On critical Ambrosetti–Prodi type problems involving mixed operator

This article contains the study of the following problem with critical growth that involves the classical Laplacian and fractional Laplacian operators precisely (Formula presented.) where Ω⊆Rn, n≥3 is a bounded domain with smooth boundary ∂Ω, u+=max{u,0}, λ>0 is a real parameter, 2∗=2nn-2 and L=-Δ+(-Δ)s,fors∈(0,1). Here φ1 is the first eigenfunction of L with homogeneous Dirichlet boundary condition, t∈R and h∈L∞(Ω) satisfies ∫Ωhφ1dx=0. We establish existence and multiplicity results for the above problem, based on different ranges of the spectrum of L, using the Linking Theorem.