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

2019/12/18 by Van Duong Dinh, Dinh, Van Duong · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.1912.08750

25 pages, to appear in Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

arxiv created 2019/12/18 · openalex publication_date 2019/12/18 · arxiv updated 2019/12/19 · openalex created_date 2019/12/26 · openalex updated_date 2026/07/28

Abstract

We consider the minimizing problem for the energy functional with prescribed mass constraint related to the fractional nonlinear Schrödinger 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.

Citations

• Nonlinear fractional Schrödinger equations in one dimension
• Hitchhikerʼs guide to the fractional Sobolev spaces
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian
• Nonlinear fractional Schrödinger equations in one dimension
• Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates

Cited by

• The uniqueness and concentration behavior of solutions for a nonlinear fractional Schrödinger system

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