and $d\mu=d\omega$ that $g(s)\leq c_{8}s^{-p^*_{\alpha}}$ and for $T>\lambda$, $$\displaystyle\begin{array}{lll}
h(\lambda)g(\lambda)+\int_\lambda^Tg(s)dh(s)\le c_{8}\lambda^{-p^*_{\alpha}} h(\lambda)+ c_{8}\int_{\lambda}^T s^{-p^*_{\alpha}}dh(s)
\\[4mm]\phantom{-----------\ \ }\displaystyle
\leq c_8T^{-p^*_\alpha}h(T)+c_{8}p^*_{\alpha}\int_{\lambda}^T
s^{-1-p^*_{\alpha}}h(s)ds,
\end{array}$$ where $c_8>0$. By the nondecreasing monotonicity of $h$, we have that $$\begin{aligned}
T^{-p^*_{\alpha}}h(T) &=& 2^{1+p_\alpha^*}h(T)(2T)^{-1-p^{*}_\alpha}\int_T^{2T}dt \\
   &\le& 2^{1+p_\alpha^*}\int_T^{2T}h(t)t^{-1-p^*_\alpha}dt,\end{aligned}$$ then it infers by (\[1.4\]) that that $$\lim_{T\to\infty}T^{-p^*_{\alpha}}h(T)= 0.$$ Therefore, by (\[1.4\]) and we take $\lambda=1$, $$\int_{{\mathbb{R}}^N}h(\mathbb{G}_{{\mathbb{R}}^N}[\nu])d\omega\leq h(1)\int_{{\mathbb{R}}^N} d\omega+c_{8}p^*_{\alpha}\int_{1}^\infty s^{-1-p^*_{\alpha}}h(s)ds<+\infty,$$ i.e. $h(\mathbb{G}_{{\mathbb{R}}^N}[\nu]) \in L^1({\mathbb{R}}^N, d\omega).$ We complete the proof.$\Box$

Basic results
-------------

This subsection is devoted to present some basic results and Comparison Principle, which are key tools in the analysis. We start it by recalling the existence of weak solution of (\[eq1.1\]) when $\Omega$ is a bounded $C^2$ domain.

[@CV2 Theorem 1.1]\[pr 2.1\] Assume that $\mathcal{O}$ is a bounded $C^2$ domain in ${\mathbb{R}}^N$, $\mu\in\mathfrak{M}^b_+(\mathcal{O})$ and $h:{\mathbb{R}}_+\to{\mathbb{R}}_+$ is a continuous nondecreasing function satisfying (\[1.4\]). Then problem $$\label{2.1}
 \arraycolsep=1pt
\begin{array}{lll}
 (-\Delta)^\alpha  u+h(u)=\mu\quad & {\rm in}\quad \mathcal{O},\\
  \phantom{   (-\Delta)^\alpha  + h(u)}
  u=0& {\rm in}\quad{\mathbb{R}}^N\setminus\mathcal{O}
\end{array}$$ admits a unique weak solution $v_\mu$ such that $$\label{2.2}
0\le v_\mu\le \mathbb{G}_{\mathcal{O}}[\mu]\quad{\rm a.e.\ in}\ \mathcal{O}.$$ Moreover, the mapping $\mu\mapsto v_\mu$ is increasing.

Next we recall the Comparison Principle from [@CFQ].

\[teo CP\][@CFQ Theorem 2.3] Suppose that $O$ is a bounded domain of ${\mathbb{R}}^N$, $p>0$, the functions $u_1$, $u_2$ 