solution of (\[1.22\]), if $u\in
L^1(\Omega)$, $h(u)\in L^1(\Omega,\rho_{\partial\Omega}^\alpha dx)$ and $$\int_\Omega [u(-\Delta)^\alpha\xi+ h(u)\xi]dx=\int_{\Omega}\xi(x)d\nu(x),\qquad \forall\xi\in \mathbb{X}_\alpha,$$ where $\mathbb{X}_{\alpha}\subset C({\mathbb{R}}^N)$ with $\alpha\in(0,1)$ denotes the space of functions $\xi$ satisfying:

-   1.  ${\rm supp}(\xi)\subset\bar\Omega$;

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    1.  $(-\Delta)^\alpha\xi(x)$ exists for all $x\in \Omega$ and $|(-\Delta)^\alpha\xi(x)|\leq C$ for some $C>0$;

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    1.  there exist $\varphi\in L^1(\Omega,\rho^\alpha_{\partial\Omega} dx)$ and $\varepsilon_0>0$ such that $|(-\Delta)_\varepsilon^\alpha\xi|\le
        \varphi$ a.e. in $\Omega$, for all $\varepsilon\in(0,\varepsilon_0]$.

A unique weak solution of (\[1.22\]) is obtained when the function $h$ is continuous, nondecreasing and satisfies $$\int_1^{\infty}(h(s)-h(-s))s^{-1-k_{\alpha,\beta}}ds<+\infty,$$ where $$k_{\alpha,\beta}=\left\{
\arraycolsep=1pt
\begin{array}{lll}
\frac{N}{N-2\alpha},\quad &{\rm if}\quad
\beta\in[0,\frac{N-2\alpha}N\alpha],\\[2mm]
\frac{N+\alpha}{N-2\alpha+\beta},\qquad &{\rm if}\quad
\beta\in(\frac{N-2\alpha}N\alpha,\alpha].
\end{array}
\right.$$

Motivated by the above results, we may approximate $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ by a sequence measures defined in $\Omega$ and consider the limit of corresponding weak solutions. To this end, for a bounded Radon measure defined in $\bar\Omega$ with support in $\partial\Omega$, we observe that $$\langle\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha},\xi\rangle=\int_{\partial\Omega}\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}d\nu(x), \quad \xi\in \mathbb{X}_\alpha,$$ and $$\frac{\partial^\alpha\xi(x)}{\partial \vec{n}_x^\alpha}=\lim_{s\to0^+}\frac{\xi(x+s\vec{n}_x)-\xi(x)}{s^\alpha}
=\lim_{s\to0^+}\xi(x+s\vec{n}_x)s^{-\alpha},$$ so $\frac{\partial^\alpha\nu}{\partial \vec{n}^\alpha}$ could be approximated by measures $\{t^{-\alpha}\nu_t\}$ with support in $\{x\in\Omega: \rho_{\partial\Omega}(x)=t\}$ generated by $\nu$, see Section 2 for details. Then we consider the limit of weak solutions as $t\to0^+$ for the problem: $$\arraycolsep=1pt
\begin{array}{lll}
(-\Delta)^\alpha  u+g(u)=t^{-\alpha}\nu_t\quad &{\rm in}\quad \Omega,
\\[2mm]\phantom{------\ }
u=0\quad&{\rm in}\quad \Omega^c.
\end{array}$$ Here 