Existence and multiplicity of positive solutions for classes of singular elliptic PDEs

We consider the boundary value problem -Delta u = phi g(u)u(-alpha) in Omega, u = 0 on partial derivative Omega, where Omega subset of R-N is a bounded domain, phi is a nonnegative function in L-infinity(Omega) such that phi > 0 on some subset of Omega of positive measure, and g : [0, infinity) -> R is continuous. We establish the existence of three positive solutions when g(0) > 0 (positone), the graph of s(alpha+1)/g(s) is roughly S-shaped, and alpha > 0. We also prove that there exists at least one positive solution when g(0) < 0 (semipositone), g(s) is eventually positive for s > 0, and 0 < alpha < 1. We employ the method of sub-super solutions to prove our results. (C) 2009 Elsevier Inc. All rights reserved.