Positive solutions for parametric $p$-Laplacian equations

Abstract

We consider parametric equations driven by the p-Laplacian and with a reaction which has a p-logistic form or is the sum of two competing nonlinearities (concave-convex nonlinearities). We look for positive solutions and how their solution set depends on the parameter λ > 0. For the plogistic equation, we examine the subdiffusive, equidiffusive and superdiffusive cases. For the equations with competing nonlinearities, we consider the case of the sum of a "concave" (i.e., (p − 1)-sublinear) term and of a "convex" (i.e., (p − 1)-superlinear) term. For the latter, we do not assume the usual in such cases Ambrosetti-Rabinowitz condition. Our approach is variational based on the critical point theory combined with suitable truncation and comparison techniques.

Solutions and positive solutions for superlinear Robin problems

“…We have u = λû 1 (ξ,β) ∈ D+. So, from (35) and the hypothesis on η [see hypothesis H 1 (iv)], we obtain μ(u) ≤λ 1 (ξ,β)∥u∥ p p , which contradicts (5). Using this lemma, we can determine the geometry near zero forφ.…”