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

TL;DR

This paper proves the existence of infinitely many solutions to a nonlinear Dirac equation modeling electron transport in graphene, using variational methods and regularization techniques to handle compactness issues.

Contribution

It introduces a novel variational approach to establish multiple solutions for a self-consistent nonlinear Dirac equation in two dimensions.

Findings

Existence of infinitely many stationary solutions.

Solutions are smooth due to regularization.

Addresses compactness issues in Sobolev embeddings.

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