On the concentration of semiclassical states for nonlinear Dirac equations

Abstract

In this paper, we study the following nonlinear Dirac equation \begin{document}$\begin{equation*}-i\varepsilonα·\nabla w+aβ w+V(x)w = g(|w|)w, x∈ \mathbb{R}^3, {\rm for} w∈ H^1(\mathbb R^3, \mathbb C^4), \end{equation*}$ \end{document} where \begin{document}$a > 0$\end{document} is a constant, \begin{document}$α = (α_1, α_2, α_3)$\end{document} , \begin{document}$α_1, α_2, α_3$\end{document} and \begin{document}$β$\end{document} are \begin{document}$4×4$\end{document} Pauli-Dirac matrices. Under the assumptions that \begin{document}$V$\end{document} and \begin{document}$g$\end{document} are continuous but are not necessarily of class \begin{document}$C^1$\end{document} , when \begin{document}$g$\end{document} is super-linear growth at infinity we obtain the existence of semiclassical solutions, which converge to the least energy solutions of its limit problem as \begin{document}$\varepsilon \to 0$\end{document} .