Boundary regularity theory of the singular Lane-Emden-Fowler equation in a Lipschitz domain
Yahong Guo, Congming Li, Chilin Zhang
TL;DR
This work studies the singular Lane-Emden-Fowler equation $- abla^2 u=f(X)u^{-eta}$ in bounded Lipschitz domains, focusing on boundary behavior as $u\to0$ and establishing well-posedness, boundary growth rates, and ratio regularity. The authors introduce a cone-frequency framework, classify boundary cones as sub-critical, critical, or super-critical relative to $\frac{2}{1+\gamma}$, and derive sharp growth estimates via barrier constructions and a Kemper-type boundary Harnack principle. A key advance is establishing a boundary Harnack principle for the singular semi-linear equation and a Schauder-type boundary estimate through Campanato iteration, enabling continuity results for the ratio $u/v$ in favorable geometries and a counterexample in general Lipschitz domains. The results extend boundary regularity theory to non-smooth domains and provide a systematic cone-based reduction technique, with implications for singular elliptic problems in Lipschitz domains.
Abstract
We study the singular Lane-Emden-Fowler equation \begin{equation} -Δu=f(X)\cdot u^{-γ} \end{equation} in a bounded Lipschitz domain $Ω$, with the Dirichlet boundary condition and a positive, bounded function $f(X)$. A distinguishing feature is that the vanishing boundary condition introduces a singularity in the equation. We focus on the well-posedness of the equation and the growth rate of solutions near the boundary. The key is to classify the limiting cone of a boundary point into three categories based on its "frequency", and obtain distinct growth rate estimates for each case. Additionally, we discuss the boundary Harnack principle for the singular Lane-Emden-Fowler equation, which is essential in deriving the boundary growth rate estimate. To our knowledge, the boundary Harnack principle we derive is the first Kemper-type estimate for singular semi-linear equations. It notably differs from the classical one for linear equations, in particular, the boundedness of the ratio \(u/v\) does not imply its continuity. To address the lack of a suitable upper barrier, we introduce new techniques, including constructing upper barriers iteratively. We also construct a subharmonic auxiliary function $V(X)$ related to the solution $u$ in the limiting cone. The growth rate of $u(X)$ is then obtained inductively from the growth rate of the auxiliary function $V(X)$. Our results and methods offer novel insights into the behavior of singular elliptic equations in non-smooth domains.