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

In this paper, we study the following nonlinear Dirac equations (NLDE) on noncompact metric graph G with localized nonlinearities {equation} D u - ω u= aχK|u|ᵖ⁻²u, {equation} where D is the Dirac operator on G, u: G → C², ω∈ R, $a > 0$, χK is the characteristic function of the compact core 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 ≥ 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 λ = -mc² is an eigenvalue of the operator 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.