Dynamical and variational properties of the NLS-[formula omitted] equation on the star graph

Abstract

We study the nonlinear Schrödinger equation with δs′ coupling of intensity β∈R∖{0} on the star graph Γ consisting of N half-lines. The nonlinearity has the form g(u)=|u|p−1u,p>1. In the first part of the paper, under certain restriction on β, we prove the existence of the ground state solution as a minimizer of the action functional Sω on the Nehari manifold. It appears that the family of critical points which contains a ground state consists of N profiles (one symmetric and N−1 asymmetric). In particular, for the attractive δs′ coupling (β<0) and the frequency ω above a certain threshold, we managed to specify the ground state.The second part is devoted to the study of orbital instability of the critical points. We prove spectral instability of the critical points using Grillakis/Jones Instability Theorem. Then orbital instability for p>2 follows from the fact that data-solution mapping associated with the equation is of class C2 in H1(Γ). Moreover, for p>5 we complete and concertize instability results showing strong instability (by blow up in finite time) for the particular critical points.