# Non-local to local transition for ground states of fractional Schr\"{o}dinger equations on bounded domains

## TL;DR

This paper investigates how ground state solutions of nonlinear fractional Schrödinger equations on bounded domains transition to local solutions as the fractional order approaches 1, revealing a convergence behavior under specific conditions.

## Contribution

It establishes the convergence of ground states from fractional to local Schrödinger equations as the fractional parameter approaches 1, under certain assumptions.

## Key findings

- Ground states converge in L^2 as s approaches 1
- Convergence occurs along subsequences under specified conditions
- Results connect fractional and classical Schrödinger equations

## Abstract

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 $s \to 1^-$.