as}\ t\to0^+,
\end{array}$$ which ends the proof. $\Box$

We note that Proposition \[pr 2.1\] shows that $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ is approximated by a sequence Radon measure with support in $\Omega$ in the distribution sense and this provides a new method to derive weak solution of (\[eq 1.1\]) by considering the limit of the weak solutions to $$\arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha   u+\epsilon g(u)=kt^{-\alpha}\nu_t\quad & {\rm in}\quad  \Omega,\\[2mm]
\phantom{(-\Delta)^\alpha   +\epsilon g(u)}
 u=0\quad  &{\rm in}\quad  \Omega^c.
\end{array}$$ To end this section, we give a upper bound for $\mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial \vec{n}^\alpha}]$ .

\[lm 2.1\] Let $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$, then there exists $c_5>0$ such that $$\mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial \vec{n}^\alpha}](x)\le \int_{\partial\Omega}\frac{c_5}{|x-y|^{N-\alpha}}d|\nu|(y),\qquad x\in\Omega.$$

[**Proof.**]{} From [@BV Theorem 1.1], there exists $c_5>0$ independent of $t$ such that for any $(x,y)\in
\Omega\times\partial\Omega_t$, $x\neq y$, $$\label{annex 01}
G_\alpha(x,y)\le c_5\frac{\rho_{\partial\Omega}^\alpha(y)}{|x-y|^{N-\alpha}}=\frac{c_5 t^{\alpha}}{|x-y|^{N-\alpha}}.$$ Then for $x\in\Omega$, $$\begin{aligned}
 \mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial \vec{n}^\alpha}](x) &=& \lim_{t\to0^+}\int_{\partial\Omega_t} G_\alpha(x,y)t^{-\alpha}d|\nu_t|(y)\\
   &\le &  \lim_{t\to0^+}\int_{\partial\Omega_t} \frac{c_5}{|x-y|^{N-\alpha}}d|\nu_t|(y)
   \\&=&\int_{\partial\Omega} \frac{c_5}{|x-y|^{N-\alpha}}d|\nu|(y).\end{aligned}$$ We complete the proof. $\Box$

Absorption Nonlinearity {#sec:existence}
=======================

In this section, our goal is to prove the existence and uniqueness of weak solution for fractional elliptic problem (\[eq 1.1\]) with $\epsilon=1$. To this end, we first consider the properties of weak solution of $$\label{2.0.6}
 \arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha   u+g_n(u)=kt^{-\alpha}\nu_t\quad & {\rm in}\quad  \Omega,\\[2mm]
\phantom{(-\Delta)^\alpha   +g_n(u)}
u=0\quad  &{\rm in}\quad  \Omega^c,
\end{array}$$ where $t\in(0,\sigma_0)$, 