Variational properties and orbital stability of standing waves for NLS equation on a star graph

We study standing waves for a model of nonlinear Schr\"odinger equation on a graph. The graph is obtained joining $N$ halflines at a vertex, i.e. it is a star graph. At the vertex an interaction occurs due to a boundary condition of delta type with strength $\alpha\leqslant 0$ (which includes the free or Kirchhoff case $\alpha=0$) there imposed. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form $ i \frac{d}{dt}\Psi_t = H \Psi_t - | \Psi_t |^{2\mu} \Psi_t $, where $H$ is the s.a. operator which generates the linear Schr\"odinger dynamics on the graph in the Hilbert space $L^2(\GG)$. We show the existence of several families of standing waves, solutions of the form $\Phi_t=e^{i\omega t}\Psi_{\omega}$ where $\Psi_{\omega} $ is a vector amplitude parametrized by a frequency $\omega$; they exist for every sign of the coupling at the vertex, attractive ($\alpha<0$) and repulsive ($\alpha>0$) and for every $\omega > \frac{\alpha^2}{N^2}$. The number of the families depends on the number $N$ of the edges of the graph. We moreover determine the ground state, which is the standing wave the amplitude of which minimizes the action on the natural (Nehari) constraint, and order the various families of standing waves according to their increasing action, thus determining a nonlinear spectrum of the problem. Finally, we show that the ground state is orbitally stable for every allowed $\omega$ if the nonlinearity is subcritical or critical; for supercritical nonlinearities there exists an $\omega^* > \frac{\alpha^2}{N^2}$ such that the ground state is orbitally stable for $\frac{\alpha^2}{N^2}<\omega<\omega^*$. Action minimization requires technical subtleties, in particular an adaptation of decreasing rearrangements on a star graph is developed with a generalization of the P\'olya-Szeg\H o inequality.