weak solution to problem (\[eq1.1\]) when $\Omega={\mathbb{R}}^N$, $h(u)=u^p$ and $\nu=k\delta_0$, that is, $$\label{eq1.3}
 \arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha  u+u^p=k\delta_0\quad   {\rm in}\quad {\mathbb{R}}^N,\\[2mm]
  \phantom{    }
  \lim_{|x|\to+\infty}u(x)=0,
\end{array}$$ where $p\in(0,\frac{N}{N-2\alpha})$, $k>0$ and $\delta_0$ denotes the Dirac mass at the origin.

\[cor 1\] Assume that $\alpha\in(0,1)$ and $p\in(0,\frac{N}{N-2\alpha})$. Then for any $k>0$, problem (\[eq1.3\]) admits a unique weak solution $u_k$ such that $$\label{1.5}
\lim_{x\to0}u_k(x)|x|^{N-2\alpha}=c_1k,$$ where $c_1>0$. Moreover,

-   1.  $\{u_k\}_{k\in(0,\infty)}$ are classical solutions of $$\label{eq1.01}
         (-\Delta)^\alpha  u+u^p=0\quad { \rm in}\quad{\mathbb{R}}^N\setminus\{0\};$$

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    1.  the mapping: $k\mapsto u_k$ is increasing.

We consider the asymptotic behavior of $u_1$ at $\infty$ when $p\in(1,\frac{N}{N-2\alpha})$.

\[th 1\] Assume that $\alpha\in(0,1)$ and $u_1$ is the solution of (\[eq1.3\]) with $k=1$. Then there exist $c_2>1$ and $R>2$ such that for $|x|\ge R$,

-   1.  if $p\in(1,1+\frac{2\alpha}{N})$, $$\label{6.1}
        \frac{1}{c_2}|x|^{-\frac{N+2\alpha}{p}}\le u_1(x)\le c_2|x|^{-\frac{N+2\alpha}{p}};$$

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    1.  if $p=1+\frac{2\alpha}{N}$, $$\label{6.3}
          \frac{1}{c_2}|x|^{-N}\log^{\frac{N}{2\alpha}}(|x|)\le u_1(x)\le c_2|x|^{-N}\log^{\frac{N}{2\alpha}}(|x|);$$

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    1.  if $p\in(1+\frac{2\alpha}{N}, \frac{N}{N-2\alpha})$, $$\label{6.2}
        \frac{1}{c_2}|x|^{-\frac{2\alpha}{p-1}}\le u_1(x)\le c_2|x|^{-\frac{2\alpha}{p-1}}.$$

According to Theorem \[th 1\], we know that the decaying power of $u_1$ shifts at the point $p=1+\frac{2\alpha}{N}$; while for $\alpha=1$ and $p\in(1,\frac{N}{N-2})$, the weak solution of (\[eq1.3\]) decays as $|x|^{-\frac{2}{p-1}}$.

From now on, we denote that $u_k$ is the weak solution of (\[eq1.3\]). Since the mapping: $k\mapsto u_k$ is increasing by Theorem 