We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order 2 s, with s ϵ (1/2, 1), and a coercive gradient term with subcritical power 0 < p < 2 s. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range 0 < p < 2 s, involving a continuum family of solutions and/or solutions blowing-up to -∞ on the boundary. This is in striking difference with the local case studied by Lasry Lions for the subquadratic case 1 < p < 2.