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

Abstract

In a certain Lipschitz domain $\Omega \subset {\mathbb {R}}^n$, we establish the boundary Harnack principle for positive superharmonic functions satisfying the nonlinear differential inequality $-\Delta u\le cu^p$, where $c>0$ and $1<p<(n+\alpha )/(n+\alpha -2)$ with constant $\alpha$ regarding the lower bound estimate of the Green function on $\Omega$. 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 $-\Delta u=a(x)u^p$ with a(x) being nonnegative and bounded on $\Omega$. 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 $\Omega$ has a smooth boundary and $a(x)\equiv 1$.