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

created by borrelli on 20 Sep 2017
modified on 29 Sep 2017

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Submitted paper

Inserted: 20 sep 2017
Last Updated: 29 sep 2017

Year: 2017

ArXiv: 1709.06387v1 PDF

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.

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