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 \(a(x)\equiv 1\).

在某个Lipschitz域\（\ Omega \ subset {\ mathbb {R}} ^ n \）中，我们建立了满足非线性微分不等式\（-\ Delta u \ le cu ^ p \的正超谐函数的边界Harnack原理），其中\（c> 0 \）和\（1 <p <（n + \ alpha）/（n + \ alpha -2）\）具有常数\（\ alpha \）关于Green函数的下界估计\（\ Omega \）。一个论点结合了对某些绿色势能的估计和迭代方法，使我们能够证明这一点。结果适用于半线性椭圆方程的正解，例如\（-\ Delta u = a（x）u ^ p \）且a（x）为非负数并限制在\（\ Omega \）上。同样，对于在边界点具有孤立奇点的正解，我们给出了一个先验估计和一个可移动性定理。在\（\ Omega \）具有光滑边界而\（a（x）\ equiv 1 \）的情况下，前者扩展了Bidaut-Véron和Vivier给出的值（Rev Mat Iberoam 16：477–513，2000 ）。