Semiclassical states for nonlinear Dirac equations with singular potentials

Abstract

In this paper, we study the following nonlinear Dirac equation

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

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. Our investigation focuses on the case in which V(x) may attain $\pm a$ at somewhere or at infinity and K(x) may approach $\pm \infty$ as x accumulating at some points and as $|x|\rightarrow \infty$. This is a case with the potentials being singular, which has not been studied before as all the works in the literature require $\limsup _{|x|\rightarrow \infty } |V(x)|< a$ and $K\in L^\infty ({\mathbb {R}}^3,{\mathbb {R}}^+)$. Under some mild assumptions on V and K, for $\varepsilon >0$ small, we construct localized bound state solutions which concentrate around the singular set of K.