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

TL;DR

This paper studies how ground state solutions of fractional Schrödinger equations transition from non-local to local behavior as the fractional parameter approaches 1, showing convergence to solutions of the classical Schrödinger equation.

Contribution

It demonstrates the local limit of ground states for fractional Schrödinger equations as the fractional order approaches 1, under specific conditions on the potential and nonlinearity.

Findings

Ground states converge in local $L^2$ to solutions of the classical Schrödinger equation as $s o 1^-$.

The convergence occurs along a subsequence under certain conditions.

The results bridge fractional and classical quantum models.

Abstract

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 $L^2_{\mathrm{loc}} (\mathbb{R}^N)$, under suitable conditions on $f$ and $V$, to a weak solution of the local problem as $s \to 1^-$.