Coupled inhomogeneous nonlinear Schrödinger system with potential in three space diemensions

Abstract

This work studies the inhomogeneous Schrödinger coupled system with potential The wave function components read for . The inhomogeneous singular term exponent is . The source term is energy sub‐critical: . Moreover, in order to avoid a singularity of the term , one assumes that . The linear Schrödinger operator reads , where is a potential satisfying some assumptions which imply that the dispersive and Strichartz estimates hold. The purpose is two‐fold. First, we develop a local well‐posedness theory in the energy space . Second, we present a dichotomy of global existence and scattering versus blow‐up of energy solutions under the ground‐state threshold in the inter‐critical focusing regime. The scattering is obtained by using the new approach of Dodson–Murphy which is based on Tao's scattering criteria and Morawetz estimates. The novelty here is the presence of the potential . The challenge is to deal with three technical problems: a coupled nonlinearity, an inhomogeneous singular term , and the presence of the potential . This note naturally extends previous works by the authors about the above problem for .