our first theorem on the existence of weak solutions for problem (\[eq1.1\]).

\[teo 1\] Assume that $\alpha\in(0,1)$, $\Omega$ is a regular domain in ${\mathbb{R}}^N (N\ge2)$ and $h:{\mathbb{R}}_+\to{\mathbb{R}}_+$ is a continuous nondecreasing function satisfying $$\label{1.4}
 \int_1^{+\infty} h(s) s^{-1-\frac{N}{N-2\alpha}}ds <+\infty.$$ Then for any $\nu\in\mathfrak{M}^b_+(\Omega)$, problem (\[eq1.1\]) admits a weak solution $u_\nu$ such that $$\label{1.5}
0\le u_\nu\le \mathbb{G}_\Omega[\nu]\quad \rm{a.e.\ in}\
\Omega.$$

If $\nu$ is a nonnegative bounded Radon measure, (\[eq 2.10wx1\]) with $\beta=0$ and (\[1.4\]) have the same critical value $\frac{N}{N-2\alpha}$.

In the case that $\Omega$ is bounded, the authors of [@CV2] took a sequence of $C^1$ functions $\{\nu_n\}$ converging to $\nu$ in the weak star sense, then they considered the solutions $u_n$ of (\[eq1.1\]) replacing $\nu$ by $\nu_n$. By the compact imbedding theorem, they showed that the limit of $\{u_n\}$ exists, up to subsequence. While for the case that $\Omega$ is unbounded, the difficulty is that Sobolev imbedding may not be compact. To overcome the difficulty, we truncate the measure $\nu$ by $\nu\chi_{B_R(0)}$ and use the increasing monotonicity of corresponding solutions $\{u_R\}$ of solutions to (\[eq1.1\]) in related bounded domains. Taking the limit as $R\to\infty$, we achieve the desired weak solution.

The second purpose in this paper is to study properties of 