A new variational characterization of the Fučik spectrum for the Kirchhoff-type problem and applications

Abstract

We focus on the equation −∫Ω|∇u|2Δu=α(u+)3+β(u−)3 in Ω with the boundary condition u = 0 on ∂Ω, where Ω is a smooth bounded domain in RN for N = 1, 2, 3, α,β∈R, u+ = max{u, 0} and u− = min{u, 0}. Denote the Fučik spectrum by Σ, which is defined as a set comprising those (α,β)∈R2 such that the above equation has a nontrivial solution. The main goal of this study is to provide a new variational characterization for a class of curves in Σ. As an application, we establish the existence results for nonresonance and resonance problems related to these curves.