Normalized solutions for a nonlinear Dirac equation

TL;DR

This work establishes the existence of a normalized stationary solution to the nonlinear Dirac equation with a Soler-type nonlinearity for exponent $\alpha$ in $(2,\tfrac{8}{3}]$ under small coupling $\gamma$. The authors cast the problem variationally on the unit $L^{2}$ sphere and perform a max–min reduction using the Dirac spectral decomposition, defining a reduced energy $\mathcal{E}_{\lambda}$ whose minimization yields a normalized solution with a Lagrange multiplier $\omega>0$. A concentration-compactness analysis ensures that minimizing sequences converge strongly (up to translations) to a nontrivial limit solving $dI(\psi)[h]=\omega(\psi|h)$, hence producing a normalized spinor $\psi$ with $|\psi|_{2}=1$. The result extends variational methods to strongly indefinite Dirac problems and provides the first normalization-based existence result for Dirac equations with Soler-type nonlinearities.

Abstract

We prove the existence of a normalized, stationary solution $Ψ\colon \mathbb{R}^{3} \to \mathbb{C}^{4}$ with frequency $w > 0$ of the nonlinear Dirac equation. The result covers the case in which the nonlinearity is the gradient of a function of the form \begin{equation*} F(Ψ) = a|(Ψ, γ^{0}Ψ)|^{\fracα{2}} + b|(Ψ, γ^{1}γ^{2} γ^{3} Ψ)|^{\fracα{2}} \end{equation*} with $α\in (2,\frac{8}{3}]$, $b \geq 0$ and $a > 0$ sufficiently small. Here $γ^{i}$, $i = 0,\ldots, 3$ are the $4 \times 4$ Dirac's matrices. We find the solution as a critical point of a suitable functional restricted to the unit sphere in $L^{2}$, and $w$ turns out to be the corresponding Lagrange multiplier.

Theorems & Definitions (25)

• Remark 1.3
• Remark 1.4
• Theorem 1.15
• Lemma 3.2
• proof
• Remark 3.6
• Lemma 3.7
• proof
• Lemma 3.8
• proof