of $\nu$ such that $\nu=\nu_+-\nu_-$.\
Furthermore, $$\label{12-08-1}
{\|g_n(u_{n})\|}_{L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)}\le k{\|\mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial\vec{n}^\alpha}]\|}_{L^1(\Omega)}$$ and $u_n$ is a classical solution of (\[1.1\]) replacing $g$ by $g_n$.

[**Proof.**]{} [*To prove the existence of weak solution.*]{} Since $\nu_t$ is a bounded Radon measure with supp$(\nu_t)\subset\partial \Omega_t$ for $t\in(0,\sigma_0)$, then by Proposition \[pr 1\], we have that problem (\[2.0.6\]) admits a unique weak solution $u_{n,t}$ such that $$\label{24-08-1}
|u_{n,t}|\le \mathbb{G}_\alpha [t^{-\alpha}|\nu_t|]\quad{\rm a.e.\ in}\quad \Omega,\qquad \int_{\Omega}|g_n(u_{n,t})| \rho_{\partial\Omega}^\alpha dx\le k{\|\mathbb{G}_\alpha [t^{-\alpha}|\nu_t|]\|}_{L^1(\Omega)}$$ and $$\label{2.1.1--}
\int_\Omega [u_{n,t}(-\Delta)^\alpha\xi+g_n(u_{n,t})\xi]dx=\int_{\partial\Omega_t}t^{-\alpha}\xi(x)d\nu_t(x),\quad \forall\xi\in \mathbb{X}_\alpha.$$

For any compact set $\mathcal{K} \subset \Omega$, there exists $t_0\in(0,\sigma_0)$ such that $$\mathcal{K} \subset \Omega_t\quad {\rm and}\quad
{\rm dist}(\mathcal{K},\partial\Omega_{t})\ge t_0,\quad \forall t\in(0,t_0].$$ By Lemma \[lm 1\], we observe that for some $\beta\in(0,\alpha)$ $${\|u_{n,t}\|}_{C^\beta(\mathcal{K})}\le c_5t_0^{-N+2\alpha} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}.$$ Therefore, up to some subsequence, there exists $u_n$ such that $$\lim_{t\to0^+}u_{n,t}=u_n\quad{\rm a.e.\ in}\quad \Omega.$$ Then $g_n(u_{n,t})$ converges to $g_n(u_n)$ almost every in $\Omega$ as $t\to0^+$. By (\[24-08-1\]) and (\[24-08-0\]), we have that $\{u_{n,t}\}_t$ is relatively compact in $L^1(\Omega)$, up to subsequence, $$u_{n,t}\to u_n\quad {\rm in}\ \ L^1(\Omega)\quad {\rm as}\ t\to0^+$$ and then $$g_n(u_{n,t})\to g_n(u_n)\quad {\rm in}\ \ L^1(\Omega,\rho^\alpha_{\partial\Omega}dx)\quad {\rm as}\ t\to0^+.$$ By Proposition \[pr 2.1\], $$\begin{aligned}
 \int_{\partial\Omega_t}t^{-\alpha}\xi(x)d\nu_t(x)   \to \int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial\vec{n}_x^\alpha}d\nu(x)\quad {\rm as}\ t\to0^+,\end{aligned}$$ Passing to the limit as $t\to 0^+$ in the identity (\[2.1.1–\]), it implies that $$\int_\Omega 