A Resonant-Superlinear Elliptic Problem Revisited

Abstract

Abstract We consider the resonant-superlinear elliptic problem -Δu = λ1u + (u+)q + f (x), with Dirichlet boundary conditions on a bounded regular domain of ℝN. We assume that with s > N satisfies ∫Ωfφ1 < 0 and (λ1, φ1) is the first eigenpair of −Δ on H1 0(Ω). We apply a non-well ordered lower and upper solution result on a family of modified problems and obtain a sequence of localized solutions of these modified problems. Thanks to this localization and a precise bootstrap argument we are abble to prove that for large modification, these solutions are, in fact, solutions of our initial problem. The problem was already considered in [2] by a totally different approach.