Small amplitude solitary waves in the Dirac-Maxwell system

We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system proving the existence of solutions in which the Dirac wave function is of the form $\varphi (x,\omega )e^{-i\omega t}$, $\omega \in (-m,{\omega }_{\ast })$, with some ${\omega }_{\ast }>-m$, such that ${\varphi }_{\omega }\in H^1({\mathbb{R}}^3,{\mathbb{C}}^4)$, $\mathrm{\parallel }{\varphi }_{\omega }{\mathrm{\parallel }}_{L^2}^2=O(m-\mathrm{\mid }\omega \mathrm{\mid })$, and $\mathrm{\parallel }{\varphi }_{\omega }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}=O(m-\mathrm{\mid }\omega \mathrm{\mid })$. The method of proof is an implicit function theorem argument based on an identification of the nonrelativistic limit as the ground state of the Choquard equation.