In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schr\"{ o}dinger (IBNLS) equation \[iu_{ t} +\Delta^{ 2} u=\lambda |x|^{ -b}|u|^{ \sigma}u,~u(0)=u_{ 0} \in H^{ s} (\mathbb R^{ d}),\] where $d\in \mathbb N$, $s\ge 0$, $0<b<4$, $\sigma>0$ and $\lambda \in \mathbb R$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is locally well-posed in $H^{ s}(\mathbb R^{ d})$ if $d\in \mathbb N$, $0\le s <\min \{ 2+\frac{ d}{ 2},\frac{ 3}{ 2}d\}$, $0<b<\min\{ 4,d,\frac{ 3}{ 2}d-s,\frac{ d}{ 2}+2-s\}$ and $0<\sigma< \sigma_{ c}(s)$. Here $\sigma_{ c}(s)=\frac{ 8-2b}{ d-2s}$ if $s<\frac{ d}{ 2}$, and $\sigma_{ c}(s)=\infty$ if $s\ge \frac{ d}{ 2}$. Our local well-posedness result improves the ones of Guzm\'{ a}n-Pastor [Nonlinear Anal. Real World Appl. 56 (2020) 103174] and Liu-Zhang [J. Differential Equations 296 (2021) 335-368] by extending the validity of $s$ and $b$.
Categories: math.AP