Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs

Abstract

We investigate the existence of local minimizers for the nonlinear Schrödinger (NLS) equation with localized nonlinearity on noncompact metric graphs. In the absence of ground states, we prove that normalized local minimizers of the NLS equation do exist under suitable topological and metric assumptions of the graphs. In particular, we provide a criterion for the existence of local minimizers for the NLS equation in this article. Our results rely on the variational method and an application of Gagliardo-Nirenberg inequalities.

Keywords

• normalized local minimizer
• nls equation
• localized nonlinearity
• noncompact metric graph
• 35r02
• 35q55
• 49j40