Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation

In this paper, we consider the inhomogeneous nonlinear Schrödinger equation \(i\partial _t u +\Delta u =K(x)|u|^\alpha u,\; u(0)=u_0\in H^1({\mathbb {R}}^N),\; N\ge 3,\; |K(x)|+|x||\nabla K(x)|\lesssim |x|^{-b},\; 0<b< \min (2, N-2),\; 0<\alpha <{(4-2b)/(N-2)}\) $i{\partial }_{t}u+\mathrm{\Delta }u={K(x)|u|}^{\alpha }u,u(0)=u_0\in H^1({\mathbb{R}}^N),N\ge 3,|K(x)|+|x||\mathrm{\nabla }K(x){|\lesssim |x|}^{-b},0<b<min(2,N-2),0<\alpha <(4-2b)/(N-2)$ . We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted \(L^2\) $L^2$ -space for a new range \(\alpha _0(b)<\alpha <(4-2b)/N\) ${\alpha }_0(b)<\alpha <(4-2b)/N$ . The value \(\alpha _0(b)\) ${\alpha }_0(b)$ is the positive root of \(N\alpha ^2+(N-2+2b)\alpha -4+2b=0,\) $N{\alpha }^2+(N-2+2b)\alpha -4+2b=0,$ which extends the Strauss exponent known for \(b=0\) $b=0$ . Our results improve the known ones for \(K(x)=\mu |x|^{-b}\) $K(x)={\mu |x|}^{-b}$ , \(\mu \in {\mathbb {C}}\) $\mu \in \mathbb{C}$ . For general potentials, we highlight the impact of the behavior at the origin and infinity on the allowed range of \(\alpha \) $\alpha$ . 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.