Multiple solutions for a self-consistent Dirac equation in two dimensions

This paper is devoted to the variational study of an effective model for the electron transport in a graphene sample. We prove the existence of infinitely many stationary solutions for a nonlinear Dirac equation which appears in the WKB limit for the Schrödinger equation describing the semi-classical electron dynamics. The interaction term is given by a mean field, self-consistent potential which is the trace of the 3D Coulomb potential. Despite the nonlinearity being 4-homogeneous, compactness issues related to the limiting Sobolev embedding H12(Ω,C)→L4(Ω,C) are avoided, thanks to the regularization property of the operator (-Δ)-12. This also allows us to prove smoothness of the solutions. Our proof follows by direct arguments.