On a singular semilinear elliptic boundary value problem and the boundary Harnack principle

Abstract

On a bounded C²-domain D⊂R^d we consider the singular boundary-value problem 1/2Δu=f(u) in D, u∂D=φ, where d≥3, f:(0,∞)→(0,∞) is a locally Holder continuous function such that f(u)→∞ as u→0 at the rate u^−α, for some α∈(0,1), and φ is a non-negative continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in ̅D. Such solutions are shown to satisfy a boundary Harnack principle.