We study both existence and the exact number of positive solutions of the problem(Pλ) fenced((λ fenced(| u′ |p - 2 u′)′ + f (u) = 0 in (0, 1),; u (0) = u (1) = 0,))where λ 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 → 1- frac(f (s), (1 - s)θ) = ω > 0 where θ ∈ (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 λ 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 RN having a flat core in a ball. © 2010 Elsevier Inc. All rights reserved.