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Existence and uniqueness of radial bound states with prescribed number of nodes for nonlinear dispersive equations on spherically symmetric manifolds
Abstract
Standing waves eiωtu(x) of nonlinear dispersive equations, such as the nonlinear Schrödinger and Klein--Gordon equations, have a profile u that solves the semilinear elliptic equation Δu+f(u)=0. We prove the existence of radial bound states with a prescribed number of nodal surfaces for this equation on a large class of spherically symmetric manifolds and for a broad class of nonlinearities f. Under further assumptions on the metric, we prove uniqueness of such bound states for the model nonlinearity f(u)=|u|p−1u−u. Both existence and uniqueness results apply to the hyperbolic spaces ℍnλ of any curvature −λ2 and dimension n≥ 3, and to some manifolds with a singularity at the origin.
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BibTeX
@misc{badalucco2026existence,
title = {{Existence and uniqueness of radial bound states with prescribed number of nodes for nonlinear dispersive equations on spherically symmetric manifolds}},
author = {Valentino Badalucco},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13813},
eprint = {2610.13813},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13813},
}
RIS
TY - UNPB
TI - Existence and uniqueness of radial bound states with prescribed number of nodes for nonlinear dispersive equations on spherically symmetric manifolds
AU - Valentino Badalucco
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.13813
UR - https://arxiv.org/abs/2610.13813
AB - Standing waves e^iωtu(x) of nonlinear dispersive equations, such as the nonlinear Schrödinger and Klein--Gordon equations, have a profile u that solves the semilinear elliptic equation Δu+f(u)=0. We prove the existence of radial bound states with a prescribed number of nodal surfaces for this equation on a large class of spherically symmetric manifolds and for a broad class of nonlinearities f. Under further assumptions on the metric, we prove uniqueness of such bound states for the model nonlinearity f(u)=|u|^p−1u−u. Both existence and uniqueness results apply to the hyperbolic spaces ℍ^n_λ of any curvature −λ^2 and dimension n≥ 3, and to some manifolds with a singularity at the origin.
ER -