Normalized solutions for a nonlinear Dirac equation with an inhomogeneous nonlinearity

We study the existence of normalized solutions for the nonlinear Dirac equation $$\{\begin{array}{l}-i{\sum }_{k=1}^3{\alpha }_k{\mathrm{\partial }}_{k}u+m\beta u-\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u=\mu u,x\in {\mathbb{R}}^3,{\int }_{{\mathbb{R}}^3}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=a,\end{array}$$ where $b\in (0,1)$, $p\in (2,3-b)$, $a>0$ is a prescribed mass, and $\mu \in \mathbb{R}$ is a Lagrange multiplier. For any $b\in (0,1)$ and $p\in (2,3-b)$, we establish the existence of a normalized solution for all sufficiently small masses $a>0$, with $\mu \in (0,m)$ and $u\in H^{\frac{1}{2}}({\mathbb{R}}^3;{\mathbb{C}}^4)$. Our results cover the full range of nonlinearities, including mass-subcritical, mass-critical, and mass-supercritical cases. The main challenges are the strongly indefinite nature of the Dirac operator and the loss of translation invariance caused by the singular weight $\mathrm{\mid }x{\mathrm{\mid }}^{-b}$. We overcome these difficulties by combining a constrained min-max reduction method with a novel weighted compact embedding in $L^p({\mathbb{R}}^3,\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{d}x;{\mathbb{C}}^4)$. This approach circumvents the singular potential at the origin and yields a unified existence theory valid in the small-mass regime.