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 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 ∧Work partially supported by project MTM2013-40846-P MINECO. The third author is also supported for the grant BES-2011-044216 associated to MTM2010-1812