Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities: the fully supercritical regime $p\ge 6$, sharp existence thresholds, and classification of the threshold eigenvalues $\pm mc^2$

For the nonlinear Dirac equation $\mathcal{D}u-\omega u=a\chi_{\mathcal{K}}|u|^{p-2}u$ on a noncompact metric graph $\mathcal{G}$ with a localized power nonlinearity supported on the compact core $\mathcal{K}$, normalized (prescribed $L^2$-mass) solutions were first constructed by He and Ji (arXiv:2505.15100, 2025; published in the Journal de Mathématiques Pures et Appliquées 206 (2026), 103837) for $2<p<6$, under sufficient smallness thresholds on the coupling $a$, a topological condition on $\mathcal{K}$, or a spectral assumption that $\lambda=-mc^2$ is an eigenvalue of the Dirac operator $\mathcal{D}$. Two follow-ups substantially advanced the picture: He and Ji (arXiv:2510.15378, 2025) established the nonrelativistic limit of these normalized solutions for $2<p<6$, and Chen and Guo (arXiv:2606.29514, 2026) proved multiplicity of normalized solutions for every $p\in(2,6)$ at large coupling and large speed of light. Three linked existence-side questions nevertheless remain open: (i) the fully supercritical regime $p\ge 6$, where the limiting Schrödinger equation is itself $L^2$-critical or $L^2$-supercritical on the graph and where, although the large-$a$ dichotomy of He–Ji (Theorem 1.5) is proved for every $p\ge 4$, no result decides which of its two branches holds and all unconditional existence constructions stop at $p<6$; (ii) whether the known sufficient thresholds on $a$ are sharp, i.e. replacing them by necessary-and-sufficient conditions for existence; and (iii) the spectral classification of the noncompact metric graphs with Kirchhoff-type vertex conditions for which $\lambda=-mc^2$ or $\lambda=+mc^2$ is an eigenvalue of $\mathcal{D}$ — the only uncontested sufficient criterion for $-mc^2$ is a simple cycle in the compact core, no graph is currently known to admit $+mc^2$, and the classification must also determine whether the terminal-edge criterion of Borrelli, Carlone and Tentarelli (arXiv:1902.00012, Proposition 2.7), which asserts that both $\pm mc^2$ are eigenvalues when the compact core contains a vertex at which at least two terminal edges are incident, holds under the Kirchhoff-type vertex conditions as defined.

Background

A metric graph is a one-dimensional singular space obtained by gluing intervals (edges) at their endpoints (vertices); a noncompact metric graph carries at least one half-line, and its compact core $\mathcal{K}$ is the subgraph of bounded edges together with their endpoints. Quantum graphs — metric graphs equipped with differential operators — model wave propagation in quasi-one-dimensional mesoscopic networks. For spinor-valued functions $u:\mathcal{G}\to\mathbb{C}^2$, the Dirac operator with Kirchhoff-type vertex conditions, introduced on graphs by Borrelli, Carlone and Tentarelli, acts on each edge as $\mathcal{D}u=-ic\sigma_1 u'+mc^2\sigma_3 u$ (with $m,c>0$ and Pauli matrices $\sigma_1,\sigma_3$) on the domain of $H^1$ spinors whose first component is continuous at every vertex and whose second components satisfy a signed Kirchhoff-type balance (sum of outgoing/incoming values zero) at every vertex. It is self-adjoint, with spectrum $(-\infty,-mc^2]\cup[mc^2,+\infty)$ and no eigenvalues inside the spectral gap $(-mc^2,mc^2)$ (point spectrum can occur only at the thresholds $\pm mc^2$ or embedded in the essential spectrum); the frequencies $\omega\in(-mc^2,mc^2)$ of the nonlinear equation are accordingly the bound-state frequencies, at which solutions decay on the half-lines. Localized nonlinearities — the nonlinearity $a\chi_{\mathcal{K}}|u|^{p-2}u$ is active only on the compact core, the attached half-lines being linear — model a nonlinear junction connected to linear leads, a by-now standard setting for the nonlinear Schrödinger equation (NLS) on graphs: Adami, Serra and Tilli showed that whether ground states of the fixed-frequency NLS energy exist depends on the graph topology, and Dovetta, Jeanjean and Serra later proved, for the mass-constrained problem with $L^2$-supercritical exponent $p>6$ on noncompact graphs, existence of positive normalized solutions for sufficiently small mass on periodic graphs and on graphs with suitable topological features, by mountain-pass, Morse-index and positive-energy compactness arguments. A normalized solution of the nonlinear Dirac equation (NLDE) is a solution of $\mathcal{D}u-\omega u=a\chi_{\mathcal{K}}|u|^{p-2}u$ with prescribed mass $\int_{\mathcal{G}}|u|^2\,dx=1$, the frequency $\omega\in(-mc^2,mc^2)$ being an unknown Lagrange multiplier; the associated energy functional is strongly indefinite (unbounded above and below on the $L^2$-sphere), since $\mathcal{D}$ has spectral subspaces of both signs. The critical exponents are dictated by the $L^2$-preserving scaling $u_\lambda(x)=\lambda^{1/2}u(\lambda x)$ of the one-dimensional line: the Dirac kinetic term and the nonlinearity balance exactly at $p=4$, so $2<p<4$ is mass-subcritical, $p=4$ is mass-critical for the Dirac problem, $4<p<6$ is mass-supercritical for Dirac but still subcritical for the limiting Schrödinger equation (whose one-dimensional $L^2$-critical exponent is $p=6$), and $p\ge 6$ is the fully supercritical regime in which even the nonrelativistic limit problem is critical or supercritical. He and Ji (2025) initiated the theory of normalized NLDE solutions on graphs: existence for $2<p<4$ and small $0<a<a_0$ (a threshold built from a Gagliardo–Nirenberg constant $C_{p,\mathcal{K}}$); existence for all $a>0$ when $2<p<4$ (under the assumptions of the paper's Theorem 1.2) and $\mathcal{K}$ is a tree with at most one leaf not incident to a half-line; a dichotomy for $p\ge 4$ and large $a$, with no upper bound on $p$; explicit graph constructions with an elongated edge for $4\le p<6$; and an extension to the sign-indefinite case $a<0$, for all $p>2$, under the spectral assumption that $\lambda=-mc^2$ is an eigenvalue of $\mathcal{D}$ — for which a simple cycle inside $\mathcal{K}$ is a known sufficient condition, realized by the explicit eigen-spinor $(0,\chi_{\text{cycle}})^T$. Eigenvalues at the edges $\pm mc^2$ of the essential spectrum are called threshold eigenvalues; deciding which graphs admit them is a linear spectral question that the whole sign-indefinite theory rests on.

Problem Statement

Let $\mathcal{G}$ be a connected noncompact metric graph with finitely many edges and nonempty compact core $\mathcal{K}$, let $m,c>0$, and let $\mathcal{D}$ be the Dirac operator with Kirchhoff-type vertex conditions (edge-wise action $\mathcal{D}u=-ic\sigma_1 u'+mc^2\sigma_3 u$; first spinor component continuous at vertices, signed Kirchhoff balance of the second component). Consider the normalized-solution problem

$\mathcal{D}u-\omega u=a\,\chi_{\mathcal{K}}|u|^{p-2}u,\qquad u:\mathcal{G}\to\mathbb{C}^2,\qquad \int_{\mathcal{G}}|u|^2\,dx=1,\qquad \omega\in(-mc^2,mc^2).$

Three linked questions, left open by He and Ji (arXiv:2505.15100) and by both follow-ups, complete the existence theory of normalized solutions for this equation:

(1) Fully supercritical regime. For $p\ge 6$ — where the limiting Schrödinger equation on the graph is itself $L^2$-critical ($p=6$) or $L^2$-supercritical ($p>6$) — determine for which graph topologies $\mathcal{G}$ and parameter values $(a,m,c)$, for both signs of $a$, normalized solutions exist or do not exist. For $p\ge 6$ the large-$a$ dichotomy of Theorem 1.5 of the source paper (which is proved for every $p\ge 4$, with no upper bound) applies, but no result determines which of its two branches holds; all unconditional existence constructions stop at $p<6$.

(2) Sharp thresholds. The known sufficient conditions — $0<a<a_0$ with $a_0=m^{(4-p)/2}c^{4-p}/(2C_{p,\mathcal{K}})$ for $2<p<4$, the tree-topology condition of Theorem 1.4 of the source paper (which, under the assumptions of Theorem 1.2, also covers $2<p<4$), and the threshold $a_{*,0}$ which, for $4\le p<6$, upgrades the large-$a$ dichotomy to unconditional existence via Lemma 7.2 (Remark 1.6; the relationship $a^{*}_{0}<a_{*,0}$ between the dichotomy threshold and $a_{*,0}$ is not guaranteed in general) — are not known to be necessary. Each listed condition is to be settled on its own: for the coupling thresholds $a_0$ (for $2<p<4$) and $a_{*,0}$ (for $4\le p<6$), determine whether the threshold is optimal, i.e. the exact boundary of the $a$-interval over which normalized solutions exist, or replace it by necessary-and-sufficient conditions on $(a,m,c,\mathcal{G},p)$; for the tree-topology condition, determine whether it is necessary, i.e. whether graphs failing it can still admit normalized solutions.

(3) Threshold eigenvalues. Characterize all noncompact metric graphs (with the above vertex conditions) for which $\lambda=-mc^2$, respectively $\lambda=+mc^2$, is an eigenvalue of $\mathcal{D}$. Known: a simple cycle in $\mathcal{K}$ implies $\lambda=-mc^2$ is an eigenvalue (explicit eigen-spinor supported on the cycle); whether any graph admits $\lambda=+mc^2$ as an eigenvalue is unknown, and the characterization must also determine whether the terminal-edge criterion of Borrelli–Carlone–Tentarelli (arXiv:1902.00012, Proposition 2.7), which asserts that both $\pm mc^2$ are eigenvalues when the compact core contains a vertex at which at least two terminal edges are incident, holds under the vertex conditions as defined; it is also unknown whether any graph simultaneously satisfies the eigenvalue assumption used for the $a<0$ theory and the tree-topology condition — the two currently known sufficient conditions being mutually exclusive.

These components jointly support one objective: a complete and sharp existence theory of mass-constrained Dirac bound states on noncompact metric graphs with localized nonlinearities, of which component (3) supplies the linear spectral foundation needed by the sign-indefinite part of (1)–(2). Each of (1)–(3) can be settled independently of the others, and the objective above is their conjunction.

References

1. Zhentao He, Chao Ji (2025), 'Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities', Journal de Mathématiques Pures et Appliquées 206 (2026), 103837, doi:10.1016/j.matpur.2025.103837; arXiv:2505.15100, https://arxiv.org/abs/2505.15100, https://doi.org/10.1016/j.matpur.2025.103837
2. Zhentao He, Chao Ji (2025), 'Nonrelativistic limit of normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities', arXiv:2510.15378, https://arxiv.org/abs/2510.15378
3. Pan Chen, Qi Guo (2026), 'Multiplicity and Nonrelativistic limit of Bound States of Nonlinear Dirac Equations on Noncompact Metric Graphs with Localized Nonlinearities', arXiv:2606.29514, https://arxiv.org/abs/2606.29514
4. William Borrelli, Raffaele Carlone, Lorenzo Tentarelli (2018), 'Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit', SIAM Journal on Mathematical Analysis 51 (2019), no. 2, 1046-1081, doi:10.1137/18M1211714; arXiv:1807.06937, https://arxiv.org/abs/1807.06937, https://doi.org/10.1137/18M1211714
5. William Borrelli, Raffaele Carlone, Lorenzo Tentarelli (2019), 'A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs', published in Mathematical Challenges of Zero-Range Physics, Springer INdAM Series 42 (2021), doi:10.1007/978-3-030-60453-0_4; arXiv:1902.00012, https://arxiv.org/abs/1902.00012, https://doi.org/10.1007/978-3-030-60453-0_4
6. Riccardo Adami, Enrico Serra, Paolo Tilli (2014), 'NLS ground states on graphs', Calculus of Variations and Partial Differential Equations 54 (2015), no. 1, 743-761, doi:10.1007/s00526-014-0804-z; arXiv:1406.4036, https://arxiv.org/abs/1406.4036, https://doi.org/10.1007/s00526-014-0804-z
7. Simone Dovetta, Louis Jeanjean, Enrico Serra (2024), 'Normalized solutions of $L^2$-supercritical NLS equations on noncompact metric graphs', arXiv:2404.15841, https://arxiv.org/abs/2404.15841
8. Guangze Gu, Ziwei Li, Michael Ruzhansky, Zhipeng Yang (2025), 'Nonlinear Dirac equations on noncompact quantum graphs with potentials: Multiplicity and Concentration', arXiv:2511.09285, https://arxiv.org/abs/2511.09285