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Radial uniqueness and nondegeneracy for the Robin Lane-Emden equation
Abstract
In this paper, we study positive radial solutions of −Δu=up in a ball BR⊂ℝN, subject to the Robin boundary condition ∂νu+βu=0, where N≥2 and β>0. For every Sobolev subcritical exponent p>1, we prove the existence of a unique positive radial solution and its nondegeneracy with respect to radial perturbations. The proof relies on the strict monotonicity of the logarithmic slope of the normalized Lane-Emden profile, which follows from the radial Pohozaev identity. When N≥3, the same argument also recovers the classical critical existence range 0<βR<N−2 and proves radial nondegeneracy throughout this interval. At βR=(N−2)/2, we exhibit N linearly independent nonradial solutions of the linearized problem.
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BibTeX
@misc{li2026radial,
title = {{Radial uniqueness and nondegeneracy for the Robin Lane-Emden equation}},
author = {Jiaguo Li and Tianqi Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27984},
eprint = {2610.27984},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27984},
}
RIS
TY - UNPB
TI - Radial uniqueness and nondegeneracy for the Robin Lane-Emden equation
AU - Jiaguo Li
AU - Tianqi Xu
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27984
UR - https://arxiv.org/abs/2610.27984
AB - In this paper, we study positive radial solutions of −Δu=u^p in a ball B_R⊂ℝ^N, subject to the Robin boundary condition ∂_νu+βu=0, where N≥2 and β>0. For every Sobolev subcritical exponent p>1, we prove the existence of a unique positive radial solution and its nondegeneracy with respect to radial perturbations. The proof relies on the strict monotonicity of the logarithmic slope of the normalized Lane-Emden profile, which follows from the radial Pohozaev identity. When N≥3, the same argument also recovers the classical critical existence range 0<βR<N−2 and proves radial nondegeneracy throughout this interval. At βR=(N−2)/2, we exhibit N linearly independent nonradial solutions of the linearized problem.
ER -