Multiple positive bound states for the subcritical NLS equation on metric graphs

We consider the Schrödinger equation with a subcritical focusing power nonlinearity on a noncompactmetricgraph,andprovethatforeveryfiniteedgethereexistsathresholdvalueof themass,beyondwhichthereexistsapositiveboundstateachievingitsmaximumonthatedge only. This bound state is characterized as a minimizer of the energy functional associated to the NLS equation, with an additional constraint (besides the mass prescription): this requires particular care in proving that the minimizer satisfies the Euler–Lagrange equation. As a consequence, for a sufficiently large mass every finite edge of the graph hosts at least one positive bound state that, owing to its minimality property, is orbitally stable.