Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in [formula omitted

Abstract

We study the local well-posedness of the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation iut+Δu=x−bf(u),u(0)=u0∈Hs(Rn),where b>0 and f(u) is a nonlinear function that behaves like λuσu with λ∈ℂ and σ>0. First, we obtain the local well-posedness result in Hs with 0≤s<n2 and 0<σ<4−2bn−2s by using the contraction mapping principle based on Strichartz estimates. We also obtain the local well-posedness result in Hs with n2≤s<minn,n2+1 and 0<σ<∞. Our results improve the local well-posedness result of Guzmán (2017) by extending the validity of not only s but also b.