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

We investigate the boundary growth of positive superharmonic functions u on a bounded domain Omega in R-n, n >= 3, satisfying the nonlinear elliptic inequality 0 <= - Delta u <= c delta(Omega)( x)(-alpha) u(p) in Omega, where c > 0, alpha >= 0 and p > 0 are constants, and delta(Omega)( x) is the distance from x to the boundary of Omega. 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) in Omega, where V and f are Borel measurable functions conditioned by the generalized Kato class.