We study the local well-posedness of the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation $i{u}_t+\Delta u={\left|x\right|}^{-b}f\left(u\right),u\left(0\right)=u_0\in H^s\left({\mathbb{R}}^n\right),$where $b>0$ and $f\left(u\right)$ is a nonlinear function that behaves like $\lambda$ ${\left|u\right|}^{\sigma }u$ with $\lambda$ $\in$ $ℂ$ and $\sigma >0$. First, we obtain the local well-posedness result in $H^s$ with $0\le s<\frac{n}{2}$ and $0<\sigma <\frac{4-2b}{n-2s}$ by using the contraction mapping principle based on Strichartz estimates. We also obtain the local well-posedness result in $H^s$ with $\frac{n}{2}\le s<min\left\{n,\frac{n}{2}+1\right\}$ and $0<\sigma <\infty$. Our results improve the local well-posedness result of Guzmán (2017) by extending the validity of not only $s$ but also $b$.

我们研究了非均匀非线性Schrödinger（INLS）方程的Cauchy问题的局部适定性 $\mathrm{一世}{ü}_{Ť}+\Delta ü={\left|X\right|}^{-b}F（ü），ü（0）={ü}_0\in H^s（{\mathbb{[R}}^{ñ}），$哪里 $b>0$ 和 $F（ü）$ 是非线性函数，其行为类似于 $\lambda$ ${\left|ü\right|}^{\sigma }ü$ 与 $\lambda$ $\in$ $ℂ$ 和 $\sigma >0$。首先，我们获得当地的适定性结果$H^s$ 与 $0\le s<\frac{ñ}{2}$ 和 $0<\sigma <\frac{4-2b}{ñ-2s}$通过使用基于Strichartz估计的收缩映射原理。我们还获得了当地的适定性结果$H^s$ 与 $\frac{ñ}{2}\le s<分\left\{ñ，\frac{ñ}{2}+\mathrm{1个}\right\}$ 和 $0<\sigma <\infty$。我们的研究结果不仅扩展了Guzmán（2017）$s$ 但是也 $b$。