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 −iα·∇ u+mβu=μu+|x|−b(|u|q−2−|u|p−2)u&in ℝ3, ∫ℝ3|u|2 dx=a, where m>0, 0<b<1, 2<q<3−b, and q<p<∞. For each fixed p>q 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 H1/2(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.
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BibTeX
@misc{wang2026normalized,
title = {{Normalized solutions and small mass asymptotics for a nonlinear Dirac equation with a singular weight and competing powers}},
author = {Hao Wang and Xiaojun Chang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39310},
eprint = {2610.39310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39310},
}
RIS
TY - UNPB
TI - Normalized solutions and small mass asymptotics for a nonlinear Dirac equation with a singular weight and competing powers
AU - Hao Wang
AU - Xiaojun Chang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39310
UR - https://arxiv.org/abs/2610.39310
AB - We study normalized solutions of the nonlinear Dirac equation −iα·∇ u+mβu=μu+|x|^−b(|u|^q−2−|u|^p−2)u&in ℝ^3, ∫_ℝ^3|u|^2 dx=a, where m>0, 0<b<1, 2<q<3−b, and q<p<∞. 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∈ H^1(ℝ^3;ℂ^4)∩ L^∞(ℝ^3;ℂ^4) and μ_a∈(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 μ_a approaches m as a→0^+. After rescaling, the upper two-component spinors converge in H^1/2(ℝ^3;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(ℝ^3;ℂ^2) for the rescaled lower two-component spinor in terms of the gradient of the limiting Schrödinger ground state.
ER -