two parts. First we show the regularity of solution $u$, then we find $u_R$ to establish the inequality (\[3.1.7\]).

[*1.Regularity of $u$.*]{} Let $\{\eta_n\}\subset C^\infty_0({\mathbb{R}}^N)$ be a sequence of radially decreasing and symmetric mollifiers such that supp$(\eta_n)\subset B_{\varepsilon_n}(0)$ with $\varepsilon_n\leq \frac1n$ and $u_n=u\ast\eta_n$. We observe that $$\label{4.1}
u_n\to u\ \ {\rm and}\ \ u_n^p\to u^p\ \ {\rm in}\  \ L^1({\mathbb{R}}^N,d\omega)\ \ {\rm as}\ n\to\infty.$$ By Fourier transformation, we have that $$\eta_n\ast(-\Delta)^{\alpha}\xi= (-\Delta)^{\alpha}(\xi\ast\eta_n),$$ then $$\int_{{\mathbb{R}}^N}\!\left(u(-\Delta)^{\alpha}(\xi\ast\eta_n) \!+\xi\ast\eta_nu^p\right)\!dx=\!
\int_{{\mathbb{R}}^N}\!\left(u\ast\eta_n(-\Delta)^{\alpha}\xi+\eta_n\ast u^p\xi\right)\! dx,$$ where $\eta_n$ is radially symmetric. It follows that $u_n$ is a classical solution of $$\label{eq1.4}
 \arraycolsep=1pt
\begin{array}{lll}
(-\Delta)^{\alpha}u_n+ u^p\ast\eta_n=k\eta_n\quad&\mbox{in }\ \ {\mathbb{R}}^N,\\[3mm]
\phantom{--}\lim_{|x|\to\infty}u_n(x)=0.
\end{array}$$ We observe that $0\le u_n\le k\mathbb{G}_{{\mathbb{R}}^N}[\eta_n]$, which implies $0\le u\le k\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]$ in ${\mathbb{R}}^N\setminus\{0\}$. Since $u^p\in L^1({\mathbb{R}}^N,d\omega)$, we have $u^p\ast\eta_n\to u^p$ in $L^1({\mathbb{R}}^N,d\omega)$ as $n\to\infty$ and that $\{k\eta_n+u_n^p-u^p\ast\eta_n\}$ converges to $k \delta_0$ in the distribution sense as $n\to\infty$.

By Lemma \[teo CP\], we have that $0\le u_n\le  \mathbb{G}_{{\mathbb{R}}^N}[k\eta_n]$ and $\mathbb{G}_{{\mathbb{R}}^N}[k\eta_n]$ converges to $\mathbb{G}_{{\mathbb{R}}^N}[k\delta_0]$ uniformly in any compact set of ${\mathbb{R}}^N\setminus \{0\}$ and in $L^1({\mathbb{R}}^N,d\omega)$. For a fixed $r>0$, there exists $N_0>0$ such that supp$(\eta_n)\subset \bar B_r(0)$ and there exists $c_{12}>0$ such that for any $n\ge N_0$, $${\|u_n\|}_{L^\infty(B^c_{r/2}(0))}\le k{\|\mathbb{G}_{{\mathbb{R}}^N}[\eta_n]\|}_{L^\infty(B^c_{r/2}(0))}\le c_{12}k{\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|}_{L^\infty( B_{r/2}^c(0))}$$ and $${\|u_n\|}_{L^1({\mathbb{R}}^N,d\omega)}\le k{\|\mathbb{G}_{{\mathbb{R}}^N}[\eta_n]\|}_{L^1({\mathbb{R}}^N,d\omega)}\le c_{12}k{\|\mathbb{G}_{{\mathbb{R}}^N}[\delta_0]\|}_{L^1({\mathbb{R}}^N,d\omega)}.$$ By Lemma \[lm 2—-1\], for 