Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials. (English) Zbl 1422.35059
Summary: 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 [R. L. Frank and E. Lenzmann, Acta Math. 210, No. 2, 261–318 (2013; Zbl 1307.35315); R. L. Frank et al., Commun. Pure Appl. Math. 69, No. 9, 1671–1726 (2016; Zbl 1365.35206)]. 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.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35R11 | Fractional partial differential equations |
| 35A15 | Variational methods applied to PDEs |
| 49J40 | Variational inequalities |
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