Dynamical and variational properties of the NLS-δs′ equation on the star graph

We study the nonlinear Schrodinger equation with delta(s)' scoupling of intensity beta is an element of R \{0} on the star graph Gamma consisting of Nhalf-lines. The nonlinearity has the form g(u) = vertical bar u vertical bar(p-1)u, p> 1. In the first part of the paper, under certain restriction on beta, we prove the existence of the ground state solution as a minimizer of the action functional S-omega on the Nehari manifold. It appears that the family of critical points which contains a ground state consists of Nprofiles (one symmetric and N-1 asymmetric). In particular, for the attractive delta(s)' scoupling (beta< 0) and the frequency.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> 2follows from the fact that data-solution mapping associated with the equation is of class C-2 in H-1(Gamma). Moreover, for p>5 we complete and concertize instability results showing strong instability (by blow up in finite time) for the particular critical points. (c) 2021 Elsevier Inc. All rights reserved.