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

TL;DR

This paper investigates the existence of normalized solutions for nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities, covering various parameter regimes and extending previous results to this geometric setting.

Contribution

It is the first study to analyze normalized solutions of NLDE on metric graphs, providing existence results across different nonlinearities and parameter conditions.

Findings

Existence of solutions for 2<p<4 using perturbation methods.

Conditions for solutions when p≥4.

Solutions exist for all p>2 when λ is an eigenvalue of the Dirac operator.

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.