We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^{n}$$\end{document} , n ≥ 3, satisfying the nonlinear elliptic inequality\documentclass[12pt]{minimal}
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\begin{document}$$0 \le - \Delta u \le c\delta_{\Omega}(x)^{-\alpha}u^p \quad {\rm in}\ \Omega,$$\end{document}where c > 0, α ≥ 0 and p > 0 are constants, and \documentclass[12pt]{minimal}
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\begin{document}$$\delta_{\Omega}(x)$$\end{document} 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
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\begin{document}$$-\Delta u + Vu = f(x, u) \quad {\rm in} \ \Omega,$$\end{document}where V and f are Borel measurable functions conditioned by the generalized Kato class.
