[u_n(-\Delta)^\alpha\xi+g_n(u_n)\xi]dx=k\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial\vec{n}_x^\alpha}d\nu(x),\quad \forall\xi\in\mathbb{ X}_\alpha.$$ This implies that $u_n$ is a weak solution of (\[10-08-0\]). We see that (\[12-08-1\]) follows by (\[12-08-0\]) and Lemma \[lm 0\]. Moreover, by the facts that $u_n=\lim_{t\to0^+} u_{n,t}$ and $$-k\mathbb{G}_\alpha[t^{-\alpha}\nu_-]\le u_{n,t}\le  k\mathbb{G}_\alpha[t^{-\alpha}\nu_+]\quad {\rm in}\quad \Omega,$$ we have that (\[10-08-1\]) holds.

[*To prove that $u_n=0$ in $\Omega^c\setminus{\rm supp}(\nu)$.* ]{} Let $x_0\in\partial\Omega\setminus{\rm supp}(\nu)$ and $x_s=x_0+s\vec{n}_{x_0}$ with $s\in(0,\sigma_0)$. We only have to prove that $\lim_{s\to0^+}u(x_s)=0$. From [@BV Theorem 1.1], for any $(x,y)\in
\Omega\times \partial\Omega_t$, $x\neq y$, $$\label{annex 010}
G_\alpha(x,y)\le c_5\frac{\rho^\alpha_{\partial\Omega}(y)\rho^\alpha_{\partial\Omega}(x)}{|x-y|^N }=c_5\frac{\rho^\alpha_{\partial\Omega}(x)t^{\alpha}}{|x-y|^N }.$$ For some $s_0>0$ and any $s\in(0,s_0)$, we observe that ${\rm dist}(x_s,{\rm supp}(\nu))\ge \frac{1}{2}{\rm dist}(x_0,{\rm supp}(\nu))$ and $$\begin{aligned}
 \mathbb{G}_\alpha[t^{-\alpha}|\nu_t|](x_s)&\le& c_5\int_{\partial\Omega}\frac{\rho^\alpha_{\partial\Omega}(x_s)}{|x_s-y|^N}d|\nu|(y)
 \\&=& c_5s^\alpha\int_{\partial\Omega\setminus{\rm supp}(\nu)}\frac{1}{|x_s-y|^N}d|\nu|(y)
 \\&\le& c_52^N s^\alpha{\rm dist}(x_0,{\rm supp}(\nu))^{-N}{\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}
 \\&\to&0\quad {\rm as}\quad s\to0^+.\end{aligned}$$ Together with the facts that $$\label{facts}
  u_n=\lim_{t\to0^+}u_{n,t}\quad {\rm and}\quad |u_{n,t}|\le \mathbb{G}_\alpha[t^{-\alpha}|\nu_t|],$$ we derive that $u_n=0$ in $\Omega^c\setminus{\rm supp}(\nu)$.

[*To prove the uniqueness of weak solution.*]{} Let $u_1,u_2$ be two weak solutions of (\[10-08-0\]) and $w=u_1-u_2$. Then $(-\Delta)^\alpha
w=g_n(u_2)-g_n(u_1)$ and $g_n(u_2)-g_n(u_1)\in L^1(\Omega,\rho^\alpha_{\partial\Omega}dx)$. By Kato’s inequatlity, see Proposition 2.4 in [@CV1], for $\xi\in\mathbb{X}_\alpha$, $\xi\ge0$, we have that $$\begin{aligned}
\int_\Omega |w|(-\Delta)^\alpha \xi dx+\int_\Omega[g_n(u_1)-g_n(u_2)]{\rm sign}(w)\xi dx\le0.\end{aligned}$$ Combining with $\int_\Omega[g_n(u_1)-g_n(u_2)]{\rm sign}(w)\xi dx\ge0$, then we have $$w=0\quad {\rm a.e.\ in}\ \ \Omega.$$

[*Regularity of $u_n$.*]{} Since $g_n$ is $C^1$ in ${\mathbb{R}}$, then by 