Boundary growth rates for superharmonic functions satisfying a nonlinear inequality in a Lipschitz domain

Abstract

In a Lipschitz domain $$\Omega \subset \mathbb {R}^N$$ , we investigate the boundary behavior of positive superharmonic functions u satisfying the nonlinear inequality $$-\Delta u(x)\le cd(x,\partial \Omega )^{-\tau }u(x)^p$$ . Based on the fact that, in some ranges of p and $$\tau $$ , the possible growth rate of u near $$\partial \Omega $$ is given by the reciprocal of $$d(x,\partial \Omega )^{{N - 2}} g_\Omega (x)$$ using the Green function $$g_\Omega$$ with a fixed pole, we estimate the size of a set of points in $$\partial \Omega $$ where a nontangential limit superior of $$d(x,\partial \Omega )^{N-2-\beta } g_\Omega (x)u(x)$$ diverges for a given $$\beta \ge 0$$ .