Relevant physical phenomena are described by nonlinear Schrödinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the (curved) energy space, for energy-subcritical nonlinearities merely satisfying Kato-type assumptions, is proven, providing the analogue of the well-established local H^1 -theory for solutions vanishing at infinity. The critical nonlinearity will be simply a byproduct of our analysis and the existing literature. Under an assumption that prevents the onset of a Benjamin–Feir type instability, global well-posedness in the energy space is proven for: (a) non-negative Hamiltonians, (b) sign-indefinite Hamiltonians under additional assumptions on the zeros of the nonlinearity, (c) generic nonlinearities and small initial data. The cases (b) and (c) only concern the 3D case.
Antonelli et al. (Tue,) studied this question.