Multiple positive bound states for NLS equations on noncompact metric graphs with an attractive potential

In this paper, we establish the existence of bounded states and geometrically distinct solutions for the subcritical NLS equation with attractive potential on metric graphs $\mathcal{G}$ when the mass $\mu$ is large enough.We show that the NLS equation exists at least as many bound states of mass $\mu$ as the number of bounded edges of $\mathcal{G}$ if the attractive potential satisfies some suitable assumptions.It is worth noting that this is different from the case of ground state, which on some graphs may fail to exist for every value of $\mu$.