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 {Lu=lambda u + u(+)(2)*(-1) + (t(phi 1)+h) in Omega, u = 0 in R-n \ Omega, where Omega subset of R-n, n >= 3 is a bounded domain with smooth boundary partial derivative Omega, u(+) = max{u, 0}, lambda > 0 is a real parameter, 2* = 2n/n-2 and L = -Delta+(-Delta)(s), for s is an element of(0,1). Here phi(1) is the first eigenfunction of L with homogeneous Dirichlet boundary condition, t is an element of R and h is an element of L-infinity(Omega) satisfies integral(Omega)h(phi 1) dx = 0. We establish existence and multiplicity results for the above problem, based on different ranges of the spectrum of L, using the Linking Theorem.