Non-local to local transition for ground states of fractional Schr\"{o}dinger equations on ${\mathbb{R}}^N$

We consider the nonlinear fractional problem \begin{align*} (-\Delta)^{s} u + V(x) u = f(x,u) &\quad \hbox{in ${\mathbb{R}}^N$} \end{align*} We show that ground state solutions converge (along a subsequence) in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2({\mathbb{R}}^N)$, under suitable conditions on $f$ and $V$, to a weak solution of the local problem as $s\to 1^{-}$.