Normalized solutions of nonlinear magnetic Schr\"odinger equations on metric graphs

Abstract

In this paper we first establish the theory of a magnetic Sobolev space $H^1_A(\mathcal{G},\mathbb{C})$ on metric graphs $\mathcal{G}$ and we prove the self-adjointness of its corresponding magnetic Schr\"odinger operator. Then, in this setting, we investigate the existence and multiplicity of normalized solutions to nonlinear magnetic Schr\"odinger equations on compact metric graphs and on noncompact metric graphs with localized nonlinearities or nonlinearities acting on whole metric graphs, covering the mass-subcritical, mass-critical, and mass-supercritical cases.