Some remarks on the inhomogeneous biharmonic NLS equation

Abstract

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation iut+Δ2u+λ|x|−b|u|αu=0,where λ=±1 and α, b>0. In the subctritical case, we improve the global well-posedness result obtained in Guzmán and Pastor (2020) for dimensions N=5,6,7 in the Sobolev space H2(RN). The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, α=8−2bN−4 are also reported. More precisely, we show well-posedness and a stability result with initial data in the critical space Ḣ2.