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

This is the authors' abstract. We don't add key points for this paper.

We study normalized solutions of the nonlinear Dirac equation \[ {cases} -iα·∇ u+mβu=μu+|x|⁻ᵇ(|u|q-2-|u|ᵖ⁻²)u&in R^3,\\ ∫R^3|u|^2\,dx=a, {cases} \] where $m>0$, $0q$ and all sufficiently small $a>0$, we prove the existence of a normalized solution (uₐ,μₐ) with uₐ∈ H¹(R³; C⁴)∩ L^∞(R³; C⁴) and μₐ∈(0,m). This covers the weighted Sobolev critical and supercritical ranges p≥3-b. Our method combines a mass-preserving spectral reduction with a convex truncation and uniform L^∞ estimates. For solutions arising from reduced minimization, we determine the energy asymptotics and the rate at which μₐ approaches m as a→0⁺. After rescaling, the upper two-component spinors converge in H¹²(R³; C²) 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 L²(R³; C²) for the rescaled lower two-component spinor in terms of the gradient of the limiting Schrödinger ground state.