Normalized solutions for a nonlinear Dirac equation

We prove the existence of a normalized, stationary solution $\psi :R^3\to C^4$ with frequency $\omega >0$ of the nonlinear Dirac equation. The result covers the case in which the nonlinearity is the gradient of a function of the form$F\left(\Psi \right)=a\left|\right(\Psi ,{\gamma }^0\Psi ){|}^{\frac{\alpha }{2}}+b|(\Psi ,{\gamma }^1{\gamma }^2{\gamma }^3\Psi ){|}^{\frac{\alpha }{2}}$ with $\alpha \in (2,\frac{8}{3}]$, $b\ge 0$ and $a>0$ sufficiently small. Here ${\gamma }^i$, $i=0,\dots ,3$ 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 ω turns out to be the corresponding Lagrange multiplier.