Normalized solutions for a nonlinear Dirac equation

Abstract

We prove the existence of a normalized, stationary solution $\hat{I}¨:{\mathbb{R}}^3\to {\mathbb{C}}^4$ 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 $\hat{I}\pm \in (2,\frac{8}{3}]$, $b\ge 0$ and $a>0$ sufficiently small. Here $γ^{i}$, $i=0,\dots ,3$ 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.