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: $$\{\begin{array}{rl}{\displaystyle }& {\displaystyle \text{al}\cdot (i\mathrm{\hslash }\mathrm{\nabla }+q(x)\text{fa}(x))w-a\text{bt}w-\omega w-q(x)\varphi (x)w=P(x)g(\text{jdz}w)w}\\ {\displaystyle }& {\displaystyle -\mathrm{\Delta }\varphi =q(x)\text{jdz}{w}^2}\\ {\displaystyle }& {\displaystyle -\mathrm{\Delta }{A}_k=q(x)({\alpha }_{k}w)\cdot \bar{w}k=1,2,3}\end{array}$$ for $x\in {\mathbb{R}}^3$, where $\text{fa}$ 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 $\mathrm{\hslash }$ small. We also describe the concentration behavior of the solutions as $\mathrm{\hslash }\to 0$.