A note on the resonance set for a semilinear elliptic equations and an application to jumping nonlinearities
Anna Maria Micheletti, Angela Pistoia · Topological Methods in Nonlinear Analysis · 1995
where Ω is a bounded smooth domain, u + =m ax(u, 0) and u − = − min(u, 0). The study of Σ turns out to be difficult except when Ω is an interval in R. Therefore it is interesting to have some information about the resonance set, as precise as possible. In [GK] the authors showed that if λk is a simple eigenvalue of − ∆t hen Σ∩]λk−1 ,λ k+1[ 2 coincides with two continuous curves through the point (λk ,λ k). In [DeFG] the authors characterized a curve γ through the point (λ2 ,λ 2 )w hich belongs to Σ such that Σ∩{(α, β) ∈ R 2 | λ1 λ 1} = ∅. Finally, in [MMP] and [M] the following result was shown: if k ≥ 2 is such that λk <λ k+1 then there exist two continuous curves (α, ϕk+1(α)), through (λk+1 ,λ k+1), and (α, ψk(α)), through (λk ,λ k), which respectively lie in the sets Σ ∩ ]λk, +∞[ 2 and