Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system

Abstract

We study semi-classical ground states of nonlinear Maxwell-Dirac system with critical/subcritical nonlinearities: α · (iℏ∇ + q(x)A(x))w − aβw − ωw − q(x)ϕ(x)w = f(x, |w|)w, − Δϕ = q(x)|w|2, and \documentclass[12pt]{minimal}\begin{document}$-\Delta {A_k}=q(x) (\alpha _k w) \cdot \bar{w},\break k=1,2,3,$\end{document}−ΔAk=q(x)(αkw)·w¯,k=1,2,3, where \documentclass[12pt]{minimal}\begin{document}$x\in \mathbb {R}^3$\end{document}x∈R3, A = (A1, A2, A3) is the magnetic field, ϕ is the electron field, and q is the changing point-wise charge distribution. We develop a variational argument to establish the existence of least energy solutions for ℏ small. We also describe the concentration phenomena of the solutions as ℏ → 0.