Let $(-\Delta)^{s}$ the fractional laplacian operator. Suppose that the problem $$ (-\Delta)^{s}u_{s}(x)= f(u_{s}(x)), x \in \Omega$$ $$ u_{s} = 0 \textrm{ in } \mathbb{R}^{N}\setminus\Omega $$
has a regular solution (i.e Hölder continuous) for each $s \in (0,1)$. What happens to this solution when we take $s \rightarrow 1^{-}$ and when $s\rightarrow 0^{+}$? I mean, does exists a function $u$ such that $\displaystyle\lim_{s\rightarrow 1^{-}}u_{s} = u$ and $-\Delta u = f(u)$? If it is true, does $u$ preserve the qualitative properties of $u_{s}$?.
Here the fractional Lpalacian is defined in the P.V sense. The set $\Omega$ is open and bounded in $\mathbb{R}^{N}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ function.