Finite energy well-posedness for nonlinear Schr\"odinger equations with non-vanishing conditions at infinity

Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the 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