Boundary regularity theory of the singular Lane-Emden-Fowler equation in a Lipschitz domain

Abstract

We study the singular Lane-Emden-Fowler equation \begin{equation} -\Delta u=f(X)\cdot u^{-\gamma} \end{equation} in a bounded Lipschitz domain $\Omega$, 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.