Existence, non-existence and blow-up behavior of minimizers for the mass-critical fractional nonlinear Schr\"odinger equations with periodic potentials

TL;DR

This paper studies the existence, non-existence, and blow-up behavior of minimizers for a fractional nonlinear Schrödinger equation with periodic potentials, providing a complete classification and detailed blow-up analysis in the mass-critical case.

Contribution

It offers a complete classification of minimizer existence and non-existence, and describes blow-up behavior in the mass-critical fractional Schrödinger equation with periodic potentials.

Findings

Complete classification of minimizer existence and non-existence.

Detailed blow-up behavior description near critical mass.

Conditions under which minimizers exist or blow up.

Abstract

We consider the minimizing problem for the energy functional with prescribed mass constraint related to the fractional nonlinear Schr\"odinger equation with periodic potentials. Using the concentration-compactness principle, we show a complete classification for the existence and non-existence of minimizers for the problem. In the mass-critical case, under a suitable assumption of the potential, we give a detailed description of blow-up behavior of minimizers once the mass tends to a critical value.