Coupled inhomogeneous nonlinear Schrödinger system with potential in three space diemensions.

This work studies the inhomogeneous Schrödinger coupled system with potential iðtuj - Hvuj = ±|x| -τ (∑m k=1 ajk|uk||p ) |uj|p-2 uj, 1 ≤ j ≤ m. The wave function components read uj: R × R3 → C for 1 ≤ j ≤ m. The inhomogeneous singular term exponent is τ > 0. The source term is energy sub-critical: 1 < p < 3 - τ. Moreover, in order to avoid a singularity of the term |uj| p-2, one assumes that p ≥ 2. The linear Schrödinger operator reads Hv = -Δ + V, where V 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 [H¹v]m:= [{ƒεL²(R3), √{ƒεL²(R³)}]m. 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 V. The challenge is to deal with three technical problems: a coupled nonlinearity, an inhomogeneous singular term |·| -τ, and the presence of the potential V. This note naturally extends previous works by the authors about the above problem for V = 0. [ABSTRACT FROM AUTHOR]