$n\to\infty$ in (\[3.6\]), we obtain that $$\label{3.10}
\int_{\Omega} [u_\nu(-\Delta)^\alpha\xi+h(u_\nu)\xi]dx=\int_{\Omega} \xi d\nu.$$ Since $\xi\in \mathbb{X}_{\Omega}$ is arbitrary, $u_\nu$ is a weak solution of (\[eq1.1\]).$\Box$

Properties of weak solutions
============================

In this section, we investigate problem (\[eq1.3\]). First we show that there is a unique solution of problem (\[eq1.3\]), then we establish the asymptotic behavior at the origin and infinity for the solution. In other words, we will prove Theorem \[cor 1\] and Theorem \[th 1\].

Properties of $u_k$
-------------------

In this subsection, we consider the properties of nonnegative weak solution to (\[eq1.3\]). To this end, we introduce an auxiliary lemma.

\[lm 2—-1\][@CV3 Lemma 3.1] Assume that $v\in C^{2\alpha+\epsilon}(\bar B_1)$ with $\epsilon>0$ satisfies $$(-\Delta)^\alpha v=\varphi\quad {\rm in}\quad B_1(0),$$ where $\varphi\in C^1(\bar B_1)$. Then for $\beta\in (0,2\alpha)$, there exists $c_{11}>0$ such that $$\|v\|_{C^\beta(\bar B_{1/4}(0))}\le {c_{11}}(\|v\|_{L^\infty(B_1(0))}+\|\varphi\|_{L^\infty(B_1(0))}+\|v\|_{L^1({\mathbb{R}}^N,d\omega)}).$$

\[lm 3.1\] Assume that $k>0$, $p\in(0,\frac{N}{N-2\alpha})$ and $u$ is a nonnegative weak solution of (\[eq1.3\]). Then $u$ is a classical solution of (\[eq1.01\]) and for any $R>1$, there exists a weak solution $u_{R}$ of $$\label{3.11}
\arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha  u+u^p=k\delta_0\quad  & {\rm in}\quad B_R(0),\\[2mm]
  \phantom{(x----}
u=0 & {\rm in}\quad B_R^c(0)
\end{array}$$ such that $$\label{3.1.7}
u-m_R\le u_{R}\le u\quad  \ {\rm in}\quad B_R(0)\setminus\{0\},$$ where $m_R=\sup_{|x|>R}u(x)$.

[**Proof.**]{} The proof is divide into 