Positive solutions for infinite semipositone problems with falling zeros

We consider the positive solutions to the singular problem {-Delta u = au - f(u) -c/u(alpha) in Omega u = 0 on partial derivative Omega where 0 < alpha < 1; a > 0 and c > 0 are constants, Omega is a bounded domain with smooth boundary and f : [0, infinity) -> R is a continuous function. We assume that there exist M > 0; A > 0; p > 1 such that au - M <= f(u) <= Au(p); for all u is an element of [0, infinity). A simple example of f satisfying these assumptions is f(u) = u(p) for any p > 1. We use the method of sub-supersolutions to prove the existence of a positive solution of (P) when a > 2 lambda(1)/1+alpha and c is small. Here lambda(1) is the first eigenvalue of operator - Lambda with Dirichlet boundary conditions. We also extend our result to classes of infinite semipositone systems. (C) 2010 Elsevier Ltd. All rights reserved.