A mathematical preprint identifies a threshold that determines whether positive global solutions can exist for a singular elliptic equation in a Lipschitz epigraphical cone. Here, singular refers to the equation's dependence on a negative power of its positive solution. If the equation's scaling exponent is below the cone's frequency, the authors construct global solutions. If it is at least that frequency, their analysis proves that no global solution exists under the stated lower-bound condition on the coefficient.
The comparison is between two quantities. The scaling exponent is set by the singularity exponent gamma: it is two divided by one plus gamma. The cone frequency is the frequency associated with the spherical domain that defines the cone. The analysis uses H, a positive homogeneous harmonic profile, as the reference function in its solution bounds.
Below the threshold, solutions form a continuum
Below the threshold, the authors construct a family indexed by every nonnegative K. The family starts with a ground solution and includes other global solutions whose asymptotic harmonic slope is represented by K. This gives the parameter a direct meaning: it describes the harmonic part of a solution's behavior at large distances.
The family is ordered point by point. When one parameter is no larger than another, the corresponding solution is no larger at every point in the cone. Their difference is nonnegative and no larger than the parameter gap multiplied by H. Each member is also bounded below by both the ground solution and K times H, and bounded above by their sum.
The ground solution grows no faster than a problem-dependent constant times spatial distance raised to the scaling exponent. More broadly, in the solvable regime, every global solution has at most harmonic-order growth in the cone, with a constant that can depend on that solution. Together, these estimates control both the baseline member and the full solution class covered by the theorem.
A classification, not an empirical study
This is theorem-level mathematical analysis, not a study of people, experiments, or measurements. It concerns positive classical global solutions in a Lipschitz epigraphical cone. The coefficient is locally Dini-continuous and stays between a positive lower bound and a finite upper bound. No empirical sample or dataset is reported.
The classification claim is limited to that framework. Under the same assumptions, every global solution in the stated class is one of the constructed K-family members. The result therefore classifies the solutions covered by the theorem, not domains, coefficients, or solution concepts outside those assumptions.
The proof combines compactness, local growth estimates, a nonlinear boundary Harnack principle, and oscillation reduction of the asymptotic slope.
What changes in a half-space
One special case makes the threshold easier to see. In a half-space, when the coefficient depends only on the coordinate normal to the boundary, the stated corollary says solutions are one-dimensional for gamma greater than one. When gamma is positive but no greater than one, it says that no global solutions exist.
The document is a versioned arXiv preprint in mathematical analysis. The authors report no conflict of interest, and the acknowledgments record partial support for Guo and Li from listed Chinese national, Shanghai, postdoctoral, CPSF, and institute programs.
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