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{align*} \left\{ \begin{array}{ll} (-Δ)^{s} u + V(x) u = f(x,u) &\quad \mathrm{in} \ Ω, \newline u = 0 &\quad \mathrm{in} \ \mathbb{R}^N \setminus Ω, \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 $s\to 1^{-}$.

References in corpus (2)

• Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities
• The Poisson equation from non-local to local

Cited by in corpus (1)

• Non-local to local transition for ground states of fractional Schrödinger equations on ${\mathbb{R}}^N$