Non-local to local transition for ground states of fractional Schrödinger equations on bounded domains

Abstract

We show that ground state solutions to the nonlinear, fractional problem \begin{equation*} \begin{cases} (-\Delta)^{s} u + V(x) u = f(x,u) & \text{in } \Omega, \\ u = 0 & \text{in } \mathbb R^N \setminus \Omega, \end{cases} \end{equation*} on a bounded domain $\Omega \subset \mathbb R^N$, converge (along a subsequence) in $L^2 (\Omega)$, under suitable conditions on $f$ and $V$, to a solution of the local problem as $s \to 1^-$.