On semi-classical limits of ground states of a nonlinear Maxwell-Dirac system

Abstract

We study the semi-classical ground states of the nonlinear Maxwell-Dirac system: \[ \left\{ \begin{aligned} &\al\cdot\big(i\hbar\nabla+ q(x)\fa(x)\big) w-a\bt w -ωw - q(x)ϕ(x) w = P(x)g(\jdz{w}) w\\ &-Δϕ=q(x)\jdz{w}^2\\ &-Δ{A_k}=q(x)(α_k w)\cdot \bar w\ \ \ \ k=1,2,3 \end{aligned}\right. \] for $x\in {\mathbb{R}}^3$, where $\fa$ is the magnetic field, $\varphi$ is the electron field and $q$ describes the changing pointwise charge distribution. We develop a variational method to establish the existence of least energy solutions for $\hslash$ small. We also describe the concentration behavior of the solutions as $\hslash \to 0$.

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• Localized concentration of semi-classical states for nonlinear Dirac equations