*Published Paper*

**Inserted:** 20 sep 2017

**Last Updated:** 29 apr 2018

**Journal:** Journal of Mathematical Physics

**Year:** 2018

**Doi:** 10.1063/1.5005998

**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 nonlinear Dirac equation which appears in the WKB limit for the Schroedinger 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 $H^{\frac{1}{2}}(\Omega,\mathbb{C}^{2})\hookrightarrow L^{4}(\Omega,\mathbb{C}^{2})$ are avoided thanks to the regularization property of the operator $(-\Delta)^{-\frac{1}{2}}$. This also allows us to prove smoothness of the solutions. Our proof follows by direct arguments.

**Keywords:**
nonlinear Dirac equation, graphene, self-consistent model