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

We consider the boundary value problem − Δ u = α u + − β u − + g ( u ) + h in Ω , − Δ u = 0 on ∂ Ω where Ω is a smooth bounded domain in R N , ( α , β ) ∈ R 2 , g : R → R is a bounded continuous function, and h ∈ L 2 ( Ω ) . We define u + : = max { u , 0 } and u − : = max { − u , 0 } . We prove existence theorems for two cases. First, the nonresonance case, where ( 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 ) 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].