Normalized solutions for a nonlinear Dirac equation

We prove the existence of a normalized, stationary solution psi : R-3 -> 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(psi) = a|(psi, gamma(0) psi)|(alpha/2) + b |(psi, gamma(1) gamma(2) gamma(3) psi)|(alpha/2) with alpha is an element of (2, 8/3], b >= 0 and a > 0 sufficiently small. Here gamma(i), i = 0, ... , 3 are the 4 x 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 omega turns out to be the corresponding Lagrange multiplier. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license.