On semiclassical states for Dirac equations

Abstract

This paper aims to study the existence and concentration of solutions for the stationary Dirac equation in $\mathbb {R}^3$ with critical nonlinearities:

$$\begin{aligned} i\varepsilon \sum \limits _{k=1}^3\alpha _k\partial _k u-a\beta u+V(x)u=P(x)f(|u|)u+Q(x)g(|u|)u, \end{aligned}$$

where $\varepsilon >0$ is a small parameter and $a>0$ is a constant. We also show the semiclassical solutions $\omega _\varepsilon$ with maximum points $x_\varepsilon$ concentrating at a special set ${\mathcal {H}}_{P}$ characterized by V(x), P(x) and Q(x), and for any sequence, $x_\varepsilon \rightarrow x_0\in {\mathcal {H}}_{P}, v_\varepsilon (x):=\omega _\varepsilon (\varepsilon x+x_\varepsilon )$ converges in $H^1(\mathbb {R}^3,\mathbb {C}^4)$ to a least energy solution u of

$$\begin{aligned} i\sum \limits _{k=1}^3\alpha _k\partial _k u-a\beta u+V(x_0)u=P(x_0)f(|u|)u+Q(x_0)g(|u|)u. \end{aligned}$$