An Infinite Sequence of Localized Semiclassical States for Nonlinear Maxwell–Dirac System

Abstract

In this paper, we study the following nonlinear Maxwell–Dirac system

$$\begin{aligned} {\left\{ \begin{array}{ll} \alpha \cdot \big (i\hbar \nabla +q(x){\textbf{A}}(x)\big )u-a\beta u+V(x)u-q(x)\phi (x)u=|u|^{p-2}u, \\ -\Delta \phi =q(x)|u|^2,\\ -\Delta A_k=q(x)(\alpha _ku)\cdot u,\ k=1,2,3, \end{array}\right. } \end{aligned}$$

for \(x\in {\mathbb {R}}^3\) and \(p\in (2,3)\), where \(a > 0\) is a constant, \(\alpha =(\alpha _1,\alpha _2,\alpha _3)\), \(\alpha _1,\alpha _2,\alpha _3\) and \(\beta \) are \(4\times 4\) Pauli–Dirac matrices, \({\textbf{A}}=(A_1,A_2,A_3)\) is the magnetic field, \(\phi \) is the electron field and q is the changing point-wise charge distribution. Under a local condition that V has a local trapping potential well, when \(\varepsilon >0\) is sufficiently small, we construct an infinite sequence of localized bound state solutions concentrating around the local minimum points of V. These solutions are of higher topological type in the sense that they are obtained from a symmetric linking structure. In the second part of this paper, we consider the case in which V(x) may approach a as \(|x|\rightarrow \infty \). This is a degenerate case as most works in the literature assume a strict gap condition \(\sup _{x\in {\mathbb {R}}^3} |V(x)|< a\), which is a key condition used in setting up the linking structure as well as in dealing with the compactness issues of the variational formulation.