Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities
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arXiv
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| Format: | Preprint |
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2025
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| author | He, Zhentao Ji, Chao |
| author_facet | He, Zhentao Ji, Chao |
| contents | 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 - ωu= aχ_{\mathcal{K}}|u|^{p-2}u, \end{equation} where $\mathcal{D}$ is the Dirac operator on $\mathcal{G}$, $u: \mathcal{G} \to \mathbb{C}^2$, $ω\in \mathbb{R}$, $a > 0$, $χ_{\mathcal{K}}$ 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 $λ= -mc^2$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_15100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities He, Zhentao Ji, Chao Analysis of PDEs 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 - ωu= aχ_{\mathcal{K}}|u|^{p-2}u, \end{equation} where $\mathcal{D}$ is the Dirac operator on $\mathcal{G}$, $u: \mathcal{G} \to \mathbb{C}^2$, $ω\in \mathbb{R}$, $a > 0$, $χ_{\mathcal{K}}$ 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 $λ= -mc^2$ 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. |
| title | Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.15100 |