Existence of solutions for a fourth order problem at resonance

Abstract

In this work, we are interested at the existence of nontrivial solutions of two fourth order problems governed by the weighted p-biharmonic operator. The first is the following Δ(ρ|Δu|p-2Δu) = λ1m(x)|u|p-2u + f(x, u) - h in Ω u = Δu = 0 on ∂Ω, where λ1 is the first eigenvalue for the eigenvalue problem Δ (ρ| Δu|p-2Δu) = λm(x)|u|p-2u in Ω u = Δu = 0 on ∂Ω. In the seconde problem, we replace λ1 by λ such that λ1 < λ < λ̄, where λ ̄ is given bellow. © Soc. Paran. de Mat.