$\lambda_1\ge \lambda_2$ and $$\lim_{\lambda\to0^+}\int_{\tilde S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx=0,$$ then there exists $\lambda_0>0$ such that $$\left(\int_{\tilde S_{\lambda_0}} \rho_{\partial\Omega}^\alpha(x)dx\right)^{-\frac{N-\alpha}{N+\alpha}}\ge\Lambda_0$$ and for $\lambda\ge \lambda_0$, we may choose $\Lambda= (\int_{\tilde S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{-\frac{N-\alpha}{N+\alpha}}\ge \Lambda_0$ and then $$\begin{aligned}
\int_{\tilde S_\lambda} \frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx\le c_{21}(\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{\frac{2\alpha}{N+\alpha}},\end{aligned}$$ where $c_{21}=c_{20}+1$. Therefore, $$\begin{aligned}
 \int_{\tilde S_\lambda}\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}](x)\rho_{\partial\Omega}^\alpha(x)dx&\le &\int_{\partial\Omega}\int_{\tilde S_\lambda}
\frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx d|\nu(y)|
\\&\le &c_{21}\int_{\partial\Omega} d|\nu(y)|(\int_{\tilde S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{\frac{2\alpha}{N+\alpha}}
\\&\le& c_{21}\|\nu\|_{\mathfrak{M}^b(\bar\Omega)} (\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx)^{\frac{2\alpha}{N+\alpha}}.\end{aligned}$$ As a consequence, $$\begin{aligned}
\lambda \tilde m(\lambda)\le c_{21}\|\nu\|_{\mathfrak{M}^b(\bar\Omega)}\tilde m(\lambda)^{\frac{2\alpha}{N+\alpha}},\end{aligned}$$ which implies (\[annex 0\]). This ends the proof. $\Box$

To estimate the nonlinearity in $L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)$, we have to introduce an auxiliary lemma as follows.

\[lm 08-09\] Assume that $g:{\mathbb{R}}_+\mapsto{\mathbb{R}}_+$ is a continuous function satisfying $$\label{p}
\int_1^{\infty} g(s)s^{-1-p}ds<+\infty$$ for some $p>0$. Then there is a sequence real positive numbers $\{T_n\}$ such that $$\lim_{n\to\infty}T_n=\infty\quad{\rm and}\quad \lim_{n\to\infty}g(T_n)T_n^{-p}=0.$$

Assume additionally that $g$ is nondecreasing, then $$\lim_{T\to\infty} g(T)T^{-p}=0.$$

[**Proof.**]{} The first argument see [@CV2 Lemma 3.1] and second see [@CV1 Lemma 3.1].$\Box$

Now we are ready to prove Theorem \[teo 1\].

[**Proof of Theorem \[teo 1\].**]{} [*To prove the existence of weak solution.*]{} Take $\{g_n\}$ a sequence of $C^1$ nondecreasing functions defined on ${\mathbb{R}}$ satisfying $g_n(0)=g(0)$ and (\[06-08-0\]). By Proposition \[pr 2.01\], problem (\[10-08-0\]) admits a unique weak solution $u_{n}$ such that $$|u_{n}|\le \mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}]\quad{\rm a.e.\ in}\quad \Omega$$ 