Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices
Abstract
We establish sharp finite-ball shooting classifications for radial nodal Dirichlet solutions of the logarithmic equation and, within the maximal simple-zero shooting class, of the sublinear scalar-field equation. Recent whole-space uniqueness and phase-transition theorems are used as external inputs, while the finite-ball zero-curve ranges, endpoint asymptotics, and spectral consequences are proved here. The two models require different singular analyses: in the logarithmic problem the nonlinearity is not locally Lipschitz at a nodal zero and the linearized potential diverges there, whereas in the sublinear problem the limiting whole-space profile reaches a double zero at a finite support radius, beyond which continuation is nonunique. For every R>0 and k≥0, the logarithmic problem has, up to sign, a unique radial Dirichlet solution with exactly k interior zeros. Its shooting height βklog(R) is a strictly decreasing C1 bijection from (0,∞) onto (αklog,∞) and satisfies βklog(R)⟶αklog (R→∞), log(βklog(R)2) =(ρk+12)/R2+κk+1,n+o(1) (R↓0), where κk+1,n>0 is given by an explicit Bessel integral. For the sublinear problem, the maximal simple-zero branch exists precisely for R∈(ρk+1,Sk), where ρk+1 is the (k+1)-st positive zero of the regular Bessel profile Φn, and Sk is the support radius of the unique compactly supported k-node whole-space bound state.
Related
- Symmetry, monotonicity, and asymptotics of singular solutions to semilinear elliptic equations
- A remark on singular solutions of semilinear elliptic equations in two dimensions
- Residual-Based Adaptive Refinement in Hybrid Variational-Collocation PINNs for Singular Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions
- Positive normalized solutions for a singular regularized p(x)-Laplacian Dirichlet problem
- Existence of positive solutions to singular elliptic problems with gradient terms
BibTeX
@misc{chen2026radial,
title = {{Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices}},
author = {Wenjing Chen and Zexi Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15283},
eprint = {2609.15283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15283},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices
AU - Wenjing Chen
AU - Zexi Wang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15283
UR - https://arxiv.org/abs/2609.15283
AB - We establish sharp finite-ball shooting classifications for radial nodal Dirichlet solutions of the logarithmic equation and, within the maximal simple-zero shooting class, of the sublinear scalar-field equation. Recent whole-space uniqueness and phase-transition theorems are used as external inputs, while the finite-ball zero-curve ranges, endpoint asymptotics, and spectral consequences are proved here. The two models require different singular analyses: in the logarithmic problem the nonlinearity is not locally Lipschitz at a nodal zero and the linearized potential diverges there, whereas in the sublinear problem the limiting whole-space profile reaches a double zero at a finite support radius, beyond which continuation is nonunique. For every R>0 and k≥0, the logarithmic problem has, up to sign, a unique radial Dirichlet solution with exactly k interior zeros. Its shooting height β_k^log(R) is a strictly decreasing C^1 bijection from (0,∞) onto (α_k^log,∞) and satisfies β_k^log(R)⟶α_k^log (R→∞), log(β_k^log(R)^2) =(ρ_k+1^2)/R^2+κ_k+1,n+o(1) (R↓0), where κ_k+1,n>0 is given by an explicit Bessel integral. For the sublinear problem, the maximal simple-zero branch exists precisely for R∈(ρ_k+1,S_k), where ρ_k+1 is the (k+1)-st positive zero of the regular Bessel profile Φ_n, and S_k is the support radius of the unique compactly supported k-node whole-space bound state.
ER -