In this talk we consider a model for a nucleon interacting with the $\sigma$ and $\omega$ mesons in the atomic nucleus. The model is relativistic, but we study it in the nuclear physics nonrelativistic limit where is described by a nonlinear Schrödinger-type equation with a mass which depends on the solution itself. After discussing some previous results on the existence of positive solutions, I will prove the uniqueness and non-degeneracy of these ones. As an application, I will construct solutions to the relativistic $\sigma$ and $\omega$ model, which consists of one Dirac equation coupled to two Klein-Gordon equations.
The talk is based on a joint work with Mathieu Lewin