Existence and multiplicity of $L^2$-normalized solutions for nonlinear Dirac equations with non-autonomous nonlinearities beyond the small-mass regime

For the stationary nonlinear Dirac equation $-i\sum_{k=1}^3\alpha_k\partial_k u+m\beta u=f(x,|u|)u+\omega u$ in $\mathbb{R}^3$ with prescribed $L^2$-mass $\int_{\mathbb{R}^3}|u|^2\,dx=a^2$ and frequency $\omega$ as an unknown Lagrange multiplier, the known existence and multiplicity results for non-autonomous nonlinearities $f(x,\cdot)$ satisfying hypotheses (f1)–(f5) of arXiv:2308.05393 — like their perturbative predecessors for special weighted or autonomous nonlinearities — are obtained by perturbation/Lyapunov–Schmidt reduction near the vacuum and are restricted to sufficiently small prescribed mass $a>0$. The open problem is to determine, for non-autonomous nonlinearities satisfying (f1)–(f5), whether normalized solutions exist for prescribed masses beyond this small-mass regime — in particular for arbitrary prescribed mass $a$ — and whether the known infinite multiplicity persists there.

Background

A solitary wave of a nonlinear Dirac equation in $3+1$ dimensions is a solution of the time-dependent equation of the form $\psi(t,x)=e^{-i\omega t}u(x)$; substituting this ansatz into the Soler-type model (the classical nonlinear Dirac field model of relativistic quantum physics) $i\partial_t\psi=-i\sum_{k=1}^3\alpha_k\partial_k\psi+m\beta\psi-f(x,|\psi|)\psi$ leads to the stationary equation $-i\sum_{k=1}^3\alpha_k\partial_k u+m\beta u=f(x,|u|)u+\omega u$ on $\mathbb{R}^3$, where $u$ takes values in $\mathbb{C}^4$, $m>0$ is the particle mass, and $\alpha_1,\alpha_2,\alpha_3,\beta$ are the $4\times 4$ Dirac matrices. A normalized solution is a pair $(u,\omega)$ solving the stationary equation subject to a prescribed $L^2$-norm (the mass or charge) $\int_{\mathbb{R}^3}|u|^2\,dx=a^2$, with the frequency $\omega\in\mathbb{R}$ arising as a Lagrange multiplier (the solutions constructed in arXiv:2308.05393 satisfy $m-Ca^{p-2}\le\omega_a\le 2c_a/a^2<m$ for small $a$); this fixed-charge formulation is the one relevant to relativistic density functional theory, where the particle number, not the frequency, is the physical datum. The natural energy space is the Sobolev space $H^{1/2}(\mathbb{R}^3;\mathbb{C}^4)$, and under the $L^2$-preserving dilation $u_\lambda(x)=\lambda^{3/2}u(\lambda x)$ a power nonlinearity $|u|^{p-2}u$ has critical exponent $p=8/3$, the mass-critical exponent for the Dirac problem (growth $2<p<8/3$ is mass-subcritical, $p>8/3$ mass-supercritical). The central analytic obstruction is strong indefiniteness: the free Dirac operator has spectrum $(-\infty,-m]\cup[m,\infty)$ with a spectral gap at zero energy, so the constrained energy functional is unbounded both above and below on the $L^2$-sphere and direct minimization is unavailable. Earlier strongly indefinite normalized problems were treated by Buffoni, Esteban and Séré (Adv. Nonlinear Stud. 2006) by penalization, and Nolasco (Ann. Inst. H. Poincaré C 2021) constructed a normalized solitary wave for the Maxwell–Dirac system. For the Dirac equation itself in $\mathbb{R}^3$, Ding, Yu and Zhao (J. Geom. Anal. 2023) proved the first $L^2$-normalized solitary-wave existence result, for a weighted single power $K(x)|u|^{p-2}u$ with $2<p<8/3$ and a compactness-restoring weight $K\in L^\infty$, by a perturbation argument. Bahrouni, Guo, Hajaiej and Yu (arXiv:2308.05393, 2023) generalized this to genuinely non-autonomous nonlinearities $f(x,|u|)u$ satisfying hypotheses (f1)–(f5): smoothness and positivity of $f(x,\cdot)$; a two-sided growth pinching $2<p\le q<3$ with $f(x,t)/t^{q-2}$ nonincreasing and $f(x,t)/t^{p-2}$ nondecreasing in $t$; decay of $r(x)=f(x,1)$ at infinity; and a lower bound $F(x,t)\ge L|x|^{-\tau}t^\alpha$ (with $F(x,t)=\int_0^t f(x,s)s\,ds$, $\alpha\in(0,8/3)$, $\tau\in(0,(8-3\alpha)/2)$) holding almost everywhere on an exterior cone $S=\{tx\mid t\ge1,\ x\in B_d(x_0)\}$ with $0<d<|x_0|$ and for $0\le t\le t_0$, which restores compactness through the nonlinearity's slow decay at small amplitudes. Their Theorems 1.1 and 1.2 establish, via Lyapunov–Schmidt reduction (splitting $u$ into spectral components of the linearized Dirac operator and solving a fixed-point equation for the infinite-dimensional part) combined with the Ljusternik–Schnirelmann category/genus theory (a topological multiplicity machine counting critical points via the genus of symmetric sets), the existence of a normalized solution, infinitely many distinct normalized solutions, and bifurcation on the left in the sense of the source's Definition 1.1 (along solutions with $\|u_n\|_{H^{1/2}}\to0$; the conclusion proved at the end of its Section 3.2 is that the fixed number $m$ is a bifurcation point on the left, reached as $a\to0^+$ with $\omega_a\to m$ and $\|u_a\|\to0$) — but all statements hold only for prescribed mass $a>0$ small enough. Existence and multiplicity for large prescribed mass lie outside the reach of the perturbative scheme.

Problem Statement

For the nonlinear Dirac equation in $\mathbb{R}^3$ with non-autonomous nonlinearity,

$$-i\sum_{k=1}^3\alpha_k\partial_k u+m\beta u=f(x,|u|)u+\omega u,\qquad \int_{\mathbb{R}^3}|u|^2\,dx=a^2,$$

where $a>0$ is a prescribed mass, $\omega\in\mathbb{R}$ is an unknown Lagrange multiplier, and $f$ satisfies the non-autonomous hypotheses (f1)–(f5) of Bahrouni–Guo–Hajaiej–Yu (positivity; two-sided pinching $2<p\le q<3$ with monotone ratios $f(x,t)/t^{q-2}$, $f(x,t)/t^{p-2}$; decay of $f(x,1)$ at infinity; and the lower bound $F(x,t)\ge L|x|^{-\tau}t^\alpha$ on $F(x,t)=\int_0^t f(x,s)s\,ds$, holding almost everywhere on an exterior cone and for $0\le t\le t_0$), determine the existence and multiplicity of $L^2$-normalized solutions $u\in H^{1/2}(\mathbb{R}^3;\mathbb{C}^4)$ beyond the perturbative regime — specifically:

1. Do $L^2$-normalized solutions exist for prescribed masses $a$ beyond the "any fixed $a>0$ small enough" regime of the source's Theorems 1.1–1.2, in particular for arbitrary prescribed mass?
2. Do the infinitely many distinct normalized solutions established by the source's Theorem 1.2 for small $a$ persist in that non-perturbative mass range?

References

1. Anouar Bahrouni, Qi Guo, Hichem Hajaiej, Yuanyang Yu, "Existence and Multiplicity of Normalized Solutions for Dirac Equations with non-autonomous nonlinearities" (2023), arXiv:2308.05393, DOI 10.48550/arXiv.2308.05393, https://arxiv.org/abs/2308.05393
2. Yanheng Ding, Yuanyang Yu, Fukun Zhao, "$L^2$-Normalized Solitary Wave Solutions of a Nonlinear Dirac Equation", The Journal of Geometric Analysis 33, article 69 (2023), DOI 10.1007/s12220-022-01117-5, https://doi.org/10.1007/s12220-022-01117-5
3. Boris Buffoni, Maria J. Esteban, Eric Séré, "Normalized Solutions to Strongly Indefinite Semilinear Equations", Advanced Nonlinear Studies 6 (2006), 323–347, DOI 10.1515/ans-2006-0212, https://doi.org/10.1515/ans-2006-0212
4. 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
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