and $$\label{2.1.1000}
\int_\Omega [u_n(-\Delta)^\alpha\xi+g_n(u_n)\xi]dx=k\int_{\partial\Omega}\frac{\partial^\alpha \xi(x)}{\partial \vec{n}_x^\alpha}d\nu(x),\quad \forall\xi\in \mathbb{X}_\alpha.$$ For any compact set $\mathcal{K} \subset \Omega$, we observe from Lemma \[lm 1\] that for some $\beta\in(0,\alpha)$, $${\|u_n\|}_{C^\beta(\mathcal{K})}\le c_{22}  {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)}.$$ Therefore, up to some subsequence, there exists $u_\nu$ such that $$\lim_{n\to\infty}u_n=u_\nu\quad{\rm a.e.\ in}\ \Omega.$$ Then $ g_n(u_n)$ converge to $g(u_\nu)$ a.e. in $\Omega$ as $n\to\infty$. By Lemma \[lm 00\] and (\[12-08-1\]), we have that $$u_n\to u_\nu\ {\rm in}\ L^1(\Omega),\quad {\|g_n(u_n)\|}_{L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)}\le c_{23}{\|\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\|}_{L^1(\Omega)}$$ and $$\tilde m(\lambda)\leq c_{18}\lambda^{-\frac{N+\alpha}{N-\alpha}} \ \quad {\rm for}\ \ \  \lambda>\lambda_0,$$ where $$\tilde m(\lambda)=\int_{\tilde S_\lambda}\rho_{\partial\Omega}^{\alpha}(x)dx
\quad{\rm  with}\quad \tilde S_\lambda=\{x\in\Omega: \mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]>\lambda\}.$$ For any Borel set $E\subset\Omega$, we have that $$\displaystyle\begin{array}{lll}
\displaystyle\int_{E}|g_n(u_n)|\rho_{\partial\Omega}^\alpha(x) dx\le \int_{E\cap\tilde S^c_{\frac{\lambda}{k}}}g\left(k\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\right)\rho_{\partial\Omega}^\alpha(x) dx+\int_{E\cap \tilde S_{\frac{\lambda}{k}}}g\left(k\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\right)\rho_{\partial\Omega}^\alpha(x) dx
\\[4mm]\phantom{\int_{E}|g(u_t)|\rho^{\alpha}_{\partial\Omega}(x)dx}
\displaystyle\leq \tilde g\left(\frac{\lambda}{k}\right)\int_E\rho^{\alpha}_{\partial\Omega}(x)dx+\int_{\tilde S_{\frac{\lambda}{k}}}\tilde g\left(k\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\right)\rho^{\alpha}_{\partial\Omega}(x)dx
\\[4mm]\phantom{\int_{E}|g(u_t)|\rho^{\alpha}_{\partial\Omega}(x)dx}
\displaystyle\leq \tilde
g\left(\frac{\lambda}{k}\right)\int_E\rho^{\alpha}_{\partial\Omega}(x)dx+\tilde m\left(\frac{\lambda}{k}\right) \tilde g\left(\frac{\lambda}{k}\right)+\int_{\frac{\lambda}{k}}^\infty\tilde m(s)d\tilde
g(s),
\end{array}$$ where $\tilde g(r)=g(|r|)-g(-|r|)$.

On the other hand, $$\int_{\frac{\lambda}{k}}^\infty \tilde g(s)d\tilde m(s)=\lim_{T\to\infty}\int_{\frac{\lambda}{k}}^T \tilde g(s)d \tilde m(s).$$ Thus, $$\displaystyle\begin{array}{lll}
\displaystyle \tilde m\left(\frac{\lambda}{k}\right) \tilde g\left(\frac{\lambda}{k}\right)+ \int_{\frac{\lambda}{k}}^T \tilde m(s)d\tilde g(s) \le c_{24}\tilde g\left(\frac{\lambda}{k}\right)\left(\frac{\lambda}{k}\right)^{-\frac{N+\alpha}{N-\alpha}}+c_{24}\int_{\frac{\lambda}{k}}^T s^{-\frac{N+\alpha}{N-\alpha}}d\tilde g(s)
\\[4mm]\phantom{-----\ \int_{\lambda}^T \tilde g(s)d\omega(s)}\displaystyle
\leq c_{25}T^{-\frac{N+\alpha}{N-\alpha}}\tilde
g(T)+\frac{c_{24}}{\frac{N+\alpha}{N-\alpha}+1}\int_{\frac{\lambda}{k}}^T
s^{-1-\frac{N+\alpha}{N-\alpha}}\tilde g(s)ds.
\end{array}$$ By assumption (\[g1\]) and Lemma \[lm 08-09\] with $p=\frac{N+\alpha}{N-\alpha}$, $T^{-\frac{N+\alpha}{N-\alpha}}\tilde g(T)\to 0$ when $T\to\infty$, therefore, $$\tilde m\left(\frac{\lambda}{k}\right) \tilde g\left(\frac{\lambda}{k}\right)+ \int_{\frac{\lambda}{k}}^\infty \tilde m(s)\ d\tilde g(s)\leq \frac{c_{24}}{\frac{N+\alpha}{N-\alpha}+1}\int_{\frac{\lambda}{k}}^\infty s^{-1-\frac{N+\alpha}{N-\alpha}}\tilde 