For the solution of $$ \begin{cases} \lambda u^\epsilon - \epsilon \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan proved that $$\lim_{\epsilon \to 0} - \epsilon \log u^\epsilon = \sqrt{2\lambda} \mathrm{dist} (x,\partial \Omega) $$

Can we prove that something similar for $$ \begin{cases} \lambda u^\epsilon + \epsilon (-\Delta)^s u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \mathbb R^N \setminus \Omega \end{cases} $$ holds? Here $(-\Delta)^s$ is the fractional Laplacian.