\varepsilon}\xi _{\varepsilon}dy}\nonumber\\
&+\begin{cases}
O\left( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-3-\varepsilon}\left| \xi _{\varepsilon} \right|(\left| \psi _{\zeta _{\varepsilon},\mu _{\varepsilon}} \right|+\left| \omega _{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right|)dy} \right) ,& 2s+1 \lt N \lt 6s,\\
O\left( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left| \xi _{\varepsilon} \right|(\left| \psi _{\zeta _{\varepsilon},\mu _{\varepsilon}} \right|^{2_{s}^{*}-2-\varepsilon}+\left| \omega _{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right|^{2_{s}^{*}-2-\varepsilon})dy} \right) ,& N=6s
\end{cases}\nonumber\\
=&(2_{s}^{*}-1-\varepsilon )(1+O(\varepsilon \ln \mu _{\varepsilon}))\mu _{\varepsilon}^{-(N-2s)}\int_{B_{\frac{r{\mu_{\varepsilon}}}{4}}(0)}{U_{0,1}^{2_{s}^{*}-2}\dot{\tilde{\xi}}_{\varepsilon}dy} +O\left( \varepsilon \mu _{\varepsilon}^{-(N-2s)}\ln \mu _{\varepsilon} \right)\nonumber\\
=&(2_{s}^{*}-1)b_{N+1}\mu _{\varepsilon}^{-(N-2s)}\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg|_{\mu =1}dy}(1+o(1))
\end{align}and
\begin{align}
C_{i,\varepsilon}=&(2_{s}^{*}-1-\varepsilon )\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{(y_i-\zeta _{\varepsilon ,i})U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-2-\varepsilon}\xi _{\varepsilon}dy}\nonumber\\
&+\begin{cases}
O\left( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left| y_i-\zeta _{\varepsilon ,i} \right|U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-3-\varepsilon}\left| \xi _{\varepsilon} \right|(\left| \psi _{\zeta _{\varepsilon},\mu _{\varepsilon}} \right|+\left| \omega _{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right|)dy} \right) ,& \!\!\!\!2s+1 \lt N \lt 6s,\\
O\left( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{\left| y_i-\zeta _{\varepsilon ,i} \right|\left| \xi _{\varepsilon} \right|(\left| \psi _{\zeta _{\varepsilon},\mu _{\varepsilon}} \right|^{2_{s}^{*}-2-\varepsilon}+\left| \omega _{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right|^{2_{s}^{*}-2-\varepsilon})dy} \right) ,& N=6s
\end{cases}\nonumber\\
=&(2_{s}^{*}-1)b_i\mu _{\varepsilon}^{-(N-2s+1)}\int_{\mathbb{R}^N}{y_iU_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial y_i}dy}(1+o(1)).
\end{align}Proof of Theorem 3.1
Firstly, we prove that 
$b_{N+1}=0$ by means of Lemma B.3.
\begin{align}
\mathrm{LHS}\, \mathrm{of}\, {eqB.3}=&\frac{N-2s}{2}\varepsilon \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-1-\varepsilon}\xi _{\varepsilon}dy}\nonumber\\
&+O\bigg( \varepsilon \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-2-\varepsilon}\left| \xi _{\varepsilon} \right|(\left| \psi _{\zeta _{\varepsilon},\mu _{\varepsilon}} \right|+\left| \omega _{\varepsilon ,\zeta _{\varepsilon},\mu _{\varepsilon}} \right|)dy} \bigg)\nonumber\\
=&\frac{N-2s}{2}\varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}}(1+O(\varepsilon \ln \mu _{\varepsilon}))\int_{B_{\mu _{\varepsilon}\frac{r}{4}}(0)}{U_{0,1}^{2_{s}^{*}-1}\dot{\tilde{\xi}}_{\varepsilon}dy}+o\left( \varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}} \right)\nonumber\\
=&o\left( \varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}} \right).
\end{align}By Lemma 3.5 and Lemma 3.6, we have
\begin{equation}
\begin{aligned}
\mathrm{RHS}\, \mathrm{of}\, {eqB.3}=&2A_{\varepsilon}B_{\varepsilon}\bigg\{\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{G}(X,\zeta _{\varepsilon})\left \lt \nabla \bar{G}(X,\zeta _{\varepsilon}),X-(\zeta _{\varepsilon},0) \right \gt dS}\\
&-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{G}(X,\zeta _{\varepsilon}) \right|^2\left \lt X-(\zeta _{\varepsilon},0),\nu \right \gt dS}\\ &+\frac{N-2s}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\bar{G}(X,\zeta _{\varepsilon})\partial _{\nu}\bar{G}(X,\zeta _{\varepsilon})dS}\bigg\}+O\left( \mu _{\varepsilon}^{-\frac{3(N-2s)}{2}-1} \right).
\end{aligned}
\end{equation}Thus, by Lemma A.2, (B.3) is equivalent to
\begin{align*}
(N-2s)A_{\varepsilon}B_{\varepsilon}h(\zeta _{\varepsilon})=o(\mu _{\varepsilon}^{-\frac{3(N-2s)}{2}}).
\end{align*}From the estimates of Aɛ and Bɛ in (3.15) and (3.21), we obtain that
\begin{equation}
\begin{aligned}
(N-2s)(2_{s}^{*}-1)A b_{N+1}{h(x_0)}\mu _{\varepsilon}^{-\frac{3(N-2s)}{2}}\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg|_{\mu =1}dy}(1+o(1))=o\left(\mu _{\varepsilon}^{-\frac{3(N-2s)}{2}}\right).
\end{aligned}
\end{equation}Since
\begin{align*}
& (2_{s}^{*}-1)\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg|_{\mu =1}dy}=\frac{\partial}{\partial \mu}\left( \int_{\mathbb{R}^N}{U_{0,\mu}^{2_{s}^{*}-1}dy} \right){\bigg|_{\mu =1}} \\ & \quad =\frac{\partial}{\partial \mu}\left( \mu ^{-\frac{N-2s}{2}}\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-1}dy} \right){\bigg|_{\mu =1}} \ne 0,
\end{align*} from (3.25), we obtain that 
$b_{N+1}=0$.
Next, we will prove that 
$b_i=0$, 
$i=1,2,\cdots,N$. By (B.7) and Lemma 3.5, we have
\begin{align*}
& A_{\varepsilon}^{2}\bigg\{\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta_\varepsilon )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{G}(X,\zeta _{\varepsilon})\partial _{x_i}\bar{G}(X,\zeta _{\varepsilon})dS}-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta_\varepsilon )}{x_{N+1}^{1-2s}\left| \nabla \bar{G}(X,\zeta _{\varepsilon}) \right|^2\nu _idS}\bigg\} \\ & \quad =O\left( \mu _{\varepsilon}^{-N} \right).
\end{align*}Using Lemma A.1, we obtain that
\begin{align*}
\frac{1}{2}A_{\varepsilon}^{2}\frac{\partial h(\zeta _{\varepsilon})}{\partial \zeta _{\varepsilon ,i}}=O\left( \mu _{\varepsilon}^{-N} \right).
\end{align*}Moreover, by (3.15), we have
\begin{align}
\frac{\partial h(\zeta _{\varepsilon})}{\partial \zeta _{\varepsilon ,i}}=O\left( \mu _{\varepsilon}^{-2s} \right).
\end{align}On the other hand, by means of Lemma 3.5 and Lemma 3.6, we can calculate (B.4) as follow,
\begin{align}
\mathrm{LHS}\, \mathrm{of}\, {eqB.4}=O\left( \mu _{\varepsilon}^{-\frac{3}{2}N+s} \right)
\end{align}and
\begin{align}
\mathrm{RHS}\, \mathrm{of}\, {eqB.4}=& 2A_{\varepsilon}B_{\varepsilon}\bigg\{\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{G}(X,\zeta _{\varepsilon})\nabla _{x_i}\bar{G}(X,\zeta _{\varepsilon})dS}\nonumber\\
& -\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| {\nabla}\bar{G}(X,\zeta _{\varepsilon}) \right|^2\nu _idS}\bigg\}\nonumber\\
&+\sum_{j=1}^N{A_{\varepsilon}C_{j,\varepsilon}} \bigg\{\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\partial _{\zeta _{\varepsilon ,j}}\bar{G}(X,\zeta _{\varepsilon})\nabla _{x_i}\bar{G}(X,\zeta _{\varepsilon})dS}\nonumber\\
&+\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{G}(X,\zeta _{\varepsilon})\nabla _{x_i}\partial _{\zeta _{\varepsilon ,j}}\bar{G}(X,\zeta _{\varepsilon})dS}\nonumber\\
&-\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{G}(X,\zeta _{\varepsilon})\nabla \partial _{\zeta _{\varepsilon ,j}}\bar{G}(X,\zeta _{\varepsilon})\nu _idS}\bigg\}\nonumber\\
&+O\left( \mu _{\varepsilon}^{-\frac{3}{2}N+s} \right).
\end{align}Using Lemmas A.1 and A.3, we obtain that (B.4) is equivalent to
\begin{align}
A_{\varepsilon}B_{\varepsilon}\frac{\partial h(\zeta _{\varepsilon})}{\partial \zeta _{\varepsilon ,i}}+\frac{1}{2}\sum_{j=1}^N{A_{\varepsilon}C_{j,\varepsilon}\frac{\partial ^2h(x)}{\partial x_i\partial x_j}\bigg| _{x=\zeta _{\varepsilon}}}=O\left( \mu _{\varepsilon}^{-\frac{3}{2}N+s} \right).
\end{align} It follows from (3.15), (3.21), (3.22) and (3.26) that, for 
$i=1,\cdots,N$,
\begin{equation*}\begin{array}{rlrlrlrlrlrl}\frac{2_s^\ast-1}2A\mu_\varepsilon^{-\frac{3(N-2s)}2-1}\sum_{j=1}^N{b_j\int_{\mathbb{R}^N}y_jU_{0,1}^{2_s^\ast-2}\frac{\partial U_{0,1}}{\partial y_j}dy\frac{\partial^2h(x)}{\partial x_i\partial x_j}\vert_{x=x_0}(1+o(1))}=O{(\mu_\varepsilon^{-\frac32N+s})}.\end{array}\end{equation*}That is to say,
\begin{equation}\begin{array}{rlrlrlrlrlrl}\sum_{j=1}^Nb_j\frac{\partial^2h(x)}{\partial x_i\partial x_j}\vert_{x=x_0}=0,{}i=1,2,\cdots,N.\end{array}\end{equation}Since x 0 is a non-degenerate critical point of h(x), from (3.30), we conclude that
\begin{align*}
b_i=0, \,\,\,i=1,2,\cdots,N.
\end{align*}Therefore, we complete the proof of Theorem 3.1.
4. Local uniqueness of single bubbling solutions
In this section, we will discuss the local uniqueness of the equation (2.2). By Theorem 2.1, the equation (2.2) has a solution of the following form
\begin{align}
\bar{u}_{\varepsilon}=\bar{P}_{\zeta _{\varepsilon},\mu _{\varepsilon}}+\bar{\omega}_{\varepsilon},
\end{align} which satisfies that as 
$\varepsilon \rightarrow0$,
\begin{align}
\zeta _{\varepsilon}\rightarrow x_0\in \varOmega, \,\mu _{\varepsilon}\rightarrow +\infty, \,\left\| \bar{\omega}_{\varepsilon}\right\|_{D_{0}^{s}(\varOmega )} \rightarrow 0.
\end{align}We will prove the following result.
Theorem 4.1. Assume that 
$\frac{1}{2} \lt s \lt 1$ and 
$2s+1 \lt N \lt 6s$. Suppose that (2.2) has solutions of the form (4.1), satisfying (4.2). If x 0 is a non-degenerate critical point of the Robin function h(x), there exists 
$\varepsilon _0 \gt 0$, such that for any 
$\varepsilon \in (0,\varepsilon _0)$, 
$\bar{u}_{\varepsilon}$ is unique.
At first, we can get a rough estimate for µɛ.
Lemma 4.2. It holds 
$\left(\mu_{\varepsilon}\right)^{\varepsilon}\rightarrow1$, as 
$\varepsilon \rightarrow0$. That is, 
$\varepsilon ln\mu_{\varepsilon} \rightarrow0$, as 
$\varepsilon \rightarrow0$.
Proof. Since 
$\bar{u}_{\varepsilon}$ is the solution of (2.2), we have
\begin{align*}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\left| \nabla \bar{u}_{\varepsilon} \right|^2dX}=\int_{\varOmega}{u_{\varepsilon}^{2_{s}^{*}-\varepsilon}dx}.
\end{align*}From (4.1) and (4.2), we can calculate
\begin{align*}
\int_{\mathbb{R}_{+}^{N+1}}{x_{N+1}^{1-2s}\left| \nabla \bar{u}_{\varepsilon} \right|^2dX}=\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}}dx}+o(1),
\end{align*}and
\begin{align*}
\int_{\varOmega}{u_{\varepsilon}^{2_{s}^{*}-\varepsilon}dx}=&\int_{\varOmega}{U_{\zeta _{\varepsilon},\mu _{\varepsilon}}^{2_{s}^{*}-\varepsilon}dx}+o(1)\\
=&\mu _{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}C_{N,s}^{2_{s}^{*}-\varepsilon}\int_{\mu _{\varepsilon}(\varOmega -\zeta _{\varepsilon})}{\frac{1}{(1+\left| z \right|^2)^{N-\frac{N-2s}{2}\varepsilon}}dz}+o(1)\\
=&\mu _{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}C_{N,s}^{2_{s}^{*}}(1+o(1))\int_{\mathbb{R}^N}{\frac{1}{(1+\left| z \right|^2)^N}dz}+o(1)\\
=&\mu _{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}}dx}(1+o(1))+o(1).
\end{align*}Thus,
\begin{align*}
\mu_{\varepsilon}^{-\frac{N-2s}{2}\varepsilon}\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}}dx}(1+o(1))=\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}}dx}+o(1).
\end{align*} Letting 
$\varepsilon\rightarrow0$, we obtain that 
$\left(\mu_{\varepsilon}\right)^{\varepsilon}\rightarrow1$.
After we have Lemma 4.2, since 
$\bar{\omega}_{\zeta_{\varepsilon},\mu_{\varepsilon}} \in E_{\bar{\zeta}_{\varepsilon},\bar{\mu}_{\varepsilon},\varepsilon}$, we can use similar arguments to those in Lemmas 2.4-2.6, and Proposition 2.7, to deduce the following estimate for 
$\|\bar{\omega}_{\zeta_{\varepsilon},\mu_{\varepsilon}}\|_{D_{0}^{s}(\varOmega )}$:
\begin{align*}
\left\| \bar{\omega}_{\zeta ,\mu ,\varepsilon} \right\| _{D_{0}^{s}(\varOmega )}\le C\left( \mu _{\varepsilon}^{-(N-2s)}+\varepsilon \ln \mu _{\varepsilon} \right),\,\,\, 2s+1 \lt N \lt 6s.
\end{align*} Next, we will get an accurate estimate of ζɛ and µɛ by means of Pohozaev identities. In 
$\S$ 3, we have
\begin{align}
\bar{u}_{\varepsilon}\le C\bar{U}_{\zeta _{\varepsilon},\mu _{\varepsilon}}.
\end{align} Thus we also have the estimates of 
$\bar{u}_\varepsilon$: For any 
$X \in \partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta_{\varepsilon})$,
\begin{equation}
\begin{aligned}
\bar{u}_{\varepsilon}(X)=A_{\varepsilon} \bar{G}(X,\zeta _{\varepsilon})+O\left(\mu _{\varepsilon}^{-\frac{N+2s}{2}}\right)+\varTheta _{\varepsilon}
\end{aligned}
\end{equation}and
\begin{equation}
\begin{aligned}
\nabla \bar{u}_{\varepsilon}(X)=&A_{\varepsilon}\nabla \bar{G}(X,\zeta _{\varepsilon})+O\left(\mu _{\varepsilon}^{-\frac{N+2s}{2}}\right)+\varTheta _{\varepsilon},
\end{aligned}
\end{equation}where
\begin{equation}
\begin{aligned}
A_{\varepsilon}=&\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon})}{u_{\varepsilon}^{2_{s}^{*}-1-\varepsilon}(y)dy}\\
=&\mu _{\varepsilon}^{-\frac{N-2s}{2}}A+O\left( \varepsilon \mu _{\varepsilon}^{-\frac{N-2s}{2}}\ln \mu _{\varepsilon}\right)+\begin{cases}
O\left( \mu _{\varepsilon}^{-\frac{3(N-2s)}{2}} \right) ,& 2s+1 \lt N \lt 4s,\\
O\left( \varepsilon \mu _{\varepsilon}^{-3s}\ln \mu _{\varepsilon} \right) ,& N=4s,\\
O\left( \mu _{\varepsilon}^{-\frac{N+2s}{2}} \right) ,& 4s \lt N \lt 6s
\end{cases}
\end{aligned}
\end{equation}and
\begin{align}
\varTheta _{\varepsilon}=\begin{cases}
O\left(\varepsilon\mu _{\varepsilon}^{-2s}ln\mu _{\varepsilon}\right), &\frac{1}{2} \lt s\leq\frac{3}{4}, 2s+1 \lt N \lt 6s, \\
O\left(\varepsilon\mu _{\varepsilon}^{-2s}ln\mu _{\varepsilon}\right), &\frac{3}{4} \lt s \lt 1, 6s-2 \lt N \lt 6s, \\
O\left(\varepsilon\mu _{\varepsilon}^{-2s}(ln\mu _{\varepsilon})^{\frac{10s-3}{6s-2}}\right), &\frac{3}{4} \lt s \lt 1, N=6s-2, \\
O\left(\varepsilon\mu _{\varepsilon}^{-\frac{N-2s}{2}-1}ln\mu _{\varepsilon}\right), &\frac{3}{4} \lt s \lt 1, 2s+1 \lt N \lt 6s-2.
\end{cases}
\end{align}Now, inserting (4.4)-(4.7) into (B.7) and (B.8), we can obtain an accurate estimate of ζɛ and µɛ.
Lemma 4.3. Assume that 
$2s+1 \lt N \lt 6s$. Suppose that 
$\bar{u}_{\varepsilon}$ is the solution of (2.2) satisfying (4.1) and (4.2). Then
\begin{align}
\nabla h(\zeta_{\varepsilon})=O\left(\mu_{\varepsilon}^{-2s} \right),
\end{align}and
\begin{align}
\varepsilon=\mu_{\varepsilon}^{-(N-2s)}\left(\frac{2NA^2}{(N-2s)B}h(x_0)+O(\varepsilon ln\mu_{\varepsilon})+O\left(\mu_{\varepsilon}^{-2s}\right)\right).
\end{align}Proof. First, from the estimation of (B.7), we get that
\begin{align*}
\frac{1}{2}A_{\varepsilon}^2\frac{\partial h(\zeta _{\varepsilon})}{\partial \zeta _{\varepsilon ,i}}=O\left(\mu_{\varepsilon}^{-N}\right)+ \mu_{\varepsilon}^{-\frac{N-2s}{2}}\varTheta _{\varepsilon}.
\end{align*}Moreover, by (4.6), we have
\begin{align}
\frac{\partial h(\zeta _{\varepsilon})}{\partial \zeta _{\varepsilon ,i}}=O\left(\mu_{\varepsilon}^{-2s}\right)+ \mu_{\varepsilon}^{\frac{N-2s}{2}}\varTheta _{\varepsilon}.
\end{align}On the other hand, we also calculate
\begin{align}
\mathrm{LHS}\, \mathrm{of}\, {eqB.8}=\left(\frac{(N-2s)^2}{4N}\varepsilon+O\left(\varepsilon^2\right)\right)\left(B+ O\left(\mu_{\varepsilon}^{-(N-2s)}\right)+O\left(\varepsilon ln\mu _{\varepsilon}\right)\right),
\end{align}and
\begin{align}
\mathrm{RHS}\, \mathrm{of}\, {eqB.8}=&\frac{N-2s}{2}A_{\varepsilon}^2 h(\zeta _{\varepsilon})+O\left(\mu_{\varepsilon}^{-N}\right)+\mu_{\varepsilon}^{-\frac{N-2s}{2}}\varTheta _{\varepsilon}\nonumber\\
=&\frac{N-2s}{2}A^2\mu_{\varepsilon}^{-(N-2s)}h(\zeta_{\varepsilon})+O\left(\mu_{\varepsilon}^{-N}\right)+\mu_{\varepsilon}^{-\frac{N-2s}{2}}\varTheta _{\varepsilon}\nonumber\\
&+O\left(\varepsilon\mu _{\varepsilon}^{-(N-2s)}ln\mu _{\varepsilon}\right)+\begin{cases}
O\left( \mu _{\varepsilon}^{-2N+4s} \right) ,& 2s+1 \lt N \lt 4s,\\
O\left( \mu _{\varepsilon}^{-4s}\ln \mu _{\varepsilon} \right) ,& N=4s,\\
O\left( \mu _{\varepsilon}^{-N} \right) ,& 4s \lt N \lt 6s.
\end{cases}
\end{align}Combining (4.11) and (4.12), we get
\begin{align}
\varepsilon\mu _{\varepsilon}^{N-2s}\rightarrow\frac{2NA^2}{(N-2s)B}h(x_0), \, \text{as}\, \varepsilon\rightarrow0,
\end{align}from (4.7). Then, from (4.10), we can see that
\begin{align*}
\frac{\partial h(\zeta _{\varepsilon})}{\partial \zeta _{\varepsilon ,i}}=O\left(\mu_{\varepsilon}^{-2s}\right).
\end{align*}By (4.13), we once again combine (4.11) and (4.12) to get
\begin{align*}
& \varepsilon\left(\frac{(N-2s)^2}{4N}+O(\varepsilon)\right)\left(B+ O\left(\varepsilon ln\mu _{\varepsilon}\right)\right) \\ & \quad =\mu_{\varepsilon}^{-(N-2s)}\left(\frac{N-2s}{2}A^2h(x_0)+ O\left(\varepsilon ln\mu _{\varepsilon}\right)+ O\left( \mu _{\varepsilon}^{-2s}\right) \right).
\end{align*}That is to say,
\begin{align*}
\varepsilon=&\frac{\mu_{\varepsilon}^{-(N-2s)}\left(\frac{2NA^2}{(N-2s)B}h(x_0)+O\left(\varepsilon ln\mu_{\varepsilon}\right)+O\left(\mu_{\varepsilon}^{-2s}\right)\right)}{\left(1+O(\varepsilon)\right)\left(1+O\left(\varepsilon ln\mu_{\varepsilon}\right)\right)}\\
=&\mu_{\varepsilon}^{-(N-2s)}\left(\frac{2NA^2}{(N-2s)B}h(x_0)+O\left(\varepsilon ln\mu_{\varepsilon}\right)+O\left(\mu_{\varepsilon}^{-2s}\right)\right).
\end{align*}As a result, we complete the proof.
We now turn to prove Theorem 4.1. Here, we argue by contradiction. Suppose that (2.2) has two different solutions as follow
\begin{align*}
\bar{u}_\varepsilon^{(1)}=\bar{P}_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}+\bar{\omega}_\varepsilon^{(1)}
\end{align*}and
\begin{align*}
\bar{u}_\varepsilon^{(2)}=\bar{P}_{\zeta_\varepsilon^{(2)},\mu_\varepsilon^{(2)}}+\bar{\omega}_\varepsilon^{(2)}.
\end{align*}Let
\begin{align*}
\bar{\xi}_{\varepsilon}=\frac{\bar{u}_\varepsilon^{(1)}-\bar{u}_\varepsilon^{(2)}}{\left\| \bar{u}_\varepsilon^{(1)}-\bar{u}_\varepsilon^{(2)}\right\|_{L^{\infty}}}.
\end{align*} Then 
$\left\|\bar{\xi}_{\varepsilon}\right\|_{L^{\infty}}=1$ and
\begin{equation}
\begin{cases}
Div( x_{N+1}^{1-2s}\nabla \bar{\xi}_{\varepsilon} ) =0,& \text{in}\,\,\mathbb{R}_{+}^{N+1},\\
\bar{\xi}_{\varepsilon}=0,& \text{on}\,\,\mathbb{R}^N\setminus \varOmega ,\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim}x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}_{\varepsilon}=C_{\varepsilon}(x)\xi_{\varepsilon}(x),& \text{on}\,\,\varOmega ,\\
\end{cases}
\end{equation}where
\begin{align*}
C_{\varepsilon}(x)=(2_s^{*}-1-\varepsilon)\int_0^1{\left(tu_\varepsilon^{(1)}+(1-t)u_\varepsilon^{(2)} \right) ^{2_{s}^{*}-2-\varepsilon}dt}.
\end{align*}Since
\begin{align}
\left|U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}} - U_{\zeta_\varepsilon^{(2)},\mu_\varepsilon^{(2)}}\right|\leq CU_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}\bigg(\mu_\varepsilon^{(1)}\left|\zeta_\varepsilon^{(1)}-\zeta_\varepsilon^{(2)} \right|+\frac{\left|\mu_\varepsilon^{(1)}-\mu_\varepsilon^{(2)}\right|}{\mu_\varepsilon^{(1)}}\bigg),
\end{align}we obtain from (4.8) and (4.9) that
\begin{align*}
\left|U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}} - U_{\zeta_\varepsilon^{(2)},\mu_\varepsilon^{(2)}}\right|\leq C\left(\mu_\varepsilon^{(1)}\right)^{1-2s}U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}},
\end{align*}which in turns implies
\begin{align}
C_\varepsilon(x)=&(2_s^{*}-1)U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-2} +O\left( (\mu_\varepsilon^{(1)})^{1-2s}U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-2} \right) +O\left((\mu_\varepsilon^{(1)})^{-\frac{N-2s}{2}}U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-3} \right)\nonumber\\
&+O\left(U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-3}\left| \omega_\varepsilon^{(1)} \right| \right)+O\left(U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-3}\left| \omega_\varepsilon^{(2)} \right| \right).
\end{align} By the similar steps in 
$\S$ 3, we will prove 
$\left\|\xi_{\varepsilon}\right\|_{L^{\infty}}\rightarrow0$ as 
$\varepsilon\rightarrow0$, which contradicts 
$\left\|\bar{\xi}_{\varepsilon}\right\|_{L^{\infty}}\rightarrow0$ as 
$\varepsilon\rightarrow0$. By (4.3) and (4.15), we know
\begin{align*}
|C_{\varepsilon}(x)|\leq CU_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-2}.
\end{align*}So as with Lemma 3.3, we have the following estimate:
\begin{align}
\left| \xi _{\varepsilon}(x) \right|\le \frac{C}{\left(1+\mu _{\varepsilon}^{(1)}\left| x-\zeta _{\varepsilon}^{(1)} \right|\right)^{N-2s}}.
\end{align} Naturally, 
$\dot{\tilde{\xi}}_{\varepsilon}(x)=\xi _{\varepsilon}((\mu _{\varepsilon}^{(1)})^{-1}x+\zeta _{\varepsilon}^{(1)})$ is also defined, and from (4.16), we also deduce that there exists some constants dj, 
$j=1,2,\cdots,N+1,$ such that
\begin{align}
\dot{\tilde{\xi}}_{\varepsilon}(x)\rightarrow \sum_{j=1}^N{d_j\frac{\partial U_{0,1}}{\partial x _j}}+d_{N+1}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg| _{\mu =1}\,\,\, as\,\, \varepsilon \rightarrow 0,
\end{align} uniformly in 
$C_{loc}^1(\mathbb{R}^N)$.
To finish the proof, just like in 
$\S$ 3, we only need to prove 
$d_j=0,\,j=1,2,\cdots,N+1$. This will be achieved by using the Pohozaev identities (B.11) and (B.12).
Similar to (4.16), we can deduce
\begin{equation}
\begin{aligned}
D_\varepsilon(x)=&{2_s^{*}}U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-1}
+O\left((\mu_\varepsilon^{(1)})^{1-2s}U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-1} \right) +O\left((\mu_\varepsilon^{(1)})^{-\frac{N-2s}{2}}U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-2} \right)\\
&+O\left(U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-2}\left| \omega_\varepsilon^{(1)} \right| \right)+O\left(U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-2}\left| \omega_\varepsilon^{(2)} \right| \right).
\end{aligned}
\end{equation}To use (B.11) and (B.12), we need the following estimates.
Lemma 4.4. For any 
$X \in \partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta_{\varepsilon}^{(1)})$, 
$i=1,2$,
\begin{align*}
\bar{u}_{\varepsilon}^{(i)}(X)=A_{\varepsilon}^{(i)} \bar{G}(X,\zeta _\varepsilon^{(1)})+O\left(\left(\mu _\varepsilon^{(1)}\right)^{-\frac{N+2s}{2}}\right),
\end{align*}and
\begin{align*}
\nabla \bar{u}_{\varepsilon}^{(i)}(X)=&A_{\varepsilon}^{(i)}\nabla \bar{G}(X,\zeta _\varepsilon^{(1)})+O\left(\left(\mu _\varepsilon^{(1)}\right)^{-\frac{N+2s}{2}}\right),
\end{align*}where
\begin{align*}
A_\varepsilon^{(i)}:=\int_{B_{\frac{r}{4}}(\zeta_\varepsilon^{(1)})}{\left( u_\varepsilon^{(i)}\right)^{2_{s}^{*}-1-\varepsilon}(y)dy}.
\end{align*}Proof. By Lemma 3.5, we can directly get an estimate of 
$\bar{u}_\varepsilon^{(1)}$ on 
$\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta_{\varepsilon}^{(1)})$. Besides, we can calculate
\begin{align*}
\bar{u}_{\varepsilon}^{(2)}(X)=&A_{\varepsilon}^{(2)} \bar{G}(X,\zeta _\varepsilon^{(1)})+\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{\left \lt \partial _y\bar{G}(X,\zeta _{\varepsilon}^{(1)}),y-\zeta _{\varepsilon}^{(1)} \right \gt \left(u_{\varepsilon}^{(2)}\right)^{2_{s}^{*}-1-\varepsilon}(y)dy}\\
&+O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{\left| y-\zeta _{\varepsilon}^{(1)} \right|^2\left(u_{\varepsilon}^{(2)}\right)^{2_{s}^{*}-1-\varepsilon}(y)dy} \bigg)\\
& +O\bigg( \int_{\varOmega \setminus \setminus B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{\bar{G}(X,y)\left(u_{\varepsilon}^{(2)}\right)^{2_{s}^{*}-1-\varepsilon}(y)dy} \bigg)\\
=&A_{\varepsilon}^{(2)} \bar{G}(X,\zeta _\varepsilon^{(1)})+O\bigg(\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}\left| y-\zeta _{\varepsilon}^{(1)}\right|U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-2}\\
& \times \left( (\mu_\varepsilon^{(1)})^{1-2s}U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}} +\left|\psi_{\zeta_\varepsilon^{(2)},\mu_\varepsilon^{(2)}} \right|+\left|\omega_\varepsilon^{(2)} \right|\right)dy\bigg)\\
&+O\left(\left(\mu _\varepsilon^{(1)}\right)^{-\frac{N+2s}{2}}\right)\\
=&A_{\varepsilon}^{(2)} \bar{G}(X,\zeta _\varepsilon^{(1)})+O\left(\left(\mu _\varepsilon^{(1)}\right)^{-\frac{N+2s}{2}}\right),
\end{align*}where
\begin{align*}
A_\varepsilon^{(2)}:=& \int_{B_{\frac{r}{4}}(\zeta_\varepsilon^{(1)})}{\left( u_\varepsilon^{(2)}\right)^{2_{s}^{*}-1-\varepsilon}(y)dy}\\
=&\int_{B_{\frac{r}{4}}(\zeta_\varepsilon^{(1)})}{U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-1-\varepsilon}dy} +O\bigg(\int_{B_{\frac{r}{4}}(\zeta_{\varepsilon}^{(1)})}U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}}^{2_s^{*}-2}\\ & \quad \times \left( (\mu_\varepsilon^{(1)})^{1-2s}U_{\zeta_\varepsilon^{(1)},\mu_\varepsilon^{(1)}} +\left|\psi_{\zeta_\varepsilon^{(2)},\mu_\varepsilon^{(2)}} \right|+\left|\omega_\varepsilon^{(2)} \right|\right)dy\bigg)\\
=&\left(\mu _\varepsilon^{(1)}\right)^{-\frac{N-2s}{2}}A+O\left( \left(\mu _\varepsilon^{(1)}\right)^{-\frac{N+2s}{2}+1} \right).
\end{align*} Similarly, we can get the estimate of 
$\nabla \bar{u}_{\varepsilon}^{(i)}(X)$, 
$i=1,2$.
Lemma 4.5. For any 
$X \in \partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta_{\varepsilon}^{(1)})$,
\begin{align*}
\bar{\xi}_\varepsilon(x)=E_\varepsilon\bar{G}(X,\zeta_\varepsilon^{(1)}) +\sum_{i=1}^{N}{F_{i,\varepsilon}\partial_{\zeta_{\varepsilon ,i}^{(1)}}\bar{G}(X,\zeta_{\varepsilon}^{(1)})} +O\left(\left(\mu_{\varepsilon}^{(1)}\right)^{-N} \right),
\end{align*}and
\begin{align*}
\nabla \bar{\xi}_\varepsilon(x)=E_\varepsilon\nabla\bar{G}(X,\zeta_\varepsilon^{(1)}) +\sum_{i=1}^{N}{F_{i,\varepsilon}\nabla\partial_{\zeta_{\varepsilon,i}^{(1)}}\bar{G}(X,\zeta_{\varepsilon}^{(1)})} +O\left(\left(\mu_{\varepsilon}^{(1)}\right)^{-N} \right),
\end{align*}where
\begin{align*}
E_{\varepsilon}=\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{C_{\varepsilon}(y)\xi_{\varepsilon}(y)dy},
\end{align*}and
\begin{align*}
F_{i,\varepsilon}=\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{\left(y_i-\xi_{\varepsilon,i}^{(1)} \right)C_{\varepsilon}(y)\xi_{\varepsilon}(y)dy}.
\end{align*}Proof. By Green’s formula, we have
\begin{align*}
\bar{\xi}_\varepsilon(x)=&\int_{\varOmega}{\bar{G}(X,y)C_{\varepsilon}(y)\xi _{\varepsilon}(y)dy}
\\
=&\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{\bar{G}(X,y)C_{\varepsilon}(y)\xi _{\varepsilon}(y)dy}+O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{\bar{G}(X,y)C_{\varepsilon}(y)\xi _{\varepsilon}(y)dy} \bigg)
\\
=&E_{\varepsilon}\bar{G}(X,\zeta _{\varepsilon}^{(1)})+\sum_{i=1}^N{F_{i,\varepsilon}\partial _{\zeta _{\varepsilon ,i}^{(1)}}\bar{G}(X,\zeta _{\varepsilon}^{(1)})}+O\left( \left( \mu _{\varepsilon}^{(1)} \right) ^{-N} \right),
\end{align*}where
\begin{align*}
E_{\varepsilon}=\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{C_{\varepsilon}(y)\xi_{\varepsilon}(y)dy}
\end{align*}and
\begin{align*}
F_{i,\varepsilon}=\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{\left(y_i-\xi_{\varepsilon,i}^{(1)} \right)C_{\varepsilon}(y)\xi_{\varepsilon}(y)dy}.
\end{align*}By (4.16), we can calculate
\begin{align*}
E_{\varepsilon}=&(2_{s}^{*}-1) \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{U_{\zeta _{\varepsilon}^{(1)},\mu _{\varepsilon}^{(1)}}^{2_{s}^{*}-2}\xi _{\varepsilon}dy}\\
&+O\bigg( \left( \mu _{\varepsilon}^{(1)} \right) ^{1-2s}\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{U_{\zeta _{\varepsilon}^{(1)},\mu _{\varepsilon}^{(1)}}^{2_{s}^{*}-2}\xi _{\varepsilon}dy} \bigg)
+O\bigg( \left( \mu _{\varepsilon}^{(1)} \right) ^{-\frac{N-2s}{2}}\int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{U_{\zeta _{\varepsilon}^{(1)},\mu _{\varepsilon}^{(1)}}^{2_{s}^{*}-3}\xi _{\varepsilon}dy} \bigg)\\
&+O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{U_{\zeta _{\varepsilon}^{(1)},\mu _{\varepsilon}^{(1)}}^{2_{s}^{*}-3}\omega _{\varepsilon}^{(1)}\xi _{\varepsilon}dy} \bigg) +O\bigg( \int_{B_{\frac{r}{4}}(\zeta _{\varepsilon}^{(1)})}{U_{\zeta _{\varepsilon}^{(1)},\mu _{\varepsilon}^{(1)}}^{2_{s}^{*}-3}\omega _{\varepsilon}^{(2)}\xi _{\varepsilon}dy} \bigg)
\\
=&(2_{s}^{*}-1)d_{N+1}\left( \mu _{\varepsilon}^{(1)} \right) ^{-(N-2s)}\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg|_{\mu =1}dy}(1+o(1))
\end{align*}and
\begin{align*}
F_{i,\varepsilon}=(2_{s}^{*}-1)d_i\left( \mu _{\varepsilon}^{(1)} \right) ^{-(N-2s+1)}\int_{\mathbb{R}^N}{y_iU_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial y_i}dy}(1+o(1)).
\end{align*}Proof of Theorem 4.1
First, we prove that 
$d_{N+1}=0$ by means of (B.12). By (4.17) and (4.19), we have
\begin{align*}
\mathrm{LHS}\, \mathrm{of}\, {eqB.12}=o\left(\varepsilon \left(\mu _\varepsilon^{(1)}\right)^{-\frac{N-2s}{2}} \right).
\end{align*}From Lemma 4.4, Lemma 4.5 and Lemma A.2 we obtain that
\begin{align*}
\mathrm{RHS}\, \mathrm{of}\, {eqB.12}=&{\frac{N-2s}{2}}\left( A_{\varepsilon}^{(1)}+A_{\varepsilon}^{(2)} \right) E_{\varepsilon}h(\zeta _{\varepsilon}^{(1)})+O\left( \left( \mu _{\varepsilon}^{(1)} \right) ^{-\frac{3(N-2s)}{2}-1} \right)
\\
=&(N+2s)d_{N+1}A\left( \mu _{\varepsilon}^{(1)}\right)^{-\frac{3(N-2s)}{2}}h(x_0) \int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg|_{\mu =1}dy}(1+o(1)).
\end{align*}Therefore,
\begin{align}
\frac{(N+2s)d_{N+1}Ah(x_0)}{\varepsilon \left( \mu _{\varepsilon}^{(1)} \right) ^{N-2s}}\int_{\mathbb{R}^N}{U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,\mu}}{\partial \mu}\bigg|_{\mu =1}dy}(1+o(1))=o(1).
\end{align} Let 
$\varepsilon\rightarrow0$ in (4.20), we get 
$d_{N+1}=0$.
Next, we will prove that 
$d_i=0, i=1, 2, \cdots, N$.
\begin{align*}
\mathrm{LHS}\, \mathrm{of}\, {eqB.11}=O\left( \left(\mu _\varepsilon^{(1)}\right)^{-\frac{3N}{2}+s} \right).
\end{align*} Using Lemmas 4.4 and 4.5 again, for 
$i=1, 2, \cdots, N$, we get
\begin{align*}
& \mathrm{RHS}\, \mathrm{of}\, {eqB.11} \\ & \quad =A_{\varepsilon}^{(2)} E_{\varepsilon}\frac{\partial h(\zeta _{\varepsilon}^{(1)})}{\partial \zeta _{\varepsilon ,i}^{(1)}}+\frac{1}{2}\sum_{j=1}^N{A_{\varepsilon}^{(2)}F_{j,\varepsilon}\frac{\partial ^2h(x)}{\partial x_i\partial x_j}\bigg| _{x=\zeta _{\varepsilon}^{(1)}}}+O\left( \left( \mu _{\varepsilon}^{(1)} \right) ^{-\frac{3}{2}N+s} \right)
\\
& \quad = \frac{2_{s}^{*}-1}{2}A\left( \mu _{\varepsilon}^{(1)} \right) ^{-\frac{3(N-2s)}{2}-1}\int_{\mathbb{R}^N}{y_1U_{0,1}^{2_{s}^{*}-2}\frac{\partial U_{0,1}}{\partial y_1}dy\sum_{j=1}^N{d_j\frac{\partial ^2h(x)}{\partial x_i\partial x_j}\bigg| _{x=\zeta _{\varepsilon}^{(1)}}}}(1+o(1)),
\end{align*}Thus, we have
\begin{align*}
\sum_{j=1}^N{d_j\frac{\partial ^2h(x)}{\partial x_i\partial x_j}\bigg|_{x=\zeta _{\varepsilon}^{(1)}}}=o(1).
\end{align*}By x 0 is a non-degenerate critical point of h(x), we conclude that
\begin{align*}
d_i=0, \,\,\,i=1,2,\cdots,N.
\end{align*}Therefore, we finish the proof of Theorem 4.1.
Acknowledgements
We would like to thank the anonymous referee for his/her useful comments and suggestions to improve this paper greatly. B. Li was supported by NSF of China (No. 12461025) and NSF of Jiangxi (Nos. 20232BAB211002, 20224ACB218001). W. Long was supported by NSF of China (No.12271223), NSF of Jiangxi (Nos. 20252BAC230002, 20242BAB26001) and Double-high talents in Jiangxi Province.
Conflict of interest
There is no conflict of interest.
Data availability statement
There is no data in this paper.
Appendix A. Identities involving the Green’s function
Recall that 
$G(x,y)$ is the Green function for 
$(-\Delta) ^s$ in a bounded domain Ω with homogeneous Dirichlet boundary condition. Its harmonic extension 
$\bar{G}(X,\zeta)$ satisfies
\begin{equation}
\begin{cases}
Div( x_{N+1}^{1-2s}\nabla \bar{G}(X,\zeta) ) =0,& \text{in}\,\,\mathbb{R}_{+}^{N+1},\\
\bar{G}(X,\zeta)=0,& \text{on}\,\,\mathbb{R}^N\setminus \varOmega ,\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim}x_{N+1}^{1-2s}\partial _{\nu}\bar{G}(X,\zeta)=\delta_\zeta,& \text{on}\,\,\varOmega .\\
\end{cases}
\end{equation}Lemma A.1. ([Reference Guo, Li, Pistoia and Yan10])
It holds that, for 
$B_r(\zeta)\subset \varOmega$,
\begin{align}&
\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{G}(X,\zeta)\partial _{x_j}\bar{G}(X,\zeta)dS}-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{G}(X,\zeta) \right|^2\nu _jdS} \nonumber\\ \quad & =\nabla_{x_j}H(x,\zeta)|_{x=\zeta}.
\end{align}Lemma A.2. ([Reference Guo, Li, Pistoia and Yan10])
For 
$B_r(\zeta)\subset \varOmega$, it holds
\begin{align}
&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{G}(X,\zeta)\left \lt \nabla \bar{G}(X,\zeta ),X-(\zeta,0) \right \gt dS}\nonumber\\
& \quad -\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{G}(X,\zeta ) \right|^2\left \lt X-(\zeta,0),\nu \right \gt dS}\nonumber\\
&+\frac{N-2s}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\bar{G}(X,\zeta)\partial _{\nu}\bar{G}(X,\zeta)dS}\nonumber\\
=&\frac{N-2s}{2}H(\zeta,\zeta).
\end{align}Lemma A.3. ([Reference Guo, Li, Pistoia and Yan10])
For 
$B_r(\zeta)\subset \varOmega$, we have
\begin{align}
&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\partial _{\zeta _{j}}\bar{G}(X,\zeta)\nabla _{x_i}\bar{G}(X,\zeta )dS}+\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{G}(X,\zeta )\nabla_{x_i}\partial _{\zeta _{\varepsilon ,j}}\bar{G}(X,\zeta)dS}\nonumber\\
&-\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{G}(X,\zeta)\nabla \partial _{\zeta _{j}}\bar{G}(X,\zeta )\nu _idS}\nonumber\\
=&\frac{1}{2}\frac{\partial h(x)}{\partial x_i\partial x_j}\bigg|_{x=\zeta}.
\end{align}Appendix B. Some Pohozaev identities
Let 
$\bar{u} \in D_0^s(\varOmega)$ solve the eqution
\begin{equation}
\begin{cases}
Div( x_{N+1}^{1-2s}\nabla \bar{u} ) =0,& \text{in}\,\,\mathbb{R}_{+}^{N+1},\\
\bar{u}=0,& \text{on}\,\,\mathbb{R}^N\setminus \varOmega ,\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim}x_{N+1}^{1-2s}\partial _{\nu}\bar{u}=u^{2_{s}^{*}-1-\varepsilon},& \text{on}\,\,\varOmega \\
\end{cases}
\end{equation} and 
$\bar{\xi}\in D_0^s(\varOmega)$ satisfies
\begin{align}
\begin{cases}
Div(x_{N+1}^{1-2s}\nabla \bar{\xi})=0,& in\,\,\mathbb{R}_{+}^{N+1},\\
\bar{\xi}=0,& on\,\,\mathbb{R}^N\setminus \varOmega ,\\
\underset{{x_{N+1}\rightarrow 0^+}}{\lim} x_{N+1}^{1-2s}\partial _{\nu} \bar{\xi} =( 2_{s}^{*}-1-\varepsilon ) u^{2_{s}^{*}-2-\varepsilon}\xi ,& on\,\,\varOmega ,\\
\end{cases}
\end{align}for bounded domain 
$\varOmega \subset \mathbb{R}^N$.
Denote 
$B_r(\zeta )=\{x\in \mathbb{R}^N,\left| x-\zeta \right| \lt r\}$, 
$\mathcal{B}_r(\zeta )=\{X\in \mathbb{R}^{N+1},\left| X-(\zeta ,0) \right| \lt r\}$, 
$\mathcal{B}_{r}^{+}(\zeta )=\mathcal{B}_r(\zeta )\cap \mathbb{R}_{+}^{N+1}$ and 
$\partial ^{\prime} \mathcal{B}_{r}^{+}(\zeta )=\partial \mathcal{B}_r(\zeta )\cap \mathbb{R}_{+}^{N+1}$.
Lemma B.1. Let r > 0 be small such that 
$B_{r}(\zeta )\subset \varOmega $. Then
\begin{equation}
\begin{aligned}
&\frac{N-2s}{2}\varepsilon \int_{B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\xi dx}\\
=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt +x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\left \lt \nabla \bar{\xi},X-(\zeta ,0) \right \gt \right\} dS}\\
&-\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}\left \lt X-(\zeta ,0),\nu \right \gt dS}\\
& +\frac{N-2s}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\bar{\xi}+x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\bar{u} \right\} dS}\\
&+\int_{\partial B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\xi \left \lt x-\zeta ,\nu \right \gt ds}
\end{aligned}
\end{equation}and
\begin{align}
-\int_{\partial B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\xi \nu _ids}& =\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\partial _i\bar{u}+x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\partial _i\bar{\xi} \right\} dS} \nonumber\\ & -\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}\nu _idS},
\end{align} for 
$i=1,2,\cdots,N$.
Proof. Multiplying the first equation of (B.1) by 
$\left \lt \nabla \bar{\xi},X-(\zeta ,0) \right \gt $ and the first equation of (B.2) by 
$\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt $, respectively, we add the two relations together to obtain
\begin{align*}
0=&Div(x_{N+1}^{1-2s}\nabla \bar{\xi})\left \lt \nabla \bar{u},X-(\zeta ,0)\right \gt +Div(x_{N+1}^{1-2s}\nabla \bar{u})\left \lt \nabla \bar{\xi},X-(\zeta ,0) \right \gt \\
=&Div\left( x_{N+1}^{1-2s}\nabla \bar{\xi}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt +x_{N+1}^{1-2s}\nabla \bar{u}\left \lt \nabla \bar{\xi},X-(\zeta ,0) \right \gt \right)\\
&-Div\left( x_{N+1}^{1-2s}\left \lt \nabla \bar{\xi},\nabla \bar{u} \right \gt ( X-(\zeta ,0) )\right) +(N-2s)x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u} .
\end{align*} Integrating the above equation in 
$\mathcal{B}_{r}^{+}(\zeta )$ yields
\begin{align}
0=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt +x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\left \lt \nabla \bar{\xi},X-(\zeta ,0) \right \gt \right\} dS}\nonumber\\
&+\int_{B_r(\zeta )\times 0}{\underset{x_{N+1}\rightarrow 0^+}{\lim}\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt +x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\left \lt \nabla \bar{\xi},X-(\zeta ,0) \right \gt \right\} dX}\nonumber\\
&-\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}\left \lt X-(\zeta ,0),\nu \right \gt dX}+(N-2s)\int_{\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}dx}\nonumber\\
=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt +x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\left \lt \nabla \bar{\xi},X-(\zeta ,0) \right \gt \right\} dS}\nonumber\\
&+\int_{B_r(\zeta )}{\left \lt \nabla (u^{2_{s}^{*}-1-\varepsilon}\xi ),x-\zeta \right \gt dx}-\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}\left \lt X-(\zeta ,0),\nu \right \gt dS}\nonumber\\
&+(N-2s)\int_{\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}dX}\nonumber\\
=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt +x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\left \lt \nabla \bar{\xi},X-(\zeta ,0) \right \gt \right\} dS}\nonumber\\
&+\int_{\partial B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\xi \left \lt x-\zeta ,\nu \right \gt ds}-\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}\left \lt X-(\zeta ,0),\nu \right \gt dS}\nonumber\\
&+(N-2s)\int_{\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}dX}-N\int_{B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\xi dx}.
\end{align}On the other hand, we have
\begin{align*}
0=&\int_{\mathcal{B}_{r}^{+}(\zeta )}{\left\{Div(x_{N+1}^{1-2s}\nabla \bar{u})\bar{\xi}+Div(x_{N+1}^{1-2s}\nabla \bar{\xi})\bar{u} \right\} dX}\\
=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\bar{u}+x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\bar{\xi} \right\} dS}+(2_{s}^{*}-\varepsilon )\int_{B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\xi dx}-2 \\ & \times \int_{\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}dX}.
\end{align*}Hence,
\begin{equation}
\begin{aligned}
&(N-2s)\int_{\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}dX}-N\int_{B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\xi dx}\\
=&\frac{N-2s}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\bar{u}+x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\bar{\xi} \right\} dS}-\frac{N-2s}{2}\varepsilon \int_{B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\xi dx}.
\end{aligned}
\end{equation}Inserting (B.6) into (B.5), we get (B.3).
Multiplying the first equation of (B.1) by 
$\partial _{i} \xi$ and the first equation of (B.2) by 
$\partial _{i} u$, for 
$i=1, 2, \cdots, N$, respectively, we then put the two relations together to obtain
\begin{align*}
0=&Div(x_{N+1}^{1-2s}\nabla \bar{u})\partial _i\bar{\xi}+Div(x_{N+1}^{1-2s}\nabla \bar{\xi})\partial _i\bar{u}\\
=&Div(x_{N+1}^{1-2s}\nabla \bar{\xi}\partial _i\bar{u}+x_{N+1}^{1-2s}\nabla \bar{u}\partial _i\bar{\xi})-\partial _i(x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}).
\end{align*} Integrating the above equation in 
$\mathcal{B}_{r}^{+}(\zeta )$ yields
\begin{align*}
0=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\partial _i\bar{u}+x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\partial _i\bar{\xi} \right\} dS}+\int_{B_r(\zeta )}{\partial _i(u^{2_{s}^{*}-1-\varepsilon}\xi )dx}\\ & -\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}\nu _idS}\\
=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\partial _i\bar{u}+x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\partial _i\bar{\xi} \right\} dS}+\int_{\partial B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\xi \nu _ids}\\ & -\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\nabla \bar{\xi}\nabla \bar{u}\nu _idS}.
\end{align*}Lemma B.2. Let r > 0 be small and 
$B_{r}(\zeta )\subset \varOmega $. Then, for 
$i=1, 2, \cdots, N$,
\begin{align}
-\frac{1}{2_{s}^{*}-\varepsilon}\int_{\partial B_r(\zeta )}{u^{2_{s}^{*}-\varepsilon}\nu _ids}=-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2\nu _idS}+\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\partial _i\bar{u}dS}
\end{align}and
\begin{equation}
\begin{aligned}
&\left( \frac{N}{2_{s}^{*}-\varepsilon}-\frac{N-2s}{2} \right) \int_{B_r(\zeta )}{u^{2_{s}^{*}-\varepsilon}dx}\\
=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt dS}-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2\left \lt X-(\zeta ,0),\nu \right \gt dS}\\
&+\frac{N-2s}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\bar{u}\partial _{\nu}\bar{u}dS}+\frac{1}{2_{s}^{*}-\varepsilon}\int_{\partial B_r(\zeta )}{u^{2_{s}^{*}-\varepsilon}\left \lt x-\zeta ,\nu \right \gt ds}.
\end{aligned}
\end{equation}Proof. Multiplying the first equation of (B.1) by 
$\partial _{i}u$, we get
\begin{align*}
0=Div(x_{N+1}^{1-2s}\nabla \bar{u})\partial _i\bar{u}=Div(x_{N+1}^{1-2s}\nabla \bar{u} \partial _i\bar{u})-\frac{1}{2}\partial _i(x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2).
\end{align*} Integrating the above equation in 
$\mathcal{B}_{r}^{+}(\zeta )$ yields
\begin{align*}
0=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\partial _i\bar{u}dS}+\int_{B_r(\zeta )}{u^{2_{s}^{*}-1-\varepsilon}\partial _iudx}-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2\nu _idS}\\
=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\partial _i\bar{u}dS}+\frac{1}{2_{s}^{*}-\varepsilon}\int_{\partial B_r(\zeta )}{u^{2_{s}^{*}-\varepsilon}\nu _ids}-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2\nu _idS}.
\end{align*} Multiplying the first equation of (B.1) by 
$\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt $, we get
\begin{align*}
0=&Div(x_{N+1}^{1-2s}\nabla \bar{u})\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt \\
=&Div(x_{N+1}^{1-2s}\nabla \bar{u}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt )-\frac{1}{2}Div(x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2(X-(\zeta ,0)))\\
&+\frac{N-2s}{2}x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2.
\end{align*} Integrating the above equation in 
$\mathcal{B}_{r}^{+}(\zeta )$ yields
\begin{align}
0=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt dS}+\frac{1}{2_{s}^{*}-\varepsilon}\int_{B_r(\zeta )}{\left \lt \nabla \left( u^{2_{s}^{*}-\varepsilon} \right) ,x-\zeta \right \gt dx}\nonumber\\
&-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2\left \lt X-(\zeta ,0),\nu \right \gt dS}+\frac{N-2s}{2}\int_{\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2dX}\nonumber\\
=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{u}\left \lt \nabla \bar{u},X-(\zeta ,0) \right \gt dS}+\frac{1}{2_{s}^{*}-\varepsilon}\int_{\partial B_r(\zeta )}{u^{2_{s}^{*}-\varepsilon}\left \lt x-\zeta ,\nu \right \gt dx}\nonumber\\
&-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2\left \lt X-(\zeta ,0),\nu \right \gt dS}+\frac{N-2s}{2}\int_{\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2dX}\nonumber\\
&-\frac{N}{2_{s}^{*}-\varepsilon}\int_{B_r(\zeta )}{u^{2_{s}^{*}-\varepsilon}dx}.
\end{align}On the other hand, we have
\begin{align*}
0=&\int_{\mathcal{B}_{r}^{+}(\zeta )}{Div\left( x_{N+1}^{1-2s}\nabla \bar{u} \right) \bar{u}dX}\\
=&-\int_{\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2dX}+\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\bar{u}\partial _{\nu}\bar{u}dS}+\int_{B_r(\zeta )}{u^{2_{s}^{*}-\varepsilon}dx}.
\end{align*}Hence,
\begin{equation}
\begin{aligned}
\int_{\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left| \nabla \bar{u} \right|^2dX}=\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\bar{u}\partial _{\nu}\bar{u}dS}+\int_{B_r(\zeta )}{u^{2_{s}^{*}-\varepsilon}dx}.
\end{aligned}
\end{equation} Let 
$\bar{u}^{(1)}$ and 
$\bar{u}^{(2)}$ be two different solutions of the equation (B.1). We define
\begin{align*}
\bar{\xi}=\frac{\bar{u}^{(1)}-\bar{u}^{(2)}}{\left\| \bar{u}^{(1)}-\bar{u}^{(2)}\right\|_{L^{\infty}}}
\end{align*}and
\begin{align*}
D(x)=(2_s^{*}-\varepsilon)\int_0^1{\left(tu^{(1)}+(1-t)u^{(2)} \right) ^{2_{s}^{*}-1-\varepsilon}dt}.
\end{align*} Applying (B.7) and (B.8) to 
$\bar{u}^{(i)}$, we obtain the following result.
Lemma B.3. It holds that, for 
$B_{r}(\zeta )\subset \varOmega $,
\begin{align}
-\frac{1}{2_s^{*}-\varepsilon}\int_{\partial B_r(\zeta^ )}{D(x)\xi\nu _ids}=&-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\left \lt \nabla \left(\bar{u}^{(1)}+\bar{u}^{(2)}\right),\nabla\bar{\xi} \right \gt \nu _idS}\nonumber\\
&+\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta)}{x_{N+1}^{1-2s}\partial _{i}\bar{u}^{(1)}\partial _{\nu}\bar{\xi}dS} +\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta)}{x_{N+1}^{1-2s}\partial _{\nu}\bar{u}^{(2)}\partial _i\bar{\xi}dS}
\end{align}and
\begin{align}
&\left( \frac{N}{2_{s}^{*}-\varepsilon}-\frac{N-2s}{2} \right) \int_{B_r(\zeta )}{D(x)\xi dx}\nonumber\\
=&\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{\xi}\left \lt \nabla \bar{u}^{(1)},X-(\zeta ,0) \right \gt dS}+\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{x_{N+1}^{1-2s}\partial _{\nu}\bar{u}^{(2)}\left \lt \nabla \bar{\xi},X-(\zeta ,0) \right \gt dS}\nonumber\\
&-\frac{1}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta)}{x_{N+1}^{1-2s}\left \lt \nabla \left( \bar{u}^{(1)}+\bar{u}^{(2)} \right) ,\nabla \bar{\xi} \right \gt \left \lt X-(\zeta ,0),\nu \right \gt dS}\nonumber\\
&+\frac{N-2s}{2}\int_{\partial ^{\prime}\mathcal{B}_{r}^{+}(\zeta )}{\left\{x_{N+1}^{1-2s}\bar{u}^{(1)}\partial _{\nu}\bar{\xi}+x_{N+1}^{1-2s}\bar{\xi}\partial _{\nu}\bar{u}^{(2)}\right\}dS}+\frac{1}{2_{s}^{*}-\varepsilon}\int_{\partial B_r(\zeta )}{D(x)\xi \left \lt x-\zeta,\nu \right \gt ds}.
\end{align}
