where lambda is a positive parameter, p > 1, the nonlinearity f is positive in ( 0,1), and f(0) = f(1) = 0. Assuming that f satisfies the condition lims(s -> 1) - f(s)/(1-s)(theta) = omega > 0 where theta is an element of (0, p - 1), we study its behavior near zero, and we obtain existence and exactness results for positive solutions. We prove the results using the shooting method. We show that there always exist solutions with a flat core for lambda sufficiently small. As an application, we prove the existence of a non-negative solution for a class of singular quasilinear elliptic problems in a bounded domain in R-N having a flat core in a ball. (C) 2010 Elsevier Inc. All rights reserved.