Bifurcation solutions for a nonlinear Dirac equation

Abstract

In this paper, we study the following nonlinear Dirac equation (NDE)−i∑k=13αk∂ku+mβu=K(x)|u|p−2u+λu,where u:R3→ℂ4, K∈L∞(R3), m>0 is the mass of the electron, λ∈(−m,m) is an unknown parameter, ħ is Planck’s constant, ∂k=∂∂xk, α1,α2,α3, β are 4 × 4 Pauli–Dirac matrices and p∈(2,83). We present a new approach which is based on some prior estimates, and show that the spectrum point m is a bifurcation point for equation (NDE) by using variational methods.