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

Abstract

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 Schr{ö}dinger 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 $H^{\frac{1}{2}}(\Omega ,\mathbb{C})\to L^4(\Omega ,\mathbb{C})$ are avoided thanks to the regular-ization property of the operator $(-Δ)^{-\frac{1}{2}$. This also allows us to prove smoothness of the solutions. Our proof follows by direct arguments.

References in corpus (4)

• The electronic properties of graphene
• Dirac materials
• Self-adjointness of two-dimensional Dirac operators on domains
• Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity

Cited by in corpus (5)

• Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit
• On the nonlinear Dirac equation on noncompact metric graphs
• Resolvent convergence to Dirac operators on planar domains
• A note on the Dirac operator with Kirchoff-type vertex conditions on noncompact metric graphs
• An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity