## Existence of semiclassical solutions for some critical Dirac equation.(English)Zbl 1456.81166

Summary: In this paper, we study the following critical Dirac equation $$- i \varepsilon \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + V(x) u = P(x) f(| u |) u + Q(x) | u | u, x \in \mathbb{R}^3$$, where $$\varepsilon > 0$$ is a small parameter; $$a > 0$$ is a constant; $$\alpha_1, \alpha_2, \alpha_3$$, and $$\beta$$ are $$4 \times 4$$ Pauli-Dirac matrices; and $$V, P, Q$$, and $$f$$ are continuous but are not necessarily of class $$\mathcal{C}^1$$. We prove the existence and concentration of semiclassical solutions under suitable assumptions on the potentials $$V(x), P(x)$$, and $$Q(x)$$ by using variational methods. We also show the semiclassical solutions $$\omega_\varepsilon$$ with maximum points $$x_\varepsilon$$ of |$$\omega_\varepsilon$$| concentrating at a special set $$\mathcal{H}_P$$ characterized by $$V(x), P(x)$$, and $$Q(x)$$ and for any sequence $$x_\varepsilon \to x_0 \in \mathcal{H}_P, v_\varepsilon(x) := \omega_\varepsilon(\varepsilon x + x_\varepsilon)$$ converges in $$W^{1, q}(\mathbb{R}^3, \mathbb{C}^4)$$ for $$q \geq 2$$ to a ground state solution $$u$$ of $$- i \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + V(x_0) u = P(x_0) f(| u |) u + Q(x_0) | u | u, \text{in} \mathbb{R}^3$$. Finally, we estimate the exponential decay properties of solutions.