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Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials

Published online by Cambridge University Press:  27 December 2018

Miao Du
Affiliation:
School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, P.R. China (dumiaomath@163.com)
Lixin Tian
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, P.R. China (tianlx@ujs.edu.cn)
Jun Wang
Affiliation:
Institute of Applied System Analysis, Jiangsu University, Zhenjiang, 212013, P.R. China (wangmath2011@126.com)
Fubao Zhang
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, P.R. China (zhangfubao@seu.edu.cn)

Abstract

In this paper, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrödinger equations. Comparingwith the Schrödinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo–Nirenberg–Sobolev inequality can be expressed by exact form, which improves the results of [17, 18]. By doing this, we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018 

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