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Existence of nonstationary bubbles in higher dimensions. (English) Zbl 1040.35116
The paper deals with existence of travelling waves for the nonlinear Schrödinger equation in \({\mathbb R}^N\) with \(\psi^3- \psi^5\)-type nonlinearity \[ \text{ i} {{\partial\psi}\over{\partial t}} +\Delta \psi - \alpha_1 \psi +\alpha_3| \psi| ^2 \psi -\alpha_5| \psi| ^4 \psi=0 \quad \text{ in}\;{\mathbb R}^N. \] It is proved, first of all, an abstract result in critical point theory which is a local variant of the classical saddle-point theorem. This result implies existence of travelling waves moving with sufficiently small velocity in space dimensions \(N\geq 4.\)

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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