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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]

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable. Our proofs rely on results in Brothers and Ziemer (J Reine Angew Math 384:153–179, 1988) and Mariş (Arch Ration Mech Anal, 192:311–330, 2009) and answer questions from Brézis and Lieb (Comm Math Phys 96:97–113, 1984) and Lions (Ann Inst H Poincaré Anal Non Linéaire 1:223–283, 1984). [ABSTRACT FROM AUTHOR]

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

We prove that the nonexistence of supersonic finite-energy traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity is a general phenomenon which holds for a large class of equations. The same is true for sonic traveling waves in two dimensions. In higher dimensions we prove that sonic traveling waves, if they exist, must approach their limit at infinity in a very rigid way. In particular, we infer that there are no sonic traveling waves with finite energy and finite momentum. [ABSTRACT FROM AUTHOR]

We are interested in the existence of travelling-wave solutions to a system which modelizes the motion of an uncharged impurity in a Bose condensate. We prove that in space dimension one, there exist travelling-waves moving with velocity c if and only if c is less than the sound velocity at infinity. In this case we investigate the structure of the set of travelling-waves and we show that it contains global subcontinua in appropriate Sobolev spaces. [ABSTRACT FROM AUTHOR]

We study the one-dimensional Gross-Pitaevskii-Schrödinger equation with a potential U moving at velocity v. For a fixed v less than the sound velocity, it is proved that there exist two time-independent solutions if the potential is not too big. [ABSTRACT FROM PUBLISHER]