Normalized solutions of the autonomous nonlinear Dirac equation beyond small coupling and Sobolev-subcritical growth

For the autonomous nonlinear Dirac equation in $\mathbb{R}^3$ with Soler-type self-interaction, Coti Zelati and Nolasco proved the first existence result for $L^2$-normalized stationary solutions (solutions of prescribed charge, the frequency arising as a Lagrange multiplier), but under two structural restrictions: the coupling constant must be small ($\gamma\in(0,\gamma_0]$) and the nonlinearity growth exponent must satisfy $\alpha\in(2,8/3]$; a convexity hypothesis present in early preprint versions was removed by the authors in the published version. Subsequent work of Chen, Ding and Guo extended the pure-power case to the mass-supercritical range $8/3<\alpha<3$, but only in an equivalent small-coupling regime (sufficiently large speed of light). The open core is to determine the admissible parameter region for normalized solutions of the autonomous equation: existence, nonexistence, or sharp coupling thresholds beyond the smallness restriction, and existence for the bilinear Soler-type nonlinearities in the mass-supercritical range $8/3<\alpha<3$ and for Sobolev-critical and supercritical exponents $\alpha\ge3$ — including the physically canonical Soler model ($\alpha=4$), which no current variational framework covers.

Background

The nonlinear Dirac equation models a relativistic self-interacting fermion, such as the electron of the Soler model of extended particles. In $\mathbb{R}^3$ one writes the free Dirac operator $H=-i\sum_{k=1}^{3}\alpha_k\partial_k+m\beta$, where $m>0$ is the particle mass and $\alpha_1,\alpha_2,\alpha_3,\beta$ are $4\times4$ hermitian Dirac matrices satisfying $\alpha_k\alpha_\ell+\alpha_\ell\alpha_k=2\delta_{k\ell}I$ and $\alpha_k\beta+\beta\alpha_k=0$; the operator acts on spinors $\psi:\mathbb{R}^3\to\mathbb{C}^4$ and has spectrum $(-\infty,-m]\cup[m,\infty)$, unbounded in both directions. A stationary solution of the equation $i\partial_t\Psi=H\Psi-\gamma\nabla F(\Psi)$ has the form $\Psi(t,x)=e^{-i\omega t}\psi(x)$ and solves $(-i\alpha\cdot\nabla+m\beta-\omega)\psi=\gamma\nabla F(\psi)$. A normalized (or prescribed-mass) solution is one satisfying $\int_{\mathbb{R}^3}|\psi|^2dx=1$; the frequency $\omega\in(0,m)$ then appears as the Lagrange multiplier of the mass constraint. Since $|\Psi(t,x)|^2$ is interpreted as the position probability density, fixing the $L^2$ norm fixes the particle number, making the normalized problem the relativistic counterpart of the mass-constrained nonlinear Schrödinger (NLS) problem.

The Soler-type nonlinearities studied here are $F(\phi)=a|(\phi,\beta\phi)|^{\alpha/2}+b|(\phi,\gamma^1\gamma^2\gamma^3\phi)|^{\alpha/2}$, where $(\cdot,\cdot)$ is the hermitian product on $\mathbb{C}^4$, $(\phi,\beta\phi)$ and $(\phi,\gamma^1\gamma^2\gamma^3\phi)$ are the scalar and pseudoscalar bilinear densities of the spinor, and the $\gamma^k$ denote the Dirac matrices $\alpha_k$ in the standard $\gamma$-matrix notation (this matrix usage of the letter $\gamma$ is unrelated to the coupling constant $\gamma$, and the growth exponent $\alpha$ below is unrelated to the matrix label $\alpha_k$); pure powers $F(\phi)\sim|\phi|^\alpha$ are also admissible. With $b=0$ and $\alpha=4$ this is the Soler model itself. Two exponents govern the analysis. Under the mass-preserving dilation $\psi_\lambda(x)=\lambda^{3/2}\psi(\lambda x)$, the interaction energy $\int F(\psi)dx$ scales as $\lambda^{3(\alpha-2)/2}$ while the Dirac kinetic energy scales as $\lambda$; the balance point $\alpha=2+2/3=8/3$ is the mass-critical exponent: for $\alpha<8/3$ concentration weakens the nonlinearity, for $\alpha>8/3$ it strengthens it. Independently, the Sobolev embedding $H^{1/2}(\mathbb{R}^3)\subset L^p$ for $p\le3$ forces $\alpha\le3$ for the energy to be defined on the natural domain $H^{1/2}(\mathbb{R}^3,\mathbb{C}^4)$; $\alpha=3$ is Sobolev-critical and the Soler exponent $\alpha=4$ is Sobolev-supercritical.

The variational theory splits into two problems. For the fixed-frequency problem, Esteban and Séré (1995) proved, for essentially the same equation, that for every assigned $\omega\in(0,m)$ a solution exists — but its mass is not prescribed. For the normalized problem the energy $I(\psi)=\frac12\|\psi_{+}\|^2-\frac12\|\psi_{-}\|^2-\gamma\int F(\psi)dx$ (where $\psi_{\pm}$ are the projections of $\psi$ onto the positive and negative spectral subspaces of $H$) is strongly indefinite — unbounded above and below on the $L^2$-sphere — so direct minimization is impossible. Ding, Yu and Zhao (2023) obtained $L^2$-normalized solitary waves only for an inhomogeneous nonlinearity $K(x)|\phi|^\alpha$ with $K(x)\to0$ at infinity. Coti Zelati and Nolasco (arXiv 2023, published 2025) proved the first existence result for the autonomous Soler-type equation: under hypotheses (H1)–(H6) — which include the growth bound $|D^2F(\phi)|\le|\phi|^{\alpha-2}$ at infinity and a lower bound $F(\phi)\ge\bar\gamma|(\phi,\beta\phi)|^{\alpha/2}$ near the origin; a convexity hypothesis $(D^2F(\phi)\xi,\xi)\ge0$ assumed in early preprint versions was removed in the published version — and for $\alpha\in(2,8/3]$ and coupling $\gamma\in(0,\gamma_0]$, with $\gamma_0$ determined by explicit smallness conditions involving the Sobolev constants of $H^{1/2}\hookrightarrow L^q$ and a growth constant of $F$, there exist $\omega\in(0,m)$ and a normalized solution $\psi\in H^{1/2}(\mathbb{R}^3,\mathbb{C}^4)$. The smallness of the coupling (equivalently, of the coefficient $a$ in $F$) is essential to their min-max construction, and the physically central Soler model has $\alpha=4$, outside the proved range.

Later work has pushed on the exponent but not on the coupling. Chen, Ding, Guo and Wang (2024) studied the nonrelativistic limit of normalized solutions for equations with potentials and nonlocal terms in mass-subcritical and mass-critical ranges. Chen, Ding and Guo (2025) then proved, for the pure-power nonlinearity $|u|^{p-2}u$, the existence of energy ground states (minimizers of the energy among positive-energy normalized critical points) for the full Sobolev-subcritical range $p\in(2,3)$ — including the mass-supercritical range $8/3<p<3$ — but only for sufficiently large speed of light $c>c_0$, which under the mass-preserving scaling corresponds exactly to the small-coupling regime of Coti Zelati and Nolasco. Chen, Guo and Yu (2026) added multiplicity results in the same large-$c$ framework, and Wang and Chang (2026) reached mass-supercritical growth only for an inhomogeneous singular weight $|x|^{-b}|u|^{p-2}u$ with small prescribed mass. What no result covers: the autonomous equation at coupling beyond the smallness threshold, the bilinear Soler-type forms in $8/3<\alpha<3$, and all exponents $\alpha\ge3$.

Problem Statement

Determine the admissible parameter region — growth exponent $\alpha$ and coupling strength $\gamma$ — for $L^2$-normalized solutions of the autonomous nonlinear Dirac equation in $\mathbb{R}^3$, $\Big(-i\sum_{k=1}^{3}\alpha_k\partial_k+m\beta-\omega\Big)\psi=\gamma\,\nabla F(\psi),\qquad\int_{\mathbb{R}^{3}}|\psi|^{2}\,dx=1,$ with $m>0$, Lagrange-multiplier frequency $\omega\in(0,m)$, Dirac matrices $\alpha_k$, $\beta$ as above (the matrix label $\alpha_k$ being unrelated to the growth exponent $\alpha$), and $F$ of Soler type, e.g. $F(\phi)=a|(\phi,\beta\phi)|^{\alpha/2}+b|(\phi,\gamma^1\gamma^2\gamma^3\phi)|^{\alpha/2}$ (where the $\gamma^k$ are the Dirac matrices $\alpha_k$ in $\gamma$-notation, unrelated to the coupling constant $\gamma$; the Soler model is $b=0$, $\alpha=4$) or a pure power $F(\phi)\sim|\phi|^{\alpha}$. Beyond the currently covered regime — all nonlinearity classes for $\alpha\in(2,8/3]$ with $\gamma\in(0,\gamma_0]$ small, and pure powers for $\alpha\in(8/3,3)$ at small effective coupling — the following are open and constitute the target. (1) Large coupling: for $\gamma$ above the smallness threshold $\gamma_0$, does a normalized solution exist — equivalently, can the restriction $\gamma\in(0,\gamma_0]$ be removed or sharpened? This asks whether a normalized solution exists for every $\gamma>0$ within the nonlinearity classes and exponent ranges under consideration, or whether there is a critical coupling $\gamma^{*}(\alpha)$ with existence for $0<\gamma<\gamma^{*}(\alpha)$ and nonexistence for $\gamma>\gamma^{*}(\alpha)$. (2) Supercritical growth: do normalized solutions exist for the bilinear Soler-type nonlinearities in the mass-supercritical range $8/3<\alpha<3$ (where only pure powers are covered, and only at small effective coupling), and for Sobolev-critical and Sobolev-supercritical exponents $\alpha\ge3$ — including the physically canonical Soler model with $\alpha=4$ — or do they not exist? For the Sobolev-critical exponent $\alpha=3$ the natural domain is $H^{1/2}(\mathbb{R}^3,\mathbb{C}^4)$; for the Sobolev-supercritical exponents $\alpha>3$, where the energy is not defined on $H^{1/2}$ alone, existence or nonexistence is understood relative to an admissible ambient function class in which the constrained problem is well posed. These two directions jointly delimit one objective: the sharp existence theory for normalized solutions of the autonomous Soler-type nonlinear Dirac equation.

References

1. Vittorio Coti Zelati, Margherita Nolasco, Normalized solutions for a nonlinear Dirac equation, Journal of Differential Equations 414 (2025) 746–772, DOI 10.1016/j.jde.2024.09.029 (arXiv:2310.07512), https://doi.org/10.1016/j.jde.2024.09.029
2. Pan Chen, Yanheng Ding, Qi Guo, Existence and nonrelativistic limit of ground states to nonlinear Dirac equation, preprint arXiv:2507.11203 (2025), https://arxiv.org/abs/2507.11203
3. Hao Wang, Xiaojun Chang, Normalized solutions for a nonlinear Dirac equation with an inhomogeneous nonlinearity, preprint arXiv:2606.22727 (2026), https://arxiv.org/abs/2606.22727
4. Pan Chen, Qi Guo, Yuanyang Yu, Limit Behavior of Multiple Bound States of Nonlinear Dirac Equations, Journal of Geometric Analysis 36 (2026), Article 131, DOI 10.1007/s12220-026-02397-x, https://doi.org/10.1007/s12220-026-02397-x
5. Anouar Bahrouni, Qi Guo, Hichem Hajaiej, Yuanyang Yu, Existence and Multiplicity of Normalized Solutions for Dirac Equations with non-autonomous nonlinearities, preprint arXiv:2308.05393 (2023), https://arxiv.org/abs/2308.05393
6. Yanheng Ding, Yuanyang Yu, Fukun Zhao, $L^2$-Normalized Solitary Wave Solutions of a Nonlinear Dirac Equation, Journal of Geometric Analysis 33 (2023), Article 69, DOI 10.1007/s12220-022-01117-5, https://doi.org/10.1007/s12220-022-01117-5
7. Pan Chen, Yanheng Ding, Qi Guo, Hua-Yang Wang, Nonrelativistic limit of normalized solutions to a class of nonlinear Dirac equations, Calculus of Variations and Partial Differential Equations 63 (2024), Article 90, DOI 10.1007/s00526-024-02702-y, https://doi.org/10.1007/s00526-024-02702-y
8. Maria J. Esteban, Eric Séré, Stationary states of the nonlinear Dirac equation: a variational approach, Communications in Mathematical Physics 171 (1995) 323–350, DOI 10.1007/BF02099273, https://doi.org/10.1007/BF02099273
9. Vittorio Coti Zelati, Margherita Nolasco, Normalized solutions for the Klein–Gordon–Dirac system, Rendiconti Lincei, Matematica e Applicazioni 34 (2023) 101–126, DOI 10.4171/RLM/999, https://doi.org/10.4171/RLM/999
10. Margherita Nolasco, A normalized solitary wave solution of the Maxwell–Dirac equations, Annales de l'Institut Henri Poincaré C, Analyse non linéaire 38 (2021) 1681–1702, DOI 10.1016/j.anihpc.2020.12.006, https://doi.org/10.1016/j.anihpc.2020.12.006
11. Zhentao He, Chao Ji, Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities, Journal de Mathématiques Pures et Appliquées 206 (2026), Article 103837, DOI 10.1016/j.matpur.2025.103837, https://doi.org/10.1016/j.matpur.2025.103837
12. Pan Chen, Qi Guo, Multiplicity and Nonrelativistic limit of Bound States of Nonlinear Dirac Equations on Noncompact Metric Graphs with Localized Nonlinearities, preprint arXiv:2606.29514 (2026), https://arxiv.org/abs/2606.29514