An infinite sequence of localized semiclassical bound states for nonlinear Dirac equations

In this paper, we study the following nonlinear Dirac equation

$$\begin{aligned} -i\varepsilon \alpha \cdot \nabla u+a\beta u+V(x)u=|u|^{p-2}u,\ x\in \mathbb {R}^3, \ \mathrm{for}\ u\in H^1({{\mathbb {R}}}^3, {{\mathbb {C}}}^4), \end{aligned}$$

where \(p\in (2,3)\), \(a > 0\) 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. Under only 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 minimax characterization of higher dimensional symmetric linking structure. The existing work in the literature give finitely many such localized solutions depending on both the local behavior of the potential function V near the local minimum points of V and the global behavior of V at infinity.