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

Part of: Elliptic equations and systems Existence theories Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  27 December 2018

Miao Du ,
Lixin Tian ,
Jun Wang  and
Fubao Zhang
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)
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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.

Keywords

Fractional Schrödinger equationnormalized solutionsmass concentration

MSC classification

Primary: 35J61: Semilinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35R11: Fractional partial differential equations 49J40: Variational methods including variational inequalities
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 617 - 653
Copyright
Copyright © Royal Society of Edinburgh 2018 

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