{\rm in}\quad  \Omega,\\[2mm]
 \phantom{(-\Delta)^\alpha   +u^p}
u=0\quad & {\rm in}\quad \Omega^c\setminus \{x_0\}
\end{array}$$ and satisfies $$\label{b k 01}
c_1 \le u_\infty(x_0+t\vec{n}_{x_0})t^{\frac{2\alpha}{p-1}}\le c_2,\qquad \forall t\in(0,\sigma_0),$$ where $c_2>c_1>0$ and $\sigma_0>0$ small enough.

$(iii)$ if we assume more that $g(s)=s^p$ with $p\in(0,1+\frac{2\alpha}{N}]$, then $$\lim_{k\to\infty}u_k(x)=+\infty,\qquad \forall x\in \Omega.$$

We notice that the limit of $\{u_k\}$ as $k\to\infty$ blows up every where in $\Omega$ when $g(u)=u^p$ with $1<p\le1+ \frac{2\alpha}{N}$. This phenomena is different from the Laplacian case, which is caused by the nonlocal characteristic of the fractional Laplacian. Theorem \[teo 1\] and Theorem \[teo 2\] show the existence and properties of weak solutions to (\[eq 1.1\]) in the subcritical case. One natural question is what happens in the critical case, i.e., $g(s)=s^p$ with $p\ge \frac{N+\alpha}{N-\alpha}$. The results are given by:

\[teo 4\]

Assume that $\epsilon=1$, $k>0$, $\Omega=B_1(e_N)$ with $e_N=(0,\cdots, 0,1)$, $\nu=\delta_{0}$ and $g(s)=s^p$ with $p= \frac{N+\alpha}{N-\alpha}$. Then problem (\[eq 1.1\]) doesn’t admit any weak solution.

In general, the nonexistence of weak solution is obtained by capacity analysis for second order differential elliptic equations involving measures, see [@V] and references therein. However, it is a very tough job to attain the nonexistence in the capacity framework by the nonlocal characteristic and the weak sense of $\frac{\partial^\alpha\delta_0}{\partial \vec{n}^\alpha}$, 