We show that ground state solutions to the nonlinear, fractional problem \begin{ align*} \left\{ \begin{ array}{ ll} (-\Delta)^{ s} u + V(x) u = f(x,u) &\quad \mathrm{ in} \ \Omega, \newline u = 0 &\quad \mathrm{ in} \ \mathbb{ R}^N \setminus \Omega, \end{ array} \right. \end{ align*} 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 $s \to 1^-$.
Categories: math.AP