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

We study the semi-classical ground states of the nonlinear Maxwell-Dirac system: $$\left\{ \begin{aligned} &\al\cdot\big(i\hbar\nabla+ q(x)\fa(x)\big) w-a\bt w -\omega w - q(x)\phi(x) w = P(x)g(\jdz{w}) w\\ &-\Delta\phi=q(x)\jdz{w}^2\\ &-\Delta{A_k}=q(x)(\alpha_k w)\cdot \bar w\ \ \ \ k=1,2,3 \end{aligned}\right.$$ for $x\in\R^3$, where $\fa$ is the magnetic field, $\phi$ 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 $\hbar$ small. We also describe the concentration behavior of the solutions as $\hbar\to 0$.