Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem $$ \begin{cases} (-\Delta)^s u +\lambda u= 0 & x \in \Omega \\ u = 1 & x \in \mathbb R^n \setminus \Omega \end{cases} $$$$ (1) \quad \begin{cases} (-\Delta)^s u +\lambda u= 0 & x \in \Omega \\ u = 1 & x \in \mathbb R^n \setminus \Omega \end{cases} $$ 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 $$ \begin{cases} (-\Delta)^s v + \lambda v = \underbrace{- \lambda \mathbf{1}_{\Omega^c}}_{=0} + c_{n,s} \int_{\mathbb R^n \setminus \Omega} |x - z|^{-n-2s} dz & x \in \Omega \\ v = 0 & x \in \mathbb R^n \setminus \Omega \end{cases} $$$$ (2)\quad \begin{cases} (-\Delta)^s v + \lambda v = \underbrace{- \lambda \mathbf{1}_{\Omega^c}}_{=0} + c_{n,s} \int_{\mathbb R^n \setminus \Omega} |x - z|^{-n-2s} dz & x \in \Omega \\ v = 0 & x \in \mathbb R^n \setminus \Omega \end{cases} $$ 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?
