We study the singular Lane–Emden–Fowler equation \(\begin{aligned} -\Delta u=f(X)\cdot u^{-\gamma } \end{aligned}\) 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 the scaling-invariant exponent of the above equation, and to 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.
