Nonrelativistic limit of normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities

Abstract

In this paper, we study the nonrelativistic limit of normalized solutions for the following nonlinear Dirac equation (NLDE) on noncompact metric graph $\G$ with finitely many edges and a non-empty compact core $\K$ \begin{equation*} \D u - \omega u= \chi_\K\abs{u}^{p-2}u, \end{equation*} under the constraint $\int_\G\abs{u}^2\,dx = 1$, where $\D$ is the Dirac operator on $\G$, $u: \G \to \mathbb{C}^2$, the frequency $\omega \in \mathbb{R}$ is part of the unknowns which arises as a Lagrange multiplier, $\chi_\K$ is the characteristic function of the compact core $\K$, and $2<p<6$. To the best of our knowledge, this is the first study to investigate the nonrelativistic limit of normalized solutions to (NLDE) on metric graphs.