Global dynamics below the standing waves for the focusing semilinear Schrödinger equation with a repulsive Dirac delta potential

where $γ \leq 0$ , $δ_{0}$ denotes the Dirac delta with the mass at the origin, and $p > 5$ . By a result of Fukuizumi, Ohta, and Ozawa (2008), it is known that the system above is locally well-posed in the energy space $H^{1} (ℝ)$ and there exist standing wave solutions $e^{i ω t} Q_{ω, γ} (x)$ when $ω > \frac{1}{2} γ^{2}$ , where $Q_{ω, γ}$ is a unique radial positive solution to $- \frac{1}{2} \partial_{x}^{2} Q + ω Q - γ δ_{0} Q = | Q |^{p - 1} Q$ . Our aim in the present paper is to find a necessary and sufficient condition on the data below the standing wave $e^{i ω t} Q_{ω, 0}$ to determine the global behavior of the solution. The similar result for NLS without potential ( $γ = 0$ ) was obtained by Akahori and Nawa (2013); the scattering result was also extended by Fang, Xie, and Cazenave (2011). Our proof of the scattering result is based on the argument of Banica and Visciglia (2016), who proved all solutions scatter in the defocusing and repulsive case ( $γ < 0$ ) by the Kenig–Merle method (2006). However, the method of Banica and Visciglia cannot be applicable to our problem because the energy may be negative in the focusing case. To overcome this difficulty, we use the variational argument based on the work of Ibrahim, Masmoudi, and Nakanishi (2011). Our proof of the blow-up result is based on the method of Du, Wu, and Zhang (2016). Moreover, we determine the global dynamics of the radial solution whose mass-energy is larger than that of the standing wave $e^{i ω t} Q_{ω, 0}$ . The difference comes from the existence of the potential.

Keywords

global dynamics, standing waves, nonlinear Schrödinger equation, Dirac delta potential

Mathematical Subject Classification 2010

Primary: 35P25, 35Q55, 47J35

Milestones

Received: 16 September 2016

Accepted: 28 November 2016

Published: 23 February 2017

Authors

Masahiro Ikeda
Department of Mathematics Graduate School of Science Kyoto University Kyoto 606-8502 Japan

Takahisa Inui
Department of Mathematics Graduate School of Science Kyoto University Kyoto 606-8502 Japan

Vol. 10, No. 2, 2017

Masahiro Ikeda and Takahisa Inui

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors