On the solvability of resonance problems with respect to the Fučík Spectrum

We consider the boundary value problem−Δu=αu+−βu−+g(u)+hin Ω,−Δu=0on ∂Ω where Ω   is a smooth bounded domain in RNRN, (α,β)∈R2(α,β)∈R2, g:R→Rg:R→R is a bounded continuous function, and h∈L2(Ω)h∈L2(Ω). We define u+:=max{u,0}u+:=max{u,0} and u−:=max{−u,0}u−:=max{−u,0}. We prove existence theorems for two cases. First, the nonresonance case, where (α,β)(α,β) is not an element of the Fučík Spectrum. In this case no further restrictions are need for g and h  . Second, the resonance case, where (α,β)(α,β) is an element of the Fučík Spectrum. In this case a generalized Landesman–Lazer condition is sufficient to prove existence. The proofs are variational and rely strongly on the variational characterization of the Fučík Spectrum developed in [3].