of the non-local character of the fractional Laplacian we are interested in the existence of weak solutions to problem (\[eq1.1\]) when $\Omega$ is a general regular domain, including $\Omega={\mathbb{R}}^N$. Before stating our main results in this paper, we introduce the definition of weak solution to (\[eq1.1\]).

\[weak definition\] A function $u\in L^1({\mathbb{R}}^N,d\omega)$ is a weak solution of (\[eq1.1\]) if $h(u)\in L^1({\mathbb{R}}^N,d\omega)$ and $$\label{weak sense}
\int_{\Omega} [u(-\Delta)^\alpha\xi+h(u)\xi]dx=\int_{\Omega} \xi d\nu,\quad \forall\xi\in \mathbb{X}_{\Omega},$$ where $\mathbb{X}_{\Omega}\subset C({\mathbb{R}}^N)$ is the space of functions $\xi$ satisfying:

\(i) the support of $\xi$ is a compact set in $\bar\Omega$;

\(ii) $(-\Delta)^\alpha\xi(x)$ exists for any $x\in\Omega$ and there exists $C>0$ such that $$|(-\Delta)^\alpha\xi(x)|\leq \frac{C}{1+|x|^{N+2\alpha}},\quad \forall x\in\Omega;$$

\(iii) there exist $\varphi\in L^1(\Omega,\rho^\alpha dx)$ and $\epsilon_0>0$ such that $|(-\Delta)_\epsilon^\alpha\xi|\le
\varphi$ a.e. in $\Omega$, for all $\epsilon\in(0,\epsilon_0]$, here $\rho(x)=\min\{1,dist(x,\partial\Omega)\}$ if $\Omega\not={\mathbb{R}}^N$ or $\rho\equiv1$ if $\Omega={\mathbb{R}}^N$.

We notice that $\mathbb{X}_{\Omega}$ coincides with the test function space $\mathbb{X}_{\alpha}$ if $\Omega$ is bounded, see [@CV2 Definition 1.1]. Moreover, the test function space $\mathbb{X}_{\Omega}$ is used as $C^{1,1}_0(\Omega)$ if $\Omega$ is bounded and $\alpha=1$, see [@V]. We denote by $G_\Omega$ the Green kernel of $(-\Delta)^\alpha$ in $\Omega\times\Omega $ and by $\mathbb{G}_\Omega[\cdot]$ the Green operator defined as $$\mathbb{G}_\Omega[\nu](x)=\int_{\Omega}G_\Omega(x,y) d\nu(y),\quad\forall \nu\in
\mathfrak{M}^b(\Omega).$$ Now we are ready to state 