Sharp Boundary Growth Rate Estimate of the Singular Equation $-\Delta u=u^{-\gamma}$ in a Critical Cone

TL;DR

This paper establishes the precise boundary growth rates of solutions to a singular PDE in critical cones, revealing different behaviors depending on the parameter gamma and resolving an open question about solvability conditions.

Contribution

It provides the sharp boundary growth estimates for solutions of the singular Lane-Emden-Fowler equation in critical cones, clarifies the solvability condition, and introduces a novel discrete integral equation approach.

Findings

Different boundary growth behaviors for gamma<2, gamma=2, gamma>2

Necessary and sufficient solvability condition for growth rate

Optimal modulus of continuity derived from growth estimates

Abstract

For $\gamma>0$, we study the sharp boundary growth rate estimate of solutions to the Dirichlet problem of the singular Lane-Emden-Fowler equation \begin{equation*} -\Delta u=u^{-\gamma} \end{equation*} in a critical $C^{1,1}$ epigraphical cone $Cone_{\Sigma}$. We show that the growth rate estimate exhibits fundamentally different behaviors in the following three cases: $1<\gamma<2$, $\gamma=2$, and $\gamma>2$. Moreover, we obtain the sharp growth rate estimate near the origin for $\gamma>1$. As a consequence, we show that when $Cone_{\Sigma}$ is a $C^{1,1}$ epigraphical cone, the additional solvability condition in \cite[Theorem 1.3]{GuLiZh25} is both sufficient and necessary to achieve the growth rate therein, thereby resolving the main open question left in that paper. With the growth rate estimate, we also derive the optimal modulus of continuity for solutions via the interior Schauder estimate. Our approach is to control the values of a solution $U(x)$ in the region $\Omega=Cone_{\Sigma}\cap B_{1}$ by introducing a sequence of reference points $p_{k}=\frac{16^{1-k}}{2}\vec{e_{n}}$. From the Green function representation of $U(x)$, we derive a discrete integral equation for the sequence $a_{k}=16^{k\phi}U(p_{k})$. Such a computation converts the original PDE problem into a recursion for a discrete integral equation, which can be effectively analyzed using basic ODE methods.