Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem 
$$
\begin{cases}
(-\Delta)^s u +\lambda u= 0 & x \in \Omega \\
u = 1 & x \in \mathbb R^n \setminus \Omega
\end{cases}
$$
where $\lambda >0$ and the fractional Laplacian 
ia $$ (-\Delta )^{s}u(x)=c_{n,s}\int \limits _{\mathbb {R} ^{n}}{{\frac {u(x)-u(y)}{|x-y|^{n+2s}}}\,dy}$$
with $$ {\displaystyle c_{n,s}={\frac {4^{s}\Gamma (n/2+s)}{\pi ^{n/2}|\Gamma (-s)|}}}$$
I know several references with data $u(x) \equiv 0$ for $x \in \mathbb R^n \setminus \Omega$, but where can I find a proof for existence and uniqueness of viscosity solutions to the problem above?
Also, is it true that the problem above is equivalent to 
$$
\begin{cases}
(-\Delta)^s v + \lambda v = \underbrace{- \lambda \mathbf{1}_{\Omega^c}}_{=0}  + c_{n,s} \int_{\mathbb R^n \setminus \Omega} |x - z|^{-n-2s} dz & x \in \Omega \\
v = 0 & x \in \mathbb R^n \setminus \Omega
\end{cases}
$$
i.e. that the change of variables $v = u-\mathbf{1}_{\Omega^c} $ can be performed to reduce the original problem to one with homogeneous data and a source term?