Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials. Issue 3 (27th December 2018)

• Record Type: Journal Article
• Authors: Du, Miao; Tian, Lixin; Wang, Jun; Zhang, Fubao
• Is Part Of: Proceedings. Volume 149:Issue 3(2019)
• Page Start: 617
• Page End: 653
• Publication Date: 2018-12-27
• Subjects: Fractional Schrödinger equation, -- normalized solutions, -- mass concentration; Primary 35J61, -- 35J20, -- 35R11, -- 49J40; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2018.41
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract: In this paper, we study the existence, nonexistence and mass concentration of L 2 -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 L 2 -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 L 2 -normalized solutions in the mass critical case.