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Uniqueness and radial nondegeneracy of nodal states for a mixed-power scalar field equation
Abstract
We consider the scalar field equation −Δu+u+λ|u|q−1u =μ|u|p−1u, & x∈ℝn, [2mm] u(x)→ 0, & |x|→∞, where n≥3, 1<q<p<(n+2)/(n−2), and λ,μ>0. We study uniqueness and radial nondegeneracy of its sign-changing radial bound states. The proof combines a comparison argument for the shooting variation across nodal intervals with uniform compactness estimates for families of bounded nodal number and continuation in a normalized parameter. We show that, for every integer k≥1, fixing the centre and requiring u(0)>0 leaves exactly one radial bound state with k nodes. Its radial linearized operator has trivial kernel in the radial space. For each fixed k, the corresponding normalized solutions form a global C1 branch.
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BibTeX
@misc{cheng2026uniqueness,
title = {{Uniqueness and radial nondegeneracy of nodal states for a mixed-power scalar field equation}},
author = {Zhibo Cheng and Sunan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38347},
eprint = {2610.38347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38347},
}
RIS
TY - UNPB
TI - Uniqueness and radial nondegeneracy of nodal states for a mixed-power scalar field equation
AU - Zhibo Cheng
AU - Sunan Wang
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38347
UR - https://arxiv.org/abs/2610.38347
AB - We consider the scalar field equation −Δu+u+λ|u|^q−1u =μ|u|^p−1u, & x∈ℝ^n, [2mm] u(x)→ 0, & |x|→∞, where n≥3, 1<q<p<(n+2)/(n−2), and λ,μ>0. We study uniqueness and radial nondegeneracy of its sign-changing radial bound states. The proof combines a comparison argument for the shooting variation across nodal intervals with uniform compactness estimates for families of bounded nodal number and continuation in a normalized parameter. We show that, for every integer k≥1, fixing the centre and requiring u(0)>0 leaves exactly one radial bound state with k nodes. Its radial linearized operator has trivial kernel in the radial space. For each fixed k, the corresponding normalized solutions form a global C^1 branch.
ER -