Global dynamics below the ground state for the focusing Schrödinger equation with a potential

Abstract

In this paper, we consider the nonlinear Schrodinger equation with a real-valued potential $$V=V(x)$$ . We study global behavior of solutions to the equation with data below the ground state under some conditions for the potential V and prove a scattering result and a blowing-up result in mass-supercritical and energy-subcritical. Our proof of the scattering result is based on an argument by Dodson–Murphy (Proc Am Math Soc 145(11):4859–4867, 2017). The proof of the blowing-up or growing-up result without radially symmetric assumption is based on the argument by Du–Wu–Zhang (Discrete Contin Dyn Syst 36(7):3639–3650, 2016). We can exclude the possibility of the growing-up result by the argument (Inui et al. in Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrodinger system without mass resonance, arXiv:1810.09153.; Nakanishi and Schlag in Calc Var Partial Differ Eq 44(1-2):1–45, 2012; Glassey in J Math Phys 18(9):1794–1797, 1977) if “the data and the potential are radially symmetric” or “the data has finite variance.”