Normalized solutions and small mass asymptotics for a nonlinear Dirac equation with a singular weight and competing powers

Abstract

We study normalized solutions of the nonlinear Dirac equation $\begin{cases} -iα\cdot\nabla u+mβu=μu+|x|^{-b}\bigl(|u|^{q-2}-|u|^{p-2}\bigr)u&\text{in }\mathbb{R}^3,\\ \displaystyle\int_{\mathbb{R}^3}|u|^2\,dx=a, \end{cases}$ where $m>0$, $0<b<1$, $2<q<3-b$, and $q<p<\infty$. For each fixed $p>q$ and all sufficiently small $a>0$, we prove the existence of a normalized solution $(u_a,μ_a)$ with $u_a\in H^1(\mathbb{R}^3;\mathbb C^4)\cap L^\infty(\mathbb{R}^3;\mathbb C^4)$ and $μ_a\in(0,m)$. This covers the weighted Sobolev critical and supercritical ranges $p\ge3-b$. Our method combines a mass-preserving spectral reduction with a convex truncation and uniform $L^\infty$ estimates. For solutions arising from reduced minimization, we determine the energy asymptotics and the rate at which $μ_a$ approaches $m$ as $a\to0^+$. After rescaling, the upper two-component spinors converge in $H^{\frac12}(\mathbb{R}^3;\mathbb C^2)$ to the set of normalized ground states of a weighted nonlinear Schrödinger equation with exponent $q$, while the lower two-component spinors converge to zero in the same space. We also derive an asymptotic formula in $L^2(\mathbb{R}^3;\mathbb C^2)$ for the rescaled lower two-component spinor in terms of the gradient of the limiting Schrödinger ground state.