& {\rm in}\quad  \Omega,\\[2mm]
 \phantom{------\ }
u=0\quad & {\rm in}\quad \Omega^c\setminus {\rm supp}(\nu).
\end{array}$$

We remark that\
$(i)$ the second equality in (\[1.1\]) is understood in the sense that $u=0$ in $\Omega^c\setminus {\rm supp}(\nu)$ and $u$ is continuous at every point in $\partial\Omega\setminus {\rm supp}(\nu)$;\
$(ii)$ the uniqueness requires the nondecreasing assumption on nonlinearity $g$, while the existence also holds without the nondecreasing assumption on $g$;\
$(iii)$ (\[g1\]) is called as integral subcritical condition with critical value $\frac{N+\alpha}{N-\alpha}$, similar integral subcritical conditions see the references [@BV; @CV1; @CV2; @V].

Applied Theorem \[teo 1\] when $\nu=\delta_{x_0}$ with $x_0\in\partial\Omega$, problem (\[eq 1.1\]) admits a unique nonnegative weak solution when $g$ satisfies the hypotheses in Theorem \[teo 1\]. Our second goal is to study the further properties of the weak solution.

\[teo 2\] Assume that $\epsilon=1$, $k>0$, $\nu=\delta_{x_0}$ with $x_0\in\partial\Omega$, $g$ is a nondecreasing function in $C^\beta$ locally in ${\mathbb{R}}$ with $\beta>0$ satisfying $g(0)\ge0$ and (\[g1\]). Let $u_k$ be the weak solution of (\[eq 1.1\]), then

$(i)$ $$\label{b k}
\lim_{t\to0^+} \frac{u_k(x_0+t\vec{n}_{x_0})}{ \mathbb{G}_\alpha[\frac{\partial^\alpha\delta_{x_0}}{\partial\vec{n}^\alpha}](x_0+t\vec{n}_{x_0})}=k.$$

$(ii)$ if additionally $g(s)=s^p$ with $p\in(1+\frac{2\alpha}{N},\frac{N+\alpha}{N-\alpha})$, then the limit of $\{u_k\}$ as $k\to\infty$ exists in ${\mathbb{R}}^N\setminus\{x_0\}$, denoting $u_\infty$. Moreover, $u_\infty$ is a classical solution of $$\label{1.3}
 \arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha   u+u^p=0\quad & 