A mathematical arXiv preprint reports a clean dividing line for a difficult class of singular equations: positive classical global solutions exist when the scaling exponent α is below the cone frequency ϕ, but none exists when α reaches or exceeds ϕ under the stated assumptions. Below the threshold, the authors construct and classify every solution as part of one ordered family.
The dividing line
The equation is studied with zero boundary data in a Lipschitz epigraphical cone. Its scaling exponent is α=2/(1+γ), and the theorem compares that number with ϕ, a quantity tied to the cone’s geometry. The coefficient f may vary with position, subject to a local Dini-continuity condition and positive upper and lower bounds.
A family below the threshold
When α<ϕ, the authors construct a ground solution Ψ0 and a family of profiles ΨK indexed by K≥0. The ground profile satisfies Ψ0(X)≤C|X|α, giving it a controlled growth rate tied to the scaling exponent.
The profiles are quantitatively ordered. Each ΨK is no smaller than either the ground profile or K times the paper’s harmonic profile H, and no larger than Ψ0+KH. For 0≤k≤K, the difference ΨK−Ψk is nonnegative and at most (K−k)H.
At large radius, ΨK(rθ)/H(rθ) tends to K in every direction θ of the cone. The index K therefore serves as the profile’s asymptotic harmonic slope.
How every solution is pinned down
The main classification theorem says that every global solution satisfying the same assumptions is one of the constructed ΨK profiles. In the existence regime, the full solution set is therefore a one-parameter ordered family.
An accompanying growth estimate says that, when α<ϕ, the maximum of any global solution inside the cone and a ball of radius R is at most C(u)Rϕ for R≥2. The constant depends on the solution rather than serving as a universal numerical limit.
The machinery behind the result
To rule out solutions in the α≥ϕ regime, the authors compare problems on truncated cones and use scaling. To build solutions below the threshold, they solve on expanding bounded domains and pass to an increasing limit.
For the classification, the argument combines harmonic replacement, the boundary Harnack principle and quantitative oscillation reduction. One reduction step contracts the allowed slope interval to at most a factor of 1−σ on a cylinder of half the original scale.
A special case, and the limits
In the paper’s half-space corollary, where f depends only on xn, global solutions are one-dimensional when γ>1. Under those special domain and coefficient assumptions, no global solutions exist when 0<γ≤1.
This is a theorem-driven mathematical analysis, not an empirical or biological study: it examines classical global solutions rather than a participant or experimental sample. Its conclusions are conditional on the stated cone, boundary and coefficient assumptions, and no statistical uncertainty or experimental validation is reported.
For a nonconstant f, the solution family is described through bounds and its asymptotic slope rather than an explicit closed-form formula. The document is arXiv version 1, dated 20 Aug 2026; a journal, DOI and peer-review status are not reported.
Paper data and sources
Original title: Classification of global solutions to the singular equation $-Δu=f(X)\cdot u^{-γ}$ in a Lipschitz epigraphical cone
Authors: Yahong Guo, Congming Li, Chilin Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text