the fact that $\{u_{k,R}\}_R$ is an increasing sequence of functions, (\[3.1.3\]) holds.$\Box$

[**Proof of Theorem \[cor 1\].**]{} By Proposition \[pr 3.1\], Lemma \[lm 3.1\] and (\[3.1.3\]), the assertion of Theorem \[cor 1\] holds except part $(ii)$.

Now, we prove part $(ii)$ of Theorem \[cor 1\]. In fact, let $k_1\le k_2$ and $u_{k_1}$, $u_{k_2}$ be the solution of (\[eq1.3\]) with $k=k_1$ and $k=k_2$, respectively. For $R>1$, we denote by $u_{k_1,R}$ and $u_{k_2,R}$ the solutions of (\[3.11\]) with $k=k_1$ and $k=k_2$, respectively. By $k_1\le k_2$ and Proposition \[pr 2.1\], we have that $$u_{k_1,R}\le u_{k_2,R}\quad {\rm in }\ \ {\mathbb{R}}^N\setminus\{0\}.$$ Similar to the proof of Theorem \[teo 1\], we know that $u_{k_i}=\lim_{R\to\infty} u_{k_i,R}$ with $i=1,2$. Therefore, $u_{k_1}\le u_{k_2}$ in ${\mathbb{R}}^N\setminus\{0\}$. $\Box$

Asymptotic behavior of $u_1$ at $\infty$
----------------------------------------

This subsection is devoted to investigate the asymptotic behavior of weak solution $u_1$ at $\infty$ to $$\label{eq 5.1}
 \arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha  u+u^p=\delta_0\quad   {\rm in}\quad {\mathbb{R}}^N,\\[2mm]
  \phantom{    }
  \lim_{|x|\to+\infty}u(x)=0,
\end{array}$$ where $p\in(1,\frac{N}{N-2\alpha})$. We observe that $$\lim_{x\to0}u_1(x)|x|^{N-2\alpha}=c_{N,\alpha}$$ and $u_1$ is a classical solution of $$\label{eq 5.2}
 (-\Delta)^\alpha  u+u^p=0\quad  {\rm in}\quad{\mathbb{R}}^N\setminus\{0\}.$$

In order to prove Theorem \[th 1\], we introduce some auxiliary lemmas. For $\tau\in(-\infty,-N+2\alpha)$, we denote by $w_\tau$ a $C^2$ nonnegative radially symmetric function in ${\mathbb{R}}^N$ such 