Uniqueness of positive radial ground states for the massive pseudo-relativistic nonlinear Schrödinger equation
Abstract
We study the semilinear pseudo-relativistic equation √(−Δ+m2) Q+ωQ=Qp−1 in Rd, with m>0, ω>0, d≥1 and 2<p<2*:=2d/(d−1) (with 2*=∞ when d=1). We prove that the positive radial variational ground state is unique. The proof rests on a mass-covariant oscillation theorem for the linearized operator, established through a purely variational reduction: the Caffarelli--Silvestre--Duffin extension for √(−Δ+m2) combined with a ground-state substitution reduces the linearized eigenvalue problem to a massless, potential-free weighted Steklov problem, whose second eigenfunction changes sign exactly once by strict superadditivity of the weighted Dirichlet energy under sign decomposition. The nondegeneracy of the linearization is then obtained from two orthogonality relations; the failure of scale invariance for m>0, which destroys the second relation used in the massless Frank--Lenzmann--Silvestre theory, is repaired by a mass-covariance identity. Global uniqueness follows by a continuation argument in the mass parameter, anchored at m=0 by the theorem of Frank--Lenzmann--Silvestre.
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BibTeX
@misc{lv2026uniqueness,
title = {{Uniqueness of positive radial ground states for the massive pseudo-relativistic nonlinear Schrödinger equation}},
author = {Zongyan Lv},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35030},
eprint = {2609.35030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35030},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniqueness of positive radial ground states for the massive pseudo-relativistic nonlinear Schrödinger equation
AU - Zongyan Lv
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35030
UR - https://arxiv.org/abs/2609.35030
AB - We study the semilinear pseudo-relativistic equation √(−Δ+m^2) Q+ωQ=Q^p−1 in R^d, with m>0, ω>0, d≥1 and 2<p<2^*:=2d/(d−1) (with 2^*=∞ when d=1). We prove that the positive radial variational ground state is unique. The proof rests on a mass-covariant oscillation theorem for the linearized operator, established through a purely variational reduction: the Caffarelli--Silvestre--Duffin extension for √(−Δ+m^2) combined with a ground-state substitution reduces the linearized eigenvalue problem to a massless, potential-free weighted Steklov problem, whose second eigenfunction changes sign exactly once by strict superadditivity of the weighted Dirichlet energy under sign decomposition. The nondegeneracy of the linearization is then obtained from two orthogonality relations; the failure of scale invariance for m>0, which destroys the second relation used in the massless Frank--Lenzmann--Silvestre theory, is repaired by a mass-covariance identity. Global uniqueness follows by a continuation argument in the mass parameter, anchored at m=0 by the theorem of Frank--Lenzmann--Silvestre.
ER -