The critical fractional Ambrosetti-Prodi problem

In this paper we focus on the following nonlocal problem with critical growth: {(-Delta)(s) u = lambda u+ u(+)(2s)*(-1) + f(x) in Omega, u = 0 in R-N \ Omega, where s is an element of (0, 1), N > 2s, Omega subset of R-N is a smooth bounded domain, lambda > 0, (-Delta)(s) is the fractional Laplacian, f = te(1) + h where t is an element of R, e(1) is the first eigenfunction of (-Delta)(s) with homogeneous Dirichlet boundary datum, and h is an element of L-infinity (Omega) is such that f(Omega) he(1) dx = 0. According to the interaction of the nonlinear term with the spectrum of (-Delta)(s), we establish some existence and multiplicity results for the above problem by means of variational methods.