Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation

Abstract

In this paper we consider the inhomogeneous nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u=K(x)|u{|}^{\alpha }u,u(0)=u_0\in H^s({\mathbb{R}}^N),s=0,1,$ $N\ge 1,$ $|K(x)|+|x{|}^s|{\nabla }^{s}K(x)|\lesssim |x{|}^{-b},$ $0<b<\min (2,N-2s),$ $0<\alpha <(4-2b)/(N-2s)$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range ${\alpha }_0(b)<\alpha <(4-2b)/N$. The value ${\alpha }_0(b)$ is the positive root of $N{\alpha }^2+(N-2+2b)\alpha -4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=\mu |x{|}^{-b}$, $\mu \in \mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $\alpha$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.