Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation

Abstract

In this paper, we consider the inhomogeneous nonlinear Schrödinger equation $i\partial _t u +\Delta u =K(x)|u|^\alpha u,\; u(0)=u_0\in H^1({\mathbb {R}}^N),\; N\ge 3,\; |K(x)|+|x||\nabla K(x)|\lesssim |x|^{-b},\; 0<b< \min (2, N-2),\; 0<\alpha <{(4-2b)/(N-2)}$. 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 $K(x)=\mu |x|^{-b}$, $\mu \in {\mathbb {C}}$. For general potentials, we highlight the impact of the behavior at the origin and infinity on the allowed range of $\alpha$. In the defocusing case, we prove decay estimates provided that the potential satisfies some rigidity-type condition which leads to a scattering result. We give also a new scattering criterion taking into account the potential K.