\[cor 1\] and then we can denote that $$\label{definition infty}
  u_{\infty}(x)=\lim_{k\to\infty}u_k(x), \quad  x\in{\mathbb{R}}^N.$$ Here we note that $u_{\infty}(x)\in {\mathbb{R}}_+\cup\{+\infty\}$ for any $x\in{\mathbb{R}}^N$. Now we state properties of $u_\infty$.

\[teo 2\] Assume that $\alpha\in(0,1)$, $p\in(0,\frac{N}{N-2\alpha})$ and $u_\infty$ is given by (\[definition infty\]). Then

-   1.  if $p\in(0,1+\frac{2\alpha}N]$, then $ u_\infty(x)=\infty, \  \forall x\in {\mathbb{R}}^N;$

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    1.  if $p\in(1+\frac{2\alpha}N, \frac{N}{N-2\alpha})$, then $u_\infty$ is a classical solution of (\[eq1.01\]) and there exists $c_3>0$ such that $$u_\infty(x)=c_3|x|^{-\frac{2\alpha}{p-1}},\quad \forall x\in{\mathbb{R}}^N\setminus\{0\}.$$

In the proof of Theorem \[teo 2\], we make use of the self-similar property of $u_\infty$.

Analogue results of Theorem \[teo 2\] in bounded domain $\Omega$ were obtained [@CV2; @CV3]. Precisely, they showed that there exists a unique weak solution $u_{k,\Omega}$ to semilinear fractional elliptic problem $$\label{eq1.001}
 \arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha  u+u^p=k\delta_0\quad & {\rm in}\quad\Omega,\\[2mm]
 \phantom{   (-\Delta)^\alpha  + u^p}
u=0\quad & {\rm in}\quad \Omega^c,
\end{array}$$ where $k>0$, $0\in \Omega$ and $p\in(0,\frac{N}{N-2\alpha})$. Moreover,\
$(i)$ the mapping $k\mapsto u_{k,\Omega}$ is increasing;\
$(ii)$ for $p\in(0,\min\{1+\frac{2\alpha}N, \frac{N}{2\alpha}\})$, $u_{\infty,\Omega}=\infty$ in $\Omega,$ where $$u_{\infty,\Omega}=\lim_{k\to\infty}u_{k,\Omega}\quad{\rm in}\ \ {\mathbb{R}}^N;$$ $(iii)$ for $p\in(1+\frac{2\alpha}N, \frac{N}{N-2\alpha})$, $u_{\infty,\Omega}$ is a classical solution of $$\label{eq1.02}
\arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha  u+u^p=0\quad  &{\rm in}\quad\Omega\setminus\{0\},\\[2mm]
 \phantom{   (-\Delta)^\alpha  + u^p}
 u=0\quad & {\rm in}\quad \Omega^c.
 \end{array}$$

Finally, we discuss properties of weak solution 