In this paper, we study the following critical Dirac equation $-i\epsilon \sum _{k=1}^3{\alpha }_k{\partial }_{k}u+a\beta u+V(x)u=P(x)f(|u|)u+Q(x)|u|u,x\in {\mathbb{R}}^3$, where ɛ > 0 is a small parameter; a > 0 is a constant; α₁, α₂, α₃, and β are 4 × 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 ω_ɛ with maximum points x_ɛ of |ω_ɛ| concentrating at a special set ${\mathcal{H}}_P$ characterized by V(x), P(x), and Q(x) and for any sequence $x_{\epsilon }\to x_0\in {\mathcal{H}}_P,v_{\epsilon }(x)≔{\omega }_{\epsilon }(\epsilon x+x_{\epsilon })$ converges in $W^{1,q}({\mathbb{R}}^3,{\mathbb{C}}^4)$ for q ≥ 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.

中文翻译：

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某些关键Dirac方程的半经典解的存在

在本文中，我们研究以下关键狄拉克方程 $-\mathrm{一世}\epsilon \sum _{ķ=\mathrm{1个}}^3{\alpha }_{ķ}{\partial }_{ķ}ü+\mathrm{一个}\beta ü+V（X）ü=P（X）F（|ü|）ü+问（X）|ü|ü，X\in {\mathbb{[R}}^3$，其中ɛ > 0是一个小参数；a > 0为常数；α ₁，α ₂，α ₃和β是4×4泡利-狄拉克矩阵; 和V，P，Q和f是连续的，但不一定是类别${\mathcal{C}}^{\mathrm{1个}}$。通过使用变分方法，我们在势V（x），P（x）和Q（x）的适当假设下证明了半经典解的存在和集中。我们还表明半经典解决方案ω _ɛ最大点X _ɛ的| ω _ɛ | 专注于一个特殊的场景${\mathcal{H}}_P$由V（x），P（x）和Q（x）以及任何序列表征$X_{\epsilon }\to X_0\in {\mathcal{H}}_P，v_{\epsilon }（X）≔{\omega }_{\epsilon }（\epsilon X+X_{\epsilon }）$ 汇合 ${\mathrm{w\; ^}}^{\mathrm{1个}，q}（{\mathbb{[R}}^3，{\mathbb{C}}^4）$为q ≥2到基态溶液Ù的$-\mathrm{一世}\sum _{ķ=\mathrm{1个}}^3{\alpha }_{ķ}{\partial }_{ķ}ü+\mathrm{一个}\beta ü+V（X_0）ü=P（X_0）F（|ü|）ü+问（X_0）|ü|ü，\text{在}{\mathbb{[R}}^3$。最后，我们估计了溶液的指数衰减性质。