Dedicated to Jean-Claude Saut, who gave me water to cross the desert. AbstractFor a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we prove the existence of nontrivial finite energy traveling waves moving with speed c in any space dimension N ≥ 3. Our results are valid as well for the Gross-Pitaevskii equation and for NLS with cubic-quintic nonlinearity. AMS subject classifications. 35Q51, 35Q55, 35Q40, 35J20, 35J15, 35B65, 37K40. 1 2 .

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 [14,6] and answer questions from [3,11].

For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we prove the existence of finite energy traveling waves moving with speed c in any space dimension N ≥ 3. Our results are valid as well for the Gross-Pitaevskii equation and for NLS with cubic-quintic nonlinearity.

Dedicated to Dorel Miheţ, for his teaching, his friendship, and the inspiration he gave to me. AbstractFor a large class of variational problems we prove that minimizers are symmetric whenever they are C 1 .