We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation iu_t +\Delta u=\lvert x\rvert^{-b} f(u),\quad u(0)=u_0 \in H^s(\mathbb{R}^n), where 0<s<\min\{n,\frac{n}{2} +1\} , 0<b<\min\{2,n-s,1+\frac{n-2s}{2}\} and f(u) is a nonlinear function that behaves like \lambda \lvert u\rvert^{\sigma} u with \lambda \in \mathbb{C} and \sigma >0 . We prove that the Cauchy problem of the INLS equation is globally well-posed in H^s(\mathbb{R}^n) if the initial data is sufficiently small and \sigma_0 <\sigma <\sigma_s , where \sigma_0 =\frac{4-2b}{n} and \sigma_s =\frac{4-2b}{n-2s} if s<\frac{n}{2} , \sigma_s =\infty if s\ge \frac{n}{2} . Our global well-posedness result improves the one of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249–286] by extending the validity of s and b . In addition, we also have the small data scattering result.

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