We prove the existence of a solution of (−∆) s u + f (u) = 0 in a smooth bounded domain Ω with a prescribed boundary value µ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and µ is a Dirac mass, we prove the existence of several critical exponents p.