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

William Borrelli
2018 Journal of Mathematical Physics  
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 nonlin-ear 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,
more » ... issues related to the limiting Sobolev embedding H^1/2(Ω,C)→ L^4 (Ω,C) are avoided thanks to the regular-ization property of the operator (-Δ)^-1/2. This also allows us to prove smoothness of the solutions. Our proof follows by direct arguments.
doi:10.1063/1.5005998 fatcat:gq5njezlubgl5cp2bkuahoitzm