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The Cauchy-Problem for 2D and 3D nonlinear Schr\"odinger equations withnon-vanishing conditions at infinity is investigated. Local well-posedness inthe energy space for energy-subcritical nonlinearities merely satisfyingKato-type assumptions is proven. Our result provides the analogue of thewell-established local $H^1$-theory for solutions vanishing at infinity, nofurther regularity assumptions on the nonlinearity are given. Globalwell-posedness is shown for defocusing nonlinearities provided that thenonlinear potential is non-negative. In addition, we also introduce globalwell-posedness in 3D for a class of nonlinearities for which the Hamiltonianenergy fails to be sign-definite such as e.g. competing focusing-defocusingnonlinearities.

Finite energy well-posedness for nonlinear Schr{\"o}dinger equations with non-vanishing conditions at infinity