Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities

Abstract

In this paper, we study the following nonlinear Dirac equations (NLDE) on noncompact metric graph $\mathcal{G}$ with localized nonlinearities \begin{equation} \mathcal{D} u - \omega u= a\chi_{\mathcal{K}}|u|^{p-2}u, \end{equation} where $\mathcal{D}$ is the Dirac operator on $\mathcal{G}$, $u: \mathcal{G} \to \mathbb{C}^2$, $\omega\in \mathbb{R}$, $a > 0$, $\chi_{\mathcal{K}}$ is the characteristic function of the compact core $\mathcal{K}$, and $p>2$. First, for $2<p<4$, we prove the existence of normalized solutions to (NLDE) using a perturbation argument. Then, for $p \geq 4$, we establish the assumption under which normalized solutions to (NLDE) exist. Finally, we extend these results to the case $a<0$ and, for all $p>2$, prove the existence of normalized solutions to (NLDE) when $\lambda = -mc^2$ is an eigenvalue of the operator $\mathcal{D}$. In the Appendix, we study the influence of the parameters $m, c > 0$ on the existence of normalized solutions to (NLDE). To the best of our knowledge, this is the first study to investigate the normalized solutions to (NLDE) on metric graphs.