We consider the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equationFirst, we give some remarks on Sobolev-Lorentz spaces and extend the chain rule under Lorentz norms for the fractional Laplacian (−∆) s/2 with s ∈ (0, 1] established by [1] to any s > 0. Applying this estimate and the contraction mapping principle based on Strichartz estimates in Lorentz spaces, we then establish the local well-posedness in H s for the IBNLS equation in both of subcritical case σ < σ c (s) and critical case σ = σ c (s). We also prove that the IBNLS equation is globally well-posed in H s , if the initial data is sufficiently small and 8−2b d≤ σ ≤ σ c (s) with σ < ∞.

In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation iu_t +\Delta^2 u=\lambda |x|^{-b}|u|^{\sigma}u,\quad 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+\nobreak\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 (2020) and Liu–Zhang (2021) by extending the validity of s and b .