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.
Categories: math.AP