Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) $$i{u}_t+{\mathrm{\Delta }}^{2}u=\lambda \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\sigma }u,u(0)=u_0\in H^s({\mathbb{R}}^d),$$ where $\lambda \in \mathbb{R}$, $d\in \mathbb{N}$, $0<s<\min \{2+\frac{d}{2},\frac{3}{2}d\}$ and $0<b<\min \{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in $H^s({\mathbb{R}}^d)$ if $\frac{8-2b}{d}<\sigma <{\sigma }_c(s)$ and the initial data is sufficiently small, where ${\sigma }_c(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and ${\sigma }_c(s)=\mathrm{\infty }$ if $s\ge \frac{d}{2}$.