In this paper, we study the following nonlinear Dirac equation (NDE)$-i\sum _{k=1}^3{\alpha }_k{\partial }_{k}u+m\beta u=K\left(x\right){\left|u\right|}^{p-2}u+\lambda u,$where $u:{\mathbb{R}}^3\to {ℂ}^4$, $K\in L^{\infty }\left({\mathbb{R}}^3\right)$, $m>0$ is the mass of the electron, $\lambda \in (-m,m)$ is an unknown parameter, $ħ$ is Planck’s constant, ${\partial }_k=\frac{\partial }{\partial x_k}$, ${\alpha }_1,{\alpha }_2,{\alpha }_3$, $\beta$ are 4 × 4 Pauli–Dirac matrices and $p\in (2,\frac{8}{3})$. We present a new approach which is based on some prior estimates, and show that the spectrum point $m$ is a bifurcation point for equation (NDE) by using variational methods.

中文翻译：

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非线性狄拉克方程的分岔解

在本文中，我们研究了以下非线性狄拉克方程（NDE）$-\mathrm{一世}\sum _{ķ=1}^3{\alpha }_{ķ}{\partial }_{ķ}你+米\beta 你=ķ\left(X\right){\left|你\right|}^{p-2}你+\lambda 你,$在哪里$你：{\mathbb{R}}^3\to {ℂ}^4$,$ķ\in {\mathrm{大号}}^{\infty }\left({\mathbb{R}}^3\right)$,$米>0$是电子的质量，$\lambda \in (-米,米)$是未知参数，$H$是普朗克常数，${\partial }_{ķ}=\frac{\partial }{\partial X_{ķ}}$,${\alpha }_1,{\alpha }_2,{\alpha }_3$,$\beta$是 4 × 4 Pauli-Dirac 矩阵和$p\in (2,\frac{8}{3})$. 我们提出了一种基于一些先前估计的新方法，并表明频谱点$米$是使用变分法的方程 (NDE) 的分岔点。