Semiclassical states of nonlinear Dirac equations with degenerate potential

Abstract

In this paper, we study the following nonlinear Dirac equation $$\begin{aligned} -\,i\varepsilon \alpha \cdot \nabla u+a\beta u+V(x)u=|u|^{p-2}u, x\in {\mathbb {R}}^3, \quad \mathrm{for} u\in H^1({\mathbb {R}}^3, {\mathbb {C}}^4), \end{aligned}$$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 approach a as $$|x|\rightarrow \infty $$. This is a degenerate case as most works in the literature assume a strict gap condition $$\sup _{x\in {\mathbb {R}}^3} |V(x)| 0$$ small, we construct bound state solutions concentrating around the local minimum points of V. As a consequence we construct an infinite sequence of localized bound state solutions as $$\varepsilon \rightarrow 0$$.