On stability and instability of standing waves for the inhomogeneous fractional Schrodinger equation

Jiayin Liu; Jiayin Liu

doi:10.3934/math.2020405

AIMS Mathematics

2020, Volume 5, Issue 6: 6298-6312. doi: 10.3934/math.2020405

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Research article

On stability and instability of standing waves for the inhomogeneous fractional Schrodinger equation

Jiayin Liu ^,

School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China

Received: 15 June 2020 Accepted: 27 July 2020 Published: 06 August 2020
MSC : 35Q55, 35B44

In this paper, we consider the stability and instability of standing waves for the inhomogeneous fractional Schrödinger equation $ i\partial_t\psi = (-\Delta)^s\psi- |x|^{-b}|\psi|^{2p}\psi. $ By applying the profile decomposition of bounded sequences in $H^s$ and variational methods, in the $L^2$-subcritical case, i.e., $0 \lt p \lt \frac{4s-2b}{N}$, we prove that the standing waves are orbitally stable. In the $L^2$-critical case, i.e., $p = \frac{4s-2b}{N}$, we show that the standing waves are strongly unstable by blow-up.
- inhomogeneous fractional Schrödinger equation,
- stability,
- instability
Citation: Jiayin Liu. On stability and instability of standing waves for the inhomogeneous fractional Schrodinger equation[J]. AIMS Mathematics, 2020, 5(6): 6298-6312. doi: 10.3934/math.2020405

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Abstract

In this paper, we consider the stability and instability of standing waves for the inhomogeneous fractional Schrödinger equation $ i\partial_t\psi = (-\Delta)^s\psi- |x|^{-b}|\psi|^{2p}\psi. $ By applying the profile decomposition of bounded sequences in $H^s$ and variational methods, in the $L^2$-subcritical case, i.e., $0 \lt p \lt \frac{4s-2b}{N}$, we prove that the standing waves are orbitally stable. In the $L^2$-critical case, i.e., $p = \frac{4s-2b}{N}$, we show that the standing waves are strongly unstable by blow-up.

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