For a non-negative and non-trivial real-valued continuous function hΩ × [0, ∞) such that h(x, 0) = 0 for all x ∈ Ω, we study the boundary-value problem

where Ω ⊆ ℝN, N ⩾ 2, is a bounded smooth domain and Δp:= div(|Du|p–2DDu) is the p-Laplacian. This work investigates growth conditions on h(x, t) that would lead to the existence or non-existence of distributional solutions to (BVP). In a major departure from past works on similar problems, in this paper we do not impose any special structure on the inhomogeneous term h(x, t), nor do we require any monotonicity condition on h in the second variable. Furthermore, h(x, t) is allowed to vanish in either of the variables.