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

In a Lipschitz domain Ω ⊂ RN , we investigate the boundary behavior of positive superharmonic functions u satisfying the nonlinear inequality -Δ u(x)≤ cd(x,∂ Ω )-τu(x)ᵖ . Based on the fact that, in some ranges of p and τ , the possible growth rate of u near ∂ Ω is given by the reciprocal of d(x,∂ Ω )^N - 2 g_Ω (x) using the Green function g_Ω with a fixed pole, we estimate the size of a set of points in ∂ Ω where a nontangential limit superior of d(x,∂ Ω )N-2-β g_Ω (x)u(x) diverges for a given β ≥ 0 .