Normalized solutions for a nonlinear Dirac equation with an inhomogeneous nonlinearity

Abstract

We study the existence of normalized solutions for the nonlinear Dirac equation \[ \begin{cases} -i\sum\limits_{k=1}^3α_k\partial_k u + mβu - |x|^{-b}|u|^{p-2}u = μu, \quad x\in\mathbb{R}^3, \int_{\mathbb{R}^3}|u|^2 dx = a, \end{cases} \] where $b\in(0,1)$, $p\in(2,3-b)$, $a>0$ is a prescribed mass, and $μ\in\mathbb{R}$ 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 $μ\in(0,m)$ and $u\in H^{\frac12}(\mathbb{R}^3;\mathbb{C}^4)$. 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 $|x|^{-b}$. We overcome these difficulties by combining a constrained min-max reduction method with a novel weighted compact embedding in $L^p(\mathbb{R}^3,|x|^{-b}\,\mathrm{d}x;\mathbb{C}^4)$. This approach circumvents the singular potential at the origin and yields a unified existence theory valid in the small-mass regime.