Analysis of the Pauli equation coupled to Poisson equations for the potentials

Question

The Pauli equation on $\mathbb{R}^3$ is given by $$i\partial_t u(x,t) = (-i\nabla +A)^2 +Vu -(\sigma \cdot B) u$$ with initial condition $u(x,0) = u_0(t)$, $A\colon \mathbb{R}^3 \rightarrow \mathbb{C}^3$ a complex-valued vector field on $\mathbb{R}^3$ (i.e. the magnetic vector potential) and $V\colon \mathbb{R}^3 \rightarrow \mathbb{C}$ a complex-valued potential (e.g. the electric potential). $B := \nabla \times A$ is the magnetic field. The expression $\sigma \cdot B$ stands for $$\sigma \cdot B := \sum\limits_{i=1}^{3} \sigma_i \cdot B_i$$ where the $\sigma_i$ are the Pauli matrices. One can couple the Pauli equation to potential equations for the magnetic and electric field, i.e. $$ \Delta V = -|u|^2$$ and $$\Delta A = i(u\nabla\bar{u} -(\bar{u}\nabla u)^T)- \nabla \times (\bar{u}\sigma u) - A|u|^2$$ My question is: Is there any theory available for this system? I know that it's a parabolic-elliptic. Note that I have stated it on whole space but maybe it is easier on a bounded domain $\Omega \subset \mathbb{R}^3$. I am happy for references on this kind of system.