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We prove the existence of nontrivial finite energy traveling waves for a large class of nonlinear Schrödinger equations with nonzero conditions at infinity (includindg the Gross-Pitaevskii and the so-called 'cubic-quintic' equations) in space dimension $${ N \geq 2}$$ . We show that minimization of the energy at fixed momentum can be used whenever the associated nonlinear potential is nonnegative and it gives a set of orbitally stable traveling waves, while minimization of the action at constant kinetic energy can be used in all cases. We also explore the relationship between the families of traveling waves obtained by different methods and we prove a sharp nonexistence result for traveling waves with small energy. [ABSTRACT FROM AUTHOR]

For a large class of variational problems we prove that minimizers are symmetric whenever they are C1. [ABSTRACT FROM AUTHOR]