Constrained energy minimization and orbital stability for the NLS equation on a star graph

On a star graph g, we consider a nonlinear Schrodinger equation with focusing nonlinearity of power type and an attractive Dirac's delta potential located at the vertex. The equation can be formally written as i partial derivative(t)Psi(t)= -Delta Psi (t) - |Psi (t)|(2 mu)Psi (t) + alpha delta(0)Psi (t), where the strength alpha of the vertex interaction is negative and the wave function Psi is supposed to be continuous at the vertex. The values of the mass and energy functionals are conserved by the flow. We show that for 0 < mu <= 2 the energy at fixed mass is bounded from below and that for every mass m below a critical mass m* it attains its minimum value at a certain <(Psi)over cap>(m) is an element of H-1 (g). Moreover, the set of minimizers has the structure M = {e(i theta) (Psi) over cap (m), theta is an element of R}. Correspondingly, for every m < m* there exists a unique omega = omega (m) such that the standing wave <(Psi)over cap>(omega)e(i omega t) al is orbitally stable. To prove the above results we adapt the concentrationcompactness method to the case of a star graph. This is nontrivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to alpha =0. (C) 2013 Elsevier Masson SAS. All rights reserved.