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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 \(N\geq 4.\)

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
Full Text: DOI
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