Let D 0 = − − i α α ⋅ ⋅ ∇ ∇ = − − i ∑ ∑ i = 1 n α α i ∂ ∂ ∂ ∂ x i be the massless Dirac operator. Let g ( s ) be continuous and real-valued, with g ( 0 ) = 0 . Denote G ( s ) = ∫ ∫ 0 s g ( t ) d t . Let ϕ ϕ ∈ ∈ L l o c ∞ ∞ ( R n , C N ) be a spinor -valued solution that satisfies the stationary form of the nonlinear Dirac equation,