On the existence and non-existence of positive solutions for a class of singular infinite semipositone problems

In this paper we consider the existence and non-existence of positive solutions of singular nonlinear semipositone problem of the form $\left\{ \begin{gathered} - div(|x|^{ - ap} |\nabla u|^{p - 2} \nabla u) = \lambda |x|^{ - (a + 1)p + b} (f(u) - \frac{1} {{u^\alpha }}),x \in \Omega , \hfill \\ u = 0,x \in \partial \Omega , \hfill \\ \end{gathered} \right. $ where Ω is a bounded smooth domain of R N with 0 ∈ Ω, 1 < p < N, 0 ≤ a < $\tfrac{{N - p}} {p} $ , α ∈ (0, 1), and b, λ are positive parameters. Here f : (0, ∞) → (0, ∞) is C 2 function. Our aim in this paper is to establish non-existence of positive solution for λ near zero and existence of positive solution for λ large. We use the method of sub-super solutions to establish our existence result.