$u_{k,\Omega}$ of (\[eq1.001\]) when $\Omega$ is an unbounded regular domain including the origin. The result is stated as follows.

\[teo 3\] Assume that $\alpha\in(0,1)$, $\Omega$ is an unbounded regular domain of ${\mathbb{R}}^N(N\ge2)$ including the origin, $p\in(0,\frac{N}{N-2\alpha})$ and $u_k$ is given by Theorem \[cor 1\]. Then

$(i)$ for any $k>0$, (\[eq1.001\]) admits a unique weak solution $u_{k,\Omega}$ such that $$u_k-m_{k,\Omega}\le  u_{k,\Omega}\le u_k\quad{\rm in}\quad \Omega$$ and the mapping $k\mapsto u_{k,\Omega}$ is increasing, where $m_{k,\Omega}=\sup_{x\in\Omega^c} u_k(x)$;

$(ii)$ 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 (\[eq1.02\]) such that $$u_\infty-m_{\infty,\Omega}\le  u_{\infty,\Omega}\le u_\infty\quad{\rm in}\quad \Omega,$$ where $u_\infty$ is defined by (\[definition infty\]) and $m_{\infty,\Omega}=\sup_{x\in\Omega^c} u_\infty(x).$

The paper is organized as follows. In Section 2 we list some properties of Marcinkiewicz spaces and establish the inequality $$\label{1.3'}
\|\mathbb{G}_{{\mathbb{R}}^N}[\nu]\|_{M^{\frac{N}{N-2\alpha}}({\mathbb{R}}^N, d\omega)}\le
c_5\|\nu\|_{\mathfrak{M}^b({\mathbb{R}}^N)},$$ which is used to obtain that $h(\mathbb{G}_{{\mathbb{R}}^N}[\nu])\in L^1({\mathbb{R}}^N, d\omega)$. In Section 3, we prove Theorem \[teo 1\]. The proofs of Theorem \[cor 1\] and Theorem \[th 1\] are addressed in Section 4. Finally, we give the proofs of Theorem \[teo 2\] and Theorem \[teo 3\] in Section 5.

Preliminary
===========

The purpose of this section is to introduce some preliminaries and prove Marcinkiewicz 