  The Cauchy-Problem for 2D and 3D nonlinear Schr\"odinger equations with
non-vanishing conditions at infinity is investigated. Local well-posedness in
the energy space for energy-subcritical nonlinearities merely satisfying
Kato-type assumptions is proven. Our result provides the analogue of the
well-established local $H^1$-theory for solutions vanishing at infinity, no
further regularity assumptions on the nonlinearity are given. Global
well-posedness is shown for defocusing nonlinearities provided that the
nonlinear potential is non-negative. In addition, we also introduce global
well-posedness in 3D for a class of nonlinearities for which the Hamiltonian
energy fails to be sign-definite such as e.g. competing focusing-defocusing
nonlinearities.
