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We consider the nonlinear Schr\"odinger equation with nonzero conditions at infinity in $\R^2$. We investigate the existence of traveling waves that are periodic in the direction transverse to the direction of propagation and minimize the energy when the momentum is kept fixed. We show that for any given value of the momentum, there is a critical value of the period such that traveling waves with period smaller than the critical value are one-dimensional, and those with larger periods depend on two variables., Comment: 56 pages

We study the standing waves for a fourth-order Schr\"odinger equation with mixed dispersion that minimize the associated energy when the $L^2-$norm (the \textit{mass}) } is kept fixed. We need some non-homogeneous Gagliardo-Nirenberg-type inequalities and we develop a method to prove such estimates that should be useful elsewhere. We prove optimal results on the existence of minimizers in the {\it mass-subcritical } and {\it mass-critical } cases. In the { \it mass supercritical} case we show that global minimizers do not exist, and we investigate the existence of local minimizers. If the mass does not exceed some threshold $ \mu_0 \in (0,+\infty)$, our results on "best" local minimizers are also optimal., Comment: Final version. The article will appear in Journal of Differential Equations 328 (2022), pp. 1-65

We prove that, if dichotomy occurs when the concentration-compactness principle is used, the dichotomizing sequence can be choosen so that a nontrivial part of it concentrates. Iterating this argument leads to a profile decomposition for arbitrary sequences of bounded Borel measures. To illustrate our results we give an application to the structure of bouded sequences in the Sobolev space $ W^{1, p}(\R^N)$., Comment: 20 pages