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This work studies an inhomogeneous Schrödinger coupled system in the mass-super-critical and energy-sub-critical regimes. In the focusing sign, a sharp dichotomy of global existence and scattering vs finite time blow-up of solutions is obtained using some variational methods, a sharp Gagliardo–Nirenberg-type inequality, and a new approach of Dodson and Murphy [Proc. Am. Math. Soc. 145(11), 4859–4867 (2017)]. In the defocusing sign, using a classical Morawetz estimate, the scattering of global solutions in the energy space is proved.

This work develops a local theory of the inhomogeneous coupled Schrödinger equations [Formula: see text]. Here, one treats the critical Sobolev regime [Formula: see text], where [Formula: see text] is the index of the invariant Sobolev norm under the dilatation [Formula: see text]. To the authors’ knowledge, the technique used in order to prove the existence of an energy local solution to the above-mentioned problem in the sub-critical regime s < s c, which consists of dividing the integrals on the unit ball of [Formula: see text] and its complementary, is no more applicable for s = s c. In order to overcome this difficulty, one uses two different methods. The first one consists of using Lorentz spaces with the fact that [Formula: see text], which allows us to handle the inhomogeneous term. In the second method, one uses some weighted Lebesgue spaces, which seem to be suitable to deal with the inhomogeneous term | x|− γ. In order to avoid a singularity of the source term, one considers the case p ≥ 2, which restricts the space dimensions to N ≤ 3.

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