Abstract

We study existence and multiplicity of positive solutions for the following problem {-Δp u = λ f(x,u) in Ω u = 0 on ∂Ω' where λ is a positive parameter, Ω is a bounded and smooth domain in ℝN, p ∈ (1, N), f(x,t) behaves, for instance, like o(|t|p-1) near 0 and +∞, and satisfies some further properties. In particular, our assumptions allow us to consider both positive and sign changing nonlinearitites f, the latter describing logistic as well as reaction-diffusion processes. By using sub- and supersolutions and variational arguments, we prove that there exists a positive constant λ such that the above problem has at least two positive solutions for λ > λ, at least one positive solution for λ = λ and no solution for λ < λ. An important rôle plays the fact that local minimizers of certain functionals in the C1-topology are also minimizers in W0 1,p(Ω). We give a short new proof of this known result. © 2008 Birkhaueser.