Threshold scattering for the focusing NLS with a repulsive Dirac delta potential

Abstract

We establish the scattering of solutions to the focusing mass supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potentiali∂tu+∂x2u+γδ(x)u+|u|p−1u=0,(t,x)∈R×R, at the mass-energy threshold, namely, when Eγ(u0)[M(u0)]σ=E0(Q)[M(Q)]σ where u0∈H1(R) is the initial data, Q is the ground state of the free NLS on the real line R, Eγ is the energy, M is the mass and σ=(p+3)/(p−5). We also prove failure of the uniform space-time bounds at the mass-energy threshold.