Boundary estimates for superharmonic functions and solutions of semilinear elliptic equations with source

In a certain Lipschitz domain Ω ⊂ R n , we establish the boundary Harnack principle for positive superharmonic functions satisfying the nonlinear differential inequality - Δ u ≤ c u p , where c > 0 and 1 < p < ( n + α ) / ( n + α - 2 ) with constant α regarding the lower bound estimate of the Green function on Ω . An argument combined estimates for certain Green potentials and iteration methods enables us to prove it. Results are applicable to positive solutions of semilinear elliptic equations like - Δ u = a ( x ) u p with a ( x ) being nonnegative and bounded on Ω . Also, we present an a priori estimate and a removability theorem for positive solutions having isolated singularities at a boundary point. The former extends one given by Bidaut-Véron and Vivier (Rev Mat Iberoam 16:477–513, 2000) in the case where Ω has a smooth boundary and a ( x ) ≡ 1 .