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

Abstract

We consider the focusing mass-supercritical semilinear Schrödinger equation with a repulsive Dirac delta potential on the real line $ℝ$:

$$\left\{\begin{array}{cc}i{\partial }_{t}u+\frac{1}{2}{\partial }_x^{2}u+\gamma {\delta }_{0}u+|u{|}^{p-1}u=0,\hfill & \left(t,x\right)\in ℝ\times ℝ,\hfill \\ u\left(0,x\right)=u_0\left(x\right)\in H^1\left(ℝ\right),\hfill \end{array}\right.$$

where $\gamma \le 0$, ${\delta }_0$ 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 $H^1\left(ℝ\right)$ and there exist standing wave solutions $e^{i\omega t}{Q}_{\omega ,\gamma }\left(x\right)$ when $\omega >\frac{1}{2}{\gamma }^2$, where $Q_{\omega ,\gamma }$ is a unique radial positive solution to $-\frac{1}{2}{\partial }_x^{2}Q+\omega Q-\gamma {\delta }_{0}Q=|Q{|}^{p-1}Q$. Our aim in the present paper is to find a necessary and sufficient condition on the data below the standing wave $e^{i\omega t}{Q}_{\omega ,0}$ to determine the global behavior of the solution. The similar result for NLS without potential ($\gamma =0$) 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 ($\gamma <0$) 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 $e^{i\omega t}{Q}_{\omega ,0}$. The difference comes from the existence of the potential.