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Sufficient conditions for multiplicity and uniqueness of positive solutions of the Hénon equation
Abstract
We establish explicit sufficient conditions for multiplicity of positive solutions of the Hénon equation in the unit ball and for uniqueness in one dimension. In the Sobolev-subcritical range, a spherical harmonic function argument shows that the radial solution has Morse index greater than one and therefore differs from a ground state. In even dimensions, a degree-four perturbation in a symmetry and monotonicity cone yields a sufficient condition for nonradiality in the supercritical range. The one-dimensional uniqueness criterion follows from a comparison of weighted Dirichlet eigenvalues and a concavity estimate.
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BibTeX
@misc{du2026sufficient,
title = {{Sufficient conditions for multiplicity and uniqueness of positive solutions of the Hénon equation}},
author = {Geyang Du},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18039},
eprint = {2609.18039},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18039},
}
RIS
TY - UNPB
TI - Sufficient conditions for multiplicity and uniqueness of positive solutions of the Hénon equation
AU - Geyang Du
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18039
UR - https://arxiv.org/abs/2609.18039
AB - We establish explicit sufficient conditions for multiplicity of positive solutions of the Hénon equation in the unit ball and for uniqueness in one dimension. In the Sobolev-subcritical range, a spherical harmonic function argument shows that the radial solution has Morse index greater than one and therefore differs from a ground state. In even dimensions, a degree-four perturbation in a symmetry and monotonicity cone yields a sufficient condition for nonradiality in the supercritical range. The one-dimensional uniqueness criterion follows from a comparison of weighted Dirichlet eigenvalues and a concavity estimate.
ER -