Local well-posedness for the inhomogeneous biharmonic nonlinear Schrödinger equation in Sobolev spaces

Abstract

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] 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 Guzmán-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 $b$.

18 pages