Semilinear problems for the fractional laplacian with a singular nonlinearity

The aim of this paper is to study the solvability of the problem (-Δ)s u = F(x,u) := λ f(x)/uγ + Mup in ω u > 0 in ω, u = 0 in RN \ ω, where Ω is a bounded smooth domain of RN, N > 2s, M ε {0, 1}, 0 < s < 1, γ > 0, λ > 0, p > 1 and f is a nonnegative function. We distinguish two cases: - For M = 0, we prove the existence of a solution for every γ > 0 and λ > 0. A1 - For M = 1, we consider f ? 1 and we find a threshold ? such that there exists a solution for every 0 < λ < ∧, and there does not for λ > ∧.