Normalized solutions for a nonlinear Dirac equation

Abstract

We prove the existence of a normalized, stationary solution $\Psi \colon \mathbb{R}^{3} \to \mathbb{C}^{4}$ 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(\Psi) = a|(\Psi, \gamma^{0}\Psi)|^{\frac{\alpha}{2}} + b|(\Psi, \gamma^{1}\gamma^{2} \gamma^{3} \Psi)|^{\frac{\alpha}{2}} \end{equation*} with $\alpha \in (2,\frac{8}{3}]$, $b \geq 0$ and $a > 0$ sufficiently small. Here $\gamma^{i}$, $i = 0,\ldots, 3$ 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.