any $x_0\in {\mathbb{R}}^N$ with $|x_0|>4r$, there exists $\beta\in(0,2\alpha)$ such that $$\arraycolsep=1pt
\begin{array}{lll}
{\|u_n\|}_{C^{\beta}(B_{2r}(x_0))} \le {c_{11}}({\|u_n\|}_{L^1({\mathbb{R}}^N,d\omega)}+{\|u^p*\eta_n\|}_{L^{\infty}(B_{3r}(x_0))}+{\|u_n\|}_{L^{\infty}(B_{3r}(x_0))})\\[2mm]
\phantom{-------} \le c_{11}(c_{12}k\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|_{L^1({\mathbb{R}}^N,d\omega)}+{\|u^p\|}_{L^{\infty}(B_{3r}(x_0))}
 \\[2mm]\phantom{---------}+c_{12} k{\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|}_{L^\infty( B_{r/2}^c(0))})\\[2mm]
\phantom{-------} \le c_{11}(c_{12}k\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|_{L^1({\mathbb{R}}^N,d\omega)}+k^p\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|^p_{L^{\infty}(B_{r/2}^c(0))}
 \\[2mm]\phantom{---------}+c_{12} k{\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|}_{L^\infty( B_{r/2}^c(0))}).
\end{array}$$ Therefore, by the definition of $u_n$ and Arzela-Ascoli Theorem, we obtan that $u\in C^{\frac\beta2}(B_{2r}(x_0))$. By Corollary 2.4 in [@RS], we deduce that $$\arraycolsep=1pt
\begin{array}{lll}
{\|u_n\|}_{C^{2\alpha+\frac\beta2}(B_r(x_0))} \le {c_{13}}({\|u_n\|}_{L^1({\mathbb{R}}^N,d\omega)}+{\|u^p*\eta_n\|}_{C^{\frac\beta2}(B_{2r}(x_0))}
\\[2mm]\phantom{----------}+{\|u_n\|}_{C^{\frac\beta2}(B_{2r}(x_0))})\\[2mm]
\phantom{--------} \le c_{14}(k\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|_{L^1({\mathbb{R}}^N,d\omega)}+{\|u\|}_{C^{\frac\beta2}(B_{2r}(x_0))}
 \\[2mm]\phantom{----------}+
k^p\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|^p_{L^{\infty}(B_{r/2}^c(0))}
+k{\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|}_{L^\infty( B_{r/2}^c(0))}),
\end{array}$$ where $c_{13},c_{14}>0$. Thus, $u\in C^{2\alpha+\frac\beta4}(B_r(x_0))$ and by arbitrary of $r>0$ and $x_0$, $u$ is $C^{2\alpha+\frac\beta4}$ locally in ${\mathbb{R}}^N\setminus\{0\}$. Therefore, $u_n\to u$ and $\eta_n\to 0$ uniformly in any compact subset of ${\mathbb{R}}^N\setminus\{0\}$ as $n\to\infty$. We conclude that $u$ is a classical solution of (\[eq1.01\]) by Corollary 4.6 in [@CS1] .

[*2. Existence of $u_{R}$.*]{} It infers from (\[eq1.4\]) that for given $R>1$, $u_n$ is a classical solution of $$\arraycolsep=1pt
\begin{array}{lll}
(-\Delta)^{\alpha}u_n+ u_n^p=k\eta_n+u_n^p-u^p\ast\eta_n\qquad&\mbox{in }\ B_R(0),\\
\phantom{------}u_n\ge 0\quad &\mbox{in }\ B_R^c(0).
\end{array}$$ We observe that for $R>1$, $$\tilde u_n:=(u-m_{R-\epsilon_n})\ast\eta_n=u_n-m_{R-\epsilon_n}\le u_n\quad{\rm in}\quad {\mathbb{R}}^N$$ and $(-\Delta)^{\alpha}\tilde u_n=(-\Delta)^{\alpha}u_n$, therefore, $$(-\Delta)^{\alpha}\tilde u_n+ |\tilde u_n|^{p-1}\tilde u_n=k\eta_n+|\tilde u_n|^{p-1}\tilde u_n-u^p\ast\eta_n\quad{\rm in}\quad  B_R(0).$$ By the definition of $m_{R-\epsilon_n}$, we have $u-m_{R-\epsilon_n}\le 0$ in $B_{R-\epsilon_n}^c(0)$, and then $$\tilde u_n\le 0\quad{\rm in}\  B_R^c(0).$$

Let $u_{n,R}$ be the solution of $$\arraycolsep=1pt
\begin{array}{lll}
(-\Delta)^{\alpha}u_{n,R}+ u_{n,R}^p=k\eta_n+u_n^p-u^p\ast \eta_n\qquad&\mbox{in }\ B_R(0),\\
\phantom{------\  }u_{n,R}=0\quad &\mbox{in 