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

Abstract

We consider the Schroedinger equation with a subcritical focusing power nonlinearity on a noncompact metric graph, and prove that for every finite edge there exists a threshold value of the mass, beyond which there exists a positive bound state achieving its maximum on that edge 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.