For the existence of classical solutions, we refer to Theorem 2.5 in [@CFQ]. The uniqueness follows by Lemma \[teo CP\].$\Box$

Existence of weak solutions
===========================

In this section, we show the existence of solutions of problem (\[eq1.1\]), that is, we will prove Theorem \[teo 1\]. We first give an auxiliary lemma as follows.

\[lm 2.1\] Assume that $\mathcal{O} $ is a bounded $C^2$ domain in ${\mathbb{R}}^N$ and $\eta\in C({\mathbb{R}}^N)$ with support in $\bar \mathcal{O}$. Then there exists $c_9>0$ such that $$\label{2.3}
 |(-\Delta)^\alpha\eta(x)|\le  \frac {c_9\|\eta\|_{L^\infty(\mathcal{O})}}{1+|x|^{N+2\alpha}},\quad x\in {\mathbb{R}}^N\setminus\mathcal{O}_d,$$ where $\mathcal{O}_d=\{x\in {\mathbb{R}}^N: dist(\mathcal{O},x)\le d\}$.

[**Proof.**]{} For $x\in \mathcal{O}_d^c$ and $y\in \mathcal{O}$, there exists $c_{10}>1$ such that $$c_{10}^{-1}(1+|x|^{N+2\alpha})\le |x-y|^{N+2\alpha}\le c_{10}( 1+|x|^{N+2\alpha}).$$ By the fact that $$(-\Delta)^\alpha\eta(x) = -\int_{\mathcal{O}}\frac{\eta(y)}{|x-y|^{N+2\alpha}}dy,\quad x\in \mathcal{O}_d^c,$$ we assert (\[2.3\]) holds, which ends the proof. $\Box$

Now we are in the position to prove Theorem \[teo 1\].

[**Proof of Theorem \[teo 1\].**]{} Let $\{\mathcal{O}_n\}_{n\in{\mathbb{N}}}$ be a sequence of $C^2$ domains in ${\mathbb{R}}^N$ such that $$\Omega\cap B_{n}(0)\subset\{\mathcal{O}_n\}\subset \Omega\cap B_{n+1}(0).$$ For $\nu\in\mathfrak{M}^b_+(\Omega)$ and $n\in{\mathbb{N}}$, we denote $\nu_n=\nu\chi_n$, where $\chi_n$ is the characteristic function in $\mathcal{O}_n$. By Proposition \[pr 2.1\], problem (\[2.1\]) with $\mathcal{O}=\mathcal{O}_n$ and $\mu=\nu_{n}$ admits a unique weak solution, denoting it by $v_n$. We divide the proof in following steps.

[*Step 1. 