The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations

Abstract

We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in $$\mathbb {R}^{n}$$ , n ≥ 3, satisfying the nonlinear elliptic inequality $$0 \le - \Delta u \le c\delta_{\Omega}(x)^{-\alpha}u^p \quad {\rm \Omega,$$ where c > 0, α ≥ 0 and p > 0 are constants, and $$\delta_{\Omega}(x)$$ is the distance from x to the boundary of Ω. The result is applied to show a Harnack inequality for such superharmonic functions. Also, we study the existence of positive solutions, with singularity on the boundary, of the nonlinear elliptic equation $$-\Delta u + Vu = f(x, u) \quad {\rm in} \Omega,$$ where V and f are Borel measurable functions conditioned by the generalized Kato class.