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.

We prove the existence of one positive, one negative and one sign-changing solution of a p-Laplacian equation on ℝN with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of ℝN have scarcely been investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.

The aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the (p, q)-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bounds and regularity up to the boundary. The positivity of the solution is shown by applying a new comparison principle, which is established here.