Sublinear and superlinear Ambrosetti–Prodi problems for the Dirichlet [formula omitted]-Laplacian

Abstract

We deal with an Ambrosetti–Prodi problem driven by the p-Laplace differential operator, with a “crossing” reaction which can be sublinear or superlinear (in the positive direction). Using variational methods based on the critical point theory, together with upper–lower solutions, truncation and comparison techniques and critical groups, we show the existence of a unique critical parameter value λ∗ such that for λ<λ∗ there are at least two nontrivial solutions, for λ=λ∗ there is at least one nontrivial solution, and for λ>λ∗ no solutions exist. We extend several recent results on this problem.